Finite Group Databasebeta

← #29661576

Subgroups — #29661576

Order 15616.

824 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#21457037C211yesmaximal, characteristic
2C2×#16840172C211yesmaximal, characteristic, complemented
3#21456201C211yesmaximal, characteristic, complemented
4#21445153C214yesmaximal, complemented
5#21445147C214yesmaximal, complemented
6#21444434C211yesmaximal, characteristic, complemented
7#21444421C211yesmaximal, characteristic, complemented
8#21444407C211yesmaximal, characteristic
9C61×#3563C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840172—22no—
12#16840172C2212yes—
13#16840172C2214yes—
14#16840132C2214yes—
15#16840121C2212yes—
16#16840114—21no—
17#16840114—21no—
18#16840090—21no—
19#16840073—21no—
20#16840057—21no—
21#16840052—21no—
22#16839819C2211yescharacteristic
23C2×#14519710—21no—
24C2×#14519708—21no—
25#16839770—22no—
26C4×#11758194C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28#16839504C2212yes—
29#16839504C2214yes—
30#16839504C2214yes—
31#16839502C2212yes—
32C61×#564C2212yes—
33C61×#563C2212yes—
34C61×#557—21no—
35C61×#554—21no—
36C61×#552—21no—
37C61×#552—22no—
38C61×#552—21no—
39C61×#544—21no—
40C61×#544—21no—
41C61×#541—21no—
42C61×#541—22no—
43C61×#539—21no—
44C61×#533C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#125×C61C2211yescharacteristic
47C22×Dic122—21no—
48C2×#11758194—21no—
49C22×#92418—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777C2312yes—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56#14519710—24no—
57#14519710—22no—
58#14519708—24no—
59#14519708—22no—
60C4⋊#92418—21no—
61#14519701—22no—
62#14519700—24no—
63#14519700—24no—
64#14519698—22no—
65C22⋊#92418—21no—
66#14519688—21no—
67#14519688C2312yes—
68#14519688C2312yes—
69#14519688C2312yes—
70C4×#92418—21no—
71C4⋊Q8×C61—22no—
72C4⋊Q8×C61—21no—
73C4⋊Q8×C61—22no—
74C4⋊Q8×C61D411yescharacteristic
75C4⋊Q8×C61D411yescharacteristic
76#126×C61—22no—
77#126×C61—22no—
78#126×C61—22no—
79#126×C61—22no—
80#126×C61—21no—
81#125×C61—22no—
82#125×C61—22no—
83#125×C61—21no—
84#125×C61—22no—
85#125×C61—22no—
86#125×C61—22no—
87#125×C61—22no—
88#125×C61—22no—
89#125×C61—22no—
90#125×C61—21no—
91#125×C61—21no—
92#125×C61D412yes—
93#125×C61C2312yes—
94#125×C61C2312yes—
95#125×C61C2311yescharacteristic
96#124×C61C2311yescharacteristic
97#123×C61—22no—
98#123×C61—22no—
99#123×C61—21no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61—21no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—21no—
105C22⋊Q8×C61—21no—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61D411yescharacteristic
109#121×C61D411yescharacteristic
110C4×Q8×C61—21no—
111C4×Q8×C61—22no—
112C4×Q8×C61—21no—
113C4×Q8×C61—21no—
114C4×Q8×C61—21no—
115C4×Q8×C61D411yescharacteristic
116C4×D4×C61—22no—
117C4×D4×C61—21no—
118C4×D4×C61D411yescharacteristic
119#117×C61—22no—
120#117×C61—21no—
121#117×C61—22no—
122#117×C61—21no—
123#117×C61D412yes—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61D411yescharacteristic
128C2×C4⋊C4×C61D411yescharacteristic
129C2×C42×C61C2311yescharacteristic
130C2×Q8×C61—22no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61C2×D411yescharacteristic
134C2×D4×C61—21no—
135C2×D4×C61C2×D411yescharacteristic
136C22×C4×C61—21no—
137C22×C4×C61—22no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×Dic61—41no—
