Finite Group Databasebeta

← #30671115

Subgroups — #30671115

Order 16768.

672 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C2×#16862222C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D131C211yesmaximal, characteristic
3#22553239C211yesmaximal, characteristic, complemented
4#22553237C211yesmaximal, characteristic, complemented
5#22553223C211yesmaximal, characteristic, complemented
6#22553201C212yesmaximal, complemented
7#22553200C212yesmaximal, complemented
8#22553174C212yesmaximal, complemented
9#22553167C212yesmaximal
10#22553159C211yesmaximal, characteristic, complemented
11#393×C131C211yesmaximal, characteristic, complemented
12C24×D131C2211yescharacteristic
13C22×C4×D131C2211yescharacteristic
14C22×C4×D131C2211yescharacteristic
15#16862222—24no—
16C2×C4⋊Dic131C2211yescharacteristic, complemented
17C2×Dic131⋊C4C2212yescomplemented
18C2×C22⋊Dic131C2212yes—
19C2×C22⋊Dic131C2211yescharacteristic
20C2×(C2×D131)⋊C4C2212yes—
21C2×(C2×D131)⋊C4C2212yes—
22C2×(C2×D131)⋊C4C2212yes—
23C2×(C2×D131)⋊C4C2211yescharacteristic
24C2×(C2×D131)⋊C4C2211yescharacteristic
25C2×(C2×D131)⋊C4C2212yes—
26C22⋊C4×D131—24no—
27C2×C4×Dic131C2212yescomplemented
28C131⋊#95C2211yescharacteristic, complemented
29C131⋊#95C2212yescomplemented
30C131⋊#95C2212yescomplemented
31C131⋊#95C2212yescomplemented
32C22×Q8×C131C2211yescharacteristic, complemented
33C2×C22⋊C4×C131C2211yescharacteristic
34C2×C22⋊C4×C131C2212yes—
35#95×C131C2211yescharacteristic, complemented
36#95×C131C2212yescomplemented
37C23×D131—22no—
38C23×D131—22no—
39C23×D131—24no—
40C23×D131—22no—
41C23×D131—42no—
42C23×D131C2311yescharacteristic
43C23×D131C2311yescharacteristic
44C22×Dic131C2311yescharacteristic
45C22×Dic131C2312yes—
46C22×Dic131C2312yes—
47C22×Dic131C2311yescharacteristic
48C2×C4×D131—22no—
49C2×C4×D131—24no—
50C2×C4×D131—42no—
51C2×C4×D131D414yes—
52C4⋊Dic131—22no—
53Dic131⋊C4—24no—
54C22⋊Dic131—22no—
55C22⋊Dic131—24no—
56(C2×D131)⋊C4—24no—
57(C2×D131)⋊C4—24no—
58(C2×D131)⋊C4—24no—
59(C2×D131)⋊C4—22no—
60(C2×D131)⋊C4—22no—
61(C2×D131)⋊C4—42no—
62C4×Dic131—42no—
63C24×C131C2311yescharacteristic
64C2×Q8×C131—22no—
65C2×Q8×C131—42no—
66C22×C4×C131C2311yescharacteristic
67C22×C4×C131C2312yes—
68C22×C4×C131C2312yes—
69C22×C4×C131C2311yescharacteristic
70C22⋊C4×C131—22no—
71C22⋊C4×C131—42no—
72C22×D131—21no—
73C22×D131—21no—
74C22×D131—22no—
75C22×D131—28no—
76C22×D131—21no—
77C22×D131—22no—
78C22×D131—21no—
79C22×D131—22no—
80C22×D131—48no—
81C22×D131—48no—
82C22×D131—44no—
83C22×D131—44no—
84C22×D131C2×D412yes—
85C22×D131#4112yes—
86C22×D131#4114yes—
87C2×Dic131—22no—
88C2×Dic131—21no—
89C2×Dic131—22no—
90C2×Dic131—22no—
91C2×Dic131—22no—
92C2×Dic131—22no—
93C2×Dic131—22no—
94C2×Dic131—22no—
95C2×Dic131—22no—
96C2×Dic131C2×D412yes—
97C4×D131—44no—
98C4×D131—44no—
99C23×C131—21no—
100C23×C131—21no—
101C23×C131—22no—
102C23×C131—22no—
103C23×C131#4112yes—
104C23×C131C2411yescharacteristic
105Q8×C131—44no—
106C2×C4×C131—22no—
107C2×C4×C131—21no—
108C2×C4×C131—22no—
109C2×C4×C131—22no—
110C2×C4×C131—22no—
111C2×C4×C131—22no—
112C2×C4×C131—22no—
113C2×C4×C131—22no—
114C2×C4×C131—22no—
115C2×C4×C131C2×D412yes—
116C22×C131—22no—
117C22×C131—24no—
118C22×C131—41no—
119C22×C131—41no—
120C22×C131—42no—
121C22×C131C22×D411yescharacteristic
122C22×C131C2×#4111yescharacteristic
123C22×C131C2×#4112yes—
124C22×C131C2×#4111yescharacteristic
125C22×C131H(5,𝔽2)12yes—
126C2×D131—22no—
127C2×D131—22no—
128C2×D131—24no—
129C2×D131—24no—
130C2×D131—22no—
131C2×D131—24no—
132C2×D131—416no—
133C2×D131—41no—
134C2×D131—41no—
135C2×D131—42no—
136C2×D131#12414yes—
137C4×C131—22no—
138C4×C131—41no—
139C4×C131—42no—
140C4×C131—42no—
141Dic131—22no—
142Dic131—42no—
143Dic131—41no—
144Dic131—42no—
145C2×C131—42no—
146C2×C131#52511yescharacteristic
147C2×C131C2×#12411yescharacteristic
148C2×C131#53411yescharacteristic
149C2×C131#55112yes—
