Finite Group Databasebeta

← #29834372

Subgroups — #29834372

Order 15616.

694 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840204C211yesmaximal, characteristic, complemented
2C2×#16840195C211yesmaximal, characteristic
3#21456307C211yesmaximal, characteristic, complemented
4#21454694C211yesmaximal, characteristic, complemented
5#21447883C211yesmaximal, characteristic, complemented
6#21447881C211yesmaximal, characteristic, complemented
7#21447877C211yesmaximal, characteristic, complemented
8#21447875C211yesmaximal, characteristic, complemented
9#21447867C211yesmaximal, characteristic, complemented
10#21447863C211yesmaximal, characteristic, complemented
11#21447861C211yesmaximal, characteristic, complemented
12#21447576C211yesmaximal, characteristic, complemented
13#21447530C211yesmaximal, characteristic, complemented
14#358×D61C211yesmaximal, characteristic, complemented
15C61×#2209C211yesmaximal, characteristic, complemented
16C2×#14519809—21no—
17C2×#14519800C2211yescharacteristic
18C2×#14519800C2211yescharacteristic
19C2×Q16×D61C2211yescharacteristic
20#16840337—21no—
21#16840244—21no—
22#16840204—24no—
23#16840195—24no—
24C2×#14519723C2211yescharacteristic
25C2×#14519722C2211yescharacteristic
26C2×#14519720C2211yescharacteristic
27C2×#14519720C2211yescharacteristic
28C2×#14519720C2211yescharacteristic
29C2×D976C2211yescharacteristic
30C2×#14519716C2211yescharacteristic
31C2×C16×D61C2211yescharacteristic
32#16840155—21no—
33#16840154—21no—
34#16840147C2211yescharacteristic
35#16840146C2211yescharacteristic
36#16840140C2211yescharacteristic
37#16839865—21no—
38#16839860—21no—
39Q16×Dic61C2211yescharacteristic
40C4×D488C2211yescharacteristic
41#16839844C2211yescharacteristic
42#107×D61C2211yescharacteristic
43#16839762—21no—
44#16839716C2211yescharacteristic
45C4⋊#11758185C2211yescharacteristic
46#16839580C2211yescharacteristic
47#16839580C2211yescharacteristic
48#16839567C2211yescharacteristic
49#16839566C2211yescharacteristic
50#16839566C2211yescharacteristic
51#16839565C2211yescharacteristic
52#16839564C2211yescharacteristic
53#16839562C2211yescharacteristic
54C2×Q32×C61C2211yescharacteristic
55C2×Q32×C61C2211yescharacteristic
56C61×#494C2211yescharacteristic
57C4×Q16×C61C2211yescharacteristic
58C61×C4⋊C16C2211yescharacteristic
59C61×#358C2211yescharacteristic
60C61×#358C2211yescharacteristic
61C2×#11758226—21no—
62C2×#11758226—21no—
63C2×Q8×D61—21no—
64#14519809—44no—
65#14519800—22no—
66#14519800—22no—
67#14519800—22no—
68#14519800D414yes—
69C2×Dic244C2311yescharacteristic
70C2×#11758202—21no—
71C2×#11758202—21no—
72Q16×D61—22no—
73Q16×D61—22no—
74C2×#11758190—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758201—21no—
78C2×D488C2311yescharacteristic
79C2×D488C2311yescharacteristic
80C2×#11758189D411yescharacteristic
81C2×C8×D61C2311yescharacteristic
82#14519775—21no—
83#14519766—21no—
84#14519765—21no—
85#14519762—41no—
86#14519762—41no—
87#14519755—41no—
88#14519755—41no—
89Q8×Dic61—21no—
90C4×D244—21no—
91C4×D244—41no—
92#14519739—21no—
93#14519739—41no—
94#14519734—41no—
95#14519734D411yescharacteristic
96C4⋊C4×D61—21no—
97C4⋊C4×D61—41no—
98#14519723—22no—
99#14519722—22no—
100#14519720—22no—
101#14519720—22no—
