Finite Group Databasebeta

← #29686943

Subgroups — #29686943

Order 15616.

906 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840028C211yesmaximal, characteristic
2C4×#14519808C211yesmaximal, characteristic, complemented
3C4×#14519805C211yesmaximal, characteristic
4C2×#16839860C211yesmaximal, characteristic, complemented
5#21454248C211yesmaximal, characteristic, complemented
6#21446723C211yesmaximal, characteristic, complemented
7#21446603C211yesmaximal, characteristic, complemented
8#21446479C211yesmaximal, characteristic, complemented
9#21446476C211yesmaximal, characteristic, complemented
10#21446331C211yesmaximal, characteristic, complemented
11#21446328C211yesmaximal, characteristic
12#21445882C211yesmaximal, characteristic, complemented
13#21445726C211yesmaximal, characteristic, complemented
14C4×#14519693C211yesmaximal, characteristic, complemented
15C4×#101×C61C211yesmaximal, characteristic, complemented
16C2×#14519808C2211yescharacteristic
17C2×#14519805C2211yescharacteristic
18C22×#11758201C2211yescharacteristic
19C2×#14519785C2211yescharacteristic
20C2×#14519785C2211yescharacteristic
21C4×#11758222C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23#16840028C418yescomplemented
24#16839875—21no—
25#16839867—21no—
26#16839864—21no—
27#16839860—21no—
28#16839860—21no—
29#16839860—21no—
30#16839860—21no—
31#16839860—21no—
32C4×#11758202—21no—
33C4×#11758190—21no—
34C4×#11758201—21no—
35C4×D488—21no—
36C2×#14519712C2211yescharacteristic
37C2×#14519712C2211yescharacteristic
38#16839804C2211yescharacteristic
39#16839802C2211yescharacteristic
40C2×#14519710C2211yescharacteristic
41#16839771C2211yescharacteristic
42#16839768C2211yescharacteristic
43#16839764C2211yescharacteristic
44#16839761C2211yescharacteristic
45C2×#14519701C2211yescharacteristic
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×Q8×C61C2211yescharacteristic
62C2×#101×C61C2211yescharacteristic
63C2×#101×C61C2211yescharacteristic
64C4×M16×C61C2211yescharacteristic
65C4×M16×C61C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C61×#343C2211yescharacteristic
68C2×#11758222C2311yescharacteristic
69C22×D244C2311yescharacteristic
70C22×C4×D61—21no—
71#14519808—22no—
72#14519808C2×C414yescomplemented
73#14519805—22no—
74#14519805C2×C414yes—
75C2×#11758202—21no—
76C2×#11758190—21no—
77C2×#11758201—21no—
78C2×#11758201—21no—
79C2×#11758201—21no—
80C2×#11758201—22no—
81C2×#11758201C2×C414yescomplemented
82C2×D488—21no—
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#14519710—21no—
115#14519710—21no—
116#14519709—21no—
117#14519708—21no—
118#14519702—21no—
119#14519701—21no—
120#14519701—21no—
121#14519701—21no—
122#14519701—21no—
123#14519700—21no—
124#14519699—21no—
125#14519699—21no—
126#14519699—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×Q8×C61C2311yescharacteristic
143C2×M16×C61C2311yescharacteristic
144C2×M16×C61C2311yescharacteristic
145C2×M16×C61C2311yescharacteristic
