Finite Group Databasebeta

← #29652447

Subgroups — #29652447

Order 15616.

1024 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455739C212yesmaximal, complemented
2#21450127C211yesmaximal, characteristic, complemented
3#21444816C214yesmaximal, complemented
4#21444810C212yesmaximal, complemented
5#21444805C212yesmaximal, complemented
6#21444362C212yesmaximal, complemented
7#21444157C211yesmaximal, characteristic, complemented
8C61×#3459C211yesmaximal, characteristic
9C2×#14519758—21no—
10C22×C4⋊Dic61C2211yescharacteristic, complemented
11#16840095—22no—
12#16840090C2214yescomplemented
13#16840078—22no—
14#16840073C2214yes—
15#16839805—22no—
16#16839763—22no—
17#16839726C2212yescomplemented
18#16839692C2212yescomplemented
19#16839687C2212yes—
20#16839660—21no—
21#16839623—21no—
22#16839622—21no—
23#16839622C2211yescharacteristic, complemented
24#16839499C2214yescomplemented
25#16839496C2214yes—
26#16839488C2212yescomplemented
27#16839486C2212yes—
28C61×#543C2211yescharacteristic
29C61×#539—22no—
30C61×#539C2212yes—
31C61×#538—22no—
32C61×#538—21no—
33C61×#534—21no—
34C61×#534C2211yescharacteristic
35C61×#529—22no—
36C61×#529—21no—
37C61×#525—22no—
38C61×#525C2212yes—
39C2×C4×D4×C61—21no—
40C2×C4×D4×C61C2211yescharacteristic
41C23×Dic61—21no—
42C2×#11758194—22no—
43#14519758—44no—
44C2×C4⋊Dic61—22no—
45C2×C4⋊Dic61—22no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61—21no—
48C2×C4⋊Dic61C2312yes—
49C2×#11758196—22no—
50C2×C22⋊Dic61—22no—
51C2×C22⋊Dic61—21no—
52#14519710—24no—
53#14519708—24no—
54#14519701—24no—
55#14519698—24no—
56C22⋊#92418—22no—
57C22⋊#92418C2314yes—
58#14519684—22no—
59#14519684—22no—
60#14519684C2312yes—
61C2×#41×C61D412yes—
62C22×D4×C61—21no—
63C22×D4×C61D411yescharacteristic
64C23×C4×C61—21no—
65C23×C4×C61D411yescharacteristic
66C23×C4×C61C2311yescharacteristic
67#126×C61—22no—
68#126×C61—22no—
69#126×C61—22no—
70#125×C61—21no—
71#125×C61—22no—
72#124×C61—22no—
73#124×C61—21no—
74#123×C61—22no—
75#123×C61—22no—
76#123×C61—22no—
77#123×C61—22no—
78#123×C61—22no—
79#123×C61—22no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—21no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61D412yes—
85C22⋊Q8×C61C2312yes—
86#121×C61—22no—
87#121×C61—22no—
88#121×C61—22no—
89#121×C61—21no—
90#121×C61D412yes—
91#121×C61C2312yes—
92#120×C61—22no—
93#120×C61—22no—
94#120×C61—22no—
95C4×Q8×C61—21no—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98C4×D4×C61—21no—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104#117×C61—22no—
105#117×C61D412yes—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61D411yescharacteristic
111C2×C22⋊C4×C61C2311yescharacteristic
112C2×C42×C61—21no—
113C2×C42×C61D411yescharacteristic
114C24×C61—21no—
115C24×C61C2×D411yescharacteristic
116C24×C61C2×D411yescharacteristic
117#41×C61—44no—
118C2×Q8×C61—21no—
119C2×Q8×C61C2×D412yes—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—21no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—41no—
128C2×D4×C61—42no—
129C2×D4×C61—42no—
130C2×D4×C61C2×D412yes—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61—21no—
136C22×C4×C61—22no—
137C22×C4×C61—21no—
138C22×C4×C61—42no—
139C22×C4×C61—42no—
140C22×C4×C61—82no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—42no—
151C22×Dic61—42no—
