Finite Group Databasebeta

← #29665354

Subgroups — #29665354

Order 15616.

644 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456943C211yesmaximal, characteristic
2#477×D61C211yesmaximal, characteristic, complemented
3#21455286C211yesmaximal, characteristic
4#21454898C211yesmaximal, characteristic, complemented
5#21445234C211yesmaximal, characteristic, complemented
6#21445233C211yesmaximal, characteristic, complemented
7#21445232C211yesmaximal, characteristic, complemented
8#21445204C211yesmaximal, characteristic, complemented
9#21445203C211yesmaximal, characteristic, complemented
10#21445202C211yesmaximal, characteristic, complemented
11#21445201C211yesmaximal, characteristic, complemented
12#21444588C211yesmaximal, characteristic, complemented
13#21444498C211yesmaximal, characteristic, complemented
14#21444465C211yesmaximal, characteristic
15C61×#1678C211yesmaximal, characteristic, complemented
16C2×#14519806—21no—
17#16840310C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19C4×Q8×D61C2211yescharacteristic
20#16840160C2211yescharacteristic
21#16840158C2211yescharacteristic
22#16840108C2211yescharacteristic
23#16840079—21no—
24#16840079—21no—
25#16840058C2211yescharacteristic
26#16840057C2211yescharacteristic
27#16840056—21no—
28#16840055—21no—
29#16840054C2211yescharacteristic
30#16840050—21no—
31#16840050—21no—
32#16839983—21no—
33#16839982—21no—
34#16839973C2211yescharacteristic
35#16839972C2211yescharacteristic
36#16839959C2211yescharacteristic
37Q8×#92418C2211yescharacteristic
38C8×Dic122—21no—
39C8×Dic122C2211yescharacteristic
40#16839886—21no—
41#16839886C2211yescharacteristic
42#16839814—21no—
43#16839814C2211yescharacteristic
44#16839813—21no—
45#16839813C2211yescharacteristic
46#107×D61—21no—
47#16839784C2211yescharacteristic
48C4⋊C8×D61C2211yescharacteristic
49#16839760—21no—
50#103×D61—21no—
51#16839513C2211yescharacteristic
52#16839512C2211yescharacteristic
53#16839512C2211yescharacteristic
54#16839511C2211yescharacteristic
55#16839510C2211yescharacteristic
56#16839510C2211yescharacteristic
57#16839509C2211yescharacteristic
58#16839507C2211yescharacteristic
59#16839506C2211yescharacteristic
60C61×#486C2211yescharacteristic
61C61×#477C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C8×Q8×C61C2211yescharacteristic
64C61×#334C2211yescharacteristic
65C61×#333C2211yescharacteristic
66C61×#331C2211yescharacteristic
67C2×#11758224—21no—
68C2×Q8×D61—21no—
69C2×Q8×D61D411yescharacteristic
70#14519806—44no—
71C2×Dic244—21no—
72C2×#11758202—21no—
73C2×#11758200—21no—
74C2×#11758189—21no—
75C2×C8×D61—21no—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519776—21no—
79#14519776—21no—
80#14519775—21no—
81#14519764—21no—
82#14519748—21no—
83#14519748—21no—
84Q8×Dic61D411yescharacteristic
85C4×Dic122—21no—
86C4×Dic122C2311yescharacteristic
87#14519742—21no—
88#14519742D411yescharacteristic
89D4×Dic61—21no—
90D4×Dic61—21no—
91C4⋊C4×D61—21no—
92C4⋊C4×D61—21no—
93C4⋊C4×D61—21no—
94C4⋊C4×D61D411yescharacteristic
95C42×D61—21no—
96C42×D61C2311yescharacteristic
97#14519711—21no—
98#14519710—21no—
99#14519710—21no—
100#14519709—21no—
101#14519709—21no—
102#14519708—21no—
103#14519708—21no—
104C4⋊#92418—21no—
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107C4⋊#92418C2311yescharacteristic
108#14519706—21no—
109#14519706—21no—
110#14519706—21no—
111#14519706—21no—
112#14519706D411yescharacteristic
113#14519706D411yescharacteristic
114#14519702—21no—
115#14519702—21no—
116#14519701—21no—
117#14519700—21no—
118#14519700—21no—
119#14519698—21no—
120#14519698—21no—
121#14519698—21no—
122#14519689—21no—
123#14519689D411yescharacteristic
124#14519689D411yescharacteristic
125C4×#92418—21no—
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C8×Dic61—21no—
129C8×Dic61—21no—
130C2×SD16×C61—21no—
131C4⋊Q8×C61C2311yescharacteristic
132#127×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134#107×C61—21no—
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×C22⋊D61—41no—
151C22×Dic61—41no—
152Q8×D61—22no—
153Q8×D61—82no—
154Q8×D61SD1614yes—
155#11758224—42no—
