Finite Group Databasebeta

← #30671116

Subgroups — #30671116

Order 16768.

672 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 149 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×C22⋊Dic131C211yesmaximal, characteristic
3#22553240C212yesmaximal, complemented
4#22553224C212yesmaximal, complemented
5#22553205C212yesmaximal, complemented
6#22553200C212yesmaximal, complemented
7#22553173C211yesmaximal, characteristic, complemented
8#22553173C211yesmaximal, characteristic, complemented
9#22553168C211yesmaximal, characteristic
10C22⋊C4×Dic131C211yesmaximal, characteristic, complemented
11#379×C131C211yesmaximal, characteristic
12C22×Dic262C2211yescharacteristic, complemented
13C23×Dic131C2211yescharacteristic
14C23×Dic131C2211yescharacteristic
15#16862223—24no—
16C2×C4⋊Dic131C2211yescharacteristic, complemented
17C2×Dic131⋊C4C2211yescharacteristic, complemented
18C2×Dic131⋊C4C2211yescharacteristic, complemented
19C2×C22⋊Dic131—24no—
20C2×C22⋊Dic131C2211yescharacteristic
21C2×C22⋊Dic131C2211yescharacteristic
22C2×C22⋊Dic131C2212yes—
23C2×C22⋊Dic131C2212yes—
24C2×C22⋊Dic131C2212yes—
25C2×C22⋊Dic131C2212yes—
26C2×C4×Dic131C2212yescomplemented
27C131⋊#95C2212yescomplemented
28C131⋊#95C2212yescomplemented
29C131⋊#95C2212yescomplemented
30C131⋊#95C2212yescomplemented
31C131⋊#95C2212yescomplemented
32C25×C131C2211yescharacteristic
33C2×C22⋊C4×C131C2211yescharacteristic
34C2×C22⋊C4×C131C2211yescharacteristic
35C2×C22⋊C4×C131C2212yes—
36C2×C22⋊C4×C131C2212yes—
37C2×Dic262—22no—
38C2×Dic262—42no—
39C22×Dic131—22no—
40C22×Dic131—22no—
41C22×Dic131—22no—
42C22×Dic131—42no—
43C22×Dic131D414yes—
44C22×Dic131C2312yes—
45C22×Dic131C2311yescharacteristic
46C22×Dic131C2312yes—
47C22×Dic131C2311yescharacteristic
48C22×Dic131C2311yescharacteristic
49C22×Dic131C2311yescharacteristic
50C4⋊Dic131—22no—
51Dic131⋊C4—22no—
52Dic131⋊C4—22no—
53C22⋊Dic131—24no—
54C22⋊Dic131—24no—
55C22⋊Dic131—22no—
56C22⋊Dic131—24no—
57C22⋊Dic131—22no—
58C22⋊Dic131—42no—
59C4×Dic131—42no—
60C24×C131—22no—
61C24×C131—24no—
62C24×C131—22no—
63C24×C131—22no—
64C24×C131—42no—
65C24×C131C2312yes—
66C24×C131C2311yescharacteristic
67C22×C4×C131C2311yescharacteristic
68C22×C4×C131C2312yes—
69C22×C4×C131C2311yescharacteristic
70C22⋊C4×C131—22no—
71C22⋊C4×C131—24no—
72C22⋊C4×C131—22no—
73C22⋊C4×C131—42no—
74Dic262—44no—
75C2×Dic131—22no—
76C2×Dic131—22no—
77C2×Dic131—22no—
78C2×Dic131—21no—
79C2×Dic131—21no—
80C2×Dic131—22no—
81C2×Dic131—22no—
82C2×Dic131—22no—
83C2×Dic131—22no—
84C2×Dic131—22no—
85C2×Dic131—22no—
86C2×Dic131—22no—
87C2×Dic131—44no—
88C2×Dic131—44no—
89C2×Dic131C2×D412yes—
90C2×Dic131C2×D412yes—
91C23×C131—22no—
92C23×C131—22no—
93C23×C131—28no—
94C23×C131—22no—
95C23×C131—22no—
96C23×C131—22no—
97C23×C131—22no—
98C23×C131—22no—
99C23×C131—22no—
100C23×C131—48no—
101C23×C131—48no—
102C23×C131—44no—
103C23×C131—44no—
104C23×C131C2×D412yes—
105C23×C131#4114yes—
106C23×C131#4112yes—
107C23×C131#4112yes—
108C23×C131C2411yescharacteristic
109C2×C4×C131—21no—
110C2×C4×C131—21no—
111C2×C4×C131—22no—
112C2×C4×C131—22no—
113C2×C4×C131—22no—
114C2×C4×C131—22no—
115C2×C4×C131—22no—
116C22×C131—22no—
117C22×C131—22no—
118C22×C131—24no—
119C22×C131—24no—
120C22×C131—24no—
121C22×C131—24no—
122C22×C131—24no—
123C22×C131—416no—
124C22×C131—42no—
125C22×C131—42no—
126C22×C131—42no—
127C22×C131—42no—
128C22×C131#12414yes—
129C22×C131C22×D411yescharacteristic
130C22×C131C2×#4112yes—
131C22×C131C2×#4111yescharacteristic
132C22×C131C2×#4111yescharacteristic
133C22×C131H(5,𝔽2)12yes—
134C4×C131—41no—
135C4×C131—41no—
136C4×C131—42no—
137Dic131—22no—
138Dic131—22no—
139Dic131—41no—
