Finite Group Databasebeta

← #16849021

Subgroups — #16849021

Order 3968.

1564 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 213 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
1C23×C22⋊D31C211yesmaximal, characteristic, complemented
2C2×#11758530C212yesmaximal, complemented
3C2×#11758514C212yesmaximal, complemented
4#14520957C214yesmaximal, complemented
5#14520929C212yesmaximal, complemented
6#14520928C212yesmaximal, complemented
7#14520905C211yesmaximal, characteristic, complemented
8C23×D4×C31C211yesmaximal, characteristic, complemented
9C25×C31C2211yescharacteristic
10C25×C31C2211yescharacteristic, complemented
11C24×D31C2211yescharacteristic, complemented
12C22×D4×C31—24no—
13C22×D4×C31—28no—
14C22×D4×C31C2212yes—
15C22×D4×C31C2212yescomplemented
16C23×C4×C31C2211yescharacteristic, complemented
17C22×C22⋊D31—24no—
18C22×C22⋊D31—28no—
19C22×C22⋊D31C2212yes—
20C22×C22⋊D31C2212yescomplemented
21C22×C22⋊D31C2214yescomplemented
22C23×Dic31C2211yescharacteristic, complemented
23#11758530—28no—
24C2×C22⋊Dic31C2212yescomplemented
25C2×C22⋊Dic31C2212yescomplemented
26C2×C22⋊Dic31C2214yescomplemented
27C2×C22⋊Dic31C2212yescomplemented
28#11758514—28no—
29C2×(C2×D31)⋊C4C2214yescomplemented
30C31⋊#95C2214yescomplemented
31C24×C31—24no—
32C24×C31—24no—
33C24×C31—22no—
34C24×C31—24no—
35C24×C31—24no—
36C24×C31—42no—
37C24×C31—42no—
38C24×C31D414yescomplemented
39C24×C31C2311yescharacteristic
40C24×C31C2312yes—
41C24×C31C2311yescharacteristic
42C24×C31C2311yescharacteristic
43C23×D31—22no—
44C23×D31—24no—
45C23×D31—24no—
46C23×D31—42no—
47C23×D31C2312yes—
48C2×D4×C31—232no—
49C2×D4×C31—24no—
50C2×D4×C31—24no—
51C2×D4×C31—24no—
52C2×D4×C31—24no—
53C2×D4×C31—24no—
54C2×D4×C31D418yescomplemented
55C22×C4×C31—22no—
56C22×C4×C31—24no—
57C22×C4×C31C2312yes—
58C2×C22⋊D31—232no—
59C2×C22⋊D31—24no—
60C2×C22⋊D31—24no—
61C2×C22⋊D31—24no—
62C2×C22⋊D31—24no—
63C2×C22⋊D31—28no—
64C2×C22⋊D31—28no—
65C2×C22⋊D31—24no—
66C2×C22⋊D31—44no—
67C2×C22⋊D31D418yescomplemented
68C22×Dic31—22no—
69C22×Dic31—24no—
70C22×Dic31C2314yes—
71C22×Dic31C2312yes—
72C22⋊Dic31—24no—
73C22⋊Dic31—24no—
74C22⋊Dic31—28no—
75C22⋊Dic31—42no—
76(C2×D31)⋊C4—28no—
77C23×C31—22no—
78C23×C31—22no—
79C23×C31—22no—
80C23×C31—22no—
81C23×C31—28no—
82C23×C31—28no—
83C23×C31—28no—
84C23×C31—24no—
85C23×C31—22no—
86C23×C31—22no—
87C23×C31—22no—
88C23×C31—22no—
89C23×C31—22no—
90C23×C31—24no—
91C23×C31—44no—
92C23×C31—416no—
93C23×C31—416no—
94C23×C31—44no—
95C23×C31—44no—
96C23×C31C2×D414yes—
97C23×C31C2×D412yes—
98C23×C31C2×D414yes—
99C23×C31C2×D412yes—
100C23×C31C2×D412yes—
101C23×C31C2×D414yescomplemented
102C23×C31C2411yescharacteristic
103C22×D31—28no—
104C22×D31—24no—
105C22×D31—22no—
106C22×D31—22no—
107C22×D31—24no—
108C22×D31—416no—
109C22×D31—44no—
110C22×D31—44no—
111C22×D31C2×D414yescomplemented
