Finite Group Databasebeta

← #30602137

Subgroups — #30602137

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—44no—
28C2×C4⋊Dic61—21no—
29C2×C4⋊Dic61—21no—
30C2×C4⋊Dic61—22no—
31C2×C4⋊Dic61C2×C412yes—
32C2×C4×Dic61—22no—
33C2×C4×Dic61C2×C411yescharacteristic
34C2×C4×Dic61D411yescharacteristic
35C2×C4×Dic61D411yescharacteristic
36C2×C4×Dic61D411yescharacteristic
37C2×C4×Dic61D411yescharacteristic
38C2×C4×Dic61C2311yescharacteristic
39#14519684—22no—
40#14519684—22no—
41C22×Q8×C61—21no—
42C4⋊Q8×C61—24no—
43C2×C4⋊C4×C61—21no—
44C2×C4⋊C4×C61C2×C412yes—
45C2×C42×C61C2×C411yescharacteristic
46C2×Q8×C61—24no—
47C2×Q8×C61—42no—
48C2×Q8×C61—42no—
49C22×C4×C61—22no—
50C22×C4×C61C22⋊C411yescharacteristic
51C22×C4×C61C22⋊C411yescharacteristic
52C22×C4×C61C22×C411yescharacteristic
53C22×Dic61—22no—
54C22×Dic61C22⋊C411yescharacteristic
55C22×Dic61C22⋊C411yescharacteristic
56C22×Dic61C2×D411yescharacteristic
57C22×Dic61C2×D411yescharacteristic
58C2×#92420—24no—
59C2×#92420—24no—
60C2×#92420—42no—
61C2×#92420—42no—
62C2×#92420—42no—
63C2×#92420—42no—
64C4⋊C4×C61—24no—
65C4⋊C4×C61—22no—
66C4⋊C4×C61M1614yes—
67C42×C61—22no—
68C4⋊Dic61—24no—
69C4⋊Dic61—44no—
70C4⋊Dic61—42no—
71C4⋊Dic61—42no—
72C4×Dic61—22no—
73C4×Dic61—22no—
74C4×Dic61—22no—
75C4×Dic61—41no—
76C4×Dic61—44no—
77C4×Dic61SD1612yes—
78C4×Dic61Q1612yes—
79C4×Dic61C2×Q812yes—
80C4×Dic61#4112yes—
81C23×C61C2×C22⋊C411yescharacteristic
82Q8×C61—48no—
83C2×C4×C61—21no—
84C2×C4×C61—21no—
85C2×C4×C61—21no—
86C2×C4×C61—21no—
87C2×C4×C61—24no—
88C2×C4×C61—42no—
89C2×C4×C61—42no—
90C2×C4×C61#10312yes—
91C2×C4×C61#10412yes—
92C2×C4×C61C4×Q812yes—
93C2×C4×C61C2×M1612yes—
94C2×C4×C61#13112yes—
95C2×Dic61—21no—
96C2×Dic61—21no—
97C2×Dic61—21no—
98C2×Dic61—21no—
99C2×Dic61—21no—
100C2×Dic61—21no—
101C2×Dic61—21no—
102C2×Dic61—42no—
103C2×Dic61—42no—
104C2×Dic61—42no—
105C2×Dic61#10312yes—
106C2×Dic61C22⋊Q812yes—
107C2×Dic61#12411yescharacteristic
108C2×Dic61C4⋊Q811yescharacteristic
109C2×Dic61C2×SD1611yescharacteristic
110C2×Dic61C2×Q1611yescharacteristic
111C2×Dic61AGL(1,ℤ/8)12yes—
112#92420—44no—
113#92420—44no—
114#9868—611nomaximal, Sylow
115C22×C61#39111yescharacteristic
116C22×C61#40711yescharacteristic
117C22×C61#40811yescharacteristic
118C22×C61C2×#10311yescharacteristic
119C22×C61#41811yescharacteristic
120C22×C61C2×#10411yescharacteristic
121C22×C61#42111yescharacteristic
122C4×C61—22no—
123C4×C61—22no—
124C4×C61—44no—
125C4×C61#44614yes—
126Dic61—21no—
127Dic61—21no—
128Dic61—22no—
129Dic61—41no—
130Dic61—82no—
131Dic61#47612yes—
132Dic61#47712yes—
133C4×C4⋊Q8—611no—
134#1767—611no—
135C4×C4⋊C8—611no—
136#1279—614no—
137C2×C61#148711yescharacteristic
138C2×C61#149011yescharacteristic
139C2×C61#149311yescharacteristic
140C2×C61#149611yescharacteristic
141C2×C61#198511yescharacteristic
142C2×C61#198611yescharacteristic
143C2×C61#198711yescharacteristic
