Finite Group Databasebeta

← #29686912

Subgroups — #29686912

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×#16840030C211yesmaximal, characteristic
2#21454632C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×Dic61C211yesmaximal, characteristic, complemented
4C2×#16839871C211yesmaximal, characteristic, complemented
5#21454245C211yesmaximal, characteristic, complemented
6#21446726C211yesmaximal, characteristic, complemented
7#21446598C211yesmaximal, characteristic, complemented
8#21446482C211yesmaximal, characteristic, complemented
9#21446478C211yesmaximal, characteristic, complemented
10#21446327C211yesmaximal, characteristic, complemented
11#21445876C211yesmaximal, characteristic, complemented
12#21445724C211yesmaximal, characteristic, complemented
13#100×Dic61C211yesmaximal, characteristic, complemented
14#21445610C211yesmaximal, characteristic, complemented
15C61×#1907C211yesmaximal, characteristic, complemented
16C2×#14519802C2211yescharacteristic
17C22×#11758199C2211yescharacteristic
18C2×#14519784C2211yescharacteristic
19#41×Dic61C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22#16840030C418yescomplemented
23#16839874—21no—
24#16839871—21no—
25#16839871—21no—
26#16839871—21no—
27#16839871—21no—
28#16839871—21no—
29#16839865—21no—
30#16839863—21no—
31SD16×Dic61—21no—
32#16839847—21no—
33D8×Dic61—21no—
34#16839842—21no—
35C2×#14519712C2211yescharacteristic
36C2×#14519712C2211yescharacteristic
37#16839804C2211yescharacteristic
38#16839802C2211yescharacteristic
39C2×#14519708C2211yescharacteristic
40#16839768C2211yescharacteristic
41#16839767C2211yescharacteristic
42#16839763C2211yescharacteristic
43C2×#14519699C2211yescharacteristic
44C2×#14519694C2211yescharacteristic
45C2×#14519693C2211yescharacteristic
46C2×#14519693C2211yescharacteristic
47#16839720—21no—
48#16839720C2211yescharacteristic
49#16839717—21no—
50#16839717C2211yescharacteristic
51M16×Dic61—21no—
52M16×Dic61C2211yescharacteristic
53M16×Dic61C2211yescharacteristic
54M16×Dic61C2211yescharacteristic
55C2×#14519687—21no—
56C2×#14519687C2211yescharacteristic
57#16839531C2211yescharacteristic
58#16839530C2211yescharacteristic
59#16839530C2211yescharacteristic
60#16839527C2211yescharacteristic
61C2×AGL(1,ℤ/8)×C61C2211yescharacteristic
62C2×#131×C61C2211yescharacteristic
63C61×#417C2211yescharacteristic
64C2×#102×C61C2211yescharacteristic
65C2×#100×C61C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C61×#341C2211yescharacteristic
68C22×D244C2311yescharacteristic
69C23×Dic61—21no—
70C22×C4×D61—21no—
71#14519802—22no—
72#14519802C2×C414yescomplemented
73C2×#11758201—21no—
74C2×#11758199—21no—
75C2×#11758199—21no—
76C2×#11758199—21no—
77C2×#11758199—22no—
78C2×#11758199C2×C414yescomplemented
79#14519784—22no—
80#14519784C2×C414yescomplemented
81C2×#11758194—21no—
82C2×#11758194C2311yescharacteristic
83C2×#11758194C2311yescharacteristic
84C2×#11758194C2311yescharacteristic
85C22×#92418—21no—
86C22×#92418C2311yescharacteristic
87Q8×Dic61D411yescharacteristic
88Q8×Dic61D411yescharacteristic
89D4×Dic61—21no—
90D4×Dic61—21no—
91D4×Dic61—21no—
92D4×Dic61—21no—
93D4×Dic61—21no—
94D4×Dic61—21no—
