Finite Group Databasebeta

← #29729845

Subgroups — #29729845

Order 15616.

690 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456208C211yesmaximal, characteristic
2C2×#16839984C211yesmaximal, characteristic
3#21454768C211yesmaximal, characteristic, complemented
4#21447183C211yesmaximal, characteristic, complemented
5#21447176C211yesmaximal, characteristic
6#21447086C211yesmaximal, characteristic, complemented
7#21447079C211yesmaximal, characteristic
8#21446733C211yesmaximal, characteristic, complemented
9#21446721C211yesmaximal, characteristic
10#21446666C211yesmaximal, characteristic
11#21446592C211yesmaximal, characteristic, complemented
12#21446480C211yesmaximal, characteristic
13#21446473C211yesmaximal, characteristic, complemented
14C4×#14519696C211yesmaximal, characteristic, complemented
15C61×#3078C211yesmaximal, characteristic, complemented
16C22×#11758202C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19#16840124—21no—
20#16840124—21no—
21#16840119—21no—
22#16840119—21no—
23#16839984—21no—
24#16839984—21no—
25#16839984—21no—
26#16839984—21no—
27#16839893—21no—
28#16839893—21no—
29#16839893—21no—
30Q8×#92418—21no—
31C2×#14519712C2211yescharacteristic
32C2×#14519712C2211yescharacteristic
33#16839789C2211yescharacteristic
34#16839789C2211yescharacteristic
35C2×C4⋊#92418C2211yescharacteristic
36#16839770C2211yescharacteristic
37#16839770C2211yescharacteristic
38#16839766C2211yescharacteristic
39#16839766C2211yescharacteristic
40C2×#14519700C2211yescharacteristic
41C2×#14519697C2211yescharacteristic
42C2×#14519696C2211yescharacteristic
43C2×#14519696C2211yescharacteristic
44C2×#14519695C2211yescharacteristic
45C2×#14519695C2211yescharacteristic
46C2×#14519693C2211yescharacteristic
47C2×#14519693C2211yescharacteristic
48#16839718C2211yescharacteristic
49#16839718C2211yescharacteristic
50C4×#11758194C2211yescharacteristic
51C4×#11758194C2211yescharacteristic
52#16839531C2211yescharacteristic
53#16839531C2211yescharacteristic
54#16839526C2211yescharacteristic
55#16839526C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C2×C4×Q8×C61C2211yescharacteristic
58C61×#479—21no—
59C61×#479—21no—
60C61×#477—21no—
61C61×#475—21no—
62C61×#428C2211yescharacteristic
63C61×#428C2211yescharacteristic
64C61×#423C2211yescharacteristic
65C61×#419C2211yescharacteristic
66C61×#419C2211yescharacteristic
67C22×Dic122C2311yescharacteristic
68C22×D244C2311yescharacteristic
69C2×#11758202—22no—
70C2×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758202—41no—
73C2×#11758201—22no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758201—41no—
77C2×#11758194C2311yescharacteristic
78C2×#11758194C2311yescharacteristic
79C2×#11758194C2311yescharacteristic
80C2×#11758194C2311yescharacteristic
81C22×#92418C2311yescharacteristic
82C22×#92418C2311yescharacteristic
83#14519765—21no—
84#14519765—21no—
85#14519765—21no—
86#14519765—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89#14519712—41no—
90#14519712—41no—
91C4⋊#92418—21no—
92C4⋊#92418—21no—
93C4⋊#92418—21no—
94C4⋊#92418—21no—
95C4⋊#92418D411yescharacteristic
96C4⋊#92418D411yescharacteristic
97C4⋊#92418D411yescharacteristic
98C4⋊#92418D411yescharacteristic
99#14519700—21no—
100#14519700—21no—
101#14519700—21no—
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—21no—
110#14519697—21no—
111#14519696—22no—
112#14519696—22no—
113#14519695—22no—
114#14519695—41no—
115#14519693—22no—
116#14519693—41no—
117#14519688—21no—
118#14519688—21no—
119#14519688—21no—
120#14519688—21no—
121C4×#92418—21no—
122C4×#92418—21no—
123C4×#92418—21no—
