Finite Group Databasebeta

← #30600763

Subgroups — #30600763

Order 15616.

422 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462612C211yesmaximal, characteristic
2#21462003C211yesmaximal, characteristic
3#21461335C211yesmaximal, characteristic
4#21461253C211yesmaximal, characteristic
5#21460673C211yesmaximal, characteristic
6#21460644C211yesmaximal, characteristic
7#21456763C211yesmaximal, characteristic
8#16840814—22no—
9#16840813—22no—
10#16840780C2211yescharacteristic
11#16840769—21no—
12#16840767C2211yescharacteristic
13#16840727—21no—
14#16840721C2211yescharacteristic
15#16840706C2211yescharacteristic
16#16840635—21no—
17#16840635—21no—
18#16840621—22no—
19#16840616—22no—
20#16840317C2211yescharacteristic
21C4⋊Q8×D61C2211yescharacteristic
22#16840141C411yescharacteristic, complemented
23#16840113C411yescharacteristic, complemented
24#16840112C411yescharacteristic, complemented
25#16839970—22no—
26#16839916—22no—
27#16839813C2211yescharacteristic
28C61×#494C411yescharacteristic, complemented
29C2×#11758213—21no—
30C2×D61⋊C8D411yescharacteristic
31C2×D61⋊C8D411yescharacteristic
32C2×C4⋊#6923—21no—
33#14519844—22no—
34#14519844—21no—
35#14519844—21no—
36#14519844D412yes—
37C4⋊#92420—22no—
38#14519834—21no—
39#14519832—22no—
40#14519832—21no—
41#14519832—21no—
42C4×#92420—22no—
43C4×#92420—21no—
44(C2×#6923)⋊C4—21no—
45(C2×#6923)⋊C4—21no—
46C2×Q8×D61—21no—
47C2×#11758202—22no—
48#14519777C2×C411yescharacteristic
49#14519777C2×C411yescharacteristic
50#14519776—21no—
51#14519776—21no—
52#14519775—21no—
53#14519768—21no—
54#14519757—21no—
55#14519753D412yes—
56Q8×Dic61D412yes—
57#14519742—21no—
58#14519735—21no—
59C4⋊C4×D61—21no—
60C4⋊C4×D61—21no—
61C42×D61C2311yescharacteristic
62#14519706—21no—
63#14519702—22no—
64#14519701—22no—
65#14519689—21no—
66C4×#92418C2×C411yescharacteristic
67C2×Q16×C61—22no—
68C4⋊Q8×C61C2×C411yescharacteristic
69C4⋊Q8×C61C2×C411yescharacteristic
70C4×C8×C61C2×C411yescharacteristic
71C22×#6923—41no—
72C2×Q8×C61—21no—
73C2×Q8×C61C22⋊C412yes—
74Q8×D61—82no—
75C2×C4×D61—41no—
76C2×C4×D61C2×D411yescharacteristic
77C2×C4×D61C2×D411yescharacteristic
78C2×C4×D61#4111yescharacteristic
79C2×Dic122—21no—
80C2×Dic122—21no—
81C2×Dic122—41no—
82C2×#92420—21no—
83C2×#92420—22no—
84C2×#92420—21no—
85C2×#92420—41no—
86C4⋊#6923—41no—
87C4⋊#6923—41no—
88#11758213—41no—
89#11758213—41no—
90D61⋊C8—21no—
91D61⋊C8—21no—
92D61⋊C8—21no—
93D61⋊C8Q1612yescomplemented
94Q16×C61—82no—
95C2×C8×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—21no—
100C42×C61C22×C411yescharacteristic
101#11758202—82no—
102#11758198—22no—
103C4⋊Dic61—21no—
104C4⋊Dic61—22no—
105C4⋊Dic61—21no—
106C4⋊Dic61—21no—
107C4⋊Dic61C22⋊C412yes—
108#11758196—22no—
109#11758196—41no—
110#11758196—41no—
111C4×Dic61—22no—
112C4×Dic61C2×D411yescharacteristic
113C4×Dic61C2×D411yescharacteristic
114C4×Dic61#4111yescharacteristic
115C2×#92418—21no—
116C22×D61#12111yescharacteristic
117C2×#6923—81no—
118C2×#6923—81no—
119C2×#6923—81no—
120Q8×C61—42no—
121Q8×C61—42no—
122C2×C4×C61—22no—
123C2×C4×C61—41no—
124C2×C4×C61C2×C22⋊C411yescharacteristic
125C2×C4×C61C4×D411yescharacteristic
126C2×C4×C61C4×D411yescharacteristic
127C2×Dic61—22no—
128C2×Dic61—41no—
129C2×Dic61#12011yescharacteristic
130C2×Dic61#12111yescharacteristic
131C2×Dic61#12311yescharacteristic
132C4×D61—41no—
133C4×D61—41no—
134C4×D61—82no—
135C4×D61C2×Q1611yescharacteristic
136C4×D61#13511yescharacteristic
137C4×D61#13711yescharacteristic
138C4×D61#13711yescharacteristic
139Dic122—42no—
140Dic122—82no—
141Dic122—81no—
142#92420—22no—
143#92420—41no—
144#92420—42no—
145#92420—81no—
146C8×C61—41no—
147#92418—41no—
148#9290—611nomaximal, Sylow
149C22×C61#38611yescharacteristic
150C2×D61—21no—
151C2×D61—21no—
152C2×D61#46211yescharacteristic
153C2×D61#46411yescharacteristic
154#6923—161no—
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—42no—
158C4×C61—81no—