146C2×M16×C61C2311yescharacteristic
147C4×Q8×C61—21no—
148C4×Q8×C61—21no—
149C4×Q8×C61—21no—
150C4×Q8×C61—21no—
151C2×C4⋊C4×C61—21no—
152C2×C4⋊C4×C61C2311yescharacteristic
153C2×C42×C61—21no—
154C2×C42×C61C2311yescharacteristic
155#101×C61—22no—
156#101×C61C2×C414yescomplemented
157#97×C61—21no—
158#97×C61—21no—
159C4×C8×C61—21no—
160C4×C8×C61—21no—
161C23×D61—21no—
162C2×Q8×C61—21no—
163C2×Q8×C61—21no—
164C2×Q8×C61—21no—
165C2×Q8×C61—21no—
166C2×Q8×C61—21no—
167C2×Q8×C61C22×C412yes—
168C22×C4×C61—21no—
169C22×C4×C61—21no—
170C22×C4×C61#4111yescharacteristic
171C22×C4×C61#4111yescharacteristic
172C22×C4×C61C2411yescharacteristic
173C2×C22⋊D61—21no—
174C22×Dic61—21no—
175#11758222—21no—
176#11758222—21no—
177#11758222C22×C412yes—
178#11758222#4112yes—
179C2×D244—21no—
180C2×D244—21no—
181C2×D244—21no—
182C2×D244—21no—
183C2×D244—21no—
184C2×D244C22×C412yes—
185C2×D244C2×D411yescharacteristic
186C2×C4×D61—21no—
187C2×C4×D61—21no—
188C2×C4×D61—21no—
189C2×C4×D61—41no—
190C2×C4×D61—41no—
191C2×C4×D61—82no—
192C2×Dic122C2×D411yescharacteristic
193M16×C61—21no—
194M16×C61—21no—
195M16×C61—21no—
196M16×C61—21no—
197M16×C61—21no—
198M16×C61C22×C412yes—
199M16×C61C22×C412yes—
200M16×C61#4112yes—
201C2×C8×C61—21no—
202C2×C8×C61—21no—
203C2×C8×C61—21no—
204C2×C8×C61—21no—
205C4⋊C4×C61—21no—
206C4⋊C4×C61—21no—
207C4⋊C4×C61—41no—
208C4⋊C4×C61—41no—
209C42×C61—41no—
210C42×C61—41no—
211C42×C61C2×D411yescharacteristic
212C42×C61C2×D411yescharacteristic
213C42×C61C2×D411yescharacteristic
214C42×C61C2×D411yescharacteristic
215C22⋊Dic61—21no—
216#11758202—24no—
217#11758201—24no—
218#11758201—24no—
219#11758201—24no—
220#11758198—21no—
221#11758198—21no—
222#11758198—41no—
223#11758198—41no—
224C4⋊Dic61—21no—
225C4⋊Dic61—21no—
226C4⋊Dic61C2×D411yescharacteristic
227C4⋊Dic61C2×D411yescharacteristic
228#11758196—21no—
229#11758196—21no—
230C4×Dic61—21no—
231C4×Dic61—21no—
232C4×Dic61—21no—
233#11758194—21no—
234#11758194—21no—
235#11758194—21no—
236#11758194C22×C412yes—
237C2×#92418—21no—
238C2×#92418—21no—
239C2×#92418—21no—
240C2×#92418—21no—
241C2×#92418C22×C412yes—
242C2×#92418C2×D411yescharacteristic
243C2×#92418C2×D411yescharacteristic
244C2×#92418C2×D411yescharacteristic
245C2×#92418C2×D411yescharacteristic
246C2×#92418#4112yes—
247D488—24no—
248#11758190—24no—
249#11758189—24no—
250#11758189—24no—
251C8×D61—24no—
252C8×D61—24no—
253C23×C61C2×#4111yescharacteristic
254C22×D61—21no—
255C22×D61—22no—
256C22×D61—41no—
257C22×D61—41no—
258C22×D61—81no—
259C22×D61—81no—
260Q8×C61—22no—
261Q8×C61—22no—
262Q8×C61—41no—
263Q8×C61—41no—
264C2×C4×C61—22no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—81no—
271C2×C4×C61#12111yescharacteristic
272C2×C4×C61#12111yescharacteristic
273C2×C4×C61#12111yescharacteristic
274C2×C4×C61#12111yescharacteristic
275C2×C4×C61C23×C411yescharacteristic
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C22×D411yescharacteristic
278C2×C4×C61C2×#4111yescharacteristic