102#14519720—22no—
103D976—22no—
104#14519716—22no—
105C2×#11758185C2311yescharacteristic
106C2×#11758185C2311yescharacteristic
107C16×D61—22no—
108#14519711C2311yescharacteristic
109#14519710—21no—
110#14519701—21no—
111#14519699—21no—
112#14519697—21no—
113#14519688D411yescharacteristic
114#14519687D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C2×Q16×C61—21no—
117C2×Q16×C61C2311yescharacteristic
118C2×Q16×C61C2311yescharacteristic
119C2×Q16×C61C2311yescharacteristic
120C4⋊Q8×C61—21no—
121C4×Q8×C61—21no—
122Q32×C61—22no—
123Q32×C61—22no—
124C2×C16×C61C2311yescharacteristic
125C2×C16×C61C2311yescharacteristic
126#107×C61C2311yescharacteristic
127#103×C61—21no—
128C4×C8×C61C2311yescharacteristic
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132#11758226—24no—
133#11758226—42no—
134#11758226—42no—
135#11758226—42no—
136Q8×D61—42no—
137Q8×D61—42no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—41no—
141C2×D244—41no—
142C2×C4×D61—41no—
143C2×C4×D61—41no—
144C2×C4×D61—41no—
145C2×C4×D61—41no—
146C2×C4×D61—41no—
147C2×C4×D61—41no—
148C2×C4×D61C2×D411yescharacteristic
149C2×Dic122—21no—
150C2×Dic122—41no—
151Q16×C61—21no—
152Q16×C61—21no—
153Q16×C61—21no—
154Q16×C61—21no—
155Q16×C61—21no—
156Q16×C61—42no—
157Q16×C61C2×D412yes—
158C2×C8×C61C2×D411yescharacteristic
159C2×C8×C61C2411yescharacteristic
160C4⋊C4×C61—21no—
161C4⋊C4×C61—41no—
162C4⋊C4×C61—41no—
163C4⋊C4×C61—41no—
164C42×C61—41no—
165C42×C61C2×D411yescharacteristic
166#11758202—42no—
167#11758202—42no—
168#11758201—24no—
169#11758201—42no—
170#11758201—42no—
171#11758198—41no—
172#11758198—41no—
173#11758198—41no—
174#11758198—41no—
175#11758198—41no—
176#11758198—41no—
177#11758198D811yescharacteristic
178#11758198D811yescharacteristic
179C4⋊Dic61—21no—
180C4⋊Dic61—41no—
181#11758196—41no—
182#11758196—41no—
183#11758196—41no—
184#11758196D811yescharacteristic
185#11758196D811yescharacteristic
186C4×Dic61—41no—
187C4×Dic61C2×D411yescharacteristic
188C2×#92418C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190Dic244—21no—
191Dic244—21no—
192D488—21no—
193D488—21no—
194D488—21no—
195D488C2×D412yes—
196#11758190—42no—
197#11758189—22no—
198C8×D61C2×D412yes—
199C8×D61#4112yes—
200C16×C61—21no—
201C16×C61—21no—
202#11758185—21no—
203#11758185—21no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61C2×D811yescharacteristic
207Q8×C61—22no—
208Q8×C61—41no—
209Q8×C61—41no—
210Q8×C61—41no—
211Q8×C61—41no—
212Q8×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61C2×D811yescharacteristic
218C2×C4×C61C2×D811yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61C2×D811yescharacteristic
223C2×Dic61C2×D811yescharacteristic
224C2×Dic61C2×D811yescharacteristic
225D244—22no—
226D244—44no—
227D244—41no—
228D244—41no—
229D244—41no—
230D244—82no—
231C4×D61—44no—
232C4×D61—82no—
233C4×D61—82no—
234C4×D61—161no—
235C4×D61—161no—
236C4×D61—161no—
237C4×D61#12112yes—
238C4×D61AGL(1,ℤ/8)12yes—
239Dic122—41no—
240Dic122—41no—
241Dic122—82no—
242C8×C61—21no—
243C8×C61C22×D411yescharacteristic
244C8×C61C2×#4111yescharacteristic
245#92418—21no—
246#92418#12112yes—
247#33590—611nomaximal, Sylow
248C22×C61C22×D811yescharacteristic
249C2×D61—21no—
250C2×D61—21no—
251C2×D61—42no—
252C2×D61—81no—
