Finite Group Databasebeta

← #29702100

Subgroups — #29702100

Order 15616.

692 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455607C211yesmaximal, characteristic
2#21454766C211yesmaximal, characteristic, complemented
3#21454387C211yesmaximal, characteristic
4#21446800C211yesmaximal, characteristic, complemented
5#21446594C211yesmaximal, characteristic, complemented
6#21446571C211yesmaximal, characteristic, complemented
7#21446566C211yesmaximal, characteristic, complemented
8#21446421C211yesmaximal, characteristic, complemented
9#21446421C211yesmaximal, characteristic, complemented
10#21446092C211yesmaximal, characteristic
11#21446092C211yesmaximal, characteristic
12#21445676C211yesmaximal, characteristic, complemented
13#21445664C211yesmaximal, characteristic
14#21445657C211yesmaximal, characteristic
15C61×#1759C211yesmaximal, 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#16839892—21no—
28#16839890—21no—
29#16839890—21no—
30C8×Dic122—21no—
31C2×#14519712C2211yescharacteristic
32C2×#14519712C2211yescharacteristic
33#16839798C2211yescharacteristic
34#16839798C2211yescharacteristic
35#16839790C2211yescharacteristic
36#16839790C2211yescharacteristic
37#16839789—21no—
38#16839789—21no—
39#16839789C2211yescharacteristic
40#16839789C2211yescharacteristic
41#16839786—21no—
42#16839786C2211yescharacteristic
43C2×C4⋊#92418—21no—
44C2×C4⋊#92418C2211yescharacteristic
45C2×C4⋊#92418C2211yescharacteristic
46#16839758C2211yescharacteristic
47#16839758C2211yescharacteristic
48#16839720C2211yescharacteristic
49#16839720C2211yescharacteristic
50C2×#14519688C2211yescharacteristic
51C2×C4×#92418C2211yescharacteristic
52#16839526C2211yescharacteristic
53#16839526C2211yescharacteristic
54#16839522C2211yescharacteristic
55#16839522C2211yescharacteristic
56#16839522C2211yescharacteristic
57#16839522C2211yescharacteristic
58#16839517C2211yescharacteristic
59#16839517C2211yescharacteristic
60C4×#41×C61C2211yescharacteristic
61C2×#104×C61C2211yescharacteristic
62C2×#104×C61C2211yescharacteristic
63C4×M16×C61C2211yescharacteristic
64C61×#376C2211yescharacteristic
65C61×#339C2211yescharacteristic
66C61×#339C2211yescharacteristic
67#14519784—22no—
68#14519784—22no—
69#14519784—41no—
70#14519784—41no—
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—41no—
92#14519712—41no—
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⋊#92418—21no—
102C4⋊#92418—21no—
103C4⋊#92418—21no—
104C4⋊#92418—21no—
105C4⋊#92418—21no—
106C4⋊#92418—21no—
107C4⋊#92418C2×C411yescharacteristic
108C4⋊#92418C2×C411yescharacteristic
109C4⋊#92418C2×C411yescharacteristic
110C4⋊#92418C2×C411yescharacteristic
111C4⋊#92418D411yescharacteristic
112C4⋊#92418D411yescharacteristic
113C4⋊#92418D411yescharacteristic
114C4⋊#92418D411yescharacteristic
115#14519706—21no—
116#14519706—21no—
117#14519706—21no—
118#14519706—21no—
119#14519701C2×C411yescharacteristic
120#14519701C2×C411yescharacteristic
121#14519701C2×C411yescharacteristic
122#14519701C2×C411yescharacteristic
123#14519698C2×C411yescharacteristic
124#14519698C2×C411yescharacteristic
125#14519698C2×C411yescharacteristic
126#14519698C2×C411yescharacteristic
127#14519688—41no—
128#14519687—21no—
129#14519687—21no—
130C4×#92418—41no—