95D4×Dic61D411yescharacteristic
96D4×Dic61D411yescharacteristic
97#14519739—21no—
98#14519739—21no—
99#14519739—21no—
100#14519739—21no—
101#14519736—21no—
102#14519736C2311yescharacteristic
103C2×C4⋊Dic61C2311yescharacteristic
104C2×C22⋊Dic61—21no—
105C2×#11758198—21no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61C2311yescharacteristic
108#14519712—22no—
109#14519712C2×C414yescomplemented
110#14519711—21no—
111#14519710—21no—
112#14519709—21no—
113#14519708—21no—
114#14519708—21no—
115#14519708—21no—
116#14519701—21no—
117#14519699—21no—
118#14519699—21no—
119#14519699—21no—
120#14519699—21no—
121#14519698—21no—
122#14519698—21no—
123#14519698—21no—
124#14519694C2×C414yescomplemented
125#14519693—22no—
126#14519693C2×C414yescomplemented
127#14519687—21no—
128#14519687—21no—
129#14519687—21no—
130#14519687—21no—
131#14519687—21no—
132#14519687D411yescharacteristic
133#14519687D411yescharacteristic
134#14519687D411yescharacteristic
135#14519687D411yescharacteristic
136C8×Dic61—21no—
137C8×Dic61—21no—
138C8×Dic61—21no—
139C8×Dic61—21no—
140C8×Dic61—21no—
141C2×#41×C61C2311yescharacteristic
142C22×D4×C61C2311yescharacteristic
143AGL(1,ℤ/8)×C61—22no—
144AGL(1,ℤ/8)×C61C2×C414yescomplemented
145C2×SD16×C61—21no—
146C2×D8×C61—21no—
147#131×C61—22no—
148#131×C61—22no—
149C2×M16×C61—21no—
150C2×M16×C61C2311yescharacteristic
151C2×M16×C61C2311yescharacteristic
152C2×M16×C61C2311yescharacteristic
153C22×C8×C61—21no—
154C22×C8×C61C2311yescharacteristic
155C2×C4⋊C4×C61C2311yescharacteristic
156#103×C61—21no—
157#102×C61—21no—
158#102×C61—21no—
159#102×C61—21no—
160#100×C61—22no—
161C24×C61—21no—
162C23×D61—21no—
163#41×C61—21no—
164#41×C61—21no—
165#41×C61C22×C412yes—
166#41×C61#4112yes—
167C2×Q8×C61C2×D411yescharacteristic
168C2×D4×C61—21no—
169C2×D4×C61—21no—
170C2×D4×C61—21no—
171C2×D4×C61—21no—
172C2×D4×C61—21no—
173C2×D4×C61—21no—
174C2×D4×C61C22×C412yes—
175C2×D4×C61C2×D411yescharacteristic
176C22×C4×C61—21no—
177C22×C4×C61—21no—
178C22×C4×C61C2411yescharacteristic
179C22×Dic61—21no—
180C22×Dic61—21no—
181C22×Dic61—21no—
182C22×Dic61—41no—
183C22×Dic61—41no—
184C22×Dic61—82no—
185C22×Dic61#4111yescharacteristic
186C22×Dic61#4111yescharacteristic
187C2×D244—21no—
188C2×D244—21no—
189C2×D244—21no—
190C2×D244—21no—
191C2×D244—21no—
192C2×D244C22×C412yes—
193C2×C4×D61—41no—
194C2×C4×D61—41no—
195C2×C4×D61—82no—
196SD16×C61—24no—
197D8×C61—24no—
198M16×C61—24no—
199M16×C61—21no—
200M16×C61—21no—
201M16×C61—21no—
202M16×C61—21no—
203M16×C61C22×C412yes—
204M16×C61#4112yes—
205C2×C8×C61—24no—
206C2×C8×C61—21no—
207C2×C8×C61—21no—
208C2×C8×C61—21no—
209C2×C8×C61—21no—
210C2×C8×C61—21no—
211C2×C8×C61C2×D411yescharacteristic
212C2×C8×C61C2×D411yescharacteristic
213C4⋊C4×C61—21no—
214C4⋊C4×C61—21no—
215C22⋊Dic61—21no—
216C22⋊Dic61—21no—
217C22⋊Dic61—21no—
218C22⋊Dic61—41no—
219C22⋊Dic61—41no—
220#11758201—24no—
221#11758199—24no—
222#11758199—24no—
223#11758199—24no—
224#11758198—41no—
225#11758198—41no—
226C4⋊Dic61—21no—
227C4⋊Dic61—21no—
228C4⋊Dic61—21no—
229C4⋊Dic61—21no—
230C4⋊Dic61C2×D411yescharacteristic
231C4⋊Dic61C2×D411yescharacteristic
232C4×Dic61—21no—
233C4×Dic61—21no—
234C4×Dic61—21no—
