Finite Group Databasebeta

← #30612730

Subgroups — #30612730

Order 15616.

586 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462723C211yesmaximal, characteristic
2#21462712C211yesmaximal, characteristic
3#21461444C211yesmaximal, characteristic, complemented
4#21461442C211yesmaximal, characteristic, complemented
5#21461432C211yesmaximal, characteristic, complemented
6#21461430C211yesmaximal, characteristic, complemented
7#21461413C211yesmaximal, characteristic, complemented
8#21461412C211yesmaximal, characteristic, complemented
9#21461406C211yesmaximal, characteristic, complemented
10Q16×#92420C211yesmaximal, characteristic, complemented
11#21461390C211yesmaximal, characteristic
12#21461161C211yesmaximal, characteristic
13#21461141C211yesmaximal, characteristic
14#21461036C211yesmaximal, characteristic
15#21455853C211yesmaximal, characteristic
16#16840814—21no—
17#16840813C2211yescharacteristic
18#16840812—21no—
19#16840812C2211yescharacteristic
20#16840811C2211yescharacteristic
21Q8×#92420—21no—
22Q8×#92420—21no—
23Q8×#92420C2211yescharacteristic
24Q8×#92420C2211yescharacteristic
25Q8×#92420C2211yescharacteristic
26Q8×#92420C2211yescharacteristic
27Q8×#92420C2211yescharacteristic
28#16840786—21no—
29#16840786—21no—
30#16840783C2211yescharacteristic
31#16840778—21no—
32#16840778—21no—
33#16840778C2211yescharacteristic
34#16840778C2211yescharacteristic
35C4×#11758217C2211yescharacteristic
36#16840626C2211yescharacteristic
37#16840625C2211yescharacteristic
38#16840624C2211yescharacteristic
39#16840623C2211yescharacteristic
40#16840612C2211yescharacteristic
41#16840612C2211yescharacteristic
42Q8⋊#92420C2211yescharacteristic
43Q8⋊#92420C2211yescharacteristic
44Q8⋊#92420C2211yescharacteristic
45Q8⋊#92420C2211yescharacteristic
46Q8⋊#92420C2211yescharacteristic
47Q8⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50#16840603C2211yescharacteristic
51C8×#92420C2211yescharacteristic
52#16840314C2211yescharacteristic
53C2×Q8×Dic61C2211yescharacteristic
54#16840098C411yescharacteristic
55#16840095C411yescharacteristic
56#16840055C411yescharacteristic
57#16840055C411yescharacteristic
58#16839970C411yescharacteristic
59#16839970C411yescharacteristic
60#16839918C411yescharacteristic
61Q16×Dic61C2211yescharacteristic
62Q16×Dic61C2211yescharacteristic
63#16839854C2211yescharacteristic
64#16839854C2211yescharacteristic
65#16839812C2211yescharacteristic
66C61×#451C411yescharacteristic
67C2×#11758217—21no—
68#14519844—21no—
69#14519844—21no—
70C4⋊#92420—21no—
71C4⋊#92420—21no—
72C4⋊#92420—21no—
73C4⋊#92420—21no—
74C4⋊#92420—21no—
75C4⋊#92420—21no—
76C4⋊#92420D411yescharacteristic
77C4⋊#92420D411yescharacteristic
78C4⋊#92420C2311yescharacteristic
79C4⋊#92420C2311yescharacteristic
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82C22⋊#92420—21no—
83C22⋊#92420—21no—
84C22⋊#92420D411yescharacteristic
85C22⋊#92420D411yescharacteristic
86#14519832—21no—
87#14519832—21no—
88#14519832—21no—
89#14519832—21no—
90C4×#92420—21no—
91C4×#92420—21no—
92C4×#92420—21no—
93C4×#92420—21no—
94C4×#92420—21no—
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C2×#11758202C2×C411yescharacteristic
