Finite Group Databasebeta

← #29687858

Subgroups — #29687858

Order 15616.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455607C211yesmaximal, characteristic
2#21454756C211yesmaximal, characteristic, complemented
3#21454247C211yesmaximal, characteristic
4#21446762C211yesmaximal, characteristic, complemented
5#21446592C211yesmaximal, characteristic, complemented
6#21446416C211yesmaximal, characteristic, complemented
7#21446416C211yesmaximal, characteristic, complemented
8#21446092C211yesmaximal, characteristic
9#21446092C211yesmaximal, characteristic
10#21445702C211yesmaximal, characteristic, complemented
11#21445697C211yesmaximal, characteristic, complemented
12#21445694C211yesmaximal, characteristic, complemented
13#21445664C211yesmaximal, characteristic
14#21445657C211yesmaximal, characteristic
15C61×#1806C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18#16840319C2211yescharacteristic
19#16840069C411yescharacteristic, complemented
20#16840069C411yescharacteristic, complemented
21#16840062C411yescharacteristic, complemented
22#16840062C411yescharacteristic, complemented
23#16840057C411yescharacteristic, complemented
24#16840052C411yescharacteristic, complemented
25#16840047C411yescharacteristic, complemented
26#16840042C411yescharacteristic, complemented
27#16839891—21no—
28#16839891—21no—
29#16839891—21no—
30#16839891—21no—
31C2×#14519712C2211yescharacteristic
32C2×#14519712C2211yescharacteristic
33#16839798C2211yescharacteristic
34#16839798C2211yescharacteristic
35#16839790C2211yescharacteristic
36#16839790C2211yescharacteristic
37C2×C4⋊#92418C2211yescharacteristic
38#16839758C2211yescharacteristic
39#16839758C2211yescharacteristic
40#16839720C2211yescharacteristic
41#16839720C2211yescharacteristic
42#16839718—21no—
43#16839718—21no—
44#16839718C2211yescharacteristic
45#16839718C2211yescharacteristic
46C4×#11758194—21no—
47C4×#11758194C2211yescharacteristic
48C2×#14519688—21no—
49C2×#14519688C2211yescharacteristic
50C2×#14519688C2211yescharacteristic
51C2×C4×#92418C2211yescharacteristic
52#16839526C2211yescharacteristic
53#16839526C2211yescharacteristic
54#16839522C2211yescharacteristic
55#16839522C2211yescharacteristic
56#16839522C2211yescharacteristic
57#16839522C2211yescharacteristic
58#16839517C2211yescharacteristic
59#16839517C2211yescharacteristic
60C4×#41×C61C2211yescharacteristic
61C61×#423C2211yescharacteristic
62C2×#104×C61C2211yescharacteristic
63C2×#104×C61C2211yescharacteristic
64C4×C4⋊C4×C61C2211yescharacteristic
65C61×#339C2211yescharacteristic
66C61×#339C2211yescharacteristic
67#14519784—22no—
68#14519784—22no—
69#14519784—22no—
70#14519784Q814yescomplemented
71C2×#11758194—21no—
72C2×#11758194—21no—
73C2×#11758194C2311yescharacteristic
74C2×#11758194C2311yescharacteristic
75C22×#92418—21no—
76C22×#92418—21no—
77C22×#92418C2311yescharacteristic
78C22×#92418C2311yescharacteristic
79C22×#92418C2311yescharacteristic
80C22×#92418C2311yescharacteristic
81#14519777C2×C411yescharacteristic
82#14519777C2×C411yescharacteristic
83#14519769C2×C411yescharacteristic
84#14519769C2×C411yescharacteristic
85#14519756—21no—
86#14519756—21no—
87C4×Dic122—21no—
88C4×Dic122—21no—
89#14519736C2311yescharacteristic
90#14519736C2311yescharacteristic
91#14519712—22no—
92#14519712—22no—
93#14519710C2×C411yescharacteristic
94#14519710C2×C411yescharacteristic
95#14519710C2×C411yescharacteristic
96#14519710C2×C411yescharacteristic
97#14519708C2×C411yescharacteristic
98#14519708C2×C411yescharacteristic
99#14519708C2×C411yescharacteristic
100#14519708C2×C411yescharacteristic
101C4⋊#92418C2×C411yescharacteristic
102C4⋊#92418C2×C411yescharacteristic