144C2×Dic122—22no—
145C2×Dic122—44no—
146C2×Dic122—41no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61C2×D412yes—
170C4⋊C4×C61C2×D412yes—
171C4⋊C4×C61C2×D412yes—
172C4⋊C4×C61C2×D412yes—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C42×C61—22no—
185C42×C61—22no—
186C42×C61—22no—
187C42×C61—22no—
188C42×C61—21no—
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2411yescharacteristic
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61—41no—
196C4⋊Dic61—41no—
197#11758194—42no—
198C2×#92418—24no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C23×C61—21no—
203C23×C61#12011yescharacteristic
204Q8×C61—41no—
205Q8×C61—42no—
206Q8×C61—81no—
207Q8×C61—82no—
208D4×C61—41no—
209D4×C61—42no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—41no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—81no—
223C2×C4×C61—82no—
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61#12011yescharacteristic
226C2×C4×C61#12011yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×Dic61—42no—
231C2×Dic61—82no—
232Dic122—44no—
233Dic122—82no—
234#92418—24no—
235#92418—42no—
236#92418—42no—
237#19029—611nomaximal, Sylow
238C22×C61—21no—
239C22×C61—21no—
240C22×C61—41no—
241C22×C61—41no—
242C22×C61—41no—
243C22×C61#45412yes—
244C22×C61C2×#12011yescharacteristic
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—42no—
251C4×C61—41no—
252C4×C61—41no—
253C4×C61—42no—
254C4×C61—42no—
255C4×C61—42no—
256C4×C61—41no—
257C4×C61#45612yes—
258Dic61—82no—
259#3563D6111yescharacteristic
260#3238—611no—
261C2×#501—611no—
262#3139—611no—
263#1664—614no—
264#1663—614no—
265#1531—611no—
266#1528—611no—
267#1524—611no—
268C2×C61—22no—
269C2×C61—81no—
270C2×C61C2×#45411yescharacteristic
271C2×C61C2×#45611yescharacteristic
272C2×C61#302211yescharacteristic
273#564C2×D6112yes—
274#563C2×D6112yes—
275#557—21no—
276#554—21no—
277#552—21no—
278#552—22no—
279#552—21no—
280#544—21no—
281#544—21no—
282#541—21no—
283#541—22no—
284#539—21no—
285#533C2×D6111yescharacteristic
286C2×C4⋊Q8—611no—
287#529C2×D6111yescharacteristic
288C2×#125C2×D6111yescharacteristic
289#501—614no—
290#501—612no—
291#501—1222no—
292#491—614no—
293#489—612no—
294#487—1221no—
295#487—1221no—
296#483—1221no—
297#480—1221no—
298#477—1221no—
299#476—1221no—
300#432—611no—
301C2×#107—1221no—
302C2×#106—1221no—
303#419—1222no—
304C4×M16—611no—
305C2×#97—611no—
306#330—612no—
307#330—614no—
308#330—614no—
309#329—612no—
310C61#1902911yescharacteristic, complemented, Sylow
311C22×Q8—1221no—
312C2×M16—1221no—
313C22×C8—1221no—
314C4⋊Q8—22no—
315C4⋊Q8—21no—
316C4⋊Q8—22no—
317C4⋊Q8—612no—
318C4⋊Q8—1221no—
319C4⋊Q8—1221no—
320C4⋊Q8—1221no—
321C4⋊Q8C22⋊D6111yescharacteristic
322C4⋊Q8C22⋊D6111yescharacteristic
323#126—22no—
324#126—22no—
325#126—22no—
326#126—22no—
327#126—21no—
328#125—22no—
329#125—22no—
330#125—21no—
331#125—22no—
332#125—22no—
333#125—22no—
334#125—22no—
335#125—22no—
336#125—22no—
337#125—21no—
338#125—21no—
339#125C22⋊D6112yes—
340#125C22×D6112yes—
341#125C22×D6111yescharacteristic
342#125C22×D6112yes—
343#124C22×D6111yescharacteristic
344#123—22no—
345#123—22no—
346#123—21no—
347C22⋊Q8—22no—
348C22⋊Q8—21no—
349C22⋊Q8—21no—
350C22⋊Q8—22no—
351C22⋊Q8—21no—
352C22⋊Q8—21no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355C22⋊Q8C22⋊D6111yescharacteristic
