Finite Group Databasebeta

← #30671118

Subgroups — #30671118

Order 16768.

672 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 145 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×#16862223C211yesmaximal, characteristic, complemented
2C22×(C2×D131)⋊C4C211yesmaximal, characteristic
3#22553241C212yesmaximal, complemented
4#22553225C212yesmaximal, complemented
5#22553204C212yesmaximal, complemented
6#22553201C212yesmaximal, complemented
7#22553175C211yesmaximal, characteristic, complemented
8#22553167C211yesmaximal, characteristic
9#22553167C211yesmaximal, characteristic
10C22⋊C4×Dic131C211yesmaximal, characteristic, complemented
11#381×C131C211yesmaximal, characteristic, complemented
12C24×D131C2211yescharacteristic
13C22×Dic262C2211yescharacteristic, complemented
14C23×Dic131C2211yescharacteristic
15#16862223—24no—
16C2×C4⋊Dic131C2211yescharacteristic, complemented
17C2×Dic131⋊C4C2211yescharacteristic, complemented
18C2×Dic131⋊C4C2211yescharacteristic, complemented
19C2×C22⋊Dic131C2211yescharacteristic
20C2×C22⋊Dic131C2211yescharacteristic
21C2×(C2×D131)⋊C4—24no—
22C2×(C2×D131)⋊C4C2212yes—
23C2×(C2×D131)⋊C4C2212yes—
24C2×(C2×D131)⋊C4C2212yes—
25C2×(C2×D131)⋊C4C2212yes—
26C2×(C2×D131)⋊C4C2212yes—
27C2×(C2×D131)⋊C4C2212yes—
28C2×C4×Dic131C2212yescomplemented
29C131⋊#95C2212yescomplemented
30C131⋊#95C2212yescomplemented
31C131⋊#95C2212yescomplemented
32C23×C4×C131C2211yescharacteristic
33C2×C22⋊C4×C131C2211yescharacteristic
34C2×C22⋊C4×C131C2211yescharacteristic
35#95×C131C2212yescomplemented
36#95×C131C2212yescomplemented
37C23×D131—24no—
38C23×D131—22no—
39C23×D131—22no—
40C23×D131—22no—
41C23×D131—42no—
42C23×D131C2312yes—
43C2×Dic262—22no—
44C2×Dic262—42no—
45C22×Dic131—42no—
46C22×Dic131D414yes—
47C22×Dic131C2311yescharacteristic
48C22×Dic131C2312yes—
49C22×Dic131C2311yescharacteristic
50C22×Dic131C2311yescharacteristic
51C22×Dic131C2311yescharacteristic
52C4⋊Dic131—22no—
53Dic131⋊C4—22no—
54Dic131⋊C4—22no—
55C22⋊Dic131—22no—
56C22⋊Dic131—22no—
57(C2×D131)⋊C4—24no—
58(C2×D131)⋊C4—24no—
59(C2×D131)⋊C4—24no—
60(C2×D131)⋊C4—24no—
61(C2×D131)⋊C4—42no—
62(C2×D131)⋊C4—42no—
63C4×Dic131—42no—
64C24×C131C2311yescharacteristic
65C22×C4×C131—22no—
66C22×C4×C131—22no—
67C22×C4×C131—22no—
68C22×C4×C131C2311yescharacteristic
69C22×C4×C131C2312yes—
70C22×C4×C131C2311yescharacteristic
71C22×C4×C131C2312yes—
72C22⋊C4×C131—22no—
73C22⋊C4×C131—22no—
74C22×D131—28no—
75C22×D131—22no—
76C22×D131—22no—
77C22×D131—22no—
78C22×D131—22no—
79C22×D131—22no—
80C22×D131—22no—
81C22×D131—48no—
82C22×D131—48no—
83C22×D131—44no—
84C22×D131—44no—
85C22×D131#4114yes—
86Dic262—44no—
87C2×Dic131—22no—
88C2×Dic131—22no—
89C2×Dic131—21no—
90C2×Dic131—21no—
91C2×Dic131—22no—
92C2×Dic131—22no—
93C2×Dic131—22no—
94C2×Dic131—22no—
95C2×Dic131—22no—
96C2×Dic131—44no—
97C2×Dic131C2×D412yes—
98C2×Dic131C2×D412yes—
99C23×C131—22no—
100C23×C131—22no—
101C23×C131C2×D412yes—
102C23×C131#4112yes—
103C23×C131#4112yes—
104C23×C131C2411yescharacteristic
105C2×C4×C131—21no—
106C2×C4×C131—21no—
107C2×C4×C131—22no—
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×C131—44no—
116C22×C131—22no—
117C22×C131—22no—
118C22×C131—24no—
119C22×C131—24no—
120C22×C131#12414yes—
121C22×C131C22×D411yescharacteristic
122C22×C131C2×#4112yes—
123C22×C131C2×#4111yescharacteristic
124C22×C131C2×#4111yescharacteristic
125C22×C131H(5,𝔽2)12yes—
126C2×D131—24no—
127C2×D131—24no—
128C2×D131—24no—
129C2×D131—416no—
130C2×D131—42no—
131C2×D131—42no—
132C2×D131—42no—
133C2×D131—42no—
134C4×C131—41no—
135C4×C131—42no—
136C4×C131—41no—
137C4×C131—42no—
138Dic131—22no—
139Dic131—22no—
140Dic131—41no—
141Dic131—41no—
142Dic131—42no—
143C2×C131—24no—
144C2×C131#52511yescharacteristic
145C2×C131C2×#12411yescharacteristic
146C2×C131#53411yescharacteristic
