Finite Group Databasebeta

← #29686913

Subgroups — #29686913

Order 15616.

938 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840031C211yesmaximal, characteristic
2C4×#14519805C211yesmaximal, characteristic, complemented
3C4×#14519803C211yesmaximal, characteristic, complemented
4C2×#16839872C211yesmaximal, characteristic, complemented
5#21454248C211yesmaximal, characteristic, complemented
6#21446726C211yesmaximal, characteristic, complemented
7#21446598C211yesmaximal, characteristic, complemented
8#21446483C211yesmaximal, characteristic, complemented
9#21446479C211yesmaximal, characteristic, complemented
10#21446328C211yesmaximal, characteristic, complemented
11#21446324C211yesmaximal, characteristic, complemented
12#21445877C211yesmaximal, characteristic, complemented
13#21445726C211yesmaximal, characteristic, complemented
14C4×#14519693C211yesmaximal, characteristic, complemented
15C4×#100×C61C211yesmaximal, characteristic, complemented
16C2×#14519805C2211yescharacteristic
17C2×#14519803C2211yescharacteristic
18C22×#11758199C2211yescharacteristic
19C2×#14519785C2211yescharacteristic
20C2×#14519785C2211yescharacteristic
21C4×#11758222C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23#16840031C418yescomplemented
24#16839875—21no—
25#16839872—21no—
26#16839872—21no—
27#16839872—21no—
28#16839872—21no—
29#16839872—21no—
30#16839864—21no—
31#16839862—21no—
32C4×#11758190—21no—
33C4×#11758200—21no—
34C4×D488—21no—
35C4×#11758199—21no—
36C2×#14519712C2211yescharacteristic
37C2×#14519712C2211yescharacteristic
38#16839804C2211yescharacteristic
39#16839802C2211yescharacteristic
40C2×#14519708C2211yescharacteristic
41#16839768C2211yescharacteristic
42#16839767C2211yescharacteristic
43#16839764C2211yescharacteristic
44#16839761C2211yescharacteristic
45C2×#14519698C2211yescharacteristic
46C2×#14519697C2211yescharacteristic
47C2×#14519693C2211yescharacteristic
48C2×#14519693C2211yescharacteristic
49#16839720C2211yescharacteristic
50#16839720C2211yescharacteristic
51#16839716—21no—
52#16839716—21no—
53C4×#11758194C2211yescharacteristic
54#16839714—21no—
55C2×#14519688C2211yescharacteristic
56#97×D61—21no—
57#16839530C2211yescharacteristic
58#16839530C2211yescharacteristic
59#16839526C2211yescharacteristic
60#16839526C2211yescharacteristic
61C2×C4×D4×C61C2211yescharacteristic
62C2×#100×C61C2211yescharacteristic
63C2×#100×C61C2211yescharacteristic
64C4×M16×C61C2211yescharacteristic
65C4×M16×C61C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C61×#343C2211yescharacteristic
68C2×#11758222C2311yescharacteristic
69C22×D244C2311yescharacteristic
70C22×C4×D61—21no—
71#14519805—22no—
72#14519805C2×C414yescomplemented
73#14519803—22no—
74#14519803C2×C414yescomplemented
75C2×#11758190—21no—
76C2×#11758200—21no—
