Finite Group Databasebeta

← #29571965

Subgroups — #29571965

Order 15616.

836 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21446747C211yesmaximal, characteristic, complemented
2#21445839C211yesmaximal, characteristic, complemented
3#21444509C211yesmaximal, characteristic, complemented
4#21444487C211yesmaximal, characteristic, complemented
5#21444318C211yesmaximal, characteristic, complemented
6#21444206C211yesmaximal, characteristic, complemented
7C61×#3508C211yesmaximal, characteristic
8#16839815C2211yescharacteristic
9#16839786C2211yescharacteristic
10C2×C4⋊#92418C2211yescharacteristic
11C2×#14519704—21no—
12C2×#14519698—22no—
13C4×C4⋊Dic61C2211yescharacteristic
14#16839521—21no—
15#16839511—21no—
16#16839508—21no—
17#16839506—22no—
18#16839499—22no—
19Q8⋊#92418—21no—
20D4⋊#92418—21no—
21C61×#559—21no—
22C61×#559C412yescomplemented
23C61×#550—21no—
24C61×#550C412yescomplemented
25C61×#546—22no—
26C61×#546—22no—
27C61×#540—22no—
28C61×#535—21no—
29C61×#533C2211yescharacteristic
30C61×#532—21no—
31C61×#532—21no—
32C2×#127×C61C2211yescharacteristic
33C61×#518—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#14519698—44no—
46C22⋊#92418—21no—
47C4×#92418—21no—
48#14519684—21no—
49C2×#41×C61—22no—
50C2×#41×C61—22no—
51C2×#41×C61D411yescharacteristic
52C22×D4×C61—21no—
53C22×D4×C61—21no—
54C22×D4×C61D412yes—
55C4⋊Q8×C61D411yescharacteristic
56C4⋊Q8×C61D411yescharacteristic
57#127×C61—22no—
58#127×C61—21no—
59#127×C61—21no—
60#127×C61C2×C412yes—
61#127×C61D412yes—
62#125×C61—21no—
63#124×C61—22no—
64#124×C61—22no—
65#124×C61—22no—
66#123×C61—24no—
67#123×C61—24no—
68C22⋊Q8×C61—21no—
69C22⋊Q8×C61D412yes—
70#121×C61—22no—
71#121×C61—22no—
72#121×C61—22no—
73#121×C61—24no—
74#121×C61—24no—
75#121×C61—22no—
76#120×C61—24no—
77C4×Q8×C61—21no—
78C4×Q8×C61C2×C412yes—
79C4×Q8×C61C2×C411yescharacteristic
80C4×Q8×C61D411yescharacteristic
81C4×D4×C61—22no—
82C4×D4×C61—22no—
83C4×D4×C61—22no—
84C4×D4×C61—21no—
85C4×D4×C61—21no—
86C4×D4×C61—22no—
87C4×D4×C61—22no—
88C4×D4×C61—22no—
89C4×D4×C61C2×C411yescharacteristic
90C4×D4×C61D411yescharacteristic
91#117×C61—21no—
92#117×C61—21no—
93#117×C61D411yescharacteristic
94C2×C22⋊C4×C61—24no—
95C2×C42×C61—21no—
96C2×C42×C61C2311yescharacteristic
97C24×C61—22no—
98#41×C61—21no—
99#41×C61—21no—
100#41×C61—21no—
101#41×C61—21no—
102#41×C61—84no—
103#41×C61—82no—
104#41×C61—82no—
105C2×Q8×C61—21no—
106C2×Q8×C61C22⋊C411yescharacteristic
107C2×Q8×C61C22⋊C412yes—
108C2×D4×C61—21no—
109C2×D4×C61—24no—
110C2×D4×C61—24no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—42no—
117C2×D4×C61—42no—
118C2×D4×C61—41no—
119C2×D4×C61—41no—
120C2×D4×C61—41no—
121C2×D4×C61—41no—
122C2×D4×C61—82no—
123C2×D4×C61C22⋊C411yescharacteristic
124C2×D4×C61C22⋊C414yes—
125C22×C4×C61—24no—
126C22×C4×C61—24no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×Dic61—41no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61C22⋊C411yescharacteristic
142C4⋊C4×C61C22⋊C411yescharacteristic
143C4⋊C4×C61C22⋊C412yes—
144C22⋊C4×C61—24no—
145C22⋊C4×C61—24no—
146C22⋊C4×C61—24no—
147C22⋊C4×C61—24no—
148C22⋊C4×C61—21no—
149C22⋊C4×C61—22no—
150C22⋊C4×C61—21no—
151C22⋊C4×C61—21no—
152C42×C61—21no—
153C42×C61—22no—
154C42×C61—21no—
155C42×C61—21no—
156C42×C61C22×C411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160C4⋊Dic61—22no—
161C4⋊Dic61—41no—
162C4×Dic61—41no—
163C4×Dic61—41no—
164#11758194—41no—
165#11758194—41no—
166C2×#92418—42no—
167C2×#92418—41no—
168C2×#92418—41no—
169C2×#92418—41no—
170C23×C61—24no—