147C2×C131#55112yes—
148C2×C131#55112yes—
149D131—44no—
150C131#256111yescharacteristic, complemented, Sylow
151#2561—1311nomaximal, Sylow
152C2×C22⋊Q8—1311no—
153C22×C22⋊C4—1311no—
154#397—1312no—
155#393—1312no—
156#388—1312no—
157#388—1312no—
158#381—1311no—
159#381D13111yescharacteristic, complemented
160#379—1311no—
161#379—1311no—
162C4×C22⋊C4—1311no—
163C25—1311no—
164C22×Q8—1311no—
165C23×C4—1311no—
166C23×C4C2×D13111yescharacteristic
167C22⋊Q8—2624no—
168C2×C4⋊C4—1311no—
169C2×C4⋊C4—1311no—
170C2×C4⋊C4—1311no—
171C2×C22⋊C4—2624no—
172C2×C22⋊C4—1311no—
173C2×C22⋊C4—1311no—
174C2×C22⋊C4—1312no—
175C2×C22⋊C4—1312no—
176C2×C22⋊C4—1312no—
177C2×C22⋊C4—1312no—
178C2×C22⋊C4—1312no—
179C2×C22⋊C4—1312no—
180C2×C22⋊C4C2×D13111yescharacteristic
181C2×C22⋊C4C2×D13111yescharacteristic
182C2×C42—1312no—
183#95—1312no—
184#95—1312no—
185#95—1312no—
186#95C2×D13112yescomplemented
187#95C2×D13112yescomplemented
188C24—2624no—
189C24—2622no—
190C24—2622no—
191C24—2622no—
192C24—5242no—
193C24—1312no—
194C24C22×D13111yescharacteristic
195C2×Q8—2622no—
196C2×Q8—5242no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—5242no—
201C22×C4—1312no—
202C22×C4—1311no—
203C22×C4—1311no—
204C22×C4—1311no—
205C22×C4—1311no—
206C22×C4—1314no—
207C22×C4C22×D13112yes—
208C22×C4C22×D13112yes—
209C22×C4C22×D13111yescharacteristic
210C22×C4C22×D13111yescharacteristic
211C4⋊C4—2622no—
212C4⋊C4—2622no—
213C4⋊C4—2622no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—2624no—
217C22⋊C4—2624no—
218C22⋊C4—2624no—
219C22⋊C4—2624no—
220C22⋊C4—2622no—
221C22⋊C4—2622no—
222C22⋊C4—5242no—
223C22⋊C4—5242no—
224C42—5242no—
225C23—22no—
226C23—22no—
227C23—2628no—
228C23—2622no—
229C23—2622no—
230C23—2622no—
231C23—2622no—
232C23—2622no—
233C23—2622no—
234C23—5248no—
235C23—5248no—
236C23—5244no—
237C23—5244no—
238C23—1314no—
239C23D4×D13112yes—
240C23#1452351212yes—
241C23#1452351312yes—
242C23C23×D13111yescharacteristic
243Q8—5244no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—44no—
255C2×C4—2622no—
256C2×C4—2622no—
257C2×C4—2621no—
258C2×C4—2621no—
259C2×C4—2622no—
260C2×C4—2622no—
261C2×C4—2622no—
262C2×C4—2622no—
263C2×C4—2622no—
264C2×C4—5244no—
265C2×C4—1312no—
266C2×C4—1312no—
267C22—22no—
268C22—24no—
269C22—24no—
270C22—22no—
271C22—2624no—
272C22—2624no—
273C22—2624no—
274C22—52416no—
275C22—5242no—
276C22—5242no—
277C22—5242no—
278C22—5242no—
279C22#1686223214yes—
280C22C2×D4×D13111yescharacteristic
281C22#41×D13112yes—
282C22C2×#1452351211yescharacteristic
283C22C2×#1452351311yescharacteristic
284C22#1686229212yes—
285C4—41no—
286C4—42no—
287C4—41no—
288C4—42no—
289C4—2622no—
290C4—2622no—
291C4—5241no—
292C4—5242no—
293C4—5241no—
294C2—24no—
295C2—5244no—
296C2#2255382011yescharacteristic
297C2C2×#1686223211yescharacteristic
298C2#2255387311yescharacteristic
299C2#2255398512yes—
300C2#2255398612yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#381×C131C2
prime radicalC24×D131C22
hypercenter#381D131
hypocenterC131#2561
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C2×#16862223C2semidirect
3. #22553241C2semidirect
4. #22553225C2semidirect
5. #22553204C2semidirect
6. #22553201C2semidirect
7. #22553175C2semidirect
10. C22⋊C4×Dic131C2semidirect
11. #381×C131C2semidirect
13. C22×Dic262C22semidirect
16. C2×C4⋊Dic131C22semidirect
17. C2×Dic131⋊C4C22semidirect
18. C2×Dic131⋊C4C22semidirect
28. C2×C4×Dic131C22semidirect
29. C131⋊#95C22semidirect
30. C131⋊#95C22semidirect
31. C131⋊#95C22semidirect
35. #95×C131C22semidirect
36. #95×C131C22semidirect
150. C131#2561semidirect
159. #381D131semidirect
186. #95C2×D131semidirect
187. #95C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622232
C22×(C2×D131)⋊C42
#225532412
#225532252
#225532042
#225532012
#225531752
#225531672
C22⋊C4×Dic1312
#381×C1312
#2561131