Finite Group Databasebeta

← #29571944

Subgroups — #29571944

Order 15616.

772 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21446743C211yesmaximal, characteristic, complemented
2#21445839C211yesmaximal, characteristic, complemented
3#21444500C211yesmaximal, characteristic, complemented
4#21444496C211yesmaximal, characteristic, complemented
5#21444314C211yesmaximal, characteristic, complemented
6#21444206C211yesmaximal, characteristic, complemented
7C61×#3511C211yesmaximal, characteristic
8#16839815C2211yescharacteristic
9#16839786C2211yescharacteristic
10C2×C4⋊#92418C2211yescharacteristic
11C2×#14519704—21no—
12C2×#14519701—22no—
13C4×C4⋊Dic61C2211yescharacteristic
14#16839521—21no—
15#16839511—22no—
16#16839508—21no—
17#16839506—21no—
18#16839496—22no—
19Q8⋊#92418—21no—
20D4⋊#92418—21no—
21C61×#556—21no—
22C61×#555C412yescomplemented
23C61×#554C412yescomplemented
24C61×#553—22no—
25C61×#548—22no—
26C61×#547—21no—
27C61×#542—22no—
28C61×#537—21no—
29C61×#533—21no—
30C61×#532—21no—
31C61×#532C2211yescharacteristic
32C2×C4⋊Q8×C61C2211yescharacteristic
33C61×#520—21no—
34C4×#41×C61C2211yescharacteristic
35C2×#11758194—21no—
36C22×#92418—21no—
37C2×C4⋊Dic61—21no—
38C2×C4×Dic61—21no—
39C4⋊#92418—21no—
40C4⋊#92418—21no—
41C4⋊#92418—21no—
42C4⋊#92418—21no—
43C4⋊#92418—21no—
44#14519704—42no—
45#14519701—44no—
46C22⋊#92418—21no—
47C4×#92418—21no—
48#14519684—21no—
49C2×#41×C61—22no—
50C2×#41×C61—21no—
51C2×#41×C61D411yescharacteristic
52C22×Q8×C61D412yes—
53C4⋊Q8×C61—21no—
54C4⋊Q8×C61—21no—
55C4⋊Q8×C61C2×C412yes—
56C4⋊Q8×C61D412yes—
57C4⋊Q8×C61D411yescharacteristic
58#127×C61D411yescharacteristic
59#126×C61—24no—
60#125×C61—22no—
61#124×C61—22no—
62#124×C61—24no—
63#124×C61—21no—
64#123×C61—24no—
65C22⋊Q8×C61—22no—
66C22⋊Q8×C61—22no—
67C22⋊Q8×C61—22no—
68C22⋊Q8×C61—24no—
69#121×C61—22no—
70#121×C61—24no—
71#121×C61—22no—
72#121×C61—22no—
73#121×C61—21no—
74#121×C61D412yes—
75C4×Q8×C61—22no—
76C4×Q8×C61—22no—
77C4×Q8×C61C2×C411yescharacteristic
78C4×Q8×C61D411yescharacteristic
79C4×D4×C61—22no—
80C4×D4×C61—24no—
81C4×D4×C61—22no—
82C4×D4×C61—21no—
83C4×D4×C61—21no—
84C4×D4×C61—21no—
85C4×D4×C61C2×C412yes—
86C4×D4×C61C2×C411yescharacteristic
87C4×D4×C61D411yescharacteristic
88#117×C61—22no—
89#117×C61—22no—
90#117×C61—21no—
91#117×C61D411yescharacteristic
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—22no—
95C2×C42×C61—21no—
96C2×C42×C61C2311yescharacteristic
97#41×C61—21no—
98#41×C61—21no—
99#41×C61—21no—
100#41×C61—21no—
101#41×C61—41no—
102#41×C61—41no—
103#41×C61—41no—
104#41×C61—41no—
105#41×C61—82no—
106#41×C61—82no—
107C2×Q8×C61—21no—
108C2×Q8×C61—22no—
109C2×Q8×C61—42no—
110C2×Q8×C61—42no—
111C2×Q8×C61C22⋊C411yescharacteristic
112C2×Q8×C61C22⋊C414yes—
113C2×D4×C61—21no—
114C2×D4×C61—22no—
115C2×D4×C61—24no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61C22⋊C411yescharacteristic
119C2×D4×C61C22⋊C412yes—
120C22×C4×C61—24no—
121C22×C4×C61—22no—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×Dic61—41no—
130C4⋊C4×C61—24no—
131C4⋊C4×C61—24no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61C22⋊C411yescharacteristic
140C4⋊C4×C61C22⋊C411yescharacteristic
141C4⋊C4×C61C22⋊C412yes—
142C22⋊C4×C61—24no—
143C22⋊C4×C61—24no—
144C22⋊C4×C61—24no—
145C22⋊C4×C61—21no—
146C22⋊C4×C61—21no—
147C22⋊C4×C61—22no—
148C22⋊C4×C61—21no—
149C22⋊C4×C61—22no—
150C42×C61—24no—
151C42×C61—21no—
152C42×C61—21no—
153C42×C61—21no—
154C42×C61C22×C411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C42×C61C2×D411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C4⋊Dic61—22no—
159C4⋊Dic61—41no—
160C4×Dic61—41no—
161C4×Dic61—41no—
