Finite Group Databasebeta

← #30598191

Subgroups — #30598191

Order 15616.

1146 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1146 conjugacy classes of subgroups — too many to draw. Only its 164 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#21461118C214yesmaximal, complemented
2#21461050C212yesmaximal, complemented
3C2×#16840237C211yesmaximal, characteristic
4#16840778C2214yescomplemented
5D4⋊#92420—28no—
6#16840607—22no—
7#16840605—22no—
8#16840237—28no—
9C2×D4×Dic61—22no—
10C2×D4×Dic61C412yes—
11C2×D4×Dic61C2212yes—
12C2×#14519736C411yescharacteristic
13C2×#14519736C2211yescharacteristic
14C22×C22⋊Dic61—22no—
15C23×D4×C61C411yescharacteristic
16C4⋊#92420—24no—
17C22⋊#92420—24no—
18C4×#92420—24no—
19C23×Dic61—22no—
20C23×Dic61—22no—
21D4×Dic61—216no—
22D4×Dic61—24no—
23D4×Dic61—24no—
24D4×Dic61—24no—
25D4×Dic61D418yes—
26#14519736—24no—
27#14519736—24no—
28C2×C4⋊Dic61C2×C412yes—
29C2×C4⋊Dic61D412yes—
30C2×C22⋊Dic61—216no—
31C2×C22⋊Dic61—21no—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61—22no—
34C2×C22⋊Dic61D412yes—
35C2×C4×Dic61—21no—
36C2×C4×Dic61C2×C411yescharacteristic
37C2×C4×Dic61C2311yescharacteristic
38C25×C61—21no—
39C22×D4×C61—24no—
40C22×D4×C61—24no—
41C22×D4×C61—21no—
42C22×D4×C61—22no—
43C22×D4×C61C814yescomplemented
44C22×D4×C61C2×C412yes—
45C23×C4×C61C2×C411yescharacteristic
46C24×C61—24no—
47C24×C61—24no—
48C24×C61—22no—
49C24×C61—24no—
50C24×C61—84no—
51C24×C61C22⋊C411yescharacteristic
52C2×D4×C61—24no—
53C2×D4×C61—216no—
54C2×D4×C61—24no—
55C2×D4×C61—216no—
56C2×D4×C61—24no—
57C2×D4×C61—22no—
58C2×D4×C61—22no—
59C2×D4×C61—22no—
60C2×D4×C61C22⋊C414yes—
61C2×D4×C61C2×C814yes—
62C2×D4×C61M1614yes—
63C22×C4×C61—22no—
64C22×C4×C61—22no—
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C2×C812yes—
67C22×C4×C61M1612yes—
68C22×C4×C61C22×C411yescharacteristic
69C22×Dic61—22no—
70C22×Dic61—28no—
71C22×Dic61—28no—
72C22×Dic61—84no—
73C22×Dic61—84no—
74C22×Dic61C22⋊C412yes—
75C22×Dic61C2×D412yes—
76C2×#92420—44no—
77C22⋊Dic61—24no—
78C22⋊Dic61—28no—
79C22⋊Dic61—24no—
80C22⋊Dic61—28no—
81C4⋊Dic61—22no—
82C4⋊Dic61—24no—
83C4⋊Dic61C22⋊C414yes—
84C4×Dic61—24no—
85C4×Dic61—22no—
86C4×Dic61C2×D414yes—
87C23×C61—216no—
88C23×C61—21no—
89C23×C61—24no—
90C23×C61—24no—
91C23×C61—21no—
92C23×C61—24no—
93C23×C61—44no—
94C23×C61—44no—
95C23×C61—42no—
96C23×C61—88no—
97C23×C61—88no—
98C23×C61—88no—
99C23×C61C22⋊C812yes—
100C23×C61C2×C22⋊C411yescharacteristic
101D4×C61—216no—
102D4×C61—24no—
103D4×C61—24no—
104D4×C61—24no—
105D4×C61C22⋊C818yes—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—24no—
109C2×C4×C61—24no—
110C2×C4×C61C22⋊C812yes—
111C2×C4×C61C2×C22⋊C412yes—
112C2×C4×C61C22×C811yescharacteristic
113C2×C4×C61C2×M1612yes—
114C2×C4×C61C2×M1611yescharacteristic
115C2×Dic61—22no—
116C2×Dic61—24no—
117C2×Dic61—42no—
118C2×Dic61—44no—
119C2×Dic61—42no—
120C2×Dic61—88no—
121C2×Dic61—88no—
122C2×Dic61#12014yes—
123#92420—84no—
124#8322—611nomaximal, Sylow
125C22×C61—22no—
126C22×C61—22no—
127C22×C61—28no—
128C22×C61—42no—
129C22×C61—48no—
130C22×C61—48no—
131C22×C61—42no—
132C22×C61—44no—
133C22×C61—44no—
134C22×C61—42no—
135C22×C61—816no—
136C22×C61#37911yescharacteristic
137C22×C61C2×C22⋊C811yescharacteristic
138C22×C61#40711yescharacteristic
139C4×C61—22no—
140C4×C61C2×C22⋊C812yes—
141C4×C61#40712yes—
142Dic61—24no—
143Dic61—44no—
144Dic61—42no—
145C2×#518—611no—
146#1478—614no—
147#1458—612no—
