Finite Group Databasebeta

← #16854054

Subgroups — #16854054

Order 3968.

924 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 201 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×#11758517C211yesmaximal, characteristic, complemented
2C2×#11758507C211yesmaximal, characteristic, complemented
3C2×#11758516C214yesmaximal, complemented
4C22×C4×Dic31C211yesmaximal, characteristic, complemented
5#14520952C211yesmaximal, characteristic, complemented
6#14520950C211yesmaximal, characteristic, complemented
7#14520949C214yesmaximal, complemented
8#14520918C211yesmaximal, characteristic, complemented
9C2×C31×#121C211yesmaximal, characteristic, complemented
10C22×D4×C31C2211yescharacteristic
11C22×D4×C31C2212yes—
12C23×C4×C31C2211yescharacteristic
13C22×C22⋊D31C2214yes—
14C22×C22⋊D31C2212yes—
15C23×Dic31C2212yes—
16C22×D124C2211yescharacteristic
17C31×#121—24no—
18C2×C4⋊C4×C31C2211yescharacteristic
19C2×C22⋊C4×C31C2212yes—
20C2×C22⋊Dic31C2214yes—
21#11758517—24no—
22#11758516—216no—
23#11758507—24no—
24C2×(C2×D31)⋊C4C2212yes—
25C2×(C2×D31)⋊C4C2214yes—
26C2×C4⋊Dic31C2211yescharacteristic
27C2×Dic31⋊C4C2214yes—
28C2×C4×Dic31—24no—
29C2×C4×Dic31C2212yes—
30C2×C4×Dic31C2211yescharacteristic
31C2×C4×Dic31C2211yescharacteristic
32C24×C31C2312yes—
33C24×C31C2311yescharacteristic
34C23×D31C2312yes—
35C2×D4×C31—28no—
36C2×D4×C31—24no—
37C2×D4×C31—22no—
38C2×D4×C31—22no—
39C2×D4×C31—22no—
40C22×C4×C31—22no—
41C22×C4×C31—22no—
42C22×C4×C31D414yescomplemented
43C22×C4×C31C2312yes—
44C22×C4×C31C2311yescharacteristic
45C22×C4×C31C2311yescharacteristic
46C2×C22⋊D31—24no—
47C2×C22⋊D31—28no—
48C2×C22⋊D31—28no—
49C2×C22⋊D31—28no—
50C2×C22⋊D31—28no—
51C22×Dic31—24no—
52C22×Dic31—24no—
53C22×Dic31D418yescomplemented
54C22×Dic31C2312yes—
55C22×Dic31C2312yes—
56C22×Dic31C2312yes—
57C2×D124—22no—
58C2×D124—22no—
59C2×D124—22no—
60C4⋊C4×C31—22no—
61C22⋊C4×C31—24no—
62C22⋊Dic31—28no—
63(C2×D31)⋊C4—24no—
64(C2×D31)⋊C4—28no—
65C4⋊Dic31—22no—
66Dic31⋊C4—28no—
67C4×Dic31—22no—
68C4×Dic31—24no—
69C4×Dic31D414yescomplemented
70C23×C31—21no—
71C23×C31—22no—
72C23×C31—21no—
73C23×C31—24no—
74C23×C31—22no—
75C23×C31—22no—
76C23×C31—24no—
77C23×C31—22no—
78C23×C31C2×D414yes—
79C23×C31C2×D412yes—
80C23×C31C2411yescharacteristic
81C22×D31—22no—
82C22×D31—24no—
83C22×D31—22no—
84C22×D31—22no—
85C22×D31—24no—
86D4×C31—48no—
87D4×C31—44no—
88C2×C4×C31—21no—
89C2×C4×C31—21no—
90C2×C4×C31—28no—
91C2×C4×C31—24no—
92C2×C4×C31—22no—
93C2×C4×C31C2×D412yes—
94C2×C4×C31C2×D412yes—
95C2×C4×C31C2×D412yes—
96C2×C4×C31#4112yes—
97C22⋊D31—48no—
98C22⋊D31—416no—
99C2×Dic31—22no—
100C2×Dic31—22no—
101C2×Dic31—216no—
102C2×Dic31—24no—
103C2×Dic31—22no—
104C2×Dic31C2×D414yes—
105C2×Dic31C2×D414yes—
106C2×Dic31C2×D414yes—
107C2×Dic31#4114yes—
108D124—44no—
109#2856—311nomaximal, Sylow
110C22×C31—24no—
111C22×C31—22no—
112C22×C31—24no—
113C22×C31—22no—
114C22×C31—44no—
115C22×C31—42no—
116C22×C31—42no—
117C22×C31—44no—
118C22×C31—42no—
119C22×C31#12714yescomplemented
120C22×C31C22×D411yescharacteristic
121C22×C31C22×D412yes—
122C22×C31C22×D411yescharacteristic
123C22×C31C2×#4111yescharacteristic
124C22×C31C2×#4112yes—
125C2×D31—42no—
126C2×D31—44no—
127C2×D31—42no—
128C2×D31—44no—
129C2×D31—42no—
130C4×C31—22no—
131C4×C31—42no—
132C4×C31#12114yescomplemented
133Dic31—24no—
134Dic31—42no—
135Dic31#12118yescomplemented
136C2×#127—311no—
137C2×#121—314no—
138C2×#121—311no—
139C2×#121D3111yescharacteristic, complemented
140C22×C42—311no—
141#390—311no—
142#390—311no—
143#390—314no—
144#384—311no—
145C2×C31—24no—
146C2×C31—82no—
147C2×C31C2×#12112yes—