144C2×C4⋊Q8#692311yescharacteristic
145C2×C4×Q8—1221no—
146C2×C4⋊C8—612no—
147C2×C4⋊C8—1221no—
148C2×#97—1221no—
149C2×C4×C8—612no—
150#391—1221no—
151#389—612no—
152#389—1221no—
153#384—611no—
154C4×C4⋊C4—612no—
155C4×C4⋊C4—1221no—
156C43—611no—
157#336—1221no—
158#336—1221no—
159C61#986811yescharacteristic, complemented, Sylow
160C22×Q8—21no—
161C22×C8—1222no—
162C22×C8—1222no—
163C4⋊Q8—24no—
164C4×Q8—2444no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—612no—
167C2×C4⋊C4—1222no—
168C2×C4⋊C4—1221no—
169C2×C4⋊C4—1221no—
170C2×C4⋊C4C2×#692312yes—
171C2×C42—611no—
172C2×C42—611no—
173C2×C42—611no—
174C2×C42—611no—
175C2×C42—611no—
176C2×C42—611no—
177C2×C42—1222no—
178C2×C42C2×#692311yescharacteristic
179C4⋊C8—614no—
180C4⋊C8—1222no—
181C4⋊C8—1224no—
182#97—2442no—
183C4×C8—1224no—
184#95—1222no—
185#95—1222no—
186C2×Q8—24no—
187C2×Q8—42no—
188C2×Q8—42no—
189C22×C4—22no—
190C22×C4—611no—
191C22×C4—611no—
192C22×C4—611no—
193C22×C4—611no—
194C22×C4—1222no—
195C22×C4C22⋊#692311yescharacteristic
196C22×C4C22⋊#692311yescharacteristic
197C22×C4C22×#692311yescharacteristic
198C2×C8—1224no—
199C2×C8—1224no—
200C2×C8—2442no—
201C2×C8—2442no—
202C2×C8—2442no—
203C2×C8—2442no—
204C4⋊C4—24no—
205C4⋊C4—22no—
206C4⋊C4—1224no—
207C4⋊C4—2442no—
208C4⋊C4—2442no—
209C4⋊C4—2444no—
210C4⋊C4#1175821714yes—
211C42—22no—
212C42—612no—
213C42—612no—
214C42—612no—
215C42—612no—
216C42—1222no—
217C42—1222no—
218C42—1222no—
219C42—2441no—
220C42—2444no—
221C23C2×C22⋊#692311yescharacteristic
222Q8—48no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—612no—
231C2×C4—611no—
232C2×C4—611no—
233C2×C4—612no—
234C2×C4—611no—
235C2×C4—611no—
236C2×C4—612no—
237C2×C4—1221no—
238C2×C4—1221no—
239C2×C4—1221no—
240C2×C4—1221no—
241C2×C4—1221no—
242C2×C4—1221no—
243C2×C4—1221no—
244C2×C4—2442no—
245C2×C4—2442no—
246C2×C4—2442no—
247C2×C4#1451984012yes—
248C2×C4#1451984112yes—
249C2×C4Q8×#692312yes—
250C2×C4C2×#1175821712yes—
251C2×C4#1451986512yes—
252C8—2444no—
253C8—2444no—
254C22(C2×#6923)⋊Q811yescharacteristic
255C22#1684073211yescharacteristic
256C22#1684073711yescharacteristic
257C22C2×#1451984011yescharacteristic
258C22#1684075411yescharacteristic
259C22C2×#1451984111yescharacteristic
260C22#1684076111yescharacteristic
261C4—22no—
262C4—22no—
263C4—44no—
264C4—612no—
265C4—612no—
266C4—1222no—
267C4—1221no—
268C4—1221no—
269C4—4882no—
270C4—2441no—
271C4#1684081314yes—
272C2#2146114711yescharacteristic
273C2#2146115811yescharacteristic
274C2#2146116911yescharacteristic
275C2#2146118011yescharacteristic
276C2#2146199311yescharacteristic
277C2#2146199511yescharacteristic
278C2#2146199711yescharacteristic
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#9868
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
159. C61#9868semidirect

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
#986861