77C2×D488—21no—
78C2×#11758199—21no—
79C2×#11758199—21no—
80C2×#11758199—21no—
81C2×#11758199—22no—
82C2×#11758199C2×C414yescomplemented
83#14519785—22no—
84#14519785—22no—
85#14519785—22no—
86#14519785C2×C414yescomplemented
87C2×#11758194C2311yescharacteristic
88C2×#11758194C2311yescharacteristic
89C2×#11758189—21no—
90C2×#11758189—21no—
91C22×#92418C2311yescharacteristic
92C22×#92418C2311yescharacteristic
93C2×C8×D61—21no—
94C2×C8×D61—21no—
95C4×Dic122D411yescharacteristic
96C4×Dic122D411yescharacteristic
97C4×D244—21no—
98C4×D244—21no—
99C4×D244—21no—
100C4×D244—21no—
101C4×D244D411yescharacteristic
102C4×D244D411yescharacteristic
103C4×C22⋊D61—21no—
104C4×C22⋊D61—21no—
105#14519736C2311yescharacteristic
106#14519734—21no—
107C2×C4⋊Dic61C2311yescharacteristic
108C2×#11758198—21no—
109C42×D61—21no—
110#14519712—22no—
111#14519712C2×C414yescomplemented
112#14519711—21no—
113#14519710—21no—
114#14519709—21no—
115#14519708—21no—
116#14519708—21no—
117#14519708—21no—
118#14519702—21no—
119#14519700—21no—
120#14519699—21no—
121#14519699—21no—
122#14519699—21no—
123#14519698—21no—
124#14519698—21no—
125#14519698—21no—
126#14519698—21no—
127#14519697—21no—
128#14519697—21no—
129#14519697—21no—
130#14519693—22no—
131#14519693C2×C414yescomplemented
132#14519688—21no—
133#14519688D411yescharacteristic
134#14519688D411yescharacteristic
135#14519688D411yescharacteristic
136#14519688D411yescharacteristic
137#14519687—21no—
138#14519687—21no—
139C4×#92418—21no—
140C8×Dic61—21no—
141C8×Dic61—21no—
142C22×D4×C61C2311yescharacteristic
143C23×C4×C61—21no—
144C2×M16×C61C2311yescharacteristic
145C2×M16×C61C2311yescharacteristic
146C2×M16×C61C2311yescharacteristic
147C2×M16×C61C2311yescharacteristic
148C4×D4×C61—21no—
149C4×D4×C61—21no—
150C4×D4×C61—21no—
151C4×D4×C61—21no—
152C2×C4⋊C4×C61C2311yescharacteristic
153C2×C22⋊C4×C61—21no—
154C2×C42×C61C2311yescharacteristic
155#100×C61—22no—
156#100×C61C2×C414yescomplemented
157#97×C61—21no—
158#97×C61—21no—
159C4×C8×C61—21no—
160C4×C8×C61—21no—
161C24×C61—21no—
162C23×D61—21no—
163C2×D4×C61—21no—
164C2×D4×C61—21no—
165C2×D4×C61—21no—
166C2×D4×C61—21no—
167C2×D4×C61—21no—
168C2×D4×C61C22×C412yes—
169C22×C4×C61—21no—
170C22×C4×C61—41no—
171C22×C4×C61—41no—
172C22×C4×C61—82no—
173C22×C4×C61#4111yescharacteristic
174C22×C4×C61#4111yescharacteristic
175C22×C4×C61C2411yescharacteristic
176C2×C22⋊D61—21no—
177C22×Dic61—21no—
178#11758222—21no—
179#11758222—21no—
180#11758222C22×C412yes—
181#11758222#4112yes—
182C2×D244—21no—
183C2×D244—21no—
184C2×D244—21no—
185C2×D244—21no—
186C2×D244—21no—
187C2×D244C22×C412yes—
188C2×D244C2×D411yescharacteristic
189C2×C4×D61—21no—
190C2×C4×D61—21no—
191C2×C4×D61—21no—
192C2×C4×D61—41no—
193C2×C4×D61—41no—
194C2×C4×D61—82no—
195C2×Dic122C2×D411yescharacteristic
