Finite Group Databasebeta

← #30668944

Subgroups — #30668944

Order 16768.

928 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 209 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×#16862214C211yesmaximal, characteristic
2C2×C4×C22⋊D131C212yesmaximal, complemented
3C2×C22⋊C4×D131C212yesmaximal, complemented
4#22553215C211yesmaximal, characteristic, complemented
5#22553206C212yesmaximal, complemented
6#22553188C212yesmaximal, complemented
7#22553186C212yesmaximal, complemented
8#22553159C212yesmaximal, complemented
9#386×C131C211yesmaximal, characteristic, complemented
10C24×D131C2211yescharacteristic
11C22×C22⋊D131C2212yes—
12C22×C4×D131C2212yes—
13C22×C4×D131C2212yes—
14#16862214C418yescomplemented
15C4×C22⋊D131—28no—
16C2×C4⋊Dic131C2211yescharacteristic, complemented
17C2×Dic131⋊C4C2212yescomplemented
18C2×C22⋊Dic131C2212yes—
19C2×C22⋊Dic131C2211yescharacteristic
20C2×C22⋊Dic131C2212yescomplemented
21C2×(C2×D131)⋊C4C2212yes—
22C2×(C2×D131)⋊C4C2212yes—
23C2×(C2×D131)⋊C4C2212yes—
24C22⋊C4×D131—28no—
25C2×C4×Dic131C2212yescomplemented
26C2×C4×Dic131C2211yescharacteristic, complemented
27C131⋊#95C2212yescomplemented
28C131⋊#95C2212yescomplemented
29C22×D4×C131C2211yescharacteristic
30C23×C4×C131C2211yescharacteristic, complemented
31C2×C22⋊C4×C131C2212yes—
32C2×C22⋊C4×C131C2211yescharacteristic, complemented
33#95×C131C2212yescomplemented
34C23×D131—24no—
35C23×D131—24no—
36C23×D131—41no—
37C23×D131—41no—
38C23×D131C2×C414yescomplemented
39C23×D131C2312yes—
40C2×C22⋊D131—24no—
41C2×C22⋊D131—24no—
42C2×C22⋊D131C2×C418yescomplemented
43C22×Dic131C2312yes—
44C22×Dic131C2312yes—
45C22×Dic131C2312yes—
46C2×C4×D131—24no—
47C2×C4×D131—24no—
48C2×C4×D131—24no—
49C2×C4×D131—24no—
50C2×C4×D131—42no—
51C2×C4×D131—42no—
52C4⋊Dic131—41no—
53Dic131⋊C4—24no—
54C22⋊Dic131—24no—
55C22⋊Dic131—24no—
56C22⋊Dic131C2×C414yescomplemented
57(C2×D131)⋊C4—24no—
58(C2×D131)⋊C4C2×C418yescomplemented
59(C2×D131)⋊C4D418yescomplemented
60C4×Dic131—24no—
61C4×Dic131—41no—
62C24×C131C2311yescharacteristic
63C24×C131C2311yescharacteristic
64C2×D4×C131—41no—
65C2×D4×C131—41no—
66C2×D4×C131C2×C414yescomplemented
67C22×C4×C131—24no—
68C22×C4×C131—41no—
69C22×C4×C131C2312yes—
70C22×C4×C131C2311yescharacteristic
71C22×C4×C131C2312yes—
72C22⋊C4×C131—24no—
73C22⋊C4×C131—41no—
74C22×D131—22no—
75C22×D131—22no—
76C22×D131—22no—
77C22×D131—22no—
78C22×D131—216no—
79C22×D131—44no—
80C22×D131—44no—
81C22×D131—48no—
82C22×D131—44no—
83C22×D131C22×C414yes—
84C22×D131C2×D414yes—
85C22×D131#4114yes—
86C22⋊D131—216no—
87C2×Dic131—22no—
88C2×Dic131—22no—
89C2×Dic131—22no—
90C2×Dic131—22no—
91C2×Dic131—22no—
92C2×Dic131—22no—
93C2×Dic131—22no—
94C2×Dic131C22×C414yes—
95C2×Dic131C2×D414yes—
