Finite Group Databasebeta

← #30623701

Subgroups — #30623701

Order 15616.

616 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 186 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#21461993C212yesmaximal
2#21461992C212yesmaximal
3#21461878C214yesmaximal
4#21461877C214yesmaximal
5#21461747C212yesmaximal
6#21448388C211yesmaximal, characteristic
7C2×#14519844C2212yes—
8C2×C4⋊#92420C2212yes—
9C2×#14519832C2212yes—
10C2×#14519832C2212yes—
11#16840628C2212yes—
12#16840628C2214yes—
13#16840628C2214yes—
14#16840628C2214yes—
15#16840628C2212yes—
16#16840628C2214yes—
17C2×#14519775C411yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20#16839668C411yescharacteristic
21#16839659C414yes—
22#16839631C411yescharacteristic
23#16839613C2214yes—
24C42×Dic61C2211yescharacteristic
25C61×#391C411yescharacteristic
26C22×#92420C2314yes—
27C22×#92420C2314yes—
28#14519844—42no—
29C4⋊#92420—42no—
30#14519832—42no—
31#14519832—42no—
32C22×Dic122C2×C411yescharacteristic
33#14519775—24no—
34Q8×Dic61—41no—
35Q8×Dic61—41no—
36#14519742—41no—
37#14519742—41no—
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×C4⋊Dic61C2×C412yes—
40C2×#11758196C2×C414yes—
41C2×#11758196C2×C412yes—
42C2×#11758196C2×C414yes—
43C2×C4×Dic61C2×C412yes—
44C2×C4×Dic61C2×C411yescharacteristic
45C2×C4×Dic61C2312yes—
46C2×C4×Dic61C2312yes—
47C2×C4×Dic61C2311yescharacteristic
48C2×C4×Dic61C2312yes—
49#14519684C2×C414yes—
50C22×Q8×C61C2×C411yescharacteristic
51C2×C4⋊C4×C61C2×C411yescharacteristic
52C2×C42×C61C2×C411yescharacteristic
53#95×C61C2×C414yes—
54C2×Q8×C61—22no—
55C2×Q8×C61—41no—
56C2×Q8×C61—41no—
57C22×C4×C61C22×C412yes—
58C22×C4×C61C22×C412yes—
59C22×C4×C61C22×C412yes—
60C22×C4×C61C22×C411yescharacteristic
61C22×Dic61C22×C412yes—
62C22×Dic61C22×C412yes—
63C22×Dic61C22×C411yescharacteristic
64C22×Dic61C22×C412yes—
65C22×Dic61C2411yescharacteristic
66C2×Dic122—22no—
67C2×Dic122—41no—
68C2×Dic122—41no—
69C2×#92420—24no—
70C2×#92420—24no—
71C2×#92420—28no—
72C2×#92420—28no—
73C4⋊C4×C61—41no—
74C42×C61—41no—
75C4⋊Dic61—42no—
76C4⋊Dic61—41no—
77#11758196—28no—
78#11758196—44no—
79#11758196—42no—
80C4×Dic61—22no—
81C4×Dic61—22no—
82C4×Dic61—22no—
83C4×Dic61—22no—
84C4×Dic61—21no—
85C4×Dic61—21no—
86C4×Dic61—24no—
87C4×Dic61—42no—
88C4×Dic61M1614yes—
89C23×C61C23×C411yescharacteristic
90Q8×C61—44no—
91C2×C4×C61—21no—
92C2×C4×C61—21no—
93C2×C4×C61—22no—
94C2×C4×C61—22no—
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—22no—
98C2×C4×C61—24no—
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61C2×M1612yes—
102C2×Dic61—22no—
103C2×Dic61—22no—
104C2×Dic61—21no—
105C2×Dic61—22no—
106C2×Dic61—22no—
107C2×Dic61—22no—
108C2×Dic61—22no—
109C2×Dic61—24no—
110C2×Dic61—22no—
111C2×Dic61C2×M1614yes—
112C2×Dic61H(5,𝔽2)11yescharacteristic
113C2×Dic61H(5,𝔽2)11yescharacteristic
114C2×Dic612−1+414yes—
115Dic122—44no—
116#92420—44no—
117#92420—44no—
118#23366—611nomaximal, Sylow
119C22×C61#51811yescharacteristic
120C22×C61#51912yes—
121C22×C61C22×M1611yescharacteristic
122C22×C61#56811yescharacteristic
123C22×C61#56812yes—
124C4×C61—22no—
125C4×C61—42no—
126C4×C61—42no—
127C4×C61—42no—