103C4⋊#92418C2×C411yescharacteristic
104C4⋊#92418C2×C411yescharacteristic
105#14519706—21no—
106#14519706—21no—
107#14519706—21no—
108#14519706—21no—
109#14519701C2×C411yescharacteristic
110#14519701C2×C411yescharacteristic
111#14519701C2×C411yescharacteristic
112#14519701C2×C411yescharacteristic
113#14519698C2×C411yescharacteristic
114#14519698C2×C411yescharacteristic
115#14519698C2×C411yescharacteristic
116#14519698C2×C411yescharacteristic
117#14519688—21no—
118#14519688—21no—
119#14519688—21no—
120#14519688—22no—
121#14519688D411yescharacteristic
122#14519688D411yescharacteristic
123#14519688D411yescharacteristic
124#14519688D411yescharacteristic
125#14519687—21no—
126#14519687—21no—
127C4×#92418—21no—
128C4×#92418—21no—
129C4×#92418—21no—
130C4×#92418—22no—
131C8×Dic61—21no—
132C8×Dic61—21no—
133C2×#41×C61C2311yescharacteristic
134C2×M16×C61C2311yescharacteristic
135C2×M16×C61C2311yescharacteristic
136C4×Q8×C61C2×C411yescharacteristic
137C4×Q8×C61C2×C411yescharacteristic
138C4×D4×C61—21no—
139C4×D4×C61—21no—
140C4×D4×C61C2×C411yescharacteristic
141C4×D4×C61C2×C411yescharacteristic
142#117×C61—21no—
143#117×C61C2311yescharacteristic
144C2×C4⋊C4×C61—21no—
145C2×C42×C61—21no—
146C2×C42×C61C2311yescharacteristic
147C2×C42×C61C2311yescharacteristic
148C2×C42×C61C2311yescharacteristic
149C4⋊C8×C61—21no—
150C4⋊C8×C61—21no—
151#104×C61—22no—
152#104×C61—22no—
153#95×C61—21no—
154#41×C61—21no—
155#41×C61—21no—
156#41×C61C2×Q812yes—
157#41×C61#4112yes—
158C2×Q8×C61C22×C411yescharacteristic
159C2×D4×C61—21no—
160C2×D4×C61C22×C411yescharacteristic
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61#4111yescharacteristic
167C22×C4×C61#4111yescharacteristic
168C22×C4×C61C2411yescharacteristic
169C2×Dic122—21no—
170C2×Dic122—21no—
171M16×C61—21no—
172M16×C61—21no—
173M16×C61—21no—
174M16×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—42no—
180C4⋊C4×C61C22×C411yescharacteristic
181C4⋊C4×C61C22×C411yescharacteristic
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61C22×C411yescharacteristic
193C42×C61C22×C411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C22⋊Dic61—21no—
197C22⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61C22×C411yescharacteristic
201C4⋊Dic61C22×C411yescharacteristic
202C4⋊Dic61C22×C411yescharacteristic
203C4⋊Dic61C22×C411yescharacteristic
204#11758196—21no—
205#11758196—21no—
206#11758196—21no—
207#11758196—21no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210#11758194—24no—
211#11758194—21no—
212#11758194—21no—
213#11758194—24no—
214#11758194C2×Q812yes—
215#11758194#4112yes—
216C2×#92418—24no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—24no—
226C2×#92418C22×C411yescharacteristic
227C2×#92418C22×C411yescharacteristic
228C2×#92418C22×C411yescharacteristic
229C2×#92418C22×C411yescharacteristic
230C2×#92418C2×D411yescharacteristic
231C2×#92418C2×D411yescharacteristic
232C2×#92418C2×D411yescharacteristic
233C2×#92418C2×D411yescharacteristic
234C2×#92418C2×Q812yes—
235C2×#92418#4112yes—
236C23×C61—21no—
237C23×C61C2×#4111yescharacteristic
238Q8×C61—21no—
239Q8×C61C4×Q812yes—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—21no—
243D4×C61C4×Q812yes—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—81no—
258C2×C4×C61—81no—
259C2×C4×C61C4×D411yescharacteristic
260C2×C4×C61C4×D411yescharacteristic
261C2×C4×C61C4×D411yescharacteristic
262C2×C4×C61C4×D411yescharacteristic
263C2×C4×C61C23×C411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×Q811yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273Dic122—81no—
274Dic122—81no—
275C8×C61—41no—
276C8×C61—41no—
