Finite Group Databasebeta

← #30601933

Subgroups — #30601933

Order 15616.

374 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 78 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#21461537C211yesmaximal, characteristic
2#21461380C211yesmaximal, characteristic
3#21461293C211yesmaximal, characteristic
4#21461292C211yesmaximal, characteristic
5#21461213C211yesmaximal, characteristic
6#21461211C211yesmaximal, characteristic
7#21456404C211yesmaximal, characteristic
8#16840784C2211yescharacteristic
9#16840784C2211yescharacteristic
10#16840721C2211yescharacteristic
11#16840698C2211yescharacteristic
12(C2×#6923)⋊C8—21no—
13(C2×#6923)⋊C8—21no—
14#16840625—21no—
15#16840623—21no—
16#16840619—22no—
17#16840619—22no—
18#16840616—22no—
19#16840615—22no—
20#16840317C2211yescharacteristic
21#16840317C2211yescharacteristic
22#16840165C411yescharacteristic, complemented
23#16840141C411yescharacteristic, complemented
24#16840113C411yescharacteristic, complemented
25#16840002—21no—
26#16840002—21no—
27#16839854—21no—
28#16839854—21no—
29C4×C8×D61C2211yescharacteristic
30C61×#487C411yescharacteristic, complemented
31C2×D61⋊C8—21no—
32C2×C4×#6923—21no—
33C4⋊#92420—21no—
34C4⋊#92420—21no—
35#14519834—21no—
36#14519834—21no—
37#14519832—21no—
38#14519832—21no—
39#14519832—21no—
40#14519832D412yes—
41(C2×#6923)⋊C4—21no—
42(C2×#6923)⋊C4D412yes—
43C2×#11758202—21no—
44C2×#11758202—21no—
45C2×C8×D61D411yescharacteristic
46C2×C8×D61D411yescharacteristic
47#14519777C2×C411yescharacteristic
48#14519777C2×C411yescharacteristic
49#14519768—21no—
50#14519768—21no—
51#14519757—21no—
52#14519757—21no—
53#14519753—21no—
54#14519753—21no—
55Q8×Dic61—21no—
56Q8×Dic61—21no—
57#14519742—21no—
58#14519742—21no—
59#14519735—21no—
60#14519735—21no—
61C42×D61C2311yescharacteristic
62#14519710—21no—
63#14519710—21no—
64#14519702—21no—
65#14519702—21no—
66C4×#92418C2×C411yescharacteristic
67C8×Dic61D411yescharacteristic
68C8×Dic61D411yescharacteristic
69C4⋊Q8×C61C2×C411yescharacteristic
70C4⋊Q8×C61C2×C411yescharacteristic
71#103×C61—21no—
72#103×C61—21no—
73C4×C8×C61C2×C411yescharacteristic
74C22×#6923—22no—
75C2×Q8×C61—21no—
76C2×Q8×C61—21no—
77C2×C4×D61C2×D411yescharacteristic
78C2×C4×D61C2×D411yescharacteristic
79C2×C4×D61#4111yescharacteristic
80C2×Dic122—21no—
81C2×Dic122—21no—
82C2×#92420—22no—
83C2×#92420—41no—
84C2×#92420—41no—
85C4×#6923—41no—
86C4×#6923—41no—
87D61⋊C8—41no—
88D61⋊C8—41no—
89C2×C8×C61C22⋊C411yescharacteristic
90C2×C8×C61C22⋊C411yescharacteristic
91C4⋊C4×C61—21no—
92C4⋊C4×C61—21no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—21no—
95C42×C61C22×C411yescharacteristic
96#11758202—42no—
97#11758202—42no—
98#11758198—41no—
99#11758198—41no—
100C4⋊Dic61—21no—
101C4⋊Dic61—21no—
102C4⋊Dic61—21no—
103C4⋊Dic61—21no—
104C4⋊Dic61—41no—
105C4⋊Dic61—41no—
106#11758196—41no—
107#11758196—41no—
108C4×Dic61—41no—
109C4×Dic61—41no—
110C4×Dic61C2×D411yescharacteristic
111C4×Dic61C2×D411yescharacteristic
112C4×Dic61#4111yescharacteristic
113C2×#92418C22⋊C411yescharacteristic
114C2×#92418C22⋊C411yescharacteristic
115C8×D61—41no—
116C8×D61—41no—
117C22×D61#12111yescharacteristic
118C2×#6923—42no—
119C2×#6923—42no—
120C2×#6923—42no—
121Q8×C61—41no—
122Q8×C61—41no—
123Q8×C61—81no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61C2×C22⋊C411yescharacteristic
127C2×C4×C61C4×D411yescharacteristic
128C2×C4×C61C4×D411yescharacteristic
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61#12111yescharacteristic
132C2×Dic61#12411yescharacteristic
133C2×Dic61#12711yescharacteristic
134C4×D61—22no—
135C4×D61—21no—
136C4×D61—21no—
137C4×D61—41no—
138Dic122—41no—
139Dic122—41no—
140Dic122—81no—
141#92420—42no—
142#92420—81no—
143#92420—81no—
144C8×C61—21no—
145C8×C61—21no—
146#92418—21no—
147#92418—21no—
148#9765—611nomaximal, Sylow
149C22×C61#39011yescharacteristic
150C2×D61—21no—
151C2×D61—21no—
152C2×D61#47312yes—