171C23×C61—21no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61—82no—
175C23×C61—82no—
176C23×C61#12011yescharacteristic
177Q8×C61—21no—
178Q8×C61—21no—
179Q8×C61—42no—
180Q8×C61—41no—
181Q8×C61—41no—
182D4×C61—21no—
183D4×C61—21no—
184D4×C61—44no—
185D4×C61—44no—
186D4×C61—41no—
187D4×C61—41no—
188D4×C61—42no—
189D4×C61—42no—
190D4×C61—42no—
191D4×C61—42no—
192D4×C61—42no—
193D4×C61—42no—
194D4×C61—82no—
195D4×C61—84no—
196D4×C61—84no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—84no—
211C2×C4×C61—84no—
212C2×C4×C61C2×C22⋊C411yescharacteristic
213C2×C4×C61C2×C22⋊C411yescharacteristic
214C2×C4×C61C2×C22⋊C411yescharacteristic
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61#12011yescharacteristic
218C2×C4×C61#13511yescharacteristic
219C2×C4×C61#13511yescharacteristic
220C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
221C2×C4×C61#13711yescharacteristic
222C2×Dic61—42no—
223C2×Dic61—81no—
224C2×Dic61—81no—
225#92418—81no—
226#92418—81no—
227#92418—81no—
228#8426—611nomaximal, Sylow
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—41no—
232C22×C61—44no—
233C22×C61—41no—
234C22×C61—44no—
235C22×C61—41no—
236C22×C61—44no—
237C22×C61—162no—
238C22×C61—162no—
239C22×C61#37911yescharacteristic
240C22×C61#44911yescharacteristic
241C22×C61#45211yescharacteristic
242C4×C61—21no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—41no—
246C4×C61—42no—
247C4×C61—41no—
248C4×C61—42no—
249C4×C61—41no—
250C4×C61#41611yescharacteristic
251C4×C61#41711yescharacteristic
252C4×C61#44911yescharacteristic
253C4×C61#45211yescharacteristic
254C4×C61#45311yescharacteristic
255C4×C61#45511yescharacteristic
256Dic61—82no—
257#3508D6111yescharacteristic
258#1982—611no—
259#1806—611no—
260#1550—611no—
261#1545—611no—
262#1503—611no—
263#1477—611no—
264C2×C61—21no—
265C2×C61—21no—
266C2×C61—81no—
267C2×C61—84no—
268C2×C61#178811yescharacteristic
269C2×C61#179211yescharacteristic
270C2×C61#179511yescharacteristic
271#559—21no—
272#559Dic6112yescomplemented
273#550—21no—
274#550Dic6112yescomplemented
275#546—22no—
276#546—22no—
277#540—22no—
278#535—21no—
279#533C2×D6111yescharacteristic
280#532—21no—
281#532—21no—
282C2×#127C2×D6111yescharacteristic
283#518—21no—
284C4×#41C2×D6111yescharacteristic
285#431—611no—
286#423—611no—
287C2×C4⋊C8—611no—
288C2×#104—1221no—
289C2×#102—1222no—
290C4×C4⋊C4—611no—
291#339—1221no—
292#333—1221no—
293#332—1221no—
294#331—1222no—
295#328—1222no—
296Q8⋊C8—1221no—
297D4⋊C8—1221no—
298C61#842611yescharacteristic, complemented, Sylow
299C2×#41—22no—
300C2×#41—22no—
301C2×#41C22⋊D6111yescharacteristic
302C22×D4—21no—
303C22×D4—21no—
304C22×D4C22⋊D6112yes—
305C2×M16—1221no—
306C22×C8—1221no—
307C4⋊Q8C22⋊D6111yescharacteristic
308C4⋊Q8C22⋊D6111yescharacteristic
309#127—22no—
310#127—21no—
311#127—21no—
312#127C2×Dic6112yes—
313#127C22⋊D6112yes—
314#125—21no—
315#124—22no—
316#124—22no—
317#124—22no—
318#123—24no—
319#123—24no—
320C22⋊Q8—21no—
321C22⋊Q8C22⋊D6112yes—
322#121—22no—
323#121—22no—
324#121—22no—
325#121—24no—
326#121—24no—
327#121—22no—
328#120—24no—
329C4×Q8—21no—
330C4×Q8C2×Dic6111yescharacteristic
331C4×Q8C2×Dic6112yes—
332C4×Q8C22⋊D6111yescharacteristic
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—22no—
341C4×D4C2×Dic6111yescharacteristic
342C4×D4C22⋊D6111yescharacteristic
343#117—21no—
344#117—21no—
345#117C22⋊D6111yescharacteristic
346C2×C4⋊C4—1221no—
347C2×C22⋊C4—24no—
348C2×C42—21no—
349C2×C42—1221no—
350C2×C42C22×D6111yescharacteristic
351C4⋊C8—1221no—
352C4⋊C8—1221no—