279C2×C4×C61C2×#4111yescharacteristic
280C2×C4×C61C2×#4111yescharacteristic
281C22⋊D61—22no—
282C22⋊D61—22no—
283C2×Dic61—21no—
284C2×Dic61—21no—
285C2×Dic61—21no—
286C2×Dic61—41no—
287C2×Dic61—81no—
288D244—22no—
289D244—22no—
290D244—21no—
291D244—41no—
292D244—41no—
293D244C4×D412yes—
294C4×D61—22no—
295C4×D61—22no—
296C4×D61—81no—
297C4×D61—81no—
298C4×D61—82no—
299C4×D61—82no—
300Dic122—21no—
301Dic122C4×D412yes—
302C8×C61—22no—
303C8×C61—22no—
304C8×C61—22no—
305C8×C61—41no—
306#92418—22no—
307#92418—21no—
308#92418—21no—
309#92418—41no—
310#92418C4×D412yes—
311#92418C4×D412yes—
312#21557—611nomaximal, Sylow
313C22×C61—21no—
314C22×C61—21no—
315C22×C61C4×#4111yescharacteristic
316C22×C61C2×#12111yescharacteristic
317C22×C61#52911yescharacteristic
318C2×D61—22no—
319C2×D61—41no—
320C2×D61—41no—
321C2×D61—42no—
322C2×D61—82no—
323C2×D61—81no—
324C2×D61—81no—
325C2×D61—82no—
326C4×C61—21no—
327C4×C61—21no—
328C4×C61—21no—
329C4×C61—21no—
330C4×C61—42no—
331C4×C61—81no—
332C4×C61C2×C4×D411yescharacteristic
333C4×C61C2×C4×D411yescharacteristic
334C4×C61#52511yescharacteristic
335C4×C61#53211yescharacteristic
336Dic61—22no—
337Dic61—41no—
338Dic61—41no—
339Dic61—81no—
340C2×#471—611no—
341C4×#137—611no—
342C4×AGL(1,ℤ/8)—611no—
343C2×#440—611no—
344#2875—611no—
345#1977—611no—
346#1952—611no—
347#1931—611no—
348#1930—611no—
349#1908—611no—
350#1907—611no—
351#1816—611no—
352#1778—611no—
353C4×#101D6111yescharacteristic, complemented
354C4×#100—611no—
355C2×C61—21no—
356C2×C61—21no—
357C2×C61C4×#12111yescharacteristic
358C2×C61#306811yescharacteristic
359C2×C61#306911yescharacteristic
360D61—42no—
361D61—82no—
362C2×#137—611no—
363C2×AGL(1,ℤ/8)—611no—
364C22×SD16—611no—
365C2×#131—611no—
366C2×#131—611no—
367C4×#41—611no—
368C2×C4×Q8C2×D6111yescharacteristic
369C2×C4×D4—611no—
370#471—618no—
371#442—1221no—
372#441—1221no—
373#440—1221no—
374#440—1221no—
375#440—1221no—
376#440—1221no—
377#440—1221no—
378#440—1221no—
379C4×Q16—1221no—
380C4×SD16—1221no—
381C4×SD16—1221no—
382C4×D8—1221no—
383C2×#108—611no—
384C2×#108—611no—
385#428—611no—
386#428—611no—
387C2×#107—611no—
388#419—611no—
389#418—611no—
390#417—611no—
391#417—611no—
392C2×#103—611no—
393C2×#102—611no—
394C2×#101C2×D6111yescharacteristic
395C2×#101C2×D6111yescharacteristic
396C2×#100—611no—
397C2×#100—611no—
398#405—611no—
399#405—611no—
400#405—1221no—
401#405—1221no—
402C4×M16—611no—
403C4×M16—1221no—
404C4×M16C2×D6111yescharacteristic
405C4×M16C2×D6111yescharacteristic
406C2×#97—611no—
407C2×#97—1221no—
408#343—611no—
409#343—611no—
410#343C2×D6111yescharacteristic
411#343C2×D6111yescharacteristic
412#341—611no—
413#341—611no—
414C61#2155711yescharacteristic, complemented, Sylow
415C2×#41—611no—
416C22×Q8C22×D6111yescharacteristic
417C22×D4—611no—
418C23×C4—1221no—
419#137—614no—
420#137—1222no—
421AGL(1,ℤ/8)—614no—
422AGL(1,ℤ/8)—1222no—
423C2×Q16—1221no—
424C2×SD16—614no—
425C2×SD16—1221no—