128Dic61—24no—
129Dic61—42no—
130Dic61—41no—
131Dic61—42no—
132Dic61#56312yes—
133Dic61#56312yes—
134C4×C4⋊Q8—611no—
135#1985—612no—
136#1985—612no—
137#1917—614no—
138#1917—614no—
139#1839—612no—
140C2×C61#238111yescharacteristic
141C2×C61#297511yescharacteristic
142C2×C61#298011yescharacteristic
143C2×C61#298312yes—
144C2×C61#298712yes—
145C2×C4⋊Q8—611no—
146C2×C4×Q8—611no—
147C2×C4×Q8—611no—
148C2×C4⋊C8—612no—
149C2×C4⋊C8—612no—
150C2×#97—612no—
151C2×#97—612no—
152#391—611no—
153#391#692311yescharacteristic
154#389—614no—
155#384—611no—
156C4×C4⋊C4—614no—
157C43—611no—
158#336—612no—
159#336—614no—
160#336—612no—
161#336—614no—
162#336—614no—
163#336—614no—
164C61#2336611yescharacteristic, complemented, Sylow
165C22×Q8—611no—
166C22×Q8C2×#692311yescharacteristic
167C22×C8—614no—
168C22×C8—614no—
169C4⋊Q8—1224no—
170C4×Q8—2441no—
171C4×Q8—2441no—
172C4×Q8—2441no—
173C4×Q8—2441no—
174C2×C4⋊C4—614no—
175C2×C4⋊C4—611no—
176C2×C4⋊C4—612no—
177C2×C4⋊C4—612no—
178C2×C4⋊C4—614no—
179C2×C4⋊C4C2×#692311yescharacteristic
180C2×C42—611no—
181C2×C42—611no—
182C2×C42—612no—
183C2×C42—612no—
184C2×C42—612no—
185C2×C42—612no—
186C2×C42C2×#692311yescharacteristic
187C4⋊C8—2442no—
188C4⋊C8—2442no—
189#97—2442no—
190#97—2442no—
191#95—614no—
192#95C2×#692314yes—
193C2×Q8—22no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—1222no—
197C2×Q8—2441no—
198C2×Q8—2441no—
199C22×C4—611no—
200C22×C4—612no—
201C22×C4—612no—
202C22×C4—612no—
203C22×C4—611no—
204C22×C4C22×#692312yes—
205C22×C4C22×#692312yes—
206C22×C4C22×#692312yes—
207C22×C4C22×#692311yescharacteristic
208C2×C8—1224no—
209C2×C8—1224no—
210C2×C8—1228no—
211C2×C8—1228no—
212C4⋊C4—41no—
213C4⋊C4—1228no—
214C4⋊C4—2444no—
215C4⋊C4—2442no—
216C4⋊C4—2441no—
217C4⋊C4—2442no—
218C42—41no—
219C42—614no—
220C42—1222no—
221C42—1222no—
222C42—1222no—
223C42—1222no—
224C42—1224no—
225C42—1221no—
226C42—1221no—
227C42—2442no—
228C23C23×#692311yescharacteristic
229Q8—44no—
230Q8—2444no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—611no—
242C2×C4—611no—
243C2×C4—614no—
244C2×C4—614no—
245C2×C4—1222no—
246C2×C4—1222no—
247C2×C4—1221no—
248C2×C4—1222no—
249C2×C4—1222no—
250C2×C4—1222no—
251C2×C4—1222no—
252C2×C4—1222no—
253C2×C4—1224no—
254C2×C4D61⋊M1612yes—
255C8—2444no—
256C8—2444no—
257C22#1684082911yescharacteristic
258C22#1684083012yes—
259C22C2×D61⋊M1611yescharacteristic
260C22#1684084011yescharacteristic
261C22#1684084112yes—
262C4—22no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—612no—
267C4—612no—
268C4—1224no—
269C4—2442no—
270C4—2441no—
271C4—2442no—
272C2#2146236411yescharacteristic
273C2#2146268311yescharacteristic
274C2#2146270511yescharacteristic
275C2#2146271612yes—
276C2#2146273112yes—
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socle, prime radicalC23×C61C23×C4
FittingC61×#391C4
cosocleC22×Dic61C24
hypercenter#391#6923
hypocenterC61#23366
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
164. C61#23366semidirect

Maximal subgroups

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

subgroupindex
#214619932
#214619922
#214618782
#214618772
#214617472
#214483882
#2336661