102C2×#11758202C2×C411yescharacteristic
103#14519776C2×C411yescharacteristic
104#14519775C2×C411yescharacteristic
105#14519767—21no—
106#14519756C2×C411yescharacteristic
107#14519754—21no—
108Q8×Dic61—21no—
109Q8×Dic61—21no—
110Q8×Dic61C2×C411yescharacteristic
111Q8×Dic61C2×C411yescharacteristic
112Q8×Dic61C2311yescharacteristic
113Q8×Dic61C2311yescharacteristic
114Q8×Dic61C2311yescharacteristic
115#14519742—21no—
116#14519742C2311yescharacteristic
117#14519733—21no—
118C2×C4⋊Dic61—21no—
119C2×C4⋊Dic61C2×C411yescharacteristic
120C2×C4×Dic61—21no—
121C2×C4×Dic61C2311yescharacteristic
122#14519711C2×C411yescharacteristic
123#14519711C2×C411yescharacteristic
124#14519710C2×C411yescharacteristic
125#14519710C2×C411yescharacteristic
126#14519706C2×C411yescharacteristic
127#14519706C2×C411yescharacteristic
128#14519702C2×C411yescharacteristic
129#14519702C2×C411yescharacteristic
130#14519701C2×C411yescharacteristic
131#14519701C2×C411yescharacteristic
132#14519701C2×C411yescharacteristic
133#14519701C2×C411yescharacteristic
134C22⋊#92418C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C22×Q8×C61C2×C411yescharacteristic
138C2×Q16×C61C2×C411yescharacteristic
139C2×Q16×C61C2×C411yescharacteristic
140C22⋊Q8×C61C2×C411yescharacteristic
141#103×C61C2×C411yescharacteristic
142#103×C61C2×C411yescharacteristic
143C22⋊C8×C61C2×C411yescharacteristic
144C2×Q8×C61—22no—
145C2×Q8×C61—22no—
146C2×Q8×C61—21no—
147C2×Q8×C61C22×C411yescharacteristic
148C2×Q8×C61C22×C411yescharacteristic
149C2×Q8×C61C22×C411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61C22×C411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61#4111yescharacteristic
155C2×Dic122—21no—
156C2×Dic122C22×C411yescharacteristic
157#11758217—42no—
158C2×#92420—21no—
159C2×#92420—21no—
160C2×#92420—21no—
161C2×#92420—21no—
162C2×#92420—21no—
163C2×#92420—21no—
164C2×#92420C2×D411yescharacteristic
165C2×#92420C2×D411yescharacteristic
166C2×#92420C2×D411yescharacteristic
167C2×#92420C2×D411yescharacteristic
168Q16×C61—22no—
169Q16×C61—22no—
170C2×C8×C61C22×C411yescharacteristic
171C2×C8×C61C22×C411yescharacteristic
172C4⋊C4×C61—21no—
173C4⋊C4×C61C22×C411yescharacteristic
174C22⋊C4×C61—21no—
175C22⋊Dic61—21no—
176#11758202—22no—
177#11758202—22no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61C22×C411yescharacteristic
185C4⋊Dic61C22×C411yescharacteristic
186C4⋊Dic61C22×C411yescharacteristic
187#11758196—21no—
188#11758196—21no—
189#11758196—21no—
190#11758196—21no—
191#11758196—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61C22×C411yescharacteristic
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61#4111yescharacteristic
201C4×Dic61C2411yescharacteristic
202C2×#92418C22×C411yescharacteristic
203C2×#92418C22×C411yescharacteristic
204C23×C61C4×D411yescharacteristic
205Q8×C61—24no—
206Q8×C61—21no—
207Q8×C61—21no—
208Q8×C61—21no—
209Q8×C61—21no—
210Q8×C61#13112yes—
211Q8×C61#13112yes—
212C2×C4×C61—24no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61C4×D411yescharacteristic
220C2×C4×C61C23×C411yescharacteristic
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—41no—
