Finite Group Databasebeta

← #30670236

Subgroups — #30670236

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×#16862204C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D131C211yesmaximal, characteristic
3#22553247C211yesmaximal, characteristic, complemented
4#22553239C211yesmaximal, characteristic, complemented
5#22553237C211yesmaximal, characteristic, complemented
6#22553206C212yesmaximal, complemented
7#22553202C212yesmaximal, complemented
8#22553183C212yesmaximal, complemented
9#22553174C211yesmaximal, characteristic, complemented
10#22553167C212yesmaximal
11#399×C131C211yesmaximal, characteristic, complemented
12C24×D131C2211yescharacteristic
13C22×C4×D131C2211yescharacteristic
14C22×C4×D131C2211yescharacteristic
15#16862204—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⋊#95C2212yescomplemented
29C131⋊#95C2211yescharacteristic, complemented
30C131⋊#95C2212yescomplemented
31C131⋊#95C2212yescomplemented
32C2×C4⋊C4×C131C2211yescharacteristic, complemented
33C2×C4⋊C4×C131C2212yescomplemented
34C2×C22⋊C4×C131C2211yescharacteristic
35C2×C22⋊C4×C131C2212yes—
36#95×C131C2211yescharacteristic, complemented
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—24no—
49C2×C4×D131—22no—
50C2×C4×D131—42no—
51C2×C4×D131D414yes—
52C4⋊Dic131—22no—
53Dic131⋊C4—42no—
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—24no—
63C24×C131C2311yescharacteristic
64C22×C4×C131C2311yescharacteristic
65C22×C4×C131C2312yes—
66C22×C4×C131C2312yes—
67C22×C4×C131C2311yescharacteristic
68C4⋊C4×C131—22no—
69C4⋊C4×C131—42no—
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—22no—
89C2×Dic131—21no—
90C2×Dic131—22no—
91C2×Dic131—22no—
92C2×Dic131—22no—
93C2×Dic131—22no—
94C2×Dic131—22no—
95C2×Dic131—22no—
96C2×Dic131C2×D412yes—
97C4×D131—28no—
98C4×D131—44no—
99C23×C131—21no—
100C23×C131—21no—
101C23×C131—22no—
102C23×C131—22no—
103C23×C131#4112yes—
104C23×C131C2411yescharacteristic
105C2×C4×C131—22no—
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×C131C2×D412yes—
115C22×C131—22no—
116C22×C131—24no—
117C22×C131—41no—
118C22×C131—41no—
119C22×C131—42no—
120C22×C131C22×D411yescharacteristic
121C22×C131C2×#4112yes—
122C22×C131C2×#4111yescharacteristic
123C22×C131C2×#4111yescharacteristic
124C22×C131H(5,𝔽2)12yes—
125C2×D131—22no—
126C2×D131—22no—
127C2×D131—24no—
128C2×D131—24no—
129C2×D131—22no—
130C2×D131—24no—
131C2×D131—416no—
132C2×D131—41no—
133C2×D131—41no—
134C2×D131—42no—
135C2×D131#12314yes—
136C4×C131—22no—
137C4×C131—42no—
138C4×C131—41no—
139C4×C131—42no—
140Dic131—22no—
141Dic131—42no—
142Dic131—41no—
143Dic131—42no—
144C2×C131—42no—
145C2×C131C2×#12311yescharacteristic
146C2×C131#52511yescharacteristic
147C2×C131#53511yescharacteristic
148C2×C131#55112yes—
149C2×C131#55112yes—
150D131—24no—
151D131—42no—
152C131#248211yescharacteristic, complemented, Sylow
153#2482—1311nomaximal, Sylow
154C2×#117—1311no—
155C22×C22⋊C4—1311no—
156#399—1311no—
157#399D13111yescharacteristic, complemented
158#397—1311no—
159#397—1311no—
160#388—1312no—
161#388—1312no—
162#385—1312no—
163#381—1311no—