426C2×SD16—1221no—
427C2×SD16—1221no—
428C2×SD16—1222no—
429C2×SD16—1221no—
430C2×D8—1221no—
431#131—614no—
432#131—1222no—
433#131—1222no—
434#131—1222no—
435C2×M16—611no—
436C2×M16—611no—
437C2×M16—1221no—
438C2×M16—1221no—
439C2×M16C22×D6111yescharacteristic
440C2×M16C22×D6111yescharacteristic
441C2×M16C22×D6111yescharacteristic
442C2×M16C22×D6111yescharacteristic
443C22×C8—611no—
444C22×C8—611no—
445C22×C8—1221no—
446C22×C8—1221no—
447C4×Q8—21no—
448C4×Q8—21no—
449C4×Q8—21no—
450C4×Q8—21no—
451C4×Q8—611no—
452C4×Q8—611no—
453C4×D4—611no—
454C4×D4—611no—
455C4×D4—1221no—
456C4×D4—1221no—
457C4×D4—1221no—
458C4×D4—1221no—
459C4×D4—1221no—
460C4×D4—1221no—
461#117—611no—
462#117—1221no—
463C2×C4⋊C4—21no—
464C2×C4⋊C4—611no—
465C2×C4⋊C4C22×D6111yescharacteristic
466C2×C22⋊C4—1221no—
467C2×C42—21no—
468C2×C42—1221no—
469C2×C42C22×D6111yescharacteristic
470#108—614no—
471#108—1222no—
472#107—1221no—
473#107—1221no—
474#107—1221no—
475#107—1221no—
476#106—1221no—
477#106—1221no—
478#103—1221no—
479#103—1221no—
480#103—1221no—
481#103—1221no—
482#103—1221no—
483#103—1221no—
484#102—1221no—
485#102—1221no—
486#102—1221no—
487#102—1221no—
488#102—1221no—
489#102—1221no—
490#101—22no—
491#101C4×D6114yescomplemented
492#100—614no—
493#100—1222no—
494#97—21no—
495#97—21no—
496#97—611no—
497#97—611no—
498#97—611no—
499#97—611no—
500#97—1221no—
501#97—1221no—
502#97—1221no—
503C4×C8—21no—
504C4×C8—21no—
505C4×C8—1221no—
506C4×C8—1221no—
507C4×C8—1221no—
508C24—1221no—
509#41—612no—
510#41—612no—
511#41—1221no—
512#41—1221no—
513C2×Q8—21no—
514C2×Q8—21no—
515C2×Q8—21no—
516C2×Q8—21no—
517C2×Q8—21no—
518C2×Q8—611no—
519C2×Q8C2×C4×D6112yes—
520C2×D4—612no—
521C2×D4—611no—
522C2×D4—1221no—
523C2×D4—1221no—
524C2×D4—1221no—
525C2×D4—1221no—
526C2×D4—1221no—
527C2×D4—1221no—
528C22×C4—21no—
529C22×C4—21no—
530C22×C4—1221no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—1221no—
534C22×C4—4882no—
535C22×C4—2441no—
536C22×C4—2441no—
537C22×C4#1175822211yescharacteristic
538C22×C4#1175822211yescharacteristic
539C22×C4C23×D6111yescharacteristic
540Q16—1224no—
541SD16—1224no—
542SD16—1224no—
543SD16—1224no—
544SD16—1224no—
545D8—1224no—
546M16—21no—
547M16—21no—
548M16—21no—
549M16—21no—
550M16—21no—
551M16—612no—
552M16—1224no—
553M16—1224no—
554M16—1221no—
555M16—1221no—
556M16—1221no—
557M16C2×C4×D6112yes—
558M16C2×C4×D6112yes—
559M16#1175822212yes—
560C2×C8—21no—
561C2×C8—21no—
562C2×C8—21no—
563C2×C8—21no—
564C2×C8—612no—
565C2×C8—611no—
566C2×C8—611no—
567C2×C8—612no—
568C2×C8—611no—
569C2×C8—611no—
570C2×C8—1224no—
571C2×C8—1224no—
572C2×C8—1221no—
573C2×C8—1221no—
574C2×C8—1221no—
575C2×C8—1221no—
576C4⋊C4—21no—
577C4⋊C4—21no—
578C4⋊C4—41no—
579C4⋊C4—41no—
580C4⋊C4—611no—
581C4⋊C4—611no—
582C4⋊C4—1221no—
583C4⋊C4—1221no—
584C4⋊C4—1221no—
585C4⋊C4—1221no—
586C22⋊C4—1221no—
587C22⋊C4—1221no—
588C22⋊C4—1221no—
589C22⋊C4—2441no—
590C22⋊C4—2441no—
591C42—41no—
592C42—41no—
593C42—1221no—