253C2×D61—81no—
254C2×D61—81no—
255C2×D61—81no—
256C2×D61—81no—
257C2×D61#45912yes—
258C4×C61—21no—
259C4×C61—21no—
260C4×C61—42no—
261C4×C61—81no—
262C4×C61—81no—
263C4×C61—81no—
264C4×C61C2×#12111yescharacteristic
265C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
266Dic61—21no—
267Dic61—21no—
268Dic61—81no—
269Dic61—81no—
270Dic61#45912yes—
271C2×#510—611no—
272C2×#508—611no—
273#3156—611no—
274#2951—611no—
275#2211—611no—
276#2211—611no—
277#2210—611no—
278#2210—611no—
279#2209—611no—
280#2209D6111yescharacteristic, complemented
281#2208—611no—
282#2208—611no—
283#2151—611no—
284#2139—611no—
285C2×#358—611no—
286C2×C61C2×#45911yescharacteristic
287C2×C61#341611yescharacteristic
288C2×C61#341711yescharacteristic
289D61—22no—
290D61—82no—
291D61—161no—
292C2×#137—1221no—
293C2×#135—611no—
294C2×#135—611no—
295C22×Q16—611no—
296#536—1221no—
297#520—1221no—
298#510—1224no—
299#508—1224no—
300C2×Q32—611no—
301C2×Q32C2×D6111yescharacteristic
302C2×Q32C2×D6111yescharacteristic
303C2×SD32—611no—
304C2×SD32—611no—
305C2×SD32—611no—
306C2×SD32—611no—
307C2×D16—611no—
308C2×M32—611no—
309C22×C16—611no—
310#497—1221no—
311#497—1221no—
312#495—611no—
313#495—611no—
314#494—611no—
315#494C2×D6111yescharacteristic
316#440—1221no—
317#440—1221no—
318C4×Q16—611no—
319C4×Q16C2×D6111yescharacteristic
320C4×D8—611no—
321C4×D8—611no—
322C2×#107—611no—
323#417—1221no—
324#405—611no—
325C4⋊C16—611no—
326C4⋊C16—611no—
327C4⋊C16—611no—
328C4⋊C16C2×D6111yescharacteristic
329#358—611no—
330#358—611no—
331#358—611no—
332#358—611no—
333#358C2×D6111yescharacteristic
334#358C2×D6111yescharacteristic
335#357—611no—
336#357—611no—
337C61#3359011yescharacteristic, complemented, Sylow
338C2×#41—1221no—
339C2×#41—1221no—
340C22×Q8—1221no—
341#137—2444no—
342#135—614no—
343#135—1222no—
344#135—1222no—
345#135—1222no—
346C2×Q16—21no—
347C2×Q16—611no—
348C2×Q16—1222no—
349C2×Q16—1222no—
350C2×Q16—1221no—
351C2×Q16—1221no—
352C2×Q16C22×D6111yescharacteristic
353C2×Q16C22×D6111yescharacteristic
354C2×Q16C22×D6111yescharacteristic
355C2×SD16—1221no—
356C2×SD16—1221no—
357C2×SD16—1221no—
358C2×SD16—1221no—
359C2×D8—611no—
360C2×D8—611no—
361C2×M16—611no—
362C22×C8—611no—
363C4⋊Q8—21no—
364C4⋊Q8—1221no—
365#124—1221no—
366#124—1221no—
367#123—2441no—
368#123—2441no—
369C22⋊Q8—2441no—
370C22⋊Q8—2441no—
371C4×Q8—21no—
372C4×Q8—1221no—
373C4×D4—1221no—
374C4×D4—1221no—
375C4×D4—2441no—
376C4×D4—2441no—
377#117—611no—
378#117—2441no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—2441no—
381Q32—22no—
382Q32—22no—
383Q32—1222no—
384SD32—1222no—
385SD32—1222no—
386SD32—1222no—
387SD32—1222no—
388D16—1222no—
389M32—1222no—
390C2×C16—611no—
391C2×C16—611no—
392C2×C16—1222no—
393C2×C16C22×D6111yescharacteristic
394C2×C16C22×D6111yescharacteristic
395#107—611no—
396#107—1221no—
397#107C22×D6111yescharacteristic
398#103—21no—
399#103—1221no—
400#102—1221no—
401#102—1221no—
402#97—611no—
403#97—611no—
404C4×C8—611no—
405C4×C8C22×D6111yescharacteristic
406#41—1224no—
407#41—2442no—
408#41—2442no—
409#41—2442no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—1221no—
414C2×Q8—2441no—
415C2×Q8—2442no—
416C2×Q8—2442no—
417C2×D4—1221no—
418C2×D4—1221no—
419C2×D4—2441no—