164#379—1312no—
165C25—1311no—
166C23×C4—1311no—
167C23×C4—1311no—
168#117—2624no—
169C2×C4⋊C4—1311no—
170C2×C4⋊C4—1312no—
171C2×C4⋊C4C2×D13111yescharacteristic, complemented
172C2×C4⋊C4C2×D13112yescomplemented
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—1312no—
186#95—1311no—
187#95—1312no—
188#95—1312no—
189#95C2×D13111yescharacteristic, complemented
190C24—2622no—
191C24—2622no—
192C24—2624no—
193C24—2622no—
194C24—5242no—
195C24—1311no—
196C24—1311no—
197C24C22×D13111yescharacteristic
198C22×C4—2624no—
199C22×C4—2622no—
200C22×C4—5242no—
201C22×C4—1312no—
202C22×C4—1312no—
203C22×C4—1311no—
204C22×C4—1311no—
205C22×C4—1314no—
206C22×C4C22×D13112yes—
207C22×C4C22×D13111yescharacteristic
208C22×C4C22×D13112yes—
209C22×C4C22×D13111yescharacteristic
210C4⋊C4—22no—
211C4⋊C4—42no—
212C4⋊C4—2622no—
213C4⋊C4—5242no—
214C22⋊C4—22no—
215C22⋊C4—42no—
216C22⋊C4—2624no—
217C22⋊C4—2624no—
218C22⋊C4—2624no—
219C22⋊C4—2622no—
220C22⋊C4—2624no—
221C22⋊C4—2622no—
222C22⋊C4—2622no—
223C22⋊C4—5242no—
224C42—2624no—
225C23—21no—
226C23—22no—
227C23—21no—
228C23—22no—
229C23—2628no—
230C23—2621no—
231C23—2621no—
232C23—2622no—
233C23—2621no—
234C23—2622no—
235C23—2621no—
236C23—2622no—
237C23—5248no—
238C23—5248no—
239C23—5244no—
240C23—5244no—
241C23—1312no—
242C23—1312no—
243C23—1314no—
244C23#1452351412yes—
245C23C23×D13111yescharacteristic
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—2628no—
256C2×C4—2622no—
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—5244no—
266C2×C4—1312no—
267C2×C4D4×D13112yes—
268C22—24no—
269C22—22no—
270C22—41no—
271C22—42no—
272C22—41no—
273C22—2622no—
274C22—2624no—
275C22—2624no—
276C22—2622no—
277C22—2624no—
278C22—2622no—
279C22—52416no—
280C22—5241no—
281C22—5242no—
282C22—5241no—
283C22—1314no—
284C22C2×D4×D13111yescharacteristic
285C22#41×D13112yes—
286C22#41×D13111yescharacteristic
287C22C2×#1452351411yescharacteristic
288C22#1686229112yes—
289C4—22no—
290C4—41no—
291C4—42no—
292C4—42no—
293C4—2622no—
294C4—5241no—
295C4—5242no—
296C4—5242no—
297C2—42no—
298C2—2624no—
299C2—5242no—
300C2#123×D13111yescharacteristic
301C2#2255382511yescharacteristic
302C2#2255387511yescharacteristic
303C2#2255398912yes—
304C2#2255399012yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#399×C131C2
prime radicalC24×D131C22
hypercenter#399D131
hypocenterC131#2482
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#16862204C2semidirect
3. #22553247C2semidirect
4. #22553239C2semidirect
5. #22553237C2semidirect
6. #22553206C2semidirect
7. #22553202C2semidirect
8. #22553183C2semidirect
9. #22553174C2semidirect
11. #399×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. C2×C4⋊C4×C131C22semidirect
33. C2×C4⋊C4×C131C22semidirect
36. #95×C131C22semidirect
152. C131#2482semidirect
157. #399D131semidirect
171. C2×C4⋊C4C2×D131semidirect
172. C2×C4⋊C4C2×D131semidirect
189. #95C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622042
C2×C22⋊C4×D1312
#225532472
#225532392
#225532372
#225532062
#225532022
#225531832
#225531742
#225531672
#399×C1312
#2482131