159C4×C61C4×Q1611yescharacteristic
160C4×C61#44111yescharacteristic
161Dic61—21no—
162Dic61—21no—
163Dic61—42no—
164Dic61—81no—
165Dic61#45111yescharacteristic
166Dic61#45211yescharacteristic
167#3209—611no—
168#2961—611no—
169#1993—611no—
170#1549—611no—
171#1518—611no—
172#1286—611no—
173#1275—611no—
174C2×C61#187811yescharacteristic
175C2×C61#190211yescharacteristic
176C2×C61#191011yescharacteristic
177D61—21no—
178D61—21no—
179#533—611no—
180C2×C4⋊Q8—611no—
181#494—611no—
182#494#692311yescharacteristic, complemented
183#487—611no—
184#487—611no—
185#462—1222no—
186#451—1222no—
187#446—1222no—
188#446—1222no—
189#431—611no—
190#431—611no—
191#424—1221no—
192#423—611no—
193#405—1221no—
194C2×#97—611no—
195#382—611no—
196#340—1221no—
197#340—1221no—
198#332—1222no—
199#328—1222no—
200C61#929011yescharacteristic, complemented, Sylow
201C22×Q8—1221no—
202C2×Q16—22no—
203C2×Q16—1222no—
204C2×M16—1221no—
205C22×C8—611no—
206C22×C8—611no—
207C4⋊Q8—611no—
208C4⋊Q8—611no—
209C4⋊Q8—1221no—
210C4⋊Q8—1221no—
211C4⋊Q8—1221no—
212C4⋊Q8C2×#692311yescharacteristic
213C4⋊Q8C2×#692311yescharacteristic
214#125—1221no—
215C22⋊Q8—612no—
216C22⋊Q8—1221no—
217C4×Q8—612no—
218C4×Q8—1221no—
219#117—1221no—
220C2×C4⋊C4—1221no—
221C2×C4⋊C4—1221no—
222C2×C4⋊C4—1221no—
223C2×C42—611no—
224C4⋊C8—612no—
225C4⋊C8—1221no—
226C4⋊C8—1221no—
227C4⋊C8—1221no—
228C4⋊C8—1222no—
229C4⋊C8—1222no—
230#103—1222no—
231#103—1222no—
232C22⋊C8—1221no—
233C22⋊C8—1221no—
234#97—1221no—
235#97—1221no—
236#97—1222no—
237C4×C8—611no—
238C4×C8—1221no—
239C4×C8—1222no—
240C4×C8C2×#692311yescharacteristic
241#95—1221no—
242#95—1221no—
243C2×Q8—21no—
244C2×Q8—1221no—
245C2×Q8—1221no—
246C2×Q8—4882no—
247C2×Q8—2441no—
248C2×Q8C22⋊#692312yes—
249C22×C4—611no—
250C22×C4—611no—
251C22×C4—611no—
252C22×C4—2441no—
253C22×C4—2441no—
254Q16—82no—
255Q16—4882no—
256M16—2441no—
257M16—2441no—
258C2×C8—21no—
259C2×C8—612no—
260C2×C8—1221no—
261C2×C8—1221no—
262C2×C8—1222no—
263C2×C8—1221no—
264C2×C8—1221no—
265C2×C8—1221no—
266C2×C8—1221no—
267C2×C8—2441no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—612no—
273C4⋊C4—1222no—
274C4⋊C4—1222no—
275C4⋊C4—1221no—
276C4⋊C4—1221no—
277C4⋊C4—1221no—
278C4⋊C4—2441no—
279C4⋊C4—2441no—
280C4⋊C4—2441no—
281C4⋊C4—2441no—
282C22⋊C4—1222no—
283C42—611no—
284C42—611no—
285C42—611no—
286C42—1222no—
287C42C22×#692311yescharacteristic
288C23—611no—
289Q8—42no—
290Q8—42no—
291Q8—4881no—
292Q8—4882no—
293Q8—2442no—
294C2×C4—22no—
295C2×C4—41no—
296C2×C4—611no—
297C2×C4—611no—
298C2×C4—611no—
299C2×C4—611no—
300C2×C4—611no—
301C2×C4—611no—
302C2×C4—611no—
303C2×C4—1222no—
304C2×C4—4882no—
305C2×C4—4881no—
306C2×C4—4881no—
307C2×C4—4881no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4C2×C22⋊#692311yescharacteristic
312C2×C4D4×#692311yescharacteristic
313C2×C4D4×#692311yescharacteristic
314C8—41no—
315C8—1222no—
316C8—4881no—
317C8—2441no—
318C8—2441no—
319C8—2442no—
320C22—611no—
321C22—611no—
322C22—1221no—
323C22—1221no—
324C22#1684071011yescharacteristic
325C4—21no—
326C4—21no—
327C4—42no—
328C4—81no—
329C4—611no—
330C4—611no—
331C4—1221no—
332C4—1221no—
333C4—9761no—
334C4—4881no—
335C4—2442no—
336C4Q16×#692311yescharacteristic
337C4#1684080211yescharacteristic
338C2—1221no—
339C2—1221no—
340C2#2146181311yescharacteristic
341C2#2146184911yescharacteristic
342C2#2146186211yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#494C4
socleC22×C61#386
prime radicalC22×D61#121
cosocleC42×D61C23
hypercenter#494#6923
hypocenterC61#9290
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
22. #16840141C4semidirect
23. #16840113C4semidirect
24. #16840112C4semidirect
28. C61×#494C4semidirect
93. D61⋊C8Q16semidirect
182. #494#6923semidirect
200. C61#9290semidirect

Maximal subgroups

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

subgroupindex
#214626122
#214620032
#214613352
#214612532
#214606732
#214606442
#214567632
#929061