277#92418—21no—
278#92418—21no—
279#92418—21no—
280#92418—21no—
281#92418—22no—
282#92418—21no—
283#92418—22no—
284#92418—21no—
285#92418C22⋊Q812yes—
286#92418C22⋊Q812yes—
287#21629—611nomaximal, Sylow
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—22no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61C2×C4×D411yescharacteristic
294C22×C61#52911yescharacteristic
295C22×C61#53311yescharacteristic
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—22no—
301C4×C61—42no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—81no—
305C4×C61C2×C4×Q811yescharacteristic
306C4×C61C4×#4111yescharacteristic
307C4×C61C2×C22⋊Q811yescharacteristic
308C4×C61#52511yescharacteristic
309Dic61—41no—
310Dic61—41no—
311Dic61—41no—
312Dic61—41no—
313#3076—611no—
314#2961—611no—
315#2875—611no—
316#1986—611no—
317#1950—611no—
318#1922—611no—
319#1922—611no—
320#1866—611no—
321#1866—611no—
322#1806D6111yescharacteristic
323#1773—611no—
324#1772—611no—
325#1771—611no—
326#1765—611no—
327#1764—611no—
328C2×C61—21no—
329C2×C61—21no—
330C2×C61—42no—
331C2×C61C4×C22⋊Q811yescharacteristic
332C2×C61#295611yescharacteristic
333C2×C61#295711yescharacteristic
334C2×#131—611no—
335C2×#131—611no—
336#533—611no—
337C4×#41C2×D6111yescharacteristic
338#479—611no—
339#479—611no—
340#478—611no—
341#478—611no—
342#477—611no—
343#476—611no—
344#475—611no—
345#474—611no—
346#446—1221no—
347#446—1221no—
348#446—1221no—
349#446—1221no—
350C2×#108—611no—
351C2×#108—611no—
352#427—611no—
353#427—611no—
354#424—611no—
355#424—611no—
356#423C2×D6111yescharacteristic
357C2×C4⋊C8—611no—
358C2×#104C2×D6111yescharacteristic
359C2×#104C2×D6111yescharacteristic
360#416—611no—
361#416—611no—
362#405—611no—
363#405—611no—
364#405—611no—
365#405—611no—
366#405—1221no—
367#405—1221no—
368C4×M16—611no—
369C4×M16—1221no—
370C2×#97—611no—
371C2×#97—611no—
372C2×#97—1221no—
373C2×C4×C8—611no—
374C4×C4⋊C4C2×D6111yescharacteristic
375#341—611no—
376#341—611no—
377#339—611no—
378#339—611no—
379#339—611no—
380#339—611no—
381#339C2×D6111yescharacteristic
382#339C2×D6111yescharacteristic
383#336—611no—
384#336—611no—
385C61#2162911yescharacteristic, complemented, Sylow
386C2×#41C22×D6111yescharacteristic
387#131—614no—
388#131—1222no—
389#131—1222no—
390#131—1222no—
391C2×M16—611no—
392C2×M16—611no—
393C2×M16—1221no—
394C2×M16—1221no—
395C2×M16C22×D6111yescharacteristic
396C2×M16C22×D6111yescharacteristic
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—611no—
400C22×C8—611no—
401C22×C8—1221no—
402C22×C8—1221no—
403C4⋊Q8—611no—
404C4⋊Q8—611no—
405#125—611no—
406#125—611no—
407C22⋊Q8—1221no—
408C22⋊Q8—1221no—
409C4×Q8—1221no—
410C4×Q8—1221no—
411C4×Q8C4×D6111yescharacteristic
412C4×Q8C4×D6111yescharacteristic
413C4×D4—21no—
414C4×D4—21no—
415C4×D4C4×D6111yescharacteristic
416C4×D4C4×D6111yescharacteristic
417#117—21no—
418#117—611no—
419#117—611no—
420#117C22×D6111yescharacteristic
421C2×C4⋊C4—21no—
422C2×C42—21no—
423C2×C42C22×D6111yescharacteristic
424C2×C42C22×D6111yescharacteristic
425C2×C42C22×D6111yescharacteristic
426#108—1222no—
427#108—1222no—
428#107—611no—
429#107—611no—
430#107—611no—
431#107—611no—
432#106—611no—
433#106—611no—
434#106—611no—
435#106—611no—
436C4⋊C8—21no—
437C4⋊C8—21no—
438C4⋊C8—611no—
439C4⋊C8—611no—
440C4⋊C8—611no—
441C4⋊C8—611no—
442C4⋊C8—1221no—
443C4⋊C8—1221no—
444C4⋊C8—1221no—
445C4⋊C8—1221no—
446#104—22no—
447#104—22no—
448#103—611no—
449#103—611no—
450#103—611no—
451#103—611no—
452#102—611no—
453#102—611no—
454#102—611no—
455#102—611no—
456#97—611no—
457#97—611no—
458#97—611no—
459#97—611no—
460#97—1222no—
461#97—1221no—
462#97—1221no—
463#97—1221no—