356#121C22⋊D6111yescharacteristic
357C4×Q8—21no—
358C4×Q8—22no—
359C4×Q8—21no—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×Q8C22⋊D6111yescharacteristic
363C4×D4—22no—
364C4×D4—21no—
365C4×D4C22⋊D6111yescharacteristic
366#117—22no—
367#117—21no—
368#117—22no—
369#117—21no—
370#117C22⋊D6112yes—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4C22⋊D6111yescharacteristic
377C2×C4⋊C4C22⋊D6111yescharacteristic
378C2×C42C22×D6111yescharacteristic
379#107—1224no—
380#107—1222no—
381#106—1224no—
382#106—1222no—
383C4⋊C8—1221no—
384#103—1222no—
385#103—1224no—
386#103—1224no—
387#102—1222no—
388C22⋊C8—1221no—
389#97—612no—
390#97—612no—
391#97—612no—
392#97—1221no—
393C4×C8—1221no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—1222no—
398C2×Q8—2444no—
399C2×Q8—2441no—
400C2×Q8C2×C22⋊D6111yescharacteristic
401C2×D4—21no—
402C2×D4C2×C22⋊D6111yescharacteristic
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—2441no—
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C2×C22⋊D6111yescharacteristic
410C22×C4C2×C22⋊D6111yescharacteristic
411M16—2442no—
412C2×C8—1224no—
413C2×C8—1222no—
414C2×C8—1222no—
415C2×C8—1222no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—1222no—
439C4⋊C4—1222no—
440C4⋊C4—2441no—
441C4⋊C4—2441no—
442C4⋊C4C2×C22⋊D6112yes—
443C4⋊C4C2×C22⋊D6112yes—
444C4⋊C4C2×C22⋊D6112yes—
445C4⋊C4C2×C22⋊D6111yescharacteristic
446C4⋊C4C2×C22⋊D6112yes—
447C4⋊C4C2×C22⋊D6111yescharacteristic
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—21no—
452C22⋊C4—22no—
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—21no—
462C42C2×C22⋊D6111yescharacteristic
463C42C2×C22⋊D6111yescharacteristic
464C42C2×C22⋊D6111yescharacteristic
465C42C23×D6111yescharacteristic
466C23—21no—
467C23#1451974611yescharacteristic
468Q8—41no—
469Q8—42no—
470Q8—81no—
471Q8—82no—
472Q8—4882no—
473Q8—2444no—
474D4—41no—
475D4—42no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—4882no—
491C2×C4—2442no—
492C2×C4#1451974611yescharacteristic
493C2×C4#1451974611yescharacteristic
494C2×C4#1451974611yescharacteristic
495C2×C4C22×C22⋊D6111yescharacteristic
496C2×C4C22×C22⋊D6111yescharacteristic
497C2×C4C22×C22⋊D6111yescharacteristic
498C8—1224no—
499C8—2442no—
500C8—2442no—
501C22—21no—
502C22—21no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22#1683993512yes—
507C22C2×#1451974611yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—42no—
514C4—41no—
515C4—41no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—41no—
520C4—4882no—
521C4#1683994512yes—
522C2—22no—
523C2—81no—
524C2C2×#1683993511yescharacteristic
525C2C2×#1683994511yescharacteristic
526C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3563C2
socleC22×C61C2×#120
prime radical#121×C61D4
hypercenter#3563D61
hypocenterC61#19029
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3563

Complements

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

NH ≅ G/Ntype
2. C2×#16840172C2semidirect
3. #21456201C2semidirect
4. #21445153C2semidirect
5. #21445147C2semidirect
6. #21444434C2semidirect
7. #21444421C2semidirect
310. C61#19029semidirect

Maximal subgroups

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

subgroupindex
#214570372
C2×#168401722
#214562012
#214451532
#214451472
#214444342
#214444212
#214444072
C61×#35632
#1902961