196M16×C61—21no—
197M16×C61—21no—
198M16×C61—21no—
199M16×C61—21no—
200M16×C61—21no—
201M16×C61C22×C412yes—
202M16×C61C22×C412yes—
203M16×C61#4112yes—
204C2×C8×C61—21no—
205C2×C8×C61—21no—
206C2×C8×C61—21no—
207C2×C8×C61—21no—
208C4⋊C4×C61—21no—
209C4⋊C4×C61—21no—
210C22⋊C4×C61—41no—
211C22⋊C4×C61—41no—
212C42×C61C2×D411yescharacteristic
213C42×C61C2×D411yescharacteristic
214C42×C61C2×D411yescharacteristic
215C42×C61C2×D411yescharacteristic
216C22⋊Dic61—21no—
217#11758200—24no—
218#11758199—24no—
219#11758199—24no—
220#11758199—24no—
221#11758198—21no—
222#11758198—21no—
223#11758198—41no—
224#11758198—41no—
225C4⋊Dic61—21no—
226C4⋊Dic61—21no—
227C4⋊Dic61C2×D411yescharacteristic
228C4⋊Dic61C2×D411yescharacteristic
229#11758196—21no—
230#11758196—21no—
231C4×Dic61—21no—
232C4×Dic61—21no—
233C4×Dic61—21no—
234#11758194—21no—
235#11758194—21no—
236#11758194—21no—
237#11758194C22×C412yes—
238C2×#92418—21no—
239C2×#92418—21no—
240C2×#92418—21no—
241C2×#92418—21no—
242C2×#92418C22×C412yes—
243C2×#92418C2×D411yescharacteristic
244C2×#92418C2×D411yescharacteristic
245C2×#92418C2×D411yescharacteristic
246C2×#92418C2×D411yescharacteristic
247C2×#92418#4112yes—
248D488—24no—
249#11758190—24no—
250#11758189—24no—
251#11758189—24no—
252C8×D61—24no—
253C8×D61—24no—
254C23×C61—22no—
255C23×C61—41no—
256C23×C61—41no—
257C23×C61—81no—
258C23×C61—81no—
259C23×C61C2×#4111yescharacteristic
260C22×D61—21no—
261C22×D61—22no—
262C22×D61—41no—
263C22×D61—41no—
264C22×D61—81no—
265C22×D61—81no—
266D4×C61—22no—
267D4×C61—22no—
268D4×C61—41no—
269D4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61—41no—
272C2×C4×C61—41no—
273C2×C4×C61—82no—
274C2×C4×C61—82no—
275C2×C4×C61—81no—
276C2×C4×C61#12111yescharacteristic
277C2×C4×C61#12111yescharacteristic
278C2×C4×C61#12111yescharacteristic
279C2×C4×C61#12111yescharacteristic
280C2×C4×C61C23×C411yescharacteristic
281C2×C4×C61C22×D411yescharacteristic
282C2×C4×C61C22×D411yescharacteristic
283C2×C4×C61C2×#4111yescharacteristic
284C2×C4×C61C2×#4111yescharacteristic
285C2×C4×C61C2×#4111yescharacteristic
286C22⋊D61—22no—
287C22⋊D61—22no—
288C2×Dic61—21no—
289C2×Dic61—21no—
290C2×Dic61—21no—
291C2×Dic61—41no—
292C2×Dic61—81no—
293D244—22no—
294D244—22no—
295D244—21no—
296D244—41no—
297D244—41no—
298D244C4×D412yes—
299C4×D61—22no—
300C4×D61—22no—
301C4×D61—81no—
302C4×D61—81no—
303C4×D61—82no—
304C4×D61—82no—
305Dic122—21no—
306Dic122C4×D412yes—
307C8×C61—22no—
308C8×C61—22no—
309C8×C61—22no—
310C8×C61—41no—
311#92418—22no—
312#92418—21no—
313#92418—21no—
314#92418—41no—
315#92418C4×D412yes—
316#92418C4×D412yes—
317#21555—611nomaximal, Sylow
318C22×C61—21no—
319C22×C61—21no—
320C22×C61—42no—
321C22×C61—82no—
322C22×C61—82no—
323C22×C61—81no—
324C22×C61—81no—
325C22×C61C4×#4111yescharacteristic