124C4×#92418—21no—
125C22×Q8×C61C2311yescharacteristic
126C2×M16×C61C2311yescharacteristic
127C2×M16×C61C2311yescharacteristic
128C4⋊Q8×C61D411yescharacteristic
129C4⋊Q8×C61D411yescharacteristic
130#125×C61D411yescharacteristic
131#125×C61D411yescharacteristic
132C22⋊Q8×C61—21no—
133C22⋊Q8×C61—21no—
134C4×Q8×C61—21no—
135C4×Q8×C61—21no—
136C4×Q8×C61—21no—
137C4×Q8×C61—21no—
138C4×Q8×C61—21no—
139C4×Q8×C61—21no—
140#117×C61C2311yescharacteristic
141#117×C61C2311yescharacteristic
142C2×C4⋊C4×C61—21no—
143C2×C4⋊C4×C61C2311yescharacteristic
144C2×C42×C61—21no—
145C2×C42×C61C2311yescharacteristic
146#107×C61—21no—
147#107×C61—21no—
148#106×C61—21no—
149#106×C61—21no—
150C4⋊C8×C61—21no—
151C4⋊C8×C61—21no—
152#103×C61—21no—
153#103×C61—21no—
154#103×C61—21no—
155#103×C61—21no—
156C23×D61—21no—
157C2×Q8×C61—21no—
158C2×Q8×C61—21no—
159C2×Q8×C61—21no—
160C2×Q8×C61—21no—
161C2×Q8×C61—21no—
162C2×Q8×C61—21no—
163C2×Q8×C61—21no—
164C2×Q8×C61#4112yes—
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61#4111yescharacteristic
168C22×C4×C61#4111yescharacteristic
169C22×C4×C61C2411yescharacteristic
170C22×Dic61—21no—
171C2×D244—21no—
172C2×D244—21no—
173C2×D244—21no—
174C2×D244—21no—
175C2×D244—21no—
176C2×D244—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183M16×C61—21no—
184M16×C61—21no—
185M16×C61—21no—
186M16×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—41no—
198C4⋊C4×C61—41no—
199C4⋊C4×C61C2×D411yescharacteristic
200C4⋊C4×C61C2×D411yescharacteristic
201C4⋊C4×C61C2×D411yescharacteristic
202C4⋊C4×C61C2×D411yescharacteristic
203C22⋊C4×C61—21no—
204C22⋊C4×C61—21no—
205C42×C61—21no—
206C42×C61—21no—
207C42×C61—41no—
208C42×C61—41no—
209C42×C61C2×D411yescharacteristic
210C42×C61C2×D411yescharacteristic
211C42×C61C2×D411yescharacteristic
212C42×C61C2×D411yescharacteristic
213#11758202—42no—
214#11758202—42no—
215#11758201—42no—
216#11758201—42no—
217#11758198—41no—
218#11758198—41no—
219#11758198—41no—
220#11758198—41no—
221#11758194—21no—
222#11758194—21no—
223#11758194—21no—
224#11758194—21no—
225#11758194—21no—
226#11758194—21no—
227#11758194—21no—
228#11758194#4112yes—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418—21no—
233C2×#92418—21no—
234C2×#92418—21no—
235C2×#92418—21no—
236C2×#92418—21no—
237C2×#92418C2×D411yescharacteristic
238C2×#92418C2×D411yescharacteristic
239C2×#92418C2×D411yescharacteristic
240C2×#92418C2×D411yescharacteristic
241C23×C61C2×#4111yescharacteristic
242C22×D61—41no—
243C22×D61—41no—
244C22×D61—41no—
245C22×D61—81no—
246C22×D61—81no—
247Q8×C61—22no—
248Q8×C61—22no—
249Q8×C61—42no—
250Q8×C61—41no—
251Q8×C61—41no—
252Q8×C61—81no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—22no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—81no—
264C2×C4×C61#12111yescharacteristic
265C2×C4×C61#12111yescharacteristic
266C2×C4×C61#12111yescharacteristic
267C2×C4×C61#12111yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C2×#4111yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C2×C4×C61H(5,𝔽2)11yescharacteristic
273C2×C4×C612−1+411yescharacteristic
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277D244—41no—
278D244—41no—
279D244—41no—
280D244—41no—
281Dic122—41no—
282Dic122—41no—
283Dic122—41no—
284Dic122—41no—
285C8×C61—41no—
286C8×C61—41no—
287#92418—21no—
288#92418—21no—
289#92418—21no—
290#92418—21no—
291#92418—22no—
292#92418—41no—
293#92418—41no—
294#92418—41no—
295#25317—611nomaximal, Sylow
296C22×C61—21no—
297C22×C61—21no—
298C22×C61C2×#12111yescharacteristic
299C22×C61#52911yescharacteristic
300C22×C61#54211yescharacteristic