156#11758224—42no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—41no—
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61#4111yescharacteristic
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—41no—
167C2×Dic122C2×D411yescharacteristic
168SD16×C61—42no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61C2×D411yescharacteristic
178C42×C61—21no—
179C42×C61C2411yescharacteristic
180C22⋊Dic61—41no—
181C22⋊Dic61—41no—
182#11758202—42no—
183#11758200—42no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61C2×D411yescharacteristic
190#11758196—21no—
191#11758196—21no—
192#11758196—21no—
193#11758196—41no—
194#11758196—41no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61C2×D411yescharacteristic
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61#4111yescharacteristic
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206Dic244—42no—
207#11758189—42no—
208C8×D61—42no—
209C23×C61—41no—
210C22×D61#12111yescharacteristic
211Q8×C61—21no—
212Q8×C61—81no—
213Q8×C61C2×SD1612yes—
214D4×C61—41no—
215D4×C61—41no—
216D4×C61—41no—
217D4×C61—41no—
218D4×C61—81no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—41no—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C22⋊D61—82no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—82no—
231C2×Dic61#12111yescharacteristic
232C2×Dic61#12111yescharacteristic
233C2×Dic61#12111yescharacteristic
234C4×D61—24no—
235C4×D61—21no—
236C4×D61—21no—
237C4×D61—41no—
238C4×D61—81no—
239C4×D61—82no—
240C4×D61C2×SD1612yes—
241C4×D61#13512yes—
242Dic122—21no—
243Dic122—24no—
244Dic122—41no—
245Dic122—41no—
246Dic122—82no—
247Dic122—81no—
248Dic122C2×SD1612yes—
249C8×C61—21no—
250C8×C61—21no—
251C8×C61—41no—
252C8×C61—41no—
253C8×C61—41no—
254#92418—21no—
255#92418—21no—
256#92418—41no—
257#92418—41no—
258#92418—41no—
259#19403—611nomaximal, Sylow
260C22×C61—81no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61C2×#12111yescharacteristic
264C2×D61—21no—
265C2×D61—21no—
266C2×D61#46512yes—
267C4×C61—22no—
268C4×C61—21no—
269C4×C61—41no—
270C4×C61—81no—
271C4×C61C22×SD1611yescharacteristic
272C4×C61C2×#13511yescharacteristic
273C4×C61#57811yescharacteristic
274C4×C61#57811yescharacteristic
275Dic61—21no—
276Dic61—22no—
277Dic61—21no—
278Dic61—41no—
279Dic61—81no—
280Dic61—81no—
281Dic61#46512yes—
282#3227—611no—
283C2×#477—611no—
284#3037—611no—
285#2982—611no—
286#1678—611no—
287#1678—611no—
288#1678—611no—
289#1678D6111yescharacteristic, complemented
290#1672—611no—
291#1672—611no—
292#1672—611no—
293#1672—611no—
294#1570—611no—
295#1548—611no—
296#1540—611no—
297C2×C61—161no—
298C2×C61#304411yescharacteristic
299C2×C61C2×#46511yescharacteristic
300C2×C61#306811yescharacteristic
301D61—22no—
302C2×#137—1221no—
303#532—611no—
304C2×C4⋊Q8—611no—
305C2×C4×Q8—611no—
306#498—611no—
307#498—611no—
308#486—611no—
309#486C2×D6111yescharacteristic
310#481—1221no—
311#481—1221no—
312#477—611no—
313#477—611no—
314#477—611no—
315#477—1221no—
316#477—1221no—
317#477C2×D6111yescharacteristic
318#476—1221no—
319#476—1221no—
320#464—1221no—
321#464—1221no—
322#462—611no—
323#462—611no—
324#460—611no—
325#460C2×D6111yescharacteristic
326C8×Q8—611no—
327C8×Q8—611no—
328C8×Q8—611no—
329C8×Q8—1221no—
330C8×Q8—1221no—
331C8×Q8C2×D6111yescharacteristic
332#431—611no—
333#431—611no—
334#431—1221no—
335#431—1221no—
336C2×#107—1221no—
337#423—611no—
338C2×C4⋊C8—611no—
339#417—1221no—
340C2×#103—1221no—
341#334—611no—
342#334—611no—
343#334—611no—
344#334C2×D6111yescharacteristic
345#333—611no—
346#333—611no—
347#333—611no—
348#333C2×D6111yescharacteristic
349#332—611no—
350#332—611no—
351#331—611no—
352#331C2×D6111yescharacteristic
353C61#1940311yescharacteristic, complemented, Sylow
354C2×#41—1221no—
355C22×Q8—611no—
356C22×Q8—1221no—
357#137—2444no—
358C2×Q16—1221no—
359C2×Q16—1221no—
360C2×SD16—21no—
361C2×SD16—1221no—
362C2×M16—1221no—
363C22×C8—1221no—
364C4⋊Q8—611no—
365C4⋊Q8—611no—
366C4⋊Q8—1221no—
367C4⋊Q8—1221no—
368C4⋊Q8—1221no—
369C4⋊Q8C22×D6111yescharacteristic
370#127C22×D6111yescharacteristic
371#124—1221no—
372#121—1221no—
373#121—1221no—
374C4×Q8—611no—