353C4⋊C8—1221no—
354C4⋊C8—1221no—
355C4⋊C8—1221no—
356#104—2442no—
357#102—2444no—
358C22⋊C8—1221no—
359C4×C8—1221no—
360#95—1221no—
361C24—22no—
362#41—21no—
363#41—21no—
364#41—21no—
365#41—21no—
366#41—84no—
367#41—82no—
368#41—82no—
369C2×Q8—21no—
370C2×Q8C22⋊Dic6111yescharacteristic
371C2×Q8C22⋊Dic6112yes—
372C2×D4—21no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—82no—
387C2×D4C22⋊Dic6111yescharacteristic
388C2×D4C22⋊Dic6114yes—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—2441no—
395C22×C4C2×C22⋊D6111yescharacteristic
396C22×C4C2×C22⋊D6111yescharacteristic
397C22×C4C2×C22⋊D6111yescharacteristic
398M16—2441no—
399M16—2441no—
400C2×C8—2442no—
401C2×C8—2441no—
402C2×C8—2441no—
403C2×C8—2441no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—1222no—
412C4⋊C4—2441no—
413C4⋊C4C22⋊Dic6111yescharacteristic
414C4⋊C4C22⋊Dic6111yescharacteristic
415C4⋊C4C22⋊Dic6112yes—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—21no—
421C22⋊C4—22no—
422C22⋊C4—21no—
423C22⋊C4—21no—
424C42—21no—
425C42—22no—
426C42—21no—
427C42—21no—
428C42—2441no—
429C42—2441no—
430C42C22×Dic6111yescharacteristic
431C42C2×C22⋊D6111yescharacteristic
432C42C2×C22⋊D6111yescharacteristic
433C42C2×C22⋊D6111yescharacteristic
434C23—24no—
435C23—21no—
436C23—42no—
437C23—42no—
438C23—82no—
439C23—82no—
440C23#1451974611yescharacteristic
441Q8—21no—
442Q8—21no—
443Q8—42no—
444Q8—41no—
445Q8—41no—
446D4—21no—
447D4—21no—
448D4—44no—
449D4—44no—
450D4—41no—
451D4—41no—
452D4—42no—
453D4—42no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—82no—
459D4—84no—
460D4—84no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—84no—
475C2×C4—84no—
476C2×C4—4881no—
477C2×C4—4881no—
478C2×C4—2442no—
479C2×C4C2×C22⋊Dic6111yescharacteristic
480C2×C4C2×C22⋊Dic6111yescharacteristic
481C2×C4C2×C22⋊Dic6111yescharacteristic
482C2×C4#1451974611yescharacteristic
483C2×C4#1451974611yescharacteristic
484C2×C4#1451974611yescharacteristic
485C2×C4#1451979711yescharacteristic
486C2×C4#1451979711yescharacteristic
487C2×C4#1451980211yescharacteristic
488C2×C4#1451980711yescharacteristic
489C8—4881no—
490C8—4881no—
491C8—4881no—
492C22—21no—
493C22—21no—
494C22—41no—
495C22—44no—
496C22—41no—
497C22—44no—
498C22—41no—
499C22—44no—
500C22—162no—
501C22—162no—
502C22#1683961811yescharacteristic
503C22#1683990711yescharacteristic
504C22#1683992411yescharacteristic
505C4—21no—
506C4—21no—
507C4—21no—
508C4—41no—
509C4—41no—
510C4—42no—
511C4—41no—
512C4—42no—
513C4—4882no—
514C4#1683975811yescharacteristic
515C4#1683976311yescharacteristic
516C4#1683990711yescharacteristic
517C4#1683992411yescharacteristic
518C4#1683993011yescharacteristic
519C4#1683994011yescharacteristic
520C2—21no—
521C2—21no—
522C2—84no—
523C2—81no—
524C2#2144575911yescharacteristic
525C2#2144577311yescharacteristic
526C2#2144578911yescharacteristic
subgroupG/H
centerC22#16839618
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#3508C2
socleC22×C61#379
cosocleC2×C42×C61C23
hypercenter#3508D61
hypocenterC61#8426
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. #21446747C2semidirect
2. #21445839C2semidirect
3. #21444509C2semidirect
4. #21444487C2semidirect
5. #21444318C2semidirect
6. #21444206C2semidirect
22. C61×#559C4semidirect
24. C61×#550C4semidirect
272. #559Dic61semidirect
274. #550Dic61semidirect
298. C61#8426semidirect

Maximal subgroups

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

subgroupindex
#214467472
#214458392
#214445092
#214444872
#214443182
#214442062
C61×#35082
#842661