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×C42×C61—21no—
145C2×C42×C61C2311yescharacteristic
146C2×C42×C61C2311yescharacteristic
147C2×C42×C61C2311yescharacteristic
148#104×C61—22no—
149#104×C61Q814yescomplemented
150#97×C61—21no—
151C4×C8×C61—21no—
152#95×C61—21no—
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×C61C2×Q812yes—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C22×C411yescharacteristic
180C4⋊C4×C61C22×C411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61C22×C411yescharacteristic
191C42×C61C22×C411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2×Q812yes—
195C22⋊Dic61—21no—
196C22⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61C22×C411yescharacteristic
200C4⋊Dic61C22×C411yescharacteristic
201C4⋊Dic61C22×C411yescharacteristic
202C4⋊Dic61C22×C411yescharacteristic
203#11758196—21no—
204#11758196—21no—
205#11758196—21no—
206#11758196—21no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209#11758194—21no—
210#11758194—21no—
211#11758194—21no—
212#11758194—21no—
213#11758194—42no—
214#11758194—42no—
215C2×#92418—21no—
216C2×#92418—21no—
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—42no—
226C2×#92418—42no—
227C2×#92418C22×C411yescharacteristic
228C2×#92418C22×C411yescharacteristic
229C2×#92418C22×C411yescharacteristic
230C2×#92418C22×C411yescharacteristic
231C2×#92418C2×D411yescharacteristic
232C2×#92418C2×D411yescharacteristic
233C2×#92418C2×D411yescharacteristic
234C2×#92418C2×D411yescharacteristic
235C23×C61—21no—
236C23×C61C2×#4111yescharacteristic
237Q8×C61—21no—
238Q8×C61C4×Q812yes—
239D4×C61—22no—
240D4×C61—22no—
241D4×C61—21no—
242D4×C61C4×Q812yes—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—42no—
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×C61C4×D411yescharacteristic
259C2×C4×C61C4×D411yescharacteristic
260C2×C4×C61C4×D411yescharacteristic
261C2×C4×C61C4×D411yescharacteristic
262C2×C4×C61C23×C411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×Q811yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×C4×C612−1+411yescharacteristic
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272Dic122—42no—
273Dic122—81no—
274C8×C61—22no—
275C8×C61—41no—
276#92418—21no—
277#92418—21no—
278#92418—21no—
279#92418—21no—
280#92418—21no—
281#92418—21no—
282#92418—21no—
283#92418—21no—
284#92418—41no—
285#92418—41no—
286#22984—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—22no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61C2×C4×D411yescharacteristic
293C22×C61#55211yescharacteristic
294C22×C61#55411yescharacteristic
295C4×C61—22no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—42no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—81no—
304C4×C61C2×C4×Q811yescharacteristic
305C4×C61#52011yescharacteristic
306C4×C61#54611yescharacteristic
307C4×C61D4×Q811yescharacteristic
308Dic61—41no—
309Dic61—41no—
310Dic61—41no—
311Dic61—41no—
312#3076—611no—
313#2962—611no—
314#2906—611no—
315#1995—611no—
316#1950—611no—
317#1946—611no—
318#1945—611no—
319#1922—611no—
320#1922—611no—
321#1866—611no—
322#1866—611no—
323#1768—611no—
324#1765—611no—
325#1764—611no—
326#1759D6111yescharacteristic
327C2×C61—21no—
328C2×C61—21no—
329C2×C61—42no—