375C4×Q8—611no—
376C4×Q8—611no—
377C4×Q8—1221no—
378C4×Q8—1221no—
379C4×Q8C22×D6111yescharacteristic
380C4×D4—1221no—
381C4×D4—1221no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C42—611no—
387C2×C42—1221no—
388#107—21no—
389#107—1221no—
390#107—1221no—
391#107—1221no—
392#106—1221no—
393#106—1221no—
394#106—1221no—
395#106—1221no—
396C4⋊C8—21no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8—1221no—
405C4⋊C8—1221no—
406C4⋊C8—1221no—
407C4⋊C8C22×D6111yescharacteristic
408C4⋊C8C22×D6111yescharacteristic
409C4⋊C8C22×D6111yescharacteristic
410#103—21no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#102—21no—
417#102—21no—
418#102—21no—
419#102—1221no—
420#102—1221no—
421#102—1221no—
422C22⋊C8—611no—
423C22⋊C8—611no—
424C22⋊C8—1221no—
425C4×C8—21no—
426C4×C8—611no—
427C4×C8—1221no—
428C4×C8—1221no—
429C4×C8—1221no—
430C4×C8—1221no—
431C4×C8C22×D6111yescharacteristic
432#41—2442no—
433#41—2442no—
434C2×Q8—21no—
435C2×Q8—614no—
436C2×Q8—611no—
437C2×Q8—1221no—
438C2×Q8—1222no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8—4882no—
442C2×Q8—2441no—
443C2×Q8D4×D6111yescharacteristic
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—2441no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—2441no—
454C22×C4—2441no—
455Q16—2442no—
456Q16—2442no—
457SD16—42no—
458SD16—2442no—
459M16—2442no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—611no—
464C2×C8—611no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—2442no—
469C2×C8D4×D6111yescharacteristic
470C2×C8D4×D6111yescharacteristic
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—611no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—2441no—
484C4⋊C4—2441no—
485C4⋊C4D4×D6111yescharacteristic
486C22⋊C4—2441no—
487C22⋊C4—2441no—
488C42—21no—
489C42—611no—
490C42—611no—
491C42—611no—
492C42—1221no—
493C42—1221no—
494C42—1221no—
495C42C23×D6111yescharacteristic
496C23—41no—
497C23—611no—
498Q8—21no—
499Q8—81no—
500Q8—612no—
501Q8—1224no—
502Q8—1221no—
503Q8—4881no—
504Q8—4882no—
505Q8—2441no—
506Q8—2441no—
507Q8SD16×D6112yes—
508D4—41no—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—81no—
513D4—4882no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—611no—
518C2×C4—612no—
519C2×C4—612no—
520C2×C4—611no—
521C2×C4—611no—
522C2×C4—1224no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—4881no—
528C2×C4—4882no—
529C2×C4—4882no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4C2×D4×D6111yescharacteristic
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4#41×D6111yescharacteristic
536C8—21no—
537C8—21no—
538C8—41no—
539C8—41no—
540C8—41no—
541C8—1221no—
542C8—1221no—
543C8—2441no—
544C8—2441no—
545C8—2441no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—612no—
550C22—1221no—
551C22—1221no—
552C22#121×D6111yescharacteristic
553C4—22no—
554C4—21no—
555C4—41no—
556C4—81no—
557C4—612no—
558C4—1222no—
559C4—1221no—
560C4—1221no—
561C4—4881no—
562C4—4881no—
563C4—2441no—
564C4C2×SD16×D6111yescharacteristic
565C4#135×D6111yescharacteristic
566C4#1684057611yescharacteristic
567C4#1684057811yescharacteristic
568C2—161no—
569C2—1222no—
570C2#2145534711yescharacteristic
571C2#465×D6111yescharacteristic
572C2#2145554511yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1678C2
socleC22×C61C2×#121
prime radical#16840310C22
hypercenter#1678D61
hypocenterC61#19403
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1678

Complements

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

NH ≅ G/Ntype
2. #477×D61C2semidirect
4. #21454898C2semidirect
5. #21445234C2semidirect
6. #21445233C2semidirect
7. #21445232C2semidirect
8. #21445204C2semidirect
9. #21445203C2semidirect
10. #21445202C2semidirect
11. #21445201C2semidirect
12. #21444588C2semidirect
13. #21444498C2semidirect
15. C61×#1678C2semidirect
289. #1678D61semidirect
353. C61#19403semidirect

Maximal subgroups

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

subgroupindex
#214569432
#477×D612
#214552862
#214548982
#214452342
#214452332
#214452322
#214452042
#214452032
#214452022
#214452012
#214445882
#214444982
#214444652
C61×#16782
#1940361