235C4×Dic61C2×D411yescharacteristic
236C4×Dic61C2×D411yescharacteristic
237C4×Dic61C2×D411yescharacteristic
238C4×Dic61C2×D411yescharacteristic
239#11758194—21no—
240#11758194—21no—
241#11758194—24no—
242#11758194—21no—
243#11758194—21no—
244#11758194C22×C412yes—
245#11758194C22×C412yes—
246C2×#92418—21no—
247C2×#92418—21no—
248C2×#92418—24no—
249C2×#92418—21no—
250C2×#92418C22×C412yes—
251C2×#92418C2×D411yescharacteristic
252C2×#92418C2×D411yescharacteristic
253C2×#92418#4112yes—
254C23×C61—21no—
255C23×C61—22no—
256C23×C61—41no—
257C23×C61—41no—
258C23×C61—81no—
259C23×C61—81no—
260C23×C61C2×#4111yescharacteristic
261C22×D61—22no—
262C22×D61—41no—
263C22×D61—41no—
264C22×D61—81no—
265C22×D61—81no—
266Q8×C61—21no—
267Q8×C61C4×D412yes—
268D4×C61—22no—
269D4×C61—22no—
270D4×C61—22no—
271D4×C61—22no—
272D4×C61—21no—
273D4×C61—41no—
274D4×C61—41no—
275D4×C61C4×D412yes—
276C2×C4×C61—21no—
277C2×C4×C61—22no—
278C2×C4×C61—22no—
279C2×C4×C61—41no—
280C2×C4×C61—81no—
281C2×C4×C61C23×C411yescharacteristic
282C2×C4×C61C22×D411yescharacteristic
283C2×C4×C61C22×D411yescharacteristic
284C2×C4×C61C2×#4111yescharacteristic
285C2×C4×C61C2×#4111yescharacteristic
286C2×C4×C61C2×#4111yescharacteristic
287C2×Dic61—21no—
288C2×Dic61—21no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—81no—
293C2×Dic61—81no—
294C2×Dic61—82no—
295C2×Dic61—82no—
296C2×Dic61—81no—
297C2×Dic61#12111yescharacteristic
298C2×Dic61#12111yescharacteristic
299C2×Dic61#12111yescharacteristic
300C2×Dic61#12111yescharacteristic
301D244—22no—
302D244—22no—
303D244—41no—
304D244—41no—
305C4×D61—82no—
306C4×D61—82no—
307C8×C61—22no—
308C8×C61—21no—
309C8×C61—22no—
310C8×C61—21no—
311C8×C61—41no—
312#92418—22no—
313#92418—22no—
314#92418—41no—
315#92418C4×D412yes—
316#92418C4×D412yes—
317#21555—611nomaximal, Sylow
318C22×C61—21no—
319C22×C61—21no—
320C22×C61—22no—
321C22×C61—41no—
322C22×C61—41no—
323C22×C61—42no—
324C22×C61—82no—
325C22×C61—82no—
326C22×C61—81no—
327C22×C61—81no—
328C22×C61C4×#4111yescharacteristic
329C22×C61C2×#12111yescharacteristic
330C22×C61#52911yescharacteristic
331C2×D61—42no—
332C2×D61—82no—
333C2×D61—81no—
334C2×D61—81no—
335C2×D61—82no—
336C4×C61—22no—
337C4×C61—81no—
338C4×C61C2×C4×D411yescharacteristic
339C4×C61C2×C4×D411yescharacteristic
340C4×C61#52511yescharacteristic
341C4×C61#53211yescharacteristic
342Dic61—21no—
343Dic61—21no—
344Dic61—21no—
345Dic61—21no—
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#1907D6111yescharacteristic, complemented
360#1815—611no—
361#1778—611no—
362C4×#100—611no—
363C4×#100—611no—
364C2×C61—21no—
365C2×C61—21no—
366C2×C61—42no—
367C2×C61—82no—
368C2×C61C4×#12111yescharacteristic
369C2×C61#306811yescharacteristic
370C2×C61#306811yescharacteristic
371D61—82no—
372C2×AGL(1,ℤ/8)—611no—
373C2×AGL(1,ℤ/8)C2×D6111yescharacteristic
374C22×D8—611no—
375C2×#131—611no—
376C2×#131C2×D6111yescharacteristic
377C4×#41—611no—
378C2×C4×D4—611no—
379C2×C4×D4—611no—
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#417C2×D6111yescharacteristic
402C2×#102—611no—
403C2×#102C2×D6111yescharacteristic
404C2×#100—611no—
405C2×#100—611no—
406C2×#100—611no—