152C22×Dic61—42no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61C2×D412yes—
163C4⋊C4×C61C2×D412yes—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C42×C61—22no—
177C42×C61—22no—
178C42×C61—21no—
179C42×C61—21no—
180C22⋊Dic61—42no—
181C22⋊Dic61—44no—
182C4⋊Dic61—24no—
183C4⋊Dic61—42no—
184C4⋊Dic61—41no—
185C4⋊Dic61—41no—
186#11758196—44no—
187#11758194—44no—
188C2×#92418—24no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—22no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—42no—
199C23×C61—81no—
200C23×C61—81no—
201C23×C61#12012yes—
202C23×C61AGL(1,ℤ/8)12yes—
203C23×C61C22×D411yescharacteristic
204Q8×C61—42no—
205Q8×C61—81no—
206D4×C61—42no—
207D4×C61—44no—
208D4×C61—44no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61—82no—
212D4×C61—82no—
213D4×C61—81no—
214C2×C4×C61—24no—
215C2×C4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—42no—
225C2×C4×C61—41no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—44no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—81no—
232C2×C4×C61—81no—
233C2×C4×C61—82no—
234C2×C4×C61—81no—
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61#12011yescharacteristic
237C2×C4×C61C22×D412yes—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—84no—
245#92418—44no—
246#18140—611nomaximal, Sylow
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—81no—
262C22×C61—84no—
263C22×C61C2×#12011yescharacteristic
264C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C61#57511yescharacteristic
266C4×C61—22no—
267C4×C61—21no—
268C4×C61—41no—
269C4×C61—42no—
270C4×C61—41no—
271C4×C61—42no—
272C4×C61—41no—
273C4×C61—42no—
274C4×C61#45812yes—
275Dic61—82no—
276Dic61—82no—
277#3459D6111yescharacteristic
278#3092—612no—
279#2486—611no—
280#1609—614no—
281#1608—612no—
282#1607—612no—
283#1513—612no—
284#1465—611no—
285C2×C61—21no—
286C2×C61—21no—
287C2×C61—42no—
288C2×C61—42no—
289C2×C61—82no—
290C2×C61#300611yescharacteristic
291C2×C61#302211yescharacteristic
292C2×C61C2×#45811yescharacteristic
293#543C2×D6111yescharacteristic
294#539—22no—
295#539C2×D6112yes—
296#538—22no—
297#538—21no—
298#534—21no—
299#534C2×D6111yescharacteristic
300#529—22no—
301#529—21no—
302#525—22no—
303#525C2×D6112yes—
304C2×#123—1221no—
305C2×C4×D4—21no—
306C2×C4×D4C2×D6111yescharacteristic
307C22×C4⋊C4—611no—
308#484—1222no—
309#483—614no—
310#481—1222no—
311#480—614no—
312#428—1222no—
313#417—1222no—
314#407—612no—
315#398—612no—
316#397—612no—
317#389—1221no—
318#381—1221no—
319#380—611no—
320#380—1221no—
321#328—614no—
322#327—614no—
323#324—612no—
324#323—612no—
325C61#1814011yescharacteristic, complemented, Sylow
326C2×#41C22⋊D6112yes—
327C22×D4—21no—
328C22×D4C22⋊D6111yescharacteristic
329C23×C4—21no—
330C23×C4—1221no—
331C23×C4C22⋊D6111yescharacteristic
332C23×C4C22×D6111yescharacteristic
333C2×M16—1222no—
334#126—22no—
335#126—22no—
336#126—22no—
337#125—21no—
338#125—22no—
339#124—22no—
340#124—21no—
341#123—22no—
342#123—22no—
343#123—22no—
344#123—22no—
345#123—22no—
346#123—22no—
347#123—2444no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350C22⋊Q8—21no—
351C22⋊Q8—22no—
352C22⋊Q8C22⋊D6112yes—
353C22⋊Q8C22×D6112yes—
354#121—22no—
355#121—22no—
356#121—22no—
357#121—21no—
358#121C22⋊D6112yes—
359#121C22×D6112yes—
360#120—22no—
361#120—22no—
362#120—22no—