153#6923—82no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—81no—
157C4×C61—81no—
158C4×C61#42111yescharacteristic
159C4×C61#42111yescharacteristic
160Dic61—21no—
161Dic61—21no—
162Dic61—21no—
163Dic61—81no—
164Dic61—81no—
165#3170—611no—
166#1746—611no—
167#1568—611no—
168#1531—611no—
169#1531—611no—
170#1506—611no—
171#1506—611no—
172C2×C61#196311yescharacteristic
173C2×C61#197611yescharacteristic
174C2×C61#197911yescharacteristic
175D61—22no—
176#533—611no—
177#533—611no—
178#500—611no—
179#494—611no—
180#487—611no—
181#487#692311yescharacteristic, complemented
182#467—1221no—
183#467—1221no—
184C4×Q16—1221no—
185C4×Q16—1221no—
186#432—611no—
187#432—611no—
188C2×#97—611no—
189C2×C4×C8—611no—
190#376—611no—
191#336—1221no—
192#336—1221no—
193#335—1221no—
194#334—1221no—
195#330—1222no—
196#330—1222no—
197#328—1222no—
198#328—1222no—
199C61#976511yescharacteristic, complemented, Sylow
200C2×Q16—1221no—
201C2×Q16—1221no—
202C22×C8—611no—
203C22×C8—611no—
204C22×C8—1221no—
205C4⋊Q8—611no—
206C4⋊Q8—611no—
207C4⋊Q8C2×#692311yescharacteristic
208C4⋊Q8C2×#692311yescharacteristic
209#125—1221no—
210#125—1221no—
211C22⋊Q8—1221no—
212C22⋊Q8—1221no—
213C22⋊Q8—1221no—
214C22⋊Q8—1221no—
215C4×Q8—1221no—
216C4×Q8—1221no—
217C4×Q8—1221no—
218C4×Q8—1221no—
219#117—1221no—
220#117—1221no—
221C2×C42—611no—
222C2×C42—1221no—
223#107—1221no—
224#107—1221no—
225C4⋊C8—1221no—
226C4⋊C8—1221no—
227#103—21no—
228#103—21no—
229#103—1221no—
230#103—1221no—
231C22⋊C8—1221no—
232C22⋊C8—1221no—
233#97—612no—
234#97—1221no—
235#97—1221no—
236#97—1221no—
237C4×C8—611no—
238C4×C8—611no—
239C4×C8—611no—
240C4×C8C2×#692311yescharacteristic
241#95—612no—
242#95—1221no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—1221no—
246C2×Q8—1221no—
247C22×C4—611no—
248C22×C4—611no—
249C22×C4—611no—
250C22×C4—1222no—
251Q16—2442no—
252Q16—2442no—
253C2×C8—611no—
254C2×C8—611no—
255C2×C8—1222no—
256C2×C8—2441no—
257C2×C8—2441no—
258C2×C8—2441no—
259C2×C8—2441no—
260C2×C8—2441no—
261C2×C8—2441no—
262C2×C8C22⋊#692311yescharacteristic
263C2×C8C22⋊#692311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—1221no—
269C4⋊C4—1221no—
270C4⋊C4—1221no—
271C4⋊C4—1221no—
272C4⋊C4—2441no—
273C4⋊C4—2441no—
274C4⋊C4—2441no—
275C4⋊C4—2441no—
276C22⋊C4—2441no—
277C22⋊C4—2441no—
278C42—611no—
279C42—611no—
280C42—611no—
281C42—2441no—
282C42—2441no—
283C42—2441no—
284C42—2441no—
285C42C22×#692311yescharacteristic
286C23—611no—
287Q8—41no—
288Q8—41no—
289Q8—81no—
290Q8—4881no—
291Q8—2441no—
292Q8—2441no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—611no—
296C2×C4—611no—
297C2×C4—611no—
298C2×C4—1222no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—2441no—
302C2×C4—2442no—
303C2×C4—2442no—
304C2×C4—2442no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4C2×C22⋊#692311yescharacteristic
308C2×C4D4×#692311yescharacteristic
309C2×C4D4×#692311yescharacteristic
310C8—21no—
311C8—21no—
312C8—1221no—
313C8—1221no—
314C8—4881no—
315C8—4881no—
316C8—2442no—
317C22—612no—
318C22—1221no—
319C22—1221no—
320C22#1684071411yescharacteristic
321C4—21no—
322C4—21no—
323C4—81no—
324C4—81no—
325C4—1221no—
326C4—1221no—
327C4—1221no—
328C4—4882no—
329C4—4881no—
330C4—4881no—
331C4#1684075911yescharacteristic
332C4#1684075911yescharacteristic
333C2—1222no—
334C2#2146195611yescharacteristic
335C2#2146197711yescharacteristic
336C2#2146198111yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#487C4
socleC22×C61#390
prime radicalC22×D61#121
cosocleC42×D61C23
hypercenter#487#6923
hypocenterC61#9765
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
22. #16840165C4semidirect
23. #16840141C4semidirect
24. #16840113C4semidirect
30. C61×#487C4semidirect
181. #487#6923semidirect
199. C61#9765semidirect

Maximal subgroups

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

subgroupindex
#214615372
#214613802
#214612932
#214612922
#214612132
#214612112
#214564042
#976561