112D4×C31—216no—
113D4×C31—216no—
114C2×C4×C31—28no—
115C2×C4×C31—22no—
116C2×C4×C31—22no—
117C2×C4×C31C2×D414yescomplemented
118C22⋊D31—216no—
119C22⋊D31—216no—
120C22⋊D31—416no—
121C2×Dic31—28no—
122C2×Dic31—22no—
123C2×Dic31—22no—
124C2×Dic31—24no—
125C2×Dic31—24no—
126C2×Dic31—24no—
127C2×Dic31C2×D414yescomplemented
128#2403—311nomaximal, Sylow
129C22×C31—22no—
130C22×C31—22no—
131C22×C31—22no—
132C22×C31—22no—
133C22×C31—28no—
134C22×C31—28no—
135C22×C31—22no—
136C22×C31—22no—
137C22×C31—24no—
138C22×C31—22no—
139C22×C31—28no—
140C22×C31—416no—
141C22×C31—416no—
142C22×C31—42no—
143C22×C31—42no—
144C22×C31—44no—
145C22×C31—44no—
146C22×C31#12014yescomplemented
147C22×C31#12014yescomplemented
148C22×C31C22×D412yes—
149C22×C31C22×D411yescharacteristic
150C22×C31C22×D412yes—
151C22×C31H(5,𝔽2)12yes—
152C2×D31—24no—
153C2×D31—28no—
154C2×D31—416no—
155C2×D31—42no—
156C2×D31—42no—
157C2×D31—44no—
158C4×C31—24no—
159Dic31—24no—
160Dic31—44no—
161C23×D4—311no—
162C23×D4D3111yescharacteristic, complemented
163C2×#120—312no—
164C2×#120—312no—
165#392—314no—
166#386—312no—
167#386—312no—
168#379—311no—
169C2×C31—24no—
170C2×C31—24no—
171C2×C31—42no—
172C2×C31—44no—
173C2×C31C2×#12011yescharacteristic
174C2×C31C2×#12011yescharacteristic
175C2×C31#53411yescharacteristic
176C2×C31D4214yes—
177D31—44no—
178C25—311no—
179C25C2×D3111yescharacteristic
180C25C2×D3111yescharacteristic, complemented
181C22×D4—24no—
182C22×D4—28no—
183C22×D4—312no—
184C22×D4—312no—
185C22×D4—314no—
186C22×D4—628no—
187C22×D4—624no—
188C22×D4C2×D3112yes—
189C22×D4C2×D3112yescomplemented
190C23×C4—311no—
191C23×C4C2×D3111yescharacteristic, complemented
192#120—628no—
193#120—628no—
194C2×C22⋊C4—312no—
195C2×C22⋊C4—312no—
196C2×C22⋊C4—314no—
197C2×C22⋊C4—314no—
198C2×C22⋊C4—312no—
199#95—314no—
200C31#240311yescharacteristic, complemented, Sylow
201C24—24no—
202C24—24no—
203C24—22no—
204C24—24no—
205C24—24no—
206C24—42no—
207C24—42no—
208C24—312no—
209C24—624no—
210C24—622no—
211C24—624no—
212C24—1242no—
213C24C22⋊D3114yescomplemented
214C24C22×D3111yescharacteristic
215C24C22×D3112yes—
216C24C22×D3111yescharacteristic
217C24C22×D3111yescharacteristic
218C2×D4—232no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—318no—
225C2×D4—624no—
226C2×D4—624no—
227C2×D4—6232no—
228C2×D4—624no—
229C2×D4—624no—
230C2×D4—624no—
231C2×D4—628no—
232C2×D4—628no—
233C2×D4—1244no—
234C2×D4C22⋊D3118yescomplemented
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—312no—
238C22×C4—314no—
239C22×C4—624no—
240C22×C4—622no—
241C22×C4C22×D3112yes—
242C22⋊C4—628no—
243C22⋊C4—624no—
244C22⋊C4—624no—
245C22⋊C4—628no—
246C22⋊C4—1242no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—28no—
252C23—28no—
253C23—28no—
254C23—24no—
255C23—22no—
256C23—22no—
257C23—24no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—44no—