228C2×Dic61—81no—
229C2×Dic61C4×D411yescharacteristic
230C2×Dic61C4×D411yescharacteristic
231C2×Dic61C22×D411yescharacteristic
232C2×Dic61C2×#4111yescharacteristic
233Dic122—21no—
234Dic122—21no—
235Dic122—81no—
236#92420—22no—
237#92420—21no—
238#92420—21no—
239#92420—21no—
240#92420—21no—
241#92420—41no—
242#92420—41no—
243#92420—41no—
244#92420—41no—
245#92420—41no—
246C8×C61—21no—
247C8×C61—21no—
248#92418—21no—
249#92418—21no—
250#17925—611nomaximal, Sylow
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61C2×C4×D411yescharacteristic
255C4×C61—22no—
256C4×C61—22no—
257C4×C61—21no—
258C4×C61—41no—
259C4×C61—41no—
260C4×C61C2×#13111yescharacteristic
261C4×C61C2×#13111yescharacteristic
262Dic61—21no—
263Dic61—21no—
264Dic61—21no—
265Dic61—41no—
266Dic61—41no—
267Dic61—41no—
268Dic61—41no—
269Dic61#57811yescharacteristic
270Dic61#57811yescharacteristic
271#3104—611no—
272#2985—611no—
273#2982—611no—
274#1587—611no—
275#1587—611no—
276#1584—611no—
277#1584—611no—
278#1579—611no—
279#1579—611no—
280C8×Q16—611no—
281C8×Q16—611no—
282#1572—611no—
283#1491—611no—
284#1485—611no—
285#1453—611no—
286C2×C61—41no—
287C2×C61#292911yescharacteristic
288C2×C61#295011yescharacteristic
289C2×C61#295611yescharacteristic
290#533—611no—
291C2×C4×Q8—611no—
292#484—611no—
293#484—611no—
294#477—611no—
295#477—611no—
296#462—611no—
297#462—611no—
298#451—611no—
299#451#692311yescharacteristic
300#446—611no—
301#446—611no—
302#446—1221no—
303#446—1221no—
304C8×Q8—611no—
305C8×Q8—611no—
306C8×Q8—611no—
307C8×Q8—611no—
308C8×Q8—611no—
309C8×Q8—611no—
310C8×Q8—1221no—
311C8×Q8—1221no—
312C4×Q16—611no—
313C4×Q16—611no—
314C4×Q16—611no—
315C4×Q16—611no—
316#433—1221no—
317#433—1221no—
318#432—611no—
319#431—611no—
320#431—611no—
321#431—611no—
322#431—1221no—
323#431—1221no—
324C4×M16—611no—
325#335—611no—
326#335—611no—
327#334—611no—
328#334—611no—
329Q8⋊C8—611no—
330Q8⋊C8—611no—
331Q8⋊C8—611no—
332Q8⋊C8—611no—
333Q8⋊C8—611no—
334Q8⋊C8—611no—
335Q8⋊C8—611no—
336Q8⋊C8—611no—
337#322—611no—
338#322—611no—
339#322—611no—
340C82—611no—
341C61#1792511yescharacteristic, complemented, Sylow
342C22×Q8C2×#692311yescharacteristic
343C2×Q16—611no—
344C2×Q16—611no—
345C2×Q16C2×#692311yescharacteristic
346C2×Q16C2×#692311yescharacteristic
347C2×M16—1221no—
348C4⋊Q8—611no—
349C4⋊Q8—611no—
350#125—1221no—
351C22⋊Q8—611no—
352C22⋊Q8—1221no—
353C22⋊Q8C2×#692311yescharacteristic
354C4×Q8—611no—
355C4×Q8—611no—
356C4×Q8—611no—
357C4×Q8—611no—
358C4×Q8—611no—
359C4×Q8—611no—
360C4×Q8—1221no—
361C4×Q8—1221no—
362C4×Q8—1221no—
363#117—1221no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—1221no—
366C2×C42—611no—
367C2×C42—1221no—
368#107—611no—
369#107—611no—
370#107—611no—
371#107—611no—
372C4⋊C8—611no—
373C4⋊C8—611no—
374C4⋊C8—611no—
375C4⋊C8—611no—
376C4⋊C8—611no—
377C4⋊C8—611no—
378C4⋊C8—611no—
379C4⋊C8—611no—
380C4⋊C8—1221no—
381C4⋊C8—1221no—
382C4⋊C8—1221no—
383C4⋊C8—1221no—
384C4⋊C8—1221no—
385C4⋊C8—1221no—
386C4⋊C8—1221no—
387C4⋊C8—1221no—
388#103—611no—
389#103—611no—
390#103—611no—
391#103—611no—
392#103—611no—
393#103—611no—
394#103C2×#692311yescharacteristic