407C2×#100C2×D6111yescharacteristic
408#405—611no—
409#405—611no—
410#405—1221no—
411#405—1221no—
412C4×M16—611no—
413C4×M16—611no—
414C4×M16—611no—
415C4×M16—1221no—
416C2×#97—611no—
417C2×#97—1221no—
418#343—611no—
419#343—611no—
420#343—611no—
421#343C2×D6111yescharacteristic
422#341—611no—
423#341C2×D6111yescharacteristic
424C61#2155511yescharacteristic, complemented, Sylow
425C2×#41C22×D6111yescharacteristic
426C22×D4—611no—
427C22×D4C22×D6111yescharacteristic
428C23×C4—1221no—
429C23×C4—1221no—
430AGL(1,ℤ/8)—22no—
431AGL(1,ℤ/8)—614no—
432AGL(1,ℤ/8)—1222no—
433AGL(1,ℤ/8)C4×D6114yescomplemented
434C2×SD16—21no—
435C2×SD16—1221no—
436C2×D8—21no—
437C2×D8—614no—
438C2×D8—1221no—
439C2×D8—1221no—
440C2×D8—1222no—
441C2×D8—1221no—
442#131—22no—
443#131—22no—
444#131—614no—
445#131—1222no—
446C2×M16—21no—
447C2×M16—611no—
448C2×M16—611no—
449C2×M16—611no—
450C2×M16—1221no—
451C2×M16C22×D6111yescharacteristic
452C2×M16C22×D6111yescharacteristic
453C2×M16C22×D6111yescharacteristic
454C22×C8—21no—
455C22×C8—611no—
456C22×C8—1221no—
457C22×C8C22×D6111yescharacteristic
458C4×Q8—611no—
459C4×Q8—611no—
460C4×D4—611no—
461C4×D4—611no—
462C4×D4—1221no—
463C4×D4—1221no—
464C4×D4—1221no—
465C4×D4—1221no—
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—1221no—
477C2×C22⋊C4—1221no—
478C2×C42—611no—
479C2×C42—1221no—
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—21no—
489#103—1221no—
490#102—21no—
491#102—21no—
492#102—21no—
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—614no—
503#100—1222no—
504#97—611no—
505#97—611no—
506#97—611no—
507#97—611no—
508#97—1221no—
509#97—1221no—
510#97—1221no—
511#97—1221no—
512#97—1221no—
513C4×C8—1221no—
514C4×C8—1221no—
515C4×C8—1221no—
516C4×C8—1221no—
517C4×C8—1221no—
518C24—21no—
519C24—1221no—
520#41—21no—
521#41—21no—
522#41C2×C4×D6112yes—
523#41#1175822412yes—
524C2×Q8D4×D6111yescharacteristic
525C2×D4—21no—
526C2×D4—21no—
527C2×D4—21no—
528C2×D4—21no—
529C2×D4—21no—
530C2×D4—21no—
531C2×D4—612no—
532C2×D4—1221no—
533C2×D4—1221no—
534C2×D4—1221no—
535C2×D4—1221no—
536C2×D4—1221no—
537C2×D4C2×C4×D6112yes—
538C2×D4D4×D6111yescharacteristic
539C22×C4—21no—
540C22×C4—21no—
541C22×C4—611no—
542C22×C4—611no—
543C22×C4—1221no—
544C22×C4—1221no—
545C22×C4—1221no—
546C22×C4—4882no—
547C22×C4—4882no—
548C22×C4—2441no—
549C22×C4—2441no—
550C22×C4—2441no—
551C22×C4—2441no—
552C22×C4C23×D6111yescharacteristic
553SD16—24no—
554SD16—1224no—
555D8—24no—
556D8—1224no—
557D8—1224no—
558D8—1224no—
559M16—24no—
560M16—21no—
561M16—21no—
562M16—21no—
563M16—21no—
564M16—612no—
565M16—612no—
566M16—1224no—
567M16—1221no—
568M16—1221no—
569M16—1221no—
570M16—1221no—
571M16C2×C4×D6112yes—
572M16#1175822412yes—
573C2×C8—24no—
574C2×C8—21no—
575C2×C8—21no—
576C2×C8—21no—
577C2×C8—21no—
578C2×C8—21no—
579C2×C8—612no—
580C2×C8—611no—
581C2×C8—611no—
582C2×C8—612no—
583C2×C8—1224no—
584C2×C8—1221no—
585C2×C8—1221no—
586C2×C8—1221no—
587C2×C8D4×D6111yescharacteristic
588C2×C8D4×D6111yescharacteristic
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—1221no—
598C22⋊C4—1221no—
599C22⋊C4—1221no—
600C22⋊C4—2441no—
601C22⋊C4—2441no—
602C22⋊C4—2441no—
603C22⋊C4—2441no—
604C42—611no—