148C2×C61—21no—
149C2×C61—21no—
150C2×C61—48no—
151C2×C61—42no—
152C2×C61#177911yescharacteristic
153C2×C61#179512yes—
154C23×D4#692311yescharacteristic
155#518—1228no—
156C2×C4×D4—612no—
157C2×C4×D4—612no—
158C2×C4×D4—1222no—
159C2×#117—611no—
160C2×#117—611no—
161C22×C22⋊C4—1222no—
162#431—614no—
163D4⋊C8—1228no—
164#324—1222no—
165#323—1222no—
166C61#832211yescharacteristic, complemented, Sylow
167C25—21no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—22no—
171C22×D4—21no—
172C22×D4#9242014yescomplemented
173C22×D4C2×#692312yes—
174C23×C4—1222no—
175C23×C4—1222no—
176C23×C4C2×#692311yescharacteristic
177C4×D4—618no—
178C4×D4—1224no—
179C4×D4—1224no—
180C4×D4—12216no—
181C4×D4—1224no—
182#117—1224no—
183#117—1224no—
184C2×C4⋊C4—612no—
185C2×C4⋊C4—612no—
186C2×C22⋊C4—612no—
187C2×C22⋊C4—1222no—
188C2×C22⋊C4—1222no—
189C2×C22⋊C4—12216no—
190C2×C22⋊C4—1221no—
191C2×C42—611no—
192C2×C42—611no—
193C2×C42—1221no—
194C4⋊C8—1224no—
195C22⋊C8—1224no—
196C4×C8—1224no—
197C24—24no—
198C24—24no—
199C24—22no—
200C24—24no—
201C24—84no—
202C24C22⋊#692311yescharacteristic
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—216no—
206C2×D4—216no—
207C2×D4—24no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4C2×#9242014yes—
212C2×D4#1175821714yes—
213C2×D4C22⋊#692314yes—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—612no—
217C22×C4—612no—
218C22×C4—1228no—
219C22×C4—1228no—
220C22×C4—1222no—
221C22×C4—4884no—
222C22×C4—4884no—
223C22×C4C2×#9242012yes—
224C22×C4#1175821712yes—
225C22×C4C22⋊#692311yescharacteristic
226C22×C4C22×#692311yescharacteristic
227C2×C8—2444no—
228C4⋊C4—614no—
229C4⋊C4—1224no—
230C4⋊C4—1222no—
231C22⋊C4—1224no—
232C22⋊C4—1228no—
233C22⋊C4—1224no—
234C22⋊C4—1228no—
235C42—614no—
236C42—1224no—
237C42—1222no—
238C23—216no—
239C23—24no—
240C23—21no—
241C23—24no—
242C23—21no—
243C23—24no—
244C23—42no—
245C23—44no—
246C23—44no—
247C23—88no—
248C23—88no—
249C23—88no—
250C23C22⋊#9242012yes—
251C23C2×C22⋊#692311yescharacteristic
252D4—216no—
253D4—24no—
254D4—24no—
255D4—24no—
256D4C22⋊#9242018yes—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—614no—
262C2×C4—1224no—
263C2×C4—1222no—
264C2×C4—4888no—
265C2×C4—4888no—
266C2×C4—2442no—
267C2×C4—2444no—
268C2×C4—2442no—
269C2×C4C22⋊#9242012yes—
270C2×C4C2×C22⋊#692312yes—
271C2×C4C22×#9242011yescharacteristic
272C2×C4C2×#1175821712yes—
273C2×C4C2×#1175821711yescharacteristic
274C8—4884no—
275C22—22no—
276C22—22no—
277C22—28no—
278C22—42no—
279C22—48no—
280C22—48no—
281C22—42no—
282C22—44no—
283C22—44no—
284C22—42no—
285C22—816no—
286C22#1684070311yescharacteristic
287C22C2×C22⋊#9242011yescharacteristic
288C22#1684073211yescharacteristic
289C4—22no—
290C4—1224no—
291C4—2442no—
292C4—2444no—
293C4C2×C22⋊#9242012yes—
294C4#1684073212yes—
295C2—21no—
296C2—21no—
297C2—48no—
298C2—42no—
299C2#2146164111yescharacteristic
300C2#2146166912yes—
subgroupG/H
centerC22#16840703
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
Fitting, prime radicalC23×D4×C61C4
socleC22×C61#379
cosocleC2×C4×Dic61C23
hypercenterC23×D4#6923
hypocenterC61#8322
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #21461118C2semidirect
2. #21461050C2semidirect
4. #16840778C22semidirect
43. C22×D4×C61C8semidirect
166. C61#8322semidirect
172. C22×D4#92420semidirect

Maximal subgroups

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

subgroupindex
#214611182
#214610502
C2×#168402372
#832261