420C2×D4—2441no—
421C22×C4—611no—
422C22×C4—2441no—
423C22×C4—2441no—
424C22×C4—2441no—
425C22×C4—2441no—
426C22×C4—2441no—
427C22×C4—2441no—
428Q16—21no—
429Q16—21no—
430Q16—21no—
431Q16—21no—
432Q16—21no—
433Q16—42no—
434Q16—1221no—
435Q16—1221no—
436Q16—2442no—
437Q16—2442no—
438Q16D4×D6112yes—
439SD16—1224no—
440SD16—2442no—
441SD16—2442no—
442SD16—2442no—
443D8—612no—
444D8—1221no—
445D8—1221no—
446D8—1221no—
447M16—1222no—
448C2×C8—611no—
449C2×C8—612no—
450C2×C8—612no—
451C2×C8—611no—
452C2×C8D4×D6111yescharacteristic
453C2×C8C23×D6111yescharacteristic
454C4⋊C4—21no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—611no—
459C4⋊C4—611no—
460C4⋊C4—1221no—
461C4⋊C4—2441no—
462C4⋊C4—2441no—
463C4⋊C4—2441no—
464C4⋊C4—2441no—
465C22⋊C4—611no—
466C22⋊C4—611no—
467C22⋊C4—2441no—
468C22⋊C4—2441no—
469C22⋊C4—2441no—
470C22⋊C4—2441no—
471C22⋊C4—2441no—
472C22⋊C4—2441no—
473C42—41no—
474C42—611no—
475C42—2441no—
476C42D4×D6111yescharacteristic
477C16—21no—
478C16—21no—
479C16—1221no—
480C16—1221no—
481C23—611no—
482C23—2441no—
483C23—2441no—
484Q8—22no—
485Q8—41no—
486Q8—41no—
487Q8—41no—
488Q8—41no—
489Q8—41no—
490Q8—4882no—
491Q8—2441no—
492Q8—2441no—
493D4—1222no—
494D4—4882no—
495D4—2444no—
496D4—2441no—
497D4—2441no—
498D4—2441no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—611no—
504C2×C4—612no—
505C2×C4—612no—
506C2×C4—611no—
507C2×C4—611no—
508C2×C4—9761no—
509C2×C4—9761no—
510C2×C4—9761no—
511C2×C4—4882no—
512C2×C4—4882no—
513C2×C4—2444no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4D8×D6111yescharacteristic
517C2×C4D8×D6111yescharacteristic
518C2×C4C2×D4×D6111yescharacteristic
519C8—21no—
520C8—612no—
521C8—1221no—
522C8C2×D4×D6111yescharacteristic
523C8#41×D6111yescharacteristic
524C22—612no—
525C22—1221no—
526C22—1221no—
527C22—4881no—
528C22—4881no—
529C22—4881no—
530C22—4881no—
531C22—4881no—
532C22—2442no—
533C22C2×D8×D6111yescharacteristic
534C4—21no—
535C4—21no—
536C4—42no—
537C4—81no—
538C4—81no—
539C4—81no—
540C4—612no—
541C4—1221no—
542C4—1221no—
543C4—4881no—
544C4—4881no—
545C4#121×D6111yescharacteristic
546C4AGL(1,ℤ/8)×D6111yescharacteristic
547C2—1222no—
548C2—9761no—
549C2—4882no—
550C2#459×D6111yescharacteristic
551C2#2145896411yescharacteristic
552C2#2145896811yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2209C2
socleC22×C61C22×D8
prime radicalC2×#16840195C2
hypercenter#2209D61
hypocenterC61#33590
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2209

Complements

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

NH ≅ G/Ntype
1. C2×#16840204C2semidirect
3. #21456307C2semidirect
4. #21454694C2semidirect
5. #21447883C2semidirect
6. #21447881C2semidirect
7. #21447877C2semidirect
8. #21447875C2semidirect
9. #21447867C2semidirect
10. #21447863C2semidirect
11. #21447861C2semidirect
12. #21447576C2semidirect
13. #21447530C2semidirect
14. #358×D61C2semidirect
15. C61×#2209C2semidirect
280. #2209D61semidirect
337. C61#33590semidirect

Maximal subgroups

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

subgroupindex
C2×#168402042
C2×#168401952
#214563072
#214546942
#214478832
#214478812
#214478772
#214478752
#214478672
#214478632
#214478612
#214475762
#214475302
#358×D612
C61×#22092
#3359061