330C2×C61#236011yescharacteristic
331C2×C61#295511yescharacteristic
332C2×C61#295711yescharacteristic
333C2×#131—611no—
334C2×#131—611no—
335#533—611no—
336C4×#41C2×D6111yescharacteristic
337#479—611no—
338#479—611no—
339#478—611no—
340#478—611no—
341#477—611no—
342#476—611no—
343#475—611no—
344#474—611no—
345#446—1221no—
346#446—1221no—
347#446—1221no—
348C8×Q8—1221no—
349C2×#108—611no—
350C2×#108—611no—
351#427—611no—
352#427—611no—
353#424—611no—
354#424—611no—
355#424—611no—
356#424—611no—
357#424—1221no—
358#424—1221no—
359#423—611no—
360#423—1221no—
361C2×C4⋊C8—611no—
362C2×C4⋊C8—611no—
363C2×C4⋊C8—1221no—
364C2×#104C2×D6111yescharacteristic
365C2×#104C2×D6111yescharacteristic
366#416—611no—
367#416—611no—
368#405—611no—
369#405—611no—
370C4×M16C2×D6111yescharacteristic
371C2×#97—611no—
372C2×C4×C8—611no—
373#376C2×D6111yescharacteristic
374#341—611no—
375#341—611no—
376#339—611no—
377#339—611no—
378#339—611no—
379#339—611no—
380#339C2×D6111yescharacteristic
381#339C2×D6111yescharacteristic
382#336—611no—
383#336—611no—
384C61#2298411yescharacteristic, complemented, Sylow
385C2×#41C22×D6111yescharacteristic
386#131—1222no—
387#131—1222no—
388#131—2441no—
389#131—2441no—
390C2×M16—611no—
391C2×M16—611no—
392C2×M16—1221no—
393C2×M16—1221no—
394C2×M16C22×D6111yescharacteristic
395C2×M16C22×D6111yescharacteristic
396C22×C8—611no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—611no—
400C22×C8—1221no—
401C22×C8—1221no—
402C4⋊Q8—611no—
403C4⋊Q8—611no—
404#125—611no—
405#125—611no—
406C22⋊Q8—1221no—
407C22⋊Q8—1221no—
408C4×Q8—1221no—
409C4×Q8—1221no—
410C4×Q8C4×D6111yescharacteristic
411C4×Q8C4×D6111yescharacteristic
412C4×D4—21no—
413C4×D4—21no—
414C4×D4C4×D6111yescharacteristic
415C4×D4C4×D6111yescharacteristic
416#117—21no—
417#117—611no—
418#117—611no—
419#117C22×D6111yescharacteristic
420C2×C42—21no—
421C2×C42C22×D6111yescharacteristic
422C2×C42C22×D6111yescharacteristic
423C2×C42C22×D6111yescharacteristic
424#108—2441no—
425#108—2441no—
426#107—611no—
427#107—611no—
428#107—611no—
429#107—611no—
430#106—611no—
431#106—611no—
432#106—611no—
433#106—611no—
434C4⋊C8—611no—
435C4⋊C8—611no—
436C4⋊C8—611no—
437C4⋊C8—611no—
438C4⋊C8—611no—
439C4⋊C8—611no—
440C4⋊C8—611no—
441C4⋊C8—611no—
442C4⋊C8—1221no—
443C4⋊C8—1221no—
444C4⋊C8—1221no—
445C4⋊C8—1221no—
446C4⋊C8—1221no—
447C4⋊C8—1221no—
448C4⋊C8—1221no—
449C4⋊C8—1221no—
450C4⋊C8—1221no—
451C4⋊C8—1221no—
452#104—22no—
453#104Dic12214yescomplemented
454#103—611no—
455#103—611no—
456#103—611no—
457#103—611no—
458#102—611no—
459#102—611no—
460#102—611no—
461#102—611no—
462#97—21no—
463#97—1221no—
464#97—1221no—
465#97—2441no—
466C4×C8—21no—
467C4×C8—1221no—
468C4×C8—1221no—
469C4×C8—2441no—
470#95—21no—
471#95—21no—
472#41—21no—
473#41—21no—
474#41C2×Dic12212yes—
475#41#1175822212yes—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8C2×C4×D6111yescharacteristic
479C2×D4—21no—
480C2×D4C2×C4×D6111yescharacteristic
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4#1175822411yescharacteristic
487C22×C4#1175822411yescharacteristic
488C22×C4C23×D6111yescharacteristic
489M16—21no—
490M16—21no—
491M16—21no—
492M16—1221no—
493M16—1221no—
494M16—1221no—
495M16—1221no—
496M16—2442no—
497M16—2442no—