326C22×C61C2×#12111yescharacteristic
327C22×C61#52911yescharacteristic
328C2×D61—22no—
329C2×D61—41no—
330C2×D61—41no—
331C2×D61—42no—
332C2×D61—82no—
333C2×D61—81no—
334C2×D61—81no—
335C2×D61—82no—
336C4×C61—21no—
337C4×C61—21no—
338C4×C61—21no—
339C4×C61—21no—
340C4×C61—81no—
341C4×C61C2×C4×D411yescharacteristic
342C4×C61C2×C4×D411yescharacteristic
343C4×C61#52511yescharacteristic
344C4×C61#53211yescharacteristic
345Dic61—22no—
346Dic61—41no—
347Dic61—41no—
348Dic61—81no—
349C2×#472—611no—
350C4×AGL(1,ℤ/8)—611no—
351C4×AGL(1,ℤ/8)—611no—
352C2×#442—611no—
353#2875—611no—
354#1978—611no—
355#1951—611no—
356#1931—611no—
357#1931—611no—
358#1907—611no—
359#1907—611no—
360#1815—611no—
361#1778—611no—
362C4×#100—611no—
363C4×#100D6111yescharacteristic, complemented
364C2×C61—21no—
365C2×C61—21no—
366C2×C61—82no—
367C2×C61C4×#12111yescharacteristic
368C2×C61#306811yescharacteristic
369C2×C61#306811yescharacteristic
370D61—42no—
371D61—82no—
372C2×AGL(1,ℤ/8)—611no—
373C2×AGL(1,ℤ/8)—611no—
374C22×D8—611no—
375C2×#131—611no—
376C2×#131—611no—
377C4×#41—611no—
378C2×C4×D4—611no—
379C2×C4×D4C2×D6111yescharacteristic
380#472—618no—
381#442—1221no—
382#442—1221no—
383#442—1221no—
384#442—1221no—
385#442—1221no—
386#442—1221no—
387#440—1221no—
388#440—1221no—
389C4×SD16—1221no—
390C4×SD16—1221no—
391C4×D8—1221no—
392C4×D8—1221no—
393C2×#108—611no—
394C2×#108—611no—
395#428—611no—
396#428—611no—
397C2×#106—611no—
398#418—611no—
399#418—611no—
400#417—611no—
401#417—611no—
402C2×#102—611no—
403C2×#102—611no—
404C2×#100—611no—
405C2×#100—611no—
406C2×#100C2×D6111yescharacteristic
407C2×#100C2×D6111yescharacteristic
408#405—611no—
409#405—611no—
410#405—1221no—
411#405—1221no—
412C4×M16—611no—
413C4×M16—1221no—
414C4×M16C2×D6111yescharacteristic
415C4×M16C2×D6111yescharacteristic
416C2×#97—611no—
417C2×#97—1221no—
418#343—611no—
419#343—611no—
420#343C2×D6111yescharacteristic
421#343C2×D6111yescharacteristic
422#341—611no—
423#341—611no—
424C61#2155511yescharacteristic, complemented, Sylow
425C2×#41—611no—
426C22×D4—611no—
427C22×D4C22×D6111yescharacteristic
428C23×C4—21no—
429C23×C4—1221no—
430AGL(1,ℤ/8)—614no—
431AGL(1,ℤ/8)—614no—
432AGL(1,ℤ/8)—1222no—
433AGL(1,ℤ/8)—1222no—
434C2×SD16—1221no—
435C2×SD16—1221no—
436C2×D8—614no—
437C2×D8—1221no—
438C2×D8—1221no—
439C2×D8—1221no—
440C2×D8—1222no—
441C2×D8—1221no—
442#131—614no—
443#131—1222no—
444#131—1222no—
445#131—1222no—
446C2×M16—611no—
447C2×M16—611no—
448C2×M16—1221no—
449C2×M16—1221no—
450C2×M16C22×D6111yescharacteristic
451C2×M16C22×D6111yescharacteristic
452C2×M16C22×D6111yescharacteristic
453C2×M16C22×D6111yescharacteristic
454C22×C8—611no—
455C22×C8—611no—
456C22×C8—1221no—
457C22×C8—1221no—
458C4×Q8—611no—
459C4×Q8—611no—
460C4×D4—21no—
461C4×D4—21no—
462C4×D4—21no—