140Dic131—42no—
141Dic131—41no—
142Dic131—42no—
143C2×C131—24no—
144C2×C131—44no—
145C2×C131#52511yescharacteristic
146C2×C131C2×#12411yescharacteristic
147C2×C131#53411yescharacteristic
148C2×C131#55112yes—
149C2×C131#55112yes—
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#381—1311no—
160#379—1311no—
161#379D13111yescharacteristic
162C4×C22⋊C4—1311no—
163C25C2×D13111yescharacteristic
164C22×Q8—1311no—
165C23×C4—1311no—
166C23×C4—1311no—
167C22⋊Q8—2624no—
168C2×C4⋊C4—1311no—
169C2×C4⋊C4—1311no—
170C2×C4⋊C4—1311no—
171C2×C22⋊C4—2624no—
172C2×C22⋊C4—1312no—
173C2×C22⋊C4—1312no—
174C2×C22⋊C4—1311no—
175C2×C22⋊C4—1311no—
176C2×C22⋊C4—1312no—
177C2×C22⋊C4—1312no—
178C2×C22⋊C4C2×D13111yescharacteristic
179C2×C22⋊C4C2×D13111yescharacteristic
180C2×C22⋊C4C2×D13112yes—
181C2×C22⋊C4C2×D13112yes—
182C2×C42—1312no—
183#95—1312no—
184#95—1312no—
185#95—1312no—
186#95—1312no—
187#95—1312no—
188C24—22no—
189C24—24no—
190C24—22no—
191C24—22no—
192C24—42no—
193C24C22×D13112yes—
194C24C22×D13111yescharacteristic
195C2×Q8—2622no—
196C2×Q8—5242no—
197C22×C4—2622no—
198C22×C4—2622no—
199C22×C4—2622no—
200C22×C4—5242no—
201C22×C4—1312no—
202C22×C4—1312no—
203C22×C4—1311no—
204C22×C4—1311no—
205C22×C4—1311no—
206C22×C4—1311no—
207C22×C4—1314no—
208C22×C4C22×D13112yes—
209C22×C4C22×D13111yescharacteristic
210C22×C4C22×D13111yescharacteristic
211C4⋊C4—2622no—
212C4⋊C4—2622no—
213C4⋊C4—2622no—
214C22⋊C4—22no—
215C22⋊C4—24no—
216C22⋊C4—22no—
217C22⋊C4—42no—
218C22⋊C4—2622no—
219C22⋊C4—2624no—
220C22⋊C4—2622no—
221C22⋊C4—2624no—
222C22⋊C4—2624no—
223C22⋊C4—5242no—
224C42—5242no—
225C23—22no—
226C23—22no—
227C23—28no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—48no—
235C23—48no—
236C23—44no—
237C23—44no—
238C23D4×D13112yes—
239C23#1452351214yes—
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—2622no—
252C2×C4—2622no—
253C2×C4—2622no—
254C2×C4—2621no—
255C2×C4—2621no—
256C2×C4—2622no—
257C2×C4—2622no—
258C2×C4—2622no—
259C2×C4—2622no—
260C2×C4—2622no—
261C2×C4—2622no—
262C2×C4—2622no—
263C2×C4—5244no—
264C2×C4—5244no—
265C2×C4—1312no—
266C2×C4—1312no—
267C22—22no—
268C22—24no—
269C22—24no—
270C22—22no—
271C22—24no—
272C22—24no—
273C22—24no—
274C22—416no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22#1686223214yes—
280C22C2×D4×D13111yescharacteristic
281C22C2×#1452351212yes—
282C22C2×#1452351211yescharacteristic
283C22C2×#1452351311yescharacteristic
284C22#1686229112yes—
285C4—41no—
286C4—42no—
287C4—41no—
288C4—2622no—
289C4—2622no—
290C4—5241no—
291C4—5242no—
292C4—5241no—
293C4—5242no—
294C2—24no—
295C2—44no—
296C2#2255381811yescharacteristic
297C2C2×#1686223211yescharacteristic
298C2#2255387211yescharacteristic
299C2#2255398812yes—
300C2#2255399312yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#379×C131C2
prime radicalC25×C131C22
hypercenter#379D131
hypocenterC131#2561
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C2×#16862223C2semidirect
3. #22553240C2semidirect
4. #22553224C2semidirect
5. #22553205C2semidirect
6. #22553200C2semidirect
7. #22553173C2semidirect
8. #22553173C2semidirect
10. C22⋊C4×Dic131C2semidirect
12. C22×Dic262C22semidirect
16. C2×C4⋊Dic131C22semidirect
17. C2×Dic131⋊C4C22semidirect
18. C2×Dic131⋊C4C22semidirect
26. C2×C4×Dic131C22semidirect
27. C131⋊#95C22semidirect
28. C131⋊#95C22semidirect
29. C131⋊#95C22semidirect
30. C131⋊#95C22semidirect
31. C131⋊#95C22semidirect
150. C131#2561semidirect

Maximal subgroups

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

subgroupindex
C2×#168622232
C22×C22⋊Dic1312
#225532402
#225532242
#225532052
#225532002
#225531732
#225531682
C22⋊C4×Dic1312
#379×C1312
#2561131