395#103C2×#692311yescharacteristic
396C22⋊C8—611no—
397C22⋊C8—611no—
398C22⋊C8—611no—
399C22⋊C8—1221no—
400C22⋊C8—1221no—
401C22⋊C8C2×#692311yescharacteristic
402#97—1221no—
403#97—1221no—
404#97—1221no—
405#97—1221no—
406C4×C8—611no—
407C4×C8—611no—
408C4×C8—611no—
409C4×C8—611no—
410C4×C8—611no—
411C4×C8—611no—
412C4×C8—1221no—
413C4×C8—1221no—
414C4×C8—1221no—
415C4×C8—1221no—
416C4×C8—1221no—
417C4×C8—1221no—
418C4×C8—1221no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—611no—
423C2×Q8—1221no—
424C2×Q8C22×#692311yescharacteristic
425C2×Q8C22×#692311yescharacteristic
426C2×Q8C22×#692311yescharacteristic
427C22×C4—21no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—1221no—
431C22×C4C22×#692311yescharacteristic
432Q16—22no—
433Q16—22no—
434Q16—1222no—
435Q16—1222no—
436M16—2442no—
437C2×C8—611no—
438C2×C8—611no—
439C2×C8—611no—
440C2×C8—611no—
441C2×C8—611no—
442C2×C8—611no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8C22×#692311yescharacteristic
450C2×C8C22×#692311yescharacteristic
451C4⋊C4—21no—
452C4⋊C4—611no—
453C4⋊C4—611no—
454C4⋊C4—611no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4C22×#692311yescharacteristic
467C22⋊C4—21no—
468C22⋊C4—1221no—
469C42—611no—
470C42—611no—
471C42—611no—
472C42—611no—
473C42—1221no—
474C42—1221no—
475C42—1221no—
476C42—1221no—
477C42—1221no—
478C42—1221no—
479C23D4×#692311yescharacteristic
480Q8—24no—
481Q8—21no—
482Q8—21no—
483Q8—21no—
484Q8—21no—
485Q8—1221no—
486Q8—1221no—
487Q8—4881no—
488Q8#1451986412yes—
489Q8#1451986412yes—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—611no—
498C2×C4—611no—
499C2×C4—611no—
500C2×C4—611no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—4881no—
508C2×C4—2441no—
509C2×C4D4×#692311yescharacteristic
510C2×C4C23×#692311yescharacteristic
511C8—21no—
512C8—21no—
513C8—1221no—
514C8—1221no—
515C8—1221no—
516C8—1221no—
517C8—1222no—
518C8—1221no—
519C8—1221no—
520C8—2441no—
521C8—2441no—
522C8—2441no—
523C8—2441no—
524C8—2441no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22C2×D4×#692311yescharacteristic
529C4—22no—
530C4—22no—
531C4—21no—
532C4—41no—
533C4—41no—
534C4—611no—
535C4—611no—
536C4—1221no—
537C4—1221no—
538C4—1221no—
539C4—2441no—
540C4—2441no—
541C4—2441no—
542C4—2441no—
543C4C2×#1451986411yescharacteristic
544C4C2×#1451986411yescharacteristic
545C2—41no—
546C2#2146253411yescharacteristic
547C2#2146258011yescharacteristic
548C2#2146259411yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#451C4
socleC22×C61C2×C4×D4
prime radicalC23×C61C4×D4
cosocleC4×Dic61C24
hypercenter#451#6923
hypocenterC61#17925
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #21461444C2semidirect
4. #21461442C2semidirect
5. #21461432C2semidirect
6. #21461430C2semidirect
7. #21461413C2semidirect
8. #21461412C2semidirect
9. #21461406C2semidirect
10. Q16×#92420C2semidirect
341. C61#17925semidirect

Maximal subgroups

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

subgroupindex
#214627232
#214627122
#214614442
#214614422
#214614322
#214614302
#214614132
#214614122
#214614062
Q16×#924202
#214613902
#214611612
#214611412
#214610362
#214558532
#1792561