262C23—416no—
263C23—416no—
264C23—44no—
265C23—44no—
266C23—314no—
267C23—628no—
268C23—622no—
269C23—624no—
270C23—624no—
271C23—622no—
272C23—12416no—
273C23—1244no—
274C23—1244no—
275C23D4×D3114yes—
276C23C2×C22⋊D3112yes—
277C23C2×C22⋊D3112yes—
278C23C2×C22⋊D3114yes—
279C23C2×C22⋊D3112yes—
280C23C2×C22⋊D3114yescomplemented
281C23C23×D3111yescharacteristic
282D4—216no—
283D4—216no—
284D4—6216no—
285D4—6216no—
286D4—12416no—
287C2×C4—28no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—314no—
291C2×C4—628no—
292C2×C4—622no—
293C2×C4—622no—
294C2×C4—624no—
295C2×C4—624no—
296C2×C4—624no—
297C2×C4C2×C22⋊D3114yescomplemented
298C22—22no—
299C22—22no—
300C22—28no—
301C22—28no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—28no—
306C22—22no—
307C22—22no—
308C22—24no—
309C22—416no—
310C22—416no—
311C22—42no—
312C22—44no—
313C22—44no—
314C22—42no—
315C22—628no—
316C22—624no—
317C22—12416no—
318C22—1242no—
319C22—1244no—
320C22—1242no—
321C22#1175851414yescomplemented
322C22#1175853014yescomplemented
323C22C2×D4×D3112yes—
324C22#1175856112yes—
325C22C22×C22⋊D3111yescharacteristic
326C22C22×C22⋊D3112yes—
327C4—24no—
328C4—624no—
329C4—1244no—
330C2—24no—
331C2—24no—
332C2—42no—
333C2—44no—
334C2—1244no—
335C2C2×#1175851411yescharacteristic
336C2C2×#1175853011yescharacteristic
337C2#1452160811yescharacteristic
338C2D4×C22⋊D3114yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC23×D4×C31C2
hypercenterC23×D4D31
hypocenterC31#2403
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D31C2semidirect
2. C2×#11758530C2semidirect
3. C2×#11758514C2semidirect
4. #14520957C2semidirect
5. #14520929C2semidirect
6. #14520928C2semidirect
7. #14520905C2semidirect
8. C23×D4×C31C2semidirect
10. C25×C31C22semidirect
11. C24×D31C22semidirect
15. C22×D4×C31C22semidirect
16. C23×C4×C31C22semidirect
20. C22×C22⋊D31C22semidirect
21. C22×C22⋊D31C22semidirect
22. C23×Dic31C22semidirect
24. C2×C22⋊Dic31C22semidirect
25. C2×C22⋊Dic31C22semidirect
26. C2×C22⋊Dic31C22semidirect
27. C2×C22⋊Dic31C22semidirect
29. C2×(C2×D31)⋊C4C22semidirect
30. C31⋊#95C22semidirect
38. C24×C31D4semidirect
54. C2×D4×C31D4semidirect
67. C2×C22⋊D31D4semidirect
101. C23×C31C2×D4semidirect
111. C22×D31C2×D4semidirect
117. C2×C4×C31C2×D4semidirect
127. C2×Dic31C2×D4semidirect
146. C22×C31#120semidirect
147. C22×C31#120semidirect
162. C23×D4D31semidirect
180. C25C2×D31semidirect
189. C22×D4C2×D31semidirect
191. C23×C4C2×D31semidirect
200. C31#2403semidirect
213. C24C22⋊D31semidirect
234. C2×D4C22⋊D31semidirect
280. C23C2×C22⋊D31semidirect
297. C2×C4C2×C22⋊D31semidirect
321. C22#11758514semidirect
322. C22#11758530semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D312
C2×#117585302
C2×#117585142
#145209572
#145209292
#145209282
#145209052
C23×D4×C312
#240331