463C4×D4—21no—
464C4×D4—611no—
465C4×D4—611no—
466C4×D4—1221no—
467C4×D4—1221no—
468C4×D4—1221no—
469C4×D4—1221no—
470C4×D4—1221no—
471C4×D4—1221no—
472#117—611no—
473#117—1221no—
474C2×C4⋊C4—611no—
475C2×C4⋊C4C22×D6111yescharacteristic
476C2×C22⋊C4—21no—
477C2×C22⋊C4—1221no—
478C2×C42—1221no—
479C2×C42C22×D6111yescharacteristic
480#108—614no—
481#108—1222no—
482#107—1221no—
483#107—1221no—
484#106—1221no—
485#106—1221no—
486#106—1221no—
487#106—1221no—
488#103—1221no—
489#103—1221no—
490#102—1221no—
491#102—1221no—
492#102—1221no—
493#102—1221no—
494#102—1221no—
495#102—1221no—
496#102—1221no—
497#102—1221no—
498#102—1221no—
499#102—1221no—
500#100—22no—
501#100—614no—
502#100—1222no—
503#100C4×D6114yescomplemented
504#97—21no—
505#97—21no—
506#97—611no—
507#97—611no—
508#97—611no—
509#97—611no—
510#97—1221no—
511#97—1221no—
512#97—1221no—
513C4×C8—21no—
514C4×C8—21no—
515C4×C8—1221no—
516C4×C8—1221no—
517C4×C8—1221no—
518C24—21no—
519C24—1221no—
520#41—612no—
521#41—612no—
522#41—1221no—
523#41—1221no—
524C2×Q8—611no—
525C2×D4—21no—
526C2×D4—21no—
527C2×D4—21no—
528C2×D4—21no—
529C2×D4—21no—
530C2×D4—612no—
531C2×D4—611no—
532C2×D4—1221no—
533C2×D4—1221no—
534C2×D4—1221no—
535C2×D4—1221no—
536C2×D4—1221no—
537C2×D4—1221no—
538C2×D4C2×C4×D6112yes—
539C22×C4—21no—
540C22×C4—41no—
541C22×C4—41no—
542C22×C4—82no—
543C22×C4—1221no—
544C22×C4—1221no—
545C22×C4—1221no—
546C22×C4—1221no—
547C22×C4—4882no—
548C22×C4—2441no—
549C22×C4—2441no—
550C22×C4#1175822211yescharacteristic
551C22×C4#1175822211yescharacteristic
552C22×C4C23×D6111yescharacteristic
553SD16—1224no—
554SD16—1224no—
555D8—1224no—
556D8—1224no—
557D8—1224no—
558D8—1224no—
559M16—21no—
560M16—21no—
561M16—21no—
562M16—21no—
563M16—21no—
564M16—612no—
565M16—1224no—
566M16—1224no—
567M16—1221no—
568M16—1221no—
569M16—1221no—
570M16C2×C4×D6112yes—
571M16C2×C4×D6112yes—
572M16#1175822212yes—
573C2×C8—21no—
574C2×C8—21no—
575C2×C8—21no—
576C2×C8—21no—
577C2×C8—612no—
578C2×C8—611no—
579C2×C8—611no—
580C2×C8—612no—
581C2×C8—611no—
582C2×C8—611no—
583C2×C8—1224no—
584C2×C8—1224no—
585C2×C8—1221no—
586C2×C8—1221no—
587C2×C8—1221no—
588C2×C8—1221no—
589C4⋊C4—21no—
590C4⋊C4—21no—
591C4⋊C4—611no—
592C4⋊C4—611no—
593C4⋊C4—1221no—
594C4⋊C4—1221no—
595C4⋊C4—1221no—
596C4⋊C4—1221no—
597C22⋊C4—41no—
598C22⋊C4—41no—
599C22⋊C4—1221no—
600C22⋊C4—1221no—
601C22⋊C4—1221no—
602C22⋊C4—2441no—
603C22⋊C4—2441no—
604C42—1221no—
605C42—1221no—
606C42—1221no—
607C42D4×D6111yescharacteristic
608C42D4×D6111yescharacteristic
609C42D4×D6111yescharacteristic
610C42D4×D6111yescharacteristic
611C23—22no—
612C23—41no—
613C23—41no—
614C23—81no—
615C23—81no—
616C23—1221no—
617C23—1222no—
618C23—4881no—
619C23—4881no—
620C23—2441no—
621C23—2441no—
622C23C2×#1175822211yescharacteristic
623Q8—612no—