498M16C2×Dic12212yes—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—611no—
502C2×C8—611no—
503C2×C8—611no—
504C2×C8—611no—
505C2×C8—611no—
506C2×C8—611no—
507C2×C8—611no—
508C2×C8—611no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—2442no—
520C2×C8—2442no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—611no—
524C4⋊C4—611no—
525C4⋊C4—611no—
526C4⋊C4—611no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4C2×C4×D6111yescharacteristic
534C4⋊C4C2×C4×D6111yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—1221no—
539C22⋊C4—1221no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—1221no—
547C42—1221no—
548C42C2×Dic12212yes—
549C42C2×C4×D6111yescharacteristic
550C42C2×C4×D6111yescharacteristic
551C42D4×D6111yescharacteristic
552C42D4×D6111yescharacteristic
553C23—21no—
554C23C2×#1175822411yescharacteristic
555Q8—21no—
556Q8—4881no—
557Q8—2442no—
558Q8C4×Dic12212yes—
559D4—22no—
560D4—22no—
561D4—21no—
562D4C4×Dic12212yes—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—81no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4#1451973711yescharacteristic
583C2×C4#1451973711yescharacteristic
584C2×C4#1451973711yescharacteristic
585C2×C4#1451973711yescharacteristic
586C2×C4C22×C4×D6111yescharacteristic
587C2×C4C2×D4×D6111yescharacteristic
588C2×C4C22×Dic12211yescharacteristic
589C2×C4C2×#1175822211yescharacteristic
590C2×C4#1451982311yescharacteristic
591C2×C4#1451982411yescharacteristic
592C8—22no—
593C8—41no—
594C8—1221no—
595C8—1221no—
596C8—1221no—
597C8—1221no—
598C8—1221no—
599C8—1221no—
600C8—1221no—
601C8—1221no—
602C8—2441no—
603C8—2441no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22C2×#1451973711yescharacteristic
610C22#1684045211yescharacteristic
611C22#1684046911yescharacteristic
612C4—22no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—42no—
618C4—41no—
619C4—41no—
620C4—81no—
621C4—2441no—
622C4—2441no—
623C4—2441no—
624C4—2441no—
625C4C2×C4×Dic12211yescharacteristic
626C4#1684024211yescharacteristic
627C4#1684040811yescharacteristic
628C4D4×Dic12211yescharacteristic
629C2—21no—
630C2—21no—
631C2—42no—
632C2#2144882211yescharacteristic
633C2#2145471811yescharacteristic
634C2#2145473211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1759C2
socleC22×C61C2×C4×D4
prime radicalC2×#41×C61C23
cosocleC22×C4×C61C24
hypercenter#1759D61
hypocenterC61#22984
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1759

Complements

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

NH ≅ G/Ntype
2. #21454766C2semidirect
4. #21446800C2semidirect
5. #21446594C2semidirect
6. #21446571C2semidirect
7. #21446566C2semidirect
8. #21446421C2semidirect
9. #21446421C2semidirect
12. #21445676C2semidirect
19. #16840069C4semidirect
20. #16840069C4semidirect
21. #16840062C4semidirect
22. #16840062C4semidirect
23. #16840057C4semidirect
24. #16840052C4semidirect
25. #16840047C4semidirect
26. #16840042C4semidirect
149. #104×C61Q8semidirect
384. C61#22984semidirect
453. #104Dic122semidirect

Maximal subgroups

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

subgroupindex
#214556072
#214547662
#214543872
#214468002
#214465942
#214465712
#214465662
#214464212
#214460922
#214456762
#214456642
#214456572
C61×#17592
#2298461