464#97—1221no—
465#97—1221no—
466C4×C8—1222no—
467C4×C8—1221no—
468C4×C8—1221no—
469C4×C8—1221no—
470C4×C8—1221no—
471C4×C8—1221no—
472#95—21no—
473#41—21no—
474#41—21no—
475#41#1175822412yes—
476#41Q8×D6112yes—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8C2×C4×D6111yescharacteristic
480C2×D4—21no—
481C2×D4C2×C4×D6111yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4#1175822411yescharacteristic
488C22×C4#1175822411yescharacteristic
489C22×C4C23×D6111yescharacteristic
490M16—21no—
491M16—21no—
492M16—21no—
493M16—21no—
494M16—612no—
495M16—612no—
496M16—1224no—
497M16—1224no—
498M16—1221no—
499M16—1221no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—611no—
503C2×C8—611no—
504C2×C8—612no—
505C2×C8—612no—
506C2×C8—611no—
507C2×C8—611no—
508C2×C8—611no—
509C2×C8—611no—
510C2×C8—611no—
511C2×C8—611no—
512C2×C8—1224no—
513C2×C8—1224no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—42no—
525C4⋊C4—611no—
526C4⋊C4—611no—
527C4⋊C4—611no—
528C4⋊C4—611no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4C2×C4×D6111yescharacteristic
536C4⋊C4C2×C4×D6111yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—1221no—
541C22⋊C4—1221no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—1221no—
550C42—1221no—
551C42C2×C4×D6111yescharacteristic
552C42C2×C4×D6111yescharacteristic
553C42D4×D6111yescharacteristic
554C42D4×D6111yescharacteristic
555C23—21no—
556C23C2×#1175822411yescharacteristic
557Q8—21no—
558Q8—4881no—
559Q8—4881no—
560Q8#1451974212yes—
561D4—22no—
562D4—22no—
563D4—21no—
564D4#1451974212yes—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4#1451973711yescharacteristic
585C2×C4#1451973711yescharacteristic
586C2×C4#1451973711yescharacteristic
587C2×C4#1451973711yescharacteristic
588C2×C4C22×C4×D6111yescharacteristic
589C2×C4C2×D4×D6111yescharacteristic
590C2×C4C2×Q8×D6111yescharacteristic
591C2×C4#41×D6111yescharacteristic
592C2×C4#41×D6111yescharacteristic
593C2×C4C2×#1175822411yescharacteristic
594C8—41no—
595C8—41no—
596C8—612no—
597C8—612no—
598C8—1221no—
599C8—1221no—
600C8—1221no—
601C8—1221no—
602C8—1222no—
603C8—1221no—
604C8—1221no—
605C8—1222no—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22C2×#1451973711yescharacteristic
612C22#1684029311yescharacteristic
613C22#1684031411yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—22no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—2441no—
624C4—2441no—
625C4—2441no—
626C4—2441no—
627C4C2×#1451974211yescharacteristic
628C4#1684023011yescharacteristic
629C4C22⋊Q8×D6111yescharacteristic
630C4#1684027211yescharacteristic
631C2—21no—
632C2—21no—
633C2—42no—
634C2#2144834711yescharacteristic
635C2#2145472511yescharacteristic
636C2#2145473211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1806C2
socleC22×C61C2×C4×D4
prime radicalC2×#41×C61C23
cosocleC22×C4×C61C24
hypercenter#1806D61
hypocenterC61#21629
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #21454756C2semidirect
4. #21446762C2semidirect
5. #21446592C2semidirect
6. #21446416C2semidirect
7. #21446416C2semidirect
10. #21445702C2semidirect
11. #21445697C2semidirect
12. #21445694C2semidirect
19. #16840069C4semidirect
20. #16840069C4semidirect
21. #16840062C4semidirect
22. #16840062C4semidirect
23. #16840057C4semidirect
24. #16840052C4semidirect
25. #16840047C4semidirect
26. #16840042C4semidirect
70. #14519784Q8semidirect
385. C61#21629semidirect

Maximal subgroups

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

subgroupindex
#214556072
#214547562
#214542472
#214467622
#214465922
#214464162
#214460922
#214457022
#214456972
#214456942
#214456642
#214456572
C61×#18062
#2162961