96C4×D131—48no—
97C4×D131—48no—
98C23×C131—22no—
99C23×C131—22no—
100C23×C131—22no—
101C23×C131—22no—
102C23×C131—21no—
103C23×C131—21no—
104C23×C131—21no—
105C23×C131—21no—
106C23×C131C22×C412yes—
107C23×C131C22×C412yes—
108C23×C131C2411yescharacteristic
109D4×C131—44no—
110C2×C4×C131—22no—
111C2×C4×C131—22no—
112C2×C4×C131—21no—
113C2×C4×C131—21no—
114C2×C4×C131—22no—
115C2×C4×C131—22no—
116C2×C4×C131—22no—
117C2×C4×C131—44no—
118C2×C4×C131C22×C412yes—
119C2×C4×C131C2×D414yes—
120C22×C131—24no—
121C22×C131—24no—
122C22×C131—22no—
123C22×C131—22no—
124C22×C131—41no—
125C22×C131—41no—
126C22×C131—42no—
127C22×C131—42no—
128C22×C131—41no—
129C22×C131—41no—
130C22×C131C23×C411yescharacteristic
131C22×C131C22×D412yes—
132C22×C131C2×#4112yes—
133C22×C131H(5,𝔽2)11yescharacteristic
134C22×C131H(5,𝔽2)11yescharacteristic
135C2×D131—24no—
136C2×D131—24no—
137C2×D131—24no—
138C2×D131—24no—
139C2×D131—24no—
140C2×D131—24no—
141C2×D131—416no—
142C2×D131C4×D418yescomplemented
143C4×C131—22no—
144C4×C131—24no—
145C4×C131—42no—
146Dic131—24no—
147Dic131—24no—
148Dic131—42no—
149C2×C131—42no—
150C2×C131—42no—
151C2×C131C2×C4×D412yes—
152C2×C131#51811yescharacteristic
153C2×C131D4211yescharacteristic
154C2×C131#54612yes—
155C2×C131#55111yescharacteristic
156D131—28no—
157C131#235811yescharacteristic, complemented, Sylow
158#2358—1311nomaximal, Sylow
159C2×#120—1311no—
160C2×C4×D4—1312no—
161C22×C22⋊C4—1312no—
162#390—1311no—
163#388—1312no—
164#386—1312no—
165#386D13111yescharacteristic, complemented
166#385—1312no—
167C4×C22⋊C4—1312no—
168C25—1311no—
169C22×D4—1312no—
170C22×D4C2×D13111yescharacteristic
171C23×C4—1312no—
172C23×C4—1312no—
173C23×C4C2×D13111yescharacteristic, complemented
174#120—1318no—
175C4×D4—2628no—
176C2×C4⋊C4—1312no—
177C2×C4⋊C4—1311no—
178C2×C22⋊C4—2628no—
179C2×C22⋊C4—1312no—
180C2×C22⋊C4—1312no—
181C2×C22⋊C4—1312no—
182C2×C22⋊C4—1311no—
183C2×C22⋊C4—1312no—
184C2×C22⋊C4—1312no—
185C2×C22⋊C4C2×D13112yes—
186C2×C22⋊C4C2×D13111yescharacteristic, complemented
187C2×C42—1312no—
188C2×C42—1311no—
189#95—1312no—
190#95—1312no—
191#95C2×D13112yescomplemented
192C24—2624no—
193C24—2624no—
194C24—5241no—
195C24—5241no—
196C24—1314no—
197C24—1312no—
198C24C22×D13111yescharacteristic
199C24C22×D13111yescharacteristic
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—2624no—
203C2×D4—2624no—
204C2×D4—1318no—
205C2×D4C4×D13114yescomplemented
206C22×C4—24no—
207C22×C4—41no—
208C22×C4—2624no—
209C22×C4—2624no—
210C22×C4—2624no—
211C22×C4—2624no—
212C22×C4—5242no—
213C22×C4—5242no—
214C22×C4—1312no—
215C22×C4—1312no—
216C22×C4—1312no—
217C22×C4C22×D13112yes—
218C22×C4C22×D13111yescharacteristic
219C22×C4C22×D13112yes—
220C4⋊C4—2624no—
221C4⋊C4—5241no—
222C22⋊C4—24no—