150C2×C131#55112yes—
151D131—24no—
152D131—42no—
153C131#256111yescharacteristic, complemented, Sylow
154#2561—1311nomaximal, Sylow
155C2×C22⋊Q8—1311no—
156C22×C22⋊C4—1311no—
157#397—1311no—
158#397—1311no—
159#393—1311no—
160#393D13111yescharacteristic, complemented
161#388—1312no—
162#388—1312no—
163#381—1312no—
164#379—1312no—
165C4×C22⋊C4—1311no—
166C25—1311no—
167C22×Q8C2×D13111yescharacteristic, complemented
168C23×C4—1311no—
169C23×C4—1311no—
170C22⋊Q8—2624no—
171C2×C4⋊C4—1312no—
172C2×C4⋊C4—1311no—
173C2×C22⋊C4—2624no—
174C2×C22⋊C4—1311no—
175C2×C22⋊C4—1312no—
176C2×C22⋊C4—1311no—
177C2×C22⋊C4—1311no—
178C2×C22⋊C4—1312no—
179C2×C22⋊C4—1312no—
180C2×C22⋊C4—1312no—
181C2×C22⋊C4—1312no—
182C2×C22⋊C4C2×D13111yescharacteristic
183C2×C22⋊C4C2×D13112yes—
184C2×C42—1312no—
185#95—1311no—
186#95—1312no—
187#95—1312no—
188#95—1312no—
189#95C2×D13111yescharacteristic, complemented
190#95C2×D13112yescomplemented
191C24—2622no—
192C24—2622no—
193C24—2624no—
194C24—2622no—
195C24—5242no—
196C24—1311no—
197C24—1311no—
198C24C22×D13111yescharacteristic
199C2×Q8—22no—
200C2×Q8—42no—
201C22×C4—2622no—
202C22×C4—2624no—
203C22×C4—5242no—
204C22×C4—1311no—
205C22×C4—1312no—
206C22×C4—1312no—
207C22×C4—1311no—
208C22×C4—1314no—
209C22×C4C22×D13111yescharacteristic
210C22×C4C22×D13112yes—
211C22×C4C22×D13112yes—
212C22×C4C22×D13111yescharacteristic
213C4⋊C4—2624no—
214C4⋊C4—2622no—
215C22⋊C4—22no—
216C22⋊C4—42no—
217C22⋊C4—2624no—
218C22⋊C4—2624no—
219C22⋊C4—2624no—
220C22⋊C4—2622no—
221C22⋊C4—2624no—
222C22⋊C4—2622no—
223C22⋊C4—2622no—
224C22⋊C4—5242no—
225C42—5242no—
226C23—21no—
227C23—22no—
228C23—21no—
229C23—22no—
230C23—2628no—
231C23—2621no—
232C23—2621no—
233C23—2622no—
234C23—2621no—
235C23—2622no—
236C23—2621no—
237C23—2622no—
238C23—5248no—
239C23—5248no—
240C23—5244no—
241C23—5244no—
242C23—1312no—
243C23—1312no—
244C23—1314no—
245C23#1452351412yes—
246C23C23×D13111yescharacteristic
247Q8—44no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—2622no—
258C2×C4—2621no—
259C2×C4—2622no—
260C2×C4—2622no—
261C2×C4—2622no—
262C2×C4—2622no—
263C2×C4—2622no—
264C2×C4—2622no—
265C2×C4—2622no—
266C2×C4—5244no—
267C2×C4—5244no—
268C2×C4—1312no—
269C2×C4D4×D13112yes—
270C22—24no—
271C22—22no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—2622no—
276C22—2624no—
277C22—2624no—
278C22—2622no—
279C22—2624no—
280C22—2622no—
281C22—52416no—
282C22—5241no—
283C22—5242no—
284C22—5241no—
285C22—1314no—
286C22C2×D4×D13111yescharacteristic
287C22#41×D13112yes—
288C22#41×D13111yescharacteristic
289C22C2×#1452351411yescharacteristic
290C22#1686229112yes—
291C4—22no—
292C4—42no—
293C4—42no—
294C4—41no—
295C4—2622no—
296C4—5242no—
297C4—5242no—
298C4—5241no—
299C2—42no—
300C2—2624no—
301C2—5242no—
302C2#2255382511yescharacteristic
303C2#124×D13111yescharacteristic
304C2#2255387211yescharacteristic
305C2#2255398912yes—
306C2#2255399212yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#393×C131C2
prime radicalC24×D131C22
hypercenter#393D131
hypocenterC131#2561
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#16862222C2semidirect
3. #22553239C2semidirect
4. #22553237C2semidirect
5. #22553223C2semidirect
6. #22553201C2semidirect
7. #22553200C2semidirect
8. #22553174C2semidirect
10. #22553159C2semidirect
11. #393×C131C2semidirect
16. C2×C4⋊Dic131C22semidirect
17. C2×Dic131⋊C4C22semidirect
27. C2×C4×Dic131C22semidirect
28. C131⋊#95C22semidirect
29. C131⋊#95C22semidirect
30. C131⋊#95C22semidirect
31. C131⋊#95C22semidirect
32. C22×Q8×C131C22semidirect
35. #95×C131C22semidirect
36. #95×C131C22semidirect
153. C131#2561semidirect
160. #393D131semidirect
167. C22×Q8C2×D131semidirect
189. #95C2×D131semidirect
190. #95C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622222
C2×C22⋊C4×D1312
#225532392
#225532372
#225532232
#225532012
#225532002
#225531742
#225531672
#225531592
#393×C1312
#2561131