Finite Group Databasebeta

← #30612729

Subgroups — #30612729

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#21462724C211yesmaximal, characteristic
2#21462713C211yesmaximal, characteristic
3#21461445C211yesmaximal, characteristic, complemented
4#21461444C211yesmaximal, characteristic, complemented
5#21461433C211yesmaximal, characteristic, complemented
6#21461432C211yesmaximal, characteristic, complemented
7#21461413C211yesmaximal, characteristic, complemented
8#21461412C211yesmaximal, characteristic, complemented
9#21461406C211yesmaximal, characteristic, complemented
10Q16×#92420C211yesmaximal, characteristic, complemented
11#21461391C211yesmaximal, characteristic
12#21461162C211yesmaximal, characteristic
13#21461142C211yesmaximal, characteristic
14#21461037C211yesmaximal, characteristic
15#21455857C211yesmaximal, characteristic
16#16840814—21no—
17#16840814—21no—
18#16840813C2211yescharacteristic
19#16840813C2211yescharacteristic
20#16840811—21no—
21#16840811—21no—
22#16840811C2211yescharacteristic
23#16840811C2211yescharacteristic
24Q8×#92420C2211yescharacteristic
25Q8×#92420C2211yescharacteristic
26Q8×#92420C2211yescharacteristic
27Q8×#92420C2211yescharacteristic
28#16840787—21no—
29#16840787—21no—
30#16840784C2211yescharacteristic
31#16840781—21no—
32#16840781C2211yescharacteristic
33#16840780—21no—
34#16840780C2211yescharacteristic
35C4×#11758213C2211yescharacteristic
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#16840317C2211yescharacteristic
53C4×Q8×D61C2211yescharacteristic
54#16840096C411yescharacteristic
55#16840096C411yescharacteristic
56#16840058C411yescharacteristic
57#16840057C411yescharacteristic
58#16839969C411yescharacteristic
59#16839916C411yescharacteristic
60#16839916C411yescharacteristic
61Q16×Dic61C2211yescharacteristic
62Q16×Dic61C2211yescharacteristic
63#16839854C2211yescharacteristic
64#16839854C2211yescharacteristic
65#16839814C2211yescharacteristic
66C61×#462C411yescharacteristic
67C2×#11758213—21no—
68#14519844—21no—
69#14519844—21no—
70#14519844—21no—
71#14519844—21no—
72#14519844D411yescharacteristic
73#14519844D411yescharacteristic
74C4⋊#92420—21no—
75C4⋊#92420—21no—
76C4⋊#92420—21no—
77C4⋊#92420—21no—
78C4⋊#92420C2311yescharacteristic
79C4⋊#92420C2311yescharacteristic
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82#14519834—21no—
83#14519834—21no—
84#14519834D411yescharacteristic
85#14519834D411yescharacteristic
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×Q8×D61C2×C411yescharacteristic
102C2×#11758202C2×C411yescharacteristic
103C2×#11758202C2×C411yescharacteristic
104#14519777C2×C411yescharacteristic
105#14519768—21no—
106#14519757—21no—
107#14519753C2×C411yescharacteristic
108#14519753C2×C411yescharacteristic
109Q8×Dic61C2311yescharacteristic
110Q8×Dic61C2311yescharacteristic
111Q8×Dic61C2311yescharacteristic
112C4×Dic122—21no—
113C4×Dic122C2×C411yescharacteristic
114#14519742—21no—
115#14519742—21no—
116#14519742C2311yescharacteristic
117#14519735—21no—
118C4⋊C4×D61—21no—
119C4⋊C4×D61C2×C411yescharacteristic
120C42×D61—21no—
121C42×D61C2311yescharacteristic
122#14519711C2×C411yescharacteristic
123#14519711C2×C411yescharacteristic
124#14519710C2×C411yescharacteristic
125#14519710C2×C411yescharacteristic
126C4⋊#92418C2×C411yescharacteristic
127#14519702C2×C411yescharacteristic
128#14519702C2×C411yescharacteristic
129#14519701C2×C411yescharacteristic
130#14519701C2×C411yescharacteristic
131#14519701C2×C411yescharacteristic
132#14519701C2×C411yescharacteristic
133#14519689C2×C411yescharacteristic
134#14519689C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C2×Q16×C61C2×C411yescharacteristic
138C2×Q16×C61C2×C411yescharacteristic
139C4⋊Q8×C61C2×C411yescharacteristic
140C4×Q8×C61C2×C411yescharacteristic
141C4⋊C8×C61C2×C411yescharacteristic
142#103×C61C2×C411yescharacteristic
143#103×C61C2×C411yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C2×Q8×C61C22×C411yescharacteristic
146C2×Q8×C61C22×C411yescharacteristic
147Q8×D61—22no—
148Q8×D61—22no—
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61C22×C411yescharacteristic
152C2×C4×D61C2×D411yescharacteristic
153C2×C4×D61#4111yescharacteristic
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122C22×C411yescharacteristic
157C2×#92420—21no—
158C2×#92420—21no—
159C2×#92420—21no—
160C2×#92420—21no—
161C2×#92420—21no—
162C2×#92420—21no—
163C2×#92420C2×D411yescharacteristic
164C2×#92420C2×D411yescharacteristic
165C2×#92420C2×D411yescharacteristic
166C2×#92420C2×D411yescharacteristic
167#11758213—42no—
168Q16×C61—22no—
169Q16×C61—22no—
170C2×C8×C61C22×C411yescharacteristic
171C2×C8×C61C22×C411yescharacteristic
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61C22×C411yescharacteristic
176C42×C61—21no—
177C42×C61C22×C411yescharacteristic