594C42—1221no—
595C42—1221no—
596C42D4×D6111yescharacteristic
597C42D4×D6111yescharacteristic
598C42D4×D6111yescharacteristic
599C42D4×D6111yescharacteristic
600C23—1221no—
601C23—1222no—
602C23—4881no—
603C23—4881no—
604C23—2441no—
605C23—2441no—
606C23C2×#1175822211yescharacteristic
607Q8—22no—
608Q8—22no—
609Q8—41no—
610Q8—41no—
611Q8—612no—
612Q8—1221no—
613D4—612no—
614D4—1222no—
615D4—1222no—
616D4—1222no—
617D4—1222no—
618D4—1221no—
619D4—2441no—
620D4—2441no—
621C2×C4—22no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—81no—
628C2×C4—1221no—
629C2×C4—1222no—
630C2×C4—1222no—
631C2×C4—1221no—
632C2×C4—1221no—
633C2×C4—4882no—
634C2×C4—4882no—
635C2×C4—4881no—
636C2×C4—4881no—
637C2×C4—4881no—
638C2×C4—2441no—
639C2×C4#1451975011yescharacteristic
640C2×C4#1451975011yescharacteristic
641C2×C4#1451975011yescharacteristic
642C2×C4#1451975011yescharacteristic
643C2×C4C22×C4×D6111yescharacteristic
644C2×C4C2×D4×D6111yescharacteristic
645C2×C4C2×D4×D6111yescharacteristic
646C2×C4#41×D6111yescharacteristic
647C2×C4#41×D6111yescharacteristic
648C2×C4C2×#1175822211yescharacteristic
649C8—22no—
650C8—22no—
651C8—22no—
652C8—41no—
653C8—612no—
654C8—612no—
655C8—1222no—
656C8—1221no—
657C8—1221no—
658C8—2441no—
659C22—21no—
660C22—21no—
661C22—1222no—
662C22—4882no—
663C22—4882no—
664C22—4881no—
665C22—4881no—
666C22—2442no—
667C22—2441no—
668C22—2441no—
669C22C4×#1175822211yescharacteristic
670C22C2×#1451975011yescharacteristic
671C22#1684029011yescharacteristic
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—42no—
677C4—81no—
678C4—1222no—
679C4—4881no—
680C4—2441no—
681C4—2441no—
682C4C4×D4×D6111yescharacteristic
683C4C4×D4×D6111yescharacteristic
684C4#1684027111yescharacteristic
685C4#1684030911yescharacteristic
686C2—21no—
687C2—21no—
688C2—4882no—
689C2—2442no—
690C2C4×#1451975011yescharacteristic
691C2#2145555011yescharacteristic
692C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#101×C61C2
socleC22×C61C2×#121
prime radicalC2×#14519805C22
cosocleC22×C4×C61C24
hypercenterC4×#101D61
hypocenterC61#21557
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
2. C4×#14519808C2semidirect
4. C2×#16839860C2semidirect
5. #21454248C2semidirect
6. #21446723C2semidirect
7. #21446603C2semidirect
8. #21446479C2semidirect
9. #21446476C2semidirect
10. #21446331C2semidirect
12. #21445882C2semidirect
13. #21445726C2semidirect
14. C4×#14519693C2semidirect
15. C4×#101×C61C2semidirect
23. #16840028C4semidirect
72. #14519808C2×C4semidirect
81. C2×#11758201C2×C4semidirect
86. #14519785C2×C4semidirect
111. #14519712C2×C4semidirect
131. #14519693C2×C4semidirect
156. #101×C61C2×C4semidirect
353. C4×#101D61semidirect
414. C61#21557semidirect
491. #101C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400282
C4×#145198082
C4×#145198052
C2×#168398602
#214542482
#214467232
#214466032
#214464792
#214464762
#214463312
#214463282
#214458822
#214457262
C4×#145196932
C4×#101×C612
#2155761