363C4×Q8—21no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—21no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4—22no—
372#117—22no—
373#117C22⋊D6112yes—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—612no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1222no—
386C2×C22⋊C4C22⋊D6111yescharacteristic
387C2×C22⋊C4C22×D6111yescharacteristic
388C2×C42—21no—
389C2×C42C22⋊D6111yescharacteristic
390#107—1224no—
391#106—1224no—
392#103—1224no—
393#102—1224no—
394C22⋊C8—614no—
395C22⋊C8—1222no—
396#95—612no—
397#95—1222no—
398#95—1222no—
399C24—21no—
400C24C2×C22⋊D6111yescharacteristic
401C24C2×C22⋊D6111yescharacteristic
402#41—44no—
403C2×Q8—21no—
404C2×Q8C2×C22⋊D6112yes—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—41no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4C2×C22⋊D6112yes—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—82no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—2442no—
429C22×C4—2442no—
430C22×C4—2442no—
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C2×C22⋊D6112yes—
433C22×C4C2×C22⋊D6111yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C2×C22⋊D6112yes—
437C22×C4C23×D6111yescharacteristic
438M16—2444no—
439C2×C8—1224no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—1224no—
450C4⋊C4—2444no—
451C4⋊C4—2442no—
452C4⋊C4—2441no—
453C4⋊C4—2441no—
454C4⋊C4C2×C22⋊D6112yes—
455C4⋊C4C2×C22⋊D6112yes—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—2442no—
469C22⋊C4—2444no—
470C42—22no—
471C42—22no—
472C42—21no—
473C42—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—81no—
485C23—81no—
486C23#1451974612yes—
487C23#1451980212yes—
488C23C22×C22⋊D6111yescharacteristic
489Q8—42no—
490Q8—81no—
491D4—42no—
492D4—44no—
493D4—44no—
494D4—42no—
495D4—42no—
496D4—82no—
497D4—82no—
498D4—81no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—4884no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4#1451974611yescharacteristic
528C2×C4#1451974611yescharacteristic
529C2×C4C22×C22⋊D6112yes—
530C8—2444no—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—81no—
546C22—84no—
547C22C2×#1451974611yescharacteristic
548C22C2×#1451980211yescharacteristic
549C22#1684056011yescharacteristic
550C4—22no—
551C4—21no—
552C4—41no—
553C4—42no—
554C4—41no—
555C4—42no—
556C4—41no—
557C4—42no—
558C4—4882no—
559C4—4882no—
560C4#1683995212yes—
561C2—21no—
562C2—21no—
563C2—42no—
564C2—42no—
565C2—82no—
566C2#2145507111yescharacteristic
567C2#2145519811yescharacteristic
568C2C2×#1683995211yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3459C2
socleC22×C61C2×#120
hypercenter#3459D61
hypocenterC61#18140
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3459

Complements

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

NH ≅ G/Ntype
1. #21455739C2semidirect
2. #21450127C2semidirect
3. #21444816C2semidirect
4. #21444810C2semidirect
5. #21444805C2semidirect
6. #21444362C2semidirect
7. #21444157C2semidirect
10. C22×C4⋊Dic61C22semidirect
12. #16840090C22semidirect
17. #16839726C22semidirect
18. #16839692C22semidirect
23. #16839622C22semidirect
24. #16839499C22semidirect
26. #16839488C22semidirect
325. C61#18140semidirect

Maximal subgroups

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

subgroupindex
#214557392
#214501272
#214448162
#214448102
#214448052
#214443622
#214441572
C61×#34592
#1814061