178#11758202—22no—
179#11758202—22no—
180#11758198—21no—
181#11758198—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61C22×C411yescharacteristic
187C4⋊Dic61C22×C411yescharacteristic
188C4⋊Dic61C22×C411yescharacteristic
189#11758196—21no—
190#11758196—21no—
191#11758196—21no—
192#11758196—21no—
193#11758196—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61#4111yescharacteristic
201C4×Dic61C2411yescharacteristic
202C2×#92418C22×C411yescharacteristic
203C2×#92418C22×C411yescharacteristic
204C22×D61C4×D411yescharacteristic
205Q8×C61—21no—
206Q8×C61—21no—
207Q8×C61—21no—
208Q8×C61—21no—
209Q8×C61—21no—
210Q8×C61#13112yes—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61C4×D411yescharacteristic
216C2×C4×C61C4×D411yescharacteristic
217C2×C4×C61C23×C411yescharacteristic
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61C4×D411yescharacteristic
223C2×Dic61C22×D411yescharacteristic
224C2×Dic61C2×#4111yescharacteristic
225C4×D61—24no—
226C4×D61—21no—
227C4×D61—21no—
228C4×D61—21no—
229C4×D61—21no—
230C4×D61—41no—
231C4×D61—81no—
232Dic122—24no—
233Dic122—21no—
234Dic122—81no—
235Dic122#13112yes—
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×C61C2×C4×D411yescharacteristic
252C2×D61—21no—
253C2×D61—21no—
254C2×D61—21no—
255C4×C61—22no—
256C4×C61—21no—
257C4×C61—21no—
258C4×C61—41no—
259C4×C61—41no—
260C4×C61—41no—
261C4×C61C2×#13111yescharacteristic
262C4×C61C2×#13111yescharacteristic
263Dic61—21no—
264Dic61—22no—
265Dic61—21no—
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#292911yescharacteristic
287C2×C61#295011yescharacteristic
288C2×C61#295611yescharacteristic
289D61—41no—
290#533—611no—
291C2×C4×Q8—611no—
292#484—611no—
293#484—611no—
294#477—611no—
295#477—611no—
296#462—611no—
297#462#692311yescharacteristic
298#451—611no—
299#451—611no—
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×Q8—611no—
343C2×Q16—611no—
344C2×Q16—611no—
345C2×Q16C2×#692311yescharacteristic
346C2×Q16C2×#692311yescharacteristic
347C2×M16—1221no—
348C4⋊Q8—611no—
349C4⋊Q8C2×#692311yescharacteristic
350#125—1221no—
351C22⋊Q8—611no—
352C22⋊Q8—611no—
353C22⋊Q8—1221no—
354C4×Q8—611no—
355C4×Q8—611no—
356C4×Q8—611no—
357C4×Q8—611no—
358C4×Q8—611no—
359C4×Q8—1221no—
360C4×Q8—1221no—
361C4×Q8—1221no—
362C4×Q8C2×#692311yescharacteristic
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—1221no—
380C4⋊C8—1221no—
381C4⋊C8—1221no—
382C4⋊C8—1221no—
383C4⋊C8—1221no—
384C4⋊C8—1221no—
385C4⋊C8—1221no—
386C4⋊C8—1221no—
387C4⋊C8C2×#692311yescharacteristic
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—611no—
400C22⋊C8—1221no—
401C22⋊C8—1221no—
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—611no—
420C2×Q8—1221no—
421C2×Q8—1222no—
422C2×Q8—1222no—
423C2×Q8—1221no—
424C2×Q8C22×#692311yescharacteristic
425C2×Q8C22×#692311yescharacteristic
426C2×Q8C22×#692311yescharacteristic
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—1221no—
431C22×C4—1221no—
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—21no—
453C4⋊C4—21no—
454C4⋊C4—611no—
455C4⋊C4—611no—
456C4⋊C4—611no—
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—1221no—
468C22⋊C4—1221no—
469C42—21no—
470C42—611no—
471C42—611no—
472C42—611no—
473C42—1221no—
474C42—1221no—
475C42—1221no—
476C42—1221no—
477C42—1221no—
478C42C22×#692311yescharacteristic
479C23—611no—
480Q8—21no—
481Q8—21no—
482Q8—21no—
483Q8—21no—
484Q8—21no—
485Q8—612no—
486Q8—1224no—
487Q8—1221no—
488Q8—4881no—
489Q8#1451986412yes—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—611no—
495C2×C4—611no—
496C2×C4—611no—
497C2×C4—1221no—
498C2×C4—1221no—
499C2×C4—1224no—
500C2×C4—1221no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—4881no—
507C2×C4—2441no—
508C2×C4D4×#692311yescharacteristic
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—1221no—
526C22—1221no—
527C22—1221no—
528C22C2×D4×#692311yescharacteristic
529C4—22no—
530C4—21no—
531C4—21no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—611no—
536C4—611no—
537C4—1221no—
538C4—1222no—
539C4—1221no—
540C4—2441no—
541C4—2441no—
542C4—2441no—
543C4C2×#1451986411yescharacteristic
544C4D61⋊#13111yescharacteristic
545C2—2441no—
546C2#2146253311yescharacteristic
547C2#2146258011yescharacteristic
548C2#2146259411yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#462C4
socleC22×C61C2×C4×D4
prime radicalC22×D61C4×D4
cosocleC4×Dic61C24
hypercenter#462#6923
hypocenterC61#17925
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
3. #21461445C2semidirect
4. #21461444C2semidirect
5. #21461433C2semidirect
6. #21461432C2semidirect
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
#214627242
#214627132
#214614452
#214614442
#214614332
#214614322
#214614132
#214614122
#214614062
Q16×#924202
#214613912
#214611622
#214611422
#214610372
#214558572
#1792561