223C22⋊C4—41no—
224C22⋊C4—2624no—
225C22⋊C4—2624no—
226C22⋊C4—2624no—
227C22⋊C4—1314no—
228C22⋊C4—1318no—
229C22⋊C4—1318no—
230C42—2624no—
231C42—5241no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—26216no—
241C23—2622no—
242C23—2622no—
243C23—2622no—
244C23—2622no—
245C23—5244no—
246C23—5244no—
247C23—5248no—
248C23—5244no—
249C23—1314no—
250C23—1314no—
251C23—1314no—
252C23C2×C4×D13112yes—
253C23C2×C4×D13112yes—
254C23C23×D13111yescharacteristic
255D4—44no—
256D4—26216no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—44no—
265C2×C4—2622no—
266C2×C4—2622no—
267C2×C4—2622no—
268C2×C4—2622no—
269C2×C4—2622no—
270C2×C4—2622no—
271C2×C4—2622no—
272C2×C4—5248no—
273C2×C4—5248no—
274C2×C4—1314no—
275C2×C4—1314no—
276C2×C4C2×C4×D13112yes—
277C2×C4D4×D13114yes—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—22no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—2624no—
289C22—2624no—
290C22—2624no—
291C22—2624no—
292C22—2624no—
293C22—2624no—
294C22—52416no—
295C22—1318no—
296C22C22×C4×D13111yescharacteristic
297C22C2×D4×D13112yes—
298C22#41×D13112yes—
299C22#1686229111yescharacteristic
300C22#1686229111yescharacteristic
301C4—24no—
302C4—22no—
303C4—42no—
304C4—2624no—
305C4—2624no—
306C4—5242no—
307C2—42no—
308C2—42no—
309C2—2628no—
310C2C4×D4×D13112yes—
311C2#2255378511yescharacteristic
312C2#2255394411yescharacteristic
313C2#2255395912yes—
314C2#2255399011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutatorC22×C131C23×C4
Fitting#386×C131C2
socle, cosocleC23×C131C24
prime radicalC2×#16862214C2
hypercenter#386D131
hypocenterC131#2358
LCSG ▷ C22×C131 ▷ C131
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D131C2semidirect
3. C2×C22⋊C4×D131C2semidirect
4. #22553215C2semidirect
5. #22553206C2semidirect
6. #22553188C2semidirect
7. #22553186C2semidirect
8. #22553159C2semidirect
9. #386×C131C2semidirect
14. #16862214C4semidirect
16. C2×C4⋊Dic131C22semidirect
17. C2×Dic131⋊C4C22semidirect
20. C2×C22⋊Dic131C22semidirect
25. C2×C4×Dic131C22semidirect
26. C2×C4×Dic131C22semidirect
27. C131⋊#95C22semidirect
28. C131⋊#95C22semidirect
30. C23×C4×C131C22semidirect
32. C2×C22⋊C4×C131C22semidirect
33. #95×C131C22semidirect
38. C23×D131C2×C4semidirect
42. C2×C22⋊D131C2×C4semidirect
56. C22⋊Dic131C2×C4semidirect
58. (C2×D131)⋊C4C2×C4semidirect
59. (C2×D131)⋊C4D4semidirect
66. C2×D4×C131C2×C4semidirect
142. C2×D131C4×D4semidirect
157. C131#2358semidirect
165. #386D131semidirect
173. C23×C4C2×D131semidirect
186. C2×C22⋊C4C2×D131semidirect
191. #95C2×D131semidirect
205. C2×D4C4×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622142
C2×C4×C22⋊D1312
C2×C22⋊C4×D1312
#225532152
#225532062
#225531882
#225531862
#225531592
#386×C1312
#2358131