162#11758194—41no—
163#11758194—41no—
164C2×#92418—42no—
165C2×#92418—41no—
166C2×#92418—41no—
167C2×#92418—41no—
168C23×C61—21no—
169C23×C61—22no—
170C23×C61#12011yescharacteristic
171Q8×C61—21no—
172Q8×C61—21no—
173Q8×C61—44no—
174Q8×C61—41no—
175Q8×C61—41no—
176Q8×C61—42no—
177Q8×C61—42no—
178D4×C61—21no—
179D4×C61—21no—
180D4×C61—42no—
181D4×C61—41no—
182D4×C61—42no—
183D4×C61—41no—
184D4×C61—42no—
185D4×C61—41no—
186D4×C61—41no—
187D4×C61—41no—
188D4×C61—41no—
189D4×C61—84no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—22no—
195C2×C4×C61—21no—
196C2×C4×C61—24no—
197C2×C4×C61—41no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—82no—
206C2×C4×C61—84no—
207C2×C4×C61C2×C22⋊C411yescharacteristic
208C2×C4×C61C2×C22⋊C411yescharacteristic
209C2×C4×C61C2×C22⋊C411yescharacteristic
210C2×C4×C61#12011yescharacteristic
211C2×C4×C61#12011yescharacteristic
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61#13511yescharacteristic
214C2×C4×C61#13511yescharacteristic
215C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
216C2×C4×C61#13711yescharacteristic
217C2×Dic61—42no—
218C2×Dic61—81no—
219C2×Dic61—81no—
220#92418—81no—
221#92418—81no—
222#92418—81no—
223#8423—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—41no—
227C22×C61—42no—
228C22×C61—41no—
229C22×C61—42no—
230C22×C61—41no—
231C22×C61—42no—
232C22×C61#37911yescharacteristic
233C22×C61#44911yescharacteristic
234C22×C61#45211yescharacteristic
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—41no—
239C4×C61—44no—
240C4×C61—41no—
241C4×C61—41no—
242C4×C61—42no—
243C4×C61#41611yescharacteristic
244C4×C61#41711yescharacteristic
245C4×C61#44911yescharacteristic
246C4×C61#45211yescharacteristic
247C4×C61#45311yescharacteristic
248C4×C61#45511yescharacteristic
249Dic61—82no—
250#3511D6111yescharacteristic
251#1981—611no—
252#1806—611no—
253#1548—611no—
254#1547—611no—
255#1502—611no—
256#1477—611no—
257C2×C61—21no—
258C2×C61—21no—
259C2×C61—81no—
260C2×C61—82no—
261C2×C61#178811yescharacteristic
262C2×C61#179211yescharacteristic
263C2×C61#179611yescharacteristic
264#556—21no—
265#555Dic6112yescomplemented
266#554Dic6112yescomplemented
267#553—22no—
268#548—22no—
269#547—21no—
270#542—22no—
271#537—21no—
272#533—21no—
273#532—21no—
274#532C2×D6111yescharacteristic
275C2×C4⋊Q8C2×D6111yescharacteristic
276#520—21no—
277C4×#41C2×D6111yescharacteristic
278#431—611no—
279#423—611no—
280C2×C4⋊C8—611no—
281C2×#104—1221no—
282C2×#103—1222no—
283C4×C4⋊C4—611no—
284#339—1221no—
285#333—1222no—
286#332—1221no—
287#331—1221no—
288#327—1222no—
289Q8⋊C8—1221no—
290D4⋊C8—1221no—
291C61#842311yescharacteristic, complemented, Sylow
292C2×#41—22no—
293C2×#41—21no—
294C2×#41C22⋊D6111yescharacteristic
295C22×Q8C22⋊D6112yes—
296C2×M16—1221no—
297C22×C8—1221no—
298C4⋊Q8—21no—
299C4⋊Q8—21no—
300C4⋊Q8C2×Dic6112yes—
301C4⋊Q8C22⋊D6112yes—
302C4⋊Q8C22⋊D6111yescharacteristic
303#127C22⋊D6111yescharacteristic
304#126—24no—
305#125—22no—
306#124—22no—
307#124—24no—
308#124—21no—
309#123—24no—
310C22⋊Q8—22no—
311C22⋊Q8—22no—
312C22⋊Q8—22no—
313C22⋊Q8—24no—
314#121—22no—
315#121—24no—
316#121—22no—
317#121—22no—
318#121—21no—
319#121C22⋊D6112yes—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×Q8C2×Dic6111yescharacteristic
323C4×Q8C22⋊D6111yescharacteristic
324C4×D4—22no—
325C4×D4—24no—
326C4×D4—22no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4C2×Dic6111yescharacteristic
331C4×D4C2×Dic6112yes—
332C4×D4C22⋊D6111yescharacteristic
333#117—22no—
334#117—22no—
335#117—21no—
336#117C22⋊D6111yescharacteristic
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—1221no—
341C2×C42—21no—
342C2×C42—1221no—