605C42—611no—
606C42—611no—
607C42—611no—
608C42—1221no—
609C42—1221no—
610C42—1221no—
611C23—21no—
612C23—22no—
613C23—41no—
614C23—41no—
615C23—81no—
616C23—81no—
617C23—1222no—
618C23—4881no—
619C23—4881no—
620C23—2441no—
621C23—2441no—
622C23#41×D6111yescharacteristic
623Q8—21no—
624Q8#1451973712yes—
625D4—22no—
626D4—22no—
627D4—22no—
628D4—22no—
629D4—21no—
630D4—41no—
631D4—41no—
632D4—1222no—
633D4—1222no—
634D4—2441no—
635D4—2441no—
636D4#1451973712yes—
637C2×C4—21no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—41no—
641C2×C4—81no—
642C2×C4—611no—
643C2×C4—611no—
644C2×C4—611no—
645C2×C4—611no—
646C2×C4—1221no—
647C2×C4—1221no—
648C2×C4—4882no—
649C2×C4—4882no—
650C2×C4—4882no—
651C2×C4—4882no—
652C2×C4—4881no—
653C2×C4—4881no—
654C2×C4—4881no—
655C2×C4—2441no—
656C2×C4—2441no—
657C2×C4—2441no—
658C2×C4C22×C4×D6111yescharacteristic
659C2×C4C2×D4×D6111yescharacteristic
660C2×C4C2×D4×D6111yescharacteristic
661C2×C4#41×D6111yescharacteristic
662C2×C4C2×#1175822411yescharacteristic
663C2×C4C2×#1175822411yescharacteristic
664C8—22no—
665C8—21no—
666C8—22no—
667C8—21no—
668C8—41no—
669C8—612no—
670C8—612no—
671C8—1222no—
672C8—1222no—
673C8—2441no—
674C22—21no—
675C22—21no—
676C22—22no—
677C22—41no—
678C22—41no—
679C22—42no—
680C22—82no—
681C22—82no—
682C22—81no—
683C22—81no—
684C22—4882no—
685C22—4882no—
686C22—4881no—
687C22—4881no—
688C22—2442no—
689C22C4×#1175822411yescharacteristic
690C22#121×D6111yescharacteristic
691C22#1684028911yescharacteristic
692C4—22no—
693C4—81no—
694C4—1221no—
695C4—1221no—
696C4—1221no—
697C4—1221no—
698C4—4881no—
699C4—2441no—
700C4—2441no—
701C4C4×D4×D6111yescharacteristic
702C4C2×#1451973711yescharacteristic
703C4#1684026811yescharacteristic
704C4#1684030711yescharacteristic
705C2—21no—
706C2—21no—
707C2—42no—
708C2—82no—
709C2—4882no—
710C2#2144832911yescharacteristic
711C2#2145554211yescharacteristic
712C2#2145554511yescharacteristic
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1907C2
socleC22×C61C2×#121
prime radicalC2×#16840030C2
cosocleC22×C4×C61C24
hypercenter#1907D61
hypocenterC61#21555
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1907

Complements

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

NH ≅ G/Ntype
2. #21454632C2semidirect
3. AGL(1,ℤ/8)×Dic61C2semidirect
4. C2×#16839871C2semidirect
5. #21454245C2semidirect
6. #21446726C2semidirect
7. #21446598C2semidirect
8. #21446482C2semidirect
9. #21446478C2semidirect
10. #21446327C2semidirect
11. #21445876C2semidirect
12. #21445724C2semidirect
13. #100×Dic61C2semidirect
14. #21445610C2semidirect
15. C61×#1907C2semidirect
22. #16840030C4semidirect
72. #14519802C2×C4semidirect
78. C2×#11758199C2×C4semidirect
80. #14519784C2×C4semidirect
109. #14519712C2×C4semidirect
124. #14519694C2×C4semidirect
126. #14519693C2×C4semidirect
144. AGL(1,ℤ/8)×C61C2×C4semidirect
359. #1907D61semidirect
424. C61#21555semidirect
433. AGL(1,ℤ/8)C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400302
#214546322
AGL(1,ℤ/8)×Dic612
C2×#168398712
#214542452
#214467262
#214465982
#214464822
#214464782
#214463272
#214458762
#214457242
#100×Dic612
#214456102
C61×#19072
#2155561