148C2×C31C2×#12111yescharacteristic
149C2×C31C2×#12711yescharacteristic
150C2×C31#53212yes—
151C2×C31#53211yescharacteristic
152D31—82no—
153C22×D4—312no—
154C22×D4—314no—
155C22×D4—311no—
156C22×D4C2×D3112yes—
157C22×D4C2×D3111yescharacteristic
158C23×C4—312no—
159C23×C4C2×D3111yescharacteristic
160#127—624no—
161#121—24no—
162#121—6216no—
163#121—624no—
164C2×C4⋊C4—314no—
165C2×C4⋊C4—311no—
166C2×C4⋊C4C2×D3111yescharacteristic
167C2×C22⋊C4—314no—
168C2×C22⋊C4—314no—
169C2×C22⋊C4—312no—
170C2×C22⋊C4C2×D3112yes—
171C2×C42—311no—
172C2×C42—312no—
173C2×C42—311no—
174C2×C42—624no—
175C31#285611yescharacteristic, complemented, Sylow
176C24—312no—
177C24C22×D3112yes—
178C24C22×D3111yescharacteristic
179C2×D4—28no—
180C2×D4—24no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—628no—
185C2×D4—628no—
186C2×D4—628no—
187C2×D4—628no—
188C2×D4—624no—
189C2×D4—622no—
190C2×D4—622no—
191C2×D4—622no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—312no—
195C22×C4—318no—
196C22×C4—312no—
197C22×C4—312no—
198C22×C4—624no—
199C22×C4—624no—
200C22×C4C22⋊D3114yescomplemented
201C22×C4C22×D3112yes—
202C22×C4C22×D3111yescharacteristic
203C22×C4C22×D3111yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—628no—
206C4⋊C4—622no—
207C22⋊C4—24no—
208C22⋊C4—628no—
209C22⋊C4—628no—
210C22⋊C4—624no—
211C42—314no—
212C42—624no—
213C42—622no—
214C23—21no—
215C23—21no—
216C23—22no—
217C23—24no—
218C23—22no—
219C23—22no—
220C23—24no—
221C23—22no—
222C23—624no—
223C23—622no—
224C23—622no—
225C23—624no—
226C23—622no—
227C23D4×D3114yes—
228C23C2×C22⋊D3112yes—
229C23C23×D3111yescharacteristic
230D4—48no—
231D4—44no—
232D4—12416no—
233D4—1248no—
234D4—1244no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—28no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—314no—
241C2×C4—314no—
242C2×C4—314no—
243C2×C4—314no—
244C2×C4—6216no—
245C2×C4—622no—
246C2×C4—622no—
247C2×C4—624no—
248C2×C4—622no—
249C2×C4D4×D3112yes—
250C2×C4#9249712yes—
251C2×C4C2×C22⋊D3112yes—
252C2×C4C2×C22⋊D3112yes—
253C22—24no—
254C22—22no—
255C22—24no—
256C22—22no—
257C22—44no—
258C22—42no—
259C22—42no—
260C22—44no—
261C22—42no—
262C22—1244no—
263C22—1242no—
264C22—1242no—
265C22—1244no—
266C22—1242no—
267C22#1175851714yescomplemented
268C22C2×D4×D3111yescharacteristic
269C22C2×D4×D3112yes—
270C22C2×#9249711yescharacteristic
271C22#41×D3112yes—
272C22C22×C22⋊D3111yescharacteristic
273C4—22no—
274C4—42no—
275C4—318no—
276C4—624no—
277C4—1242no—
278C4#1175851614yescomplemented
279C2—24no—
280C2—82no—
281C2—2482no—
282C2#121×D3112yes—
283C2C2×#1175851611yescharacteristic
284C2C2×#1175851711yescharacteristic
285C2#1452159412yes—
286C2#1452159511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC2×C31×#121C2
hypercenterC2×#121D31
hypocenterC31#2856
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#11758517C2semidirect
2. C2×#11758507C2semidirect
3. C2×#11758516C2semidirect
4. C22×C4×Dic31C2semidirect
5. #14520952C2semidirect
6. #14520950C2semidirect
7. #14520949C2semidirect
8. #14520918C2semidirect
9. C2×C31×#121C2semidirect
42. C22×C4×C31D4semidirect
53. C22×Dic31D4semidirect
69. C4×Dic31D4semidirect
119. C22×C31#127semidirect
132. C4×C31#121semidirect
135. Dic31#121semidirect
139. C2×#121D31semidirect
175. C31#2856semidirect
200. C22×C4C22⋊D31semidirect
267. C22#11758517semidirect
278. C4#11758516semidirect

Maximal subgroups

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

subgroupindex
C2×#117585172
C2×#117585072
C2×#117585162
C22×C4×Dic312
#145209522
#145209502
#145209492
#145209182
C2×C31×#1212
#285631