301C2×D61—81no—
302C2×D61—81no—
303C2×D61—81no—
304C2×D61—161no—
305C2×D61—161no—
306C4×C61—21no—
307C4×C61—21no—
308C4×C61—21no—
309C4×C61—21no—
310C4×C61—42no—
311C4×C61—41no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61—41no—
315C4×C61—81no—
316C4×C61#54611yescharacteristic
317C4×C61#54811yescharacteristic
318C4×C61#54811yescharacteristic
319C4×C61#55011yescharacteristic
320Dic61—81no—
321#3140—611no—
322#3078D6111yescharacteristic, complemented
323C2×#464—611no—
324#2962—611no—
325#2064—611no—
326#2063—611no—
327#2047—611no—
328#2046—611no—
329#1979—611no—
330#1977—611no—
331#1967—611no—
332#1950—611no—
333#1931—611no—
334#1930—611no—
335C4×#101—611no—
336C2×C61—21no—
337C2×C61—21no—
338C2×C61#242611yescharacteristic
339C2×C61#306711yescharacteristic
340C2×C61#306911yescharacteristic
341D61—161no—
342C22×Q16—611no—
343C22×SD16—611no—
344#533C2×D6111yescharacteristic
345C2×#124—611no—
346C2×C4×Q8C2×D6111yescharacteristic
347#489—1221no—
348#489—1221no—
349#488—1221no—
350#488—1221no—
351#479—21no—
352#479—21no—
353#477—21no—
354#475—21no—
355#464—1221no—
356#464—1221no—
357#464—1221no—
358#464—1221no—
359#446—1221no—
360#446—1221no—
361#446—1221no—
362C8×Q8—1221no—
363C2×#108—611no—
364C2×#108—611no—
365#428C2×D6111yescharacteristic
366#428C2×D6111yescharacteristic
367#424—611no—
368#424—611no—
369#423C2×D6111yescharacteristic
370C2×C4⋊C8—611no—
371#419—611no—
372#419—611no—
373#419C2×D6111yescharacteristic
374#419C2×D6111yescharacteristic
375#418—611no—
376#418—611no—
377C2×#103—611no—
378C2×#102—611no—
379C2×#101—611no—
380C2×#101—611no—
381C2×#101—611no—
382C2×#101—611no—
383C2×#100—611no—
384C2×#100—611no—
385#405—611no—
386#405—611no—
387C4×M16—611no—
388C4×M16—611no—
389#343—611no—
390#343—611no—
391#341—611no—
392#341—611no—
393C61#2531711yescharacteristic, complemented, Sylow
394C22×Q8—611no—
395C22×Q8C22×D6111yescharacteristic
396C22×D4—611no—
397C2×Q16—1221no—
398C2×Q16—1221no—
399C2×Q16—1222no—
400C2×Q16—2441no—
401C2×SD16—1221no—
402C2×SD16—1221no—
403C2×SD16—1222no—
404C2×SD16—2441no—
405C2×M16—611no—
406C2×M16—611no—
407C2×M16—611no—
408C2×M16—611no—
409C2×M16C22×D6111yescharacteristic
410C2×M16C22×D6111yescharacteristic
411C22×C8—611no—
412C22×C8—611no—
413C4⋊Q8C22⋊D6111yescharacteristic
414C4⋊Q8C22⋊D6111yescharacteristic
415#125C22⋊D6111yescharacteristic
416#125C22⋊D6111yescharacteristic
417#124—1221no—
418#124—1221no—
419#124—1221no—
420#124—1221no—
421C22⋊Q8—21no—
422C22⋊Q8—21no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×Q8—21no—
427C4×Q8—21no—
428C4×Q8—21no—
429#117C22×D6111yescharacteristic
430#117C22×D6111yescharacteristic
431C2×C4⋊C4—21no—
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C42—21no—
436C2×C42C22×D6111yescharacteristic
437#108—2441no—
438#108—2441no—
439#107—21no—
440#107—21no—
441#106—21no—
442#106—21no—
443C4⋊C8—21no—
444C4⋊C8—21no—
445C4⋊C8—611no—
446C4⋊C8—611no—
447C4⋊C8—611no—
448C4⋊C8—611no—
449C4⋊C8—1221no—
450C4⋊C8—1221no—
451C4⋊C8—1221no—
452C4⋊C8—1221no—
453#103—21no—
454#103—21no—
455#103—21no—
456#103—21no—
457#103—1221no—
458#103—1221no—
459#103—1221no—
460#103—1221no—
461#103—1221no—
462#103—1221no—
463#102—1221no—
464#102—1221no—
465#102—1221no—
466#102—1221no—
467#102—1221no—
468#102—1221no—
469#101—1222no—
470#101—1222no—
471#101—1222no—
472#101—2441no—
473#100—1222no—
474#100—2441no—
475#97—1221no—
476#97—1221no—
477#97—1221no—
478#97—1221no—
479C4×C8—1221no—
480C4×C8—1221no—
481C4×C8—1221no—
482C4×C8—1221no—
483C24—1221no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—1221no—