343C2×C42C22×D6111yescharacteristic
344C4⋊C8—1221no—
345C4⋊C8—1221no—
346C4⋊C8—1221no—
347C4⋊C8—1221no—
348C4⋊C8—1221no—
349#104—2442no—
350#103—2444no—
351C22⋊C8—1221no—
352C4×C8—1221no—
353#95—1221no—
354#41—21no—
355#41—21no—
356#41—21no—
357#41—21no—
358#41—41no—
359#41—41no—
360#41—41no—
361#41—41no—
362#41—82no—
363#41—82no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8C22⋊Dic6111yescharacteristic
369C2×Q8C22⋊Dic6114yes—
370C2×D4—21no—
371C2×D4—22no—
372C2×D4—24no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4C22⋊Dic6111yescharacteristic
376C2×D4C22⋊Dic6112yes—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—2441no—
384C22×C4C2×C22⋊D6111yescharacteristic
385C22×C4C2×C22⋊D6111yescharacteristic
386C22×C4C2×C22⋊D6111yescharacteristic
387M16—2441no—
388M16—2441no—
389C2×C8—2442no—
390C2×C8—2441no—
391C2×C8—2441no—
392C2×C8—2441no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—1222no—
403C4⋊C4—2441no—
404C4⋊C4C22⋊Dic6111yescharacteristic
405C4⋊C4C22⋊Dic6111yescharacteristic
406C4⋊C4C22⋊Dic6112yes—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—22no—
413C22⋊C4—21no—
414C22⋊C4—22no—
415C42—24no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—2441no—
420C42—2441no—
421C42C22×Dic6111yescharacteristic
422C42C2×C22⋊D6111yescharacteristic
423C42C2×C22⋊D6111yescharacteristic
424C42C2×C22⋊D6111yescharacteristic
425C23—21no—
426C23—22no—
427C23#1451974611yescharacteristic
428Q8—21no—
429Q8—21no—
430Q8—44no—
431Q8—42no—
432Q8—41no—
433Q8—41no—
434Q8—42no—
435D4—21no—
436D4—21no—
437D4—42no—
438D4—41no—
439D4—42no—
440D4—41no—
441D4—42no—
442D4—41no—
443D4—41no—
444D4—41no—
445D4—41no—
446D4—84no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—24no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—82no—
463C2×C4—84no—
464C2×C4—4881no—
465C2×C4—4881no—
466C2×C4—2442no—
467C2×C4C2×C22⋊Dic6111yescharacteristic
468C2×C4C2×C22⋊Dic6111yescharacteristic
469C2×C4C2×C22⋊Dic6111yescharacteristic
470C2×C4#1451974611yescharacteristic
471C2×C4#1451974611yescharacteristic
472C2×C4#1451974611yescharacteristic
473C2×C4#1451979711yescharacteristic
474C2×C4#1451979711yescharacteristic
475C2×C4#1451980211yescharacteristic
476C2×C4#1451980711yescharacteristic
477C8—4881no—
478C8—4881no—
479C8—4881no—
480C22—21no—
481C22—21no—
482C22—41no—
483C22—42no—
484C22—41no—
485C22—42no—
486C22—41no—
487C22—42no—
488C22#1683961811yescharacteristic
489C22#1683990711yescharacteristic
490C22#1683992411yescharacteristic
491C4—21no—
492C4—21no—
493C4—21no—
494C4—41no—
495C4—42no—
496C4—41no—
497C4—44no—
498C4—41no—
499C4—4882no—
500C4#1683975811yescharacteristic
501C4#1683976311yescharacteristic
502C4#1683990711yescharacteristic
503C4#1683992411yescharacteristic
504C4#1683993011yescharacteristic
505C4#1683994011yescharacteristic
506C2—21no—
507C2—21no—
508C2—81no—
509C2—82no—
510C2#2144575911yescharacteristic
511C2#2144577311yescharacteristic
512C2#2144579411yescharacteristic
subgroupG/H
centerC22#16839618
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#3511C2
socleC22×C61#379
cosocleC2×C42×C61C23
hypercenter#3511D61
hypocenterC61#8423
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. #21446743C2semidirect
2. #21445839C2semidirect
3. #21444500C2semidirect
4. #21444496C2semidirect
5. #21444314C2semidirect
6. #21444206C2semidirect
22. C61×#555C4semidirect
23. C61×#554C4semidirect
265. #555Dic61semidirect
266. #554Dic61semidirect
291. C61#8423semidirect

Maximal subgroups

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

subgroupindex
#214467432
#214458392
#214445002
#214444962
#214443142
#214442062
C61×#35112
#842361