624Q8—1221no—
625D4—22no—
626D4—22no—
627D4—41no—
628D4—41no—
629D4—612no—
630D4—1222no—
631D4—1222no—
632D4—1222no—
633D4—1222no—
634D4—1221no—
635D4—2441no—
636D4—2441no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—81no—
643C2×C4—1221no—
644C2×C4—1222no—
645C2×C4—1222no—
646C2×C4—1221no—
647C2×C4—1221no—
648C2×C4—4882no—
649C2×C4—4882no—
650C2×C4—4881no—
651C2×C4—4881no—
652C2×C4—4881no—
653C2×C4—2441no—
654C2×C4#1451975011yescharacteristic
655C2×C4#1451975011yescharacteristic
656C2×C4#1451975011yescharacteristic
657C2×C4#1451975011yescharacteristic
658C2×C4C22×C4×D6111yescharacteristic
659C2×C4C2×D4×D6111yescharacteristic
660C2×C4C2×D4×D6111yescharacteristic
661C2×C4#41×D6111yescharacteristic
662C2×C4#41×D6111yescharacteristic
663C2×C4C2×#1175822211yescharacteristic
664C8—22no—
665C8—22no—
666C8—22no—
667C8—41no—
668C8—612no—
669C8—612no—
670C8—1222no—
671C8—1221no—
672C8—1221no—
673C8—2441no—
674C22—21no—
675C22—21no—
676C22—42no—
677C22—82no—
678C22—82no—
679C22—81no—
680C22—81no—
681C22—1222no—
682C22—4882no—
683C22—4882no—
684C22—4881no—
685C22—4881no—
686C22—2442no—
687C22—2441no—
688C22—2441no—
689C22C4×#1175822211yescharacteristic
690C22C2×#1451975011yescharacteristic
691C22#1684029011yescharacteristic
692C4—21no—
693C4—21no—
694C4—21no—
695C4—21no—
696C4—81no—
697C4—1222no—
698C4—4881no—
699C4—2441no—
700C4—2441no—
701C4C4×D4×D6111yescharacteristic
702C4C4×D4×D6111yescharacteristic
703C4#1684027111yescharacteristic
704C4#1684030911yescharacteristic
705C2—21no—
706C2—21no—
707C2—82no—
708C2—4882no—
709C2—2442no—
710C2C4×#1451975011yescharacteristic
711C2#2145554611yescharacteristic
712C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#100×C61C2
socleC22×C61C2×#121
prime radicalC2×#16840031C2
cosocleC22×C4×C61C24
hypercenterC4×#100D61
hypocenterC61#21555
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C4×#14519805C2semidirect
3. C4×#14519803C2semidirect
4. C2×#16839872C2semidirect
5. #21454248C2semidirect
6. #21446726C2semidirect
7. #21446598C2semidirect
8. #21446483C2semidirect
9. #21446479C2semidirect
10. #21446328C2semidirect
11. #21446324C2semidirect
12. #21445877C2semidirect
13. #21445726C2semidirect
14. C4×#14519693C2semidirect
15. C4×#100×C61C2semidirect
23. #16840031C4semidirect
72. #14519805C2×C4semidirect
74. #14519803C2×C4semidirect
82. C2×#11758199C2×C4semidirect
86. #14519785C2×C4semidirect
111. #14519712C2×C4semidirect
131. #14519693C2×C4semidirect
156. #100×C61C2×C4semidirect
363. C4×#100D61semidirect
424. C61#21555semidirect
503. #100C4×D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168400312
C4×#145198052
C4×#145198032
C2×#168398722
#214542482
#214467262
#214465982
#214464832
#214464792
#214463282
#214463242
#214458772
#214457262
C4×#145196932
C4×#100×C612
#2155561