492C2×Q8—1221no—
493C2×Q8—1221no—
494C2×Q8—1221no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8#1175822612yes—
498C2×D4—1221no—
499C2×D4—1221no—
500C2×D4—1221no—
501C2×D4—1221no—
502C2×D4—1221no—
503C2×D4—1221no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—1221no—
507C22×C4#1175822411yescharacteristic
508C22×C4#1175822411yescharacteristic
509C22×C4C23×D6111yescharacteristic
510Q16—2442no—
511Q16—2442no—
512SD16—2442no—
513SD16—2442no—
514M16—21no—
515M16—21no—
516M16—21no—
517M16—21no—
518M16—612no—
519M16—1221no—
520M16—1221no—
521M16—1221no—
522M16—1221no—
523M16—1221no—
524M16—1221no—
525M16—1221no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—611no—
529C2×C8—611no—
530C2×C8—611no—
531C2×C8—611no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1221no—
539C2×C8—1221no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—41no—
549C4⋊C4—41no—
550C4⋊C4C2×C22⋊D6111yescharacteristic
551C4⋊C4C2×C22⋊D6111yescharacteristic
552C4⋊C4C2×C22⋊D6111yescharacteristic
553C4⋊C4C2×C22⋊D6111yescharacteristic
554C22⋊C4—21no—
555C22⋊C4—21no—
556C22⋊C4—2441no—
557C22⋊C4—2441no—
558C22⋊C4—2441no—
559C22⋊C4—2441no—
560C42—21no—
561C42—21no—
562C42—41no—
563C42—41no—
564C42D4×D6111yescharacteristic
565C42D4×D6111yescharacteristic
566C42C2×C22⋊D6111yescharacteristic
567C42C2×C22⋊D6111yescharacteristic
568C23—4881no—
569C23—4881no—
570C23—2441no—
571C23—2441no—
572C23—2441no—
573C23C2×#1175822411yescharacteristic
574Q8—22no—
575Q8—22no—
576Q8—42no—
577Q8—41no—
578Q8—41no—
579Q8—81no—
580Q8—2441no—
581Q8—2441no—
582Q8—2441no—
583Q8—2441no—
584D4—2441no—
585D4—2441no—
586D4—2441no—
587D4—2441no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—81no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4#1451974911yescharacteristic
603C2×C4#1451974911yescharacteristic
604C2×C4#1451974911yescharacteristic
605C2×C4#1451974911yescharacteristic
606C2×C4C2×D4×D6111yescharacteristic
607C2×C4C22×C22⋊D6111yescharacteristic
608C2×C4#41×D6111yescharacteristic
609C2×C4C2×#1175822611yescharacteristic
610C2×C4#1451982311yescharacteristic
611C2×C4#1451982511yescharacteristic
612C8—41no—
613C8—41no—
614C8—1221no—
615C8—1221no—
616C8—1221no—
617C8—1221no—
618C8—1222no—
619C8—2441no—
620C8—2441no—
621C8—2441no—
622C22—21no—
623C22—21no—
624C22—9761no—
625C22—9761no—
626C22—4881no—
627C22—4881no—
628C22—4881no—
629C22C2×#1451974911yescharacteristic
630C22#1684029811yescharacteristic
631C22#1684038511yescharacteristic
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—42no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—81no—
642C4—4881no—
643C4#1684040311yescharacteristic
644C4#1684042211yescharacteristic
645C4#1684042311yescharacteristic
646C4#1684043111yescharacteristic
647C2—21no—
648C2—21no—
649C2—9761no—
650C2#2144938711yescharacteristic
651C2#2145553611yescharacteristic
652C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3078C2
socleC22×C61C2×#121
prime radicalC22×D244C23
hypercenter#3078D61
hypocenterC61#25317
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
3. #21454768C2semidirect
4. #21447183C2semidirect
6. #21447086C2semidirect
8. #21446733C2semidirect
11. #21446592C2semidirect
13. #21446473C2semidirect
14. C4×#14519696C2semidirect
15. C61×#3078C2semidirect
322. #3078D61semidirect
393. C61#25317semidirect

Maximal subgroups

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

subgroupindex
#214562082
C2×#168399842
#214547682
#214471832
#214471762
#214470862
#214470792
#214467332
#214467212
#214466662
#214465922
#214464802
#214464732
C4×#145196962
C61×#30782
#2531761