Finite Group Databasebeta

← #30602140

Subgroups — #30602140

Order 15616.

512 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 138 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#21461605C211yesmaximal, characteristic
2C4×#14519844C211yesmaximal, characteristic
3#21460655C214yesmaximal
4C4⋊Q8×Dic61C211yesmaximal, characteristic
5C2×#14519844—21no—
6C2×C4⋊#92420C2212yes—
7C2×#14519832—21no—
8C2×C4×#92420C2212yes—
9#16840628—21no—
10#16840628—21no—
11C2×Q8×Dic61—21no—
12#16839669—21no—
13#16839661—21no—
14#16839661C412yes—
15#16839632C411yescharacteristic
16C4⋊C4×Dic61—21no—
17C4⋊C4×Dic61C2212yes—
18C42×Dic61C2211yescharacteristic
19C2×C4⋊Q8×C61C411yescharacteristic
20C22×#92420—22no—
21C22×#92420—22no—
22#14519844—24no—
23C4⋊#92420—22no—
24C4⋊#92420Q814yescomplemented
25#14519832—42no—
26C4×#92420—24no—
27Q8×Dic61—42no—
28Q8×Dic61—42no—
29C2×C4⋊Dic61—21no—
30C2×C4⋊Dic61—21no—
31C2×C4⋊Dic61—22no—
32C2×C4⋊Dic61C2×C412yes—
33C2×C4×Dic61—22no—
34C2×C4×Dic61C2×C411yescharacteristic
35C2×C4×Dic61D411yescharacteristic
36C2×C4×Dic61D411yescharacteristic
37C2×C4×Dic61D411yescharacteristic
38C2×C4×Dic61D411yescharacteristic
39C2×C4×Dic61C2311yescharacteristic
40#14519684—22no—
41#14519684—22no—
42C22×Q8×C61—21no—
43C4⋊Q8×C61—24no—
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61C2×C412yes—
46C2×C42×C61C2×C411yescharacteristic
47C2×Q8×C61—24no—
48C2×Q8×C61—42no—
49C2×Q8×C61—42no—
50C22×C4×C61—22no—
51C22×C4×C61C22⋊C411yescharacteristic
52C22×C4×C61C22⋊C411yescharacteristic
53C22×C4×C61C22×C411yescharacteristic
54C22×Dic61—22no—
55C22×Dic61C22⋊C411yescharacteristic
56C22×Dic61C22⋊C411yescharacteristic
57C22×Dic61C2×D411yescharacteristic
58C22×Dic61C2×D411yescharacteristic
59C2×#92420—24no—
60C2×#92420—24no—
61C2×#92420—42no—
62C2×#92420—42no—
63C2×#92420—42no—
64C2×#92420—42no—
65C4⋊C4×C61—24no—
66C4⋊C4×C61—22no—
67C4⋊C4×C61M1614yes—
68C42×C61—22no—
69C4⋊Dic61—24no—
70C4⋊Dic61—42no—
71C4⋊Dic61—42no—
72C4⋊Dic61—44no—
73C4×Dic61—22no—
74C4×Dic61—22no—
75C4×Dic61—22no—
76C4×Dic61—41no—
77C4×Dic61—42no—
78C4×Dic61—42no—
79C4×Dic61D812yes—
80C4×Dic61SD1612yes—
81C4×Dic61C2×Q812yes—
82C4×Dic61#4112yes—
83C23×C61C2×C22⋊C411yescharacteristic
84Q8×C61—48no—
85C2×C4×C61—21no—
86C2×C4×C61—21no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—24no—
90C2×C4×C61—42no—
91C2×C4×C61—42no—
92C2×C4×C61#10212yes—
93C2×C4×C61#10412yes—
94C2×C4×C61C4×Q812yes—
95C2×C4×C61C2×M1612yes—
96C2×C4×C61#13112yes—
97C2×Dic61—21no—
98C2×Dic61—21no—
99C2×Dic61—21no—
100C2×Dic61—21no—
101C2×Dic61—21no—
102C2×Dic61—21no—
103C2×Dic61—21no—
104C2×Dic61—42no—
105C2×Dic61—42no—
106C2×Dic61—42no—
107C2×Dic61#10212yes—
108C2×Dic61C22⋊Q812yes—
109C2×Dic61#12411yescharacteristic
110C2×Dic61C4⋊Q811yescharacteristic
111C2×Dic61C2×D811yescharacteristic
112C2×Dic61C2×SD1611yescharacteristic
113C2×Dic61#13712yes—
114#92420—44no—
115#92420—44no—
116#9869—611nomaximal, Sylow
117C22×C61#39111yescharacteristic
118C22×C61#40711yescharacteristic
119C22×C61#40811yescharacteristic
120C22×C61C2×#10211yescharacteristic
121C22×C61#41911yescharacteristic
122C22×C61C2×#10411yescharacteristic
123C22×C61#42111yescharacteristic
124C4×C61—22no—
125C4×C61—22no—
126C4×C61—44no—
127C4×C61#44614yes—
128Dic61—21no—
129Dic61—21no—
130Dic61—22no—
131Dic61—41no—
132Dic61—82no—
133Dic61#47412yes—
134Dic61#47512yes—
135C4×C4⋊Q8—611no—
136#1767—611no—
137C4×C4⋊C8—611no—
138#1279—614no—
139C2×C61#148611yescharacteristic
140C2×C61#149111yescharacteristic
141C2×C61#149211yescharacteristic
142C2×C61#149711yescharacteristic
143C2×C61#198511yescharacteristic
144C2×C61#198611yescharacteristic
145C2×C61#198811yescharacteristic
146C2×C4⋊Q8#692311yescharacteristic
147C2×C4×Q8—1221no—
148C2×C4⋊C8—612no—
149C2×C4⋊C8—1221no—
150C2×#97—1221no—
151C2×C4×C8—612no—
152#391—1221no—
153#389—612no—
154#389—1221no—
155#384—611no—
156C4×C4⋊C4—612no—
157C4×C4⋊C4—1221no—
158C43—611no—
159#336—1221no—
160#336—1221no—
161C61#986911yescharacteristic, complemented, Sylow
162C22×Q8—21no—
163C22×C8—1222no—
164C22×C8—1222no—
165C4⋊Q8—24no—
166C4×Q8—2442no—
167C4×Q8—2442no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—612no—
170C2×C4⋊C4—1222no—
171C2×C4⋊C4—1221no—
172C2×C4⋊C4—1221no—
173C2×C4⋊C4C2×#692312yes—
174C2×C42—611no—
175C2×C42—611no—
176C2×C42—611no—
177C2×C42—611no—
178C2×C42—611no—
179C2×C42—611no—
180C2×C42—1222no—
181C2×C42C2×#692311yescharacteristic
182C4⋊C8—614no—
183C4⋊C8—1222no—
184C4⋊C8—1224no—
185#97—2442no—
186C4×C8—1224no—
187#95—1222no—
188#95—1222no—
189C2×Q8—24no—
190C2×Q8—42no—
191C2×Q8—42no—
192C22×C4—22no—
193C22×C4—611no—
194C22×C4—611no—
195C22×C4—611no—
196C22×C4—611no—
197C22×C4—1222no—
198C22×C4C22⋊#692311yescharacteristic
199C22×C4C22⋊#692311yescharacteristic
200C22×C4C22×#692311yescharacteristic
201C2×C8—1224no—
202C2×C8—1224no—
203C2×C8—2442no—
204C2×C8—2442no—
205C2×C8—2442no—
206C2×C8—2442no—
207C4⋊C4—24no—
208C4⋊C4—22no—
209C4⋊C4—1224no—
210C4⋊C4—2444no—
211C4⋊C4—2442no—
212C4⋊C4—2442no—
213C4⋊C4#1175821714yes—
214C42—22no—
215C42—612no—
216C42—612no—
217C42—612no—
218C42—612no—
219C42—1222no—
220C42—1222no—
221C42—1222no—
222C42—2441no—
223C42—2442no—
224C42—2442no—
225C23C2×C22⋊#692311yescharacteristic
226Q8—48no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—24no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—612no—
235C2×C4—611no—
236C2×C4—611no—
237C2×C4—612no—
238C2×C4—611no—
239C2×C4—611no—
240C2×C4—612no—
241C2×C4—1221no—
242C2×C4—1221no—
243C2×C4—1221no—
244C2×C4—1221no—
245C2×C4—1221no—
246C2×C4—1221no—
247C2×C4—1221no—
248C2×C4—2442no—
249C2×C4—2442no—
250C2×C4—2442no—
251C2×C4#1451983912yes—
252C2×C4#1451984212yes—
253C2×C4Q8×#692312yes—
254C2×C4C2×#1175821712yes—
255C2×C4#1451986512yes—
256C8—2444no—
257C8—2444no—
258C22(C2×#6923)⋊Q811yescharacteristic
259C22#1684073211yescharacteristic
260C22#1684073711yescharacteristic
261C22C2×#1451983911yescharacteristic
262C22#1684075511yescharacteristic
263C22C2×#1451984211yescharacteristic
264C22#1684076011yescharacteristic
265C4—22no—
266C4—22no—
267C4—44no—
268C4—612no—
269C4—612no—
270C4—1222no—
271C4—1221no—
272C4—1221no—
273C4—4882no—
274C4—2441no—
275C4#1684081314yes—
276C2#2146114411yescharacteristic
277C2#2146116111yescharacteristic
278C2#2146116611yescharacteristic
279C2#2146118311yescharacteristic
280C2#2146199311yescharacteristic
281C2#2146199411yescharacteristic
282C2#2146199811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4⋊Q8×C61C4
socle, prime radicalC23×C61C2×C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×C4⋊Q8#6923
hypocenterC61#9869
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
24. C4⋊#92420Q8semidirect
161. C61#9869semidirect

Maximal subgroups

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

subgroupindex
#214616052
C4×#145198442
#214606552
C4⋊Q8×Dic612
#986961