Finite Group Databasebeta

← #16848290

Subgroups — #16848290

Order 3968.

1104 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C22×D4×D31C211yesmaximal, characteristic
2C2×#11758469C211yesmaximal, characteristic, complemented
3C2×#11758449C211yesmaximal, characteristic, complemented
4#14520952C211yesmaximal, characteristic, complemented
5#14520951C211yesmaximal, characteristic, complemented
6#14520950C211yesmaximal, characteristic, complemented
7#14520949C214yesmaximal, complemented
8#14520904C214yesmaximal, complemented
9C31×#390C211yesmaximal, characteristic, complemented
10C24×D31C2212yes—
11C22×D4×C31C2211yescharacteristic
12C22×C22⋊D31C2212yes—
13C2×D4×D31—24no—
14C2×D4×D31—24no—
15C2×D4×D31C418yescomplemented
16C22×D124C2211yescharacteristic
17C22×C4×D31C2211yescharacteristic
18C2×C4⋊C4×C31C2211yescharacteristic, complemented
19C2×C22⋊C4×C31C2214yescomplemented
20C2×C42×C31C2211yescharacteristic, complemented
21C2×C22⋊Dic31C2214yescomplemented
22C2×(C2×D31)⋊C4C2212yes—
23C2×(C2×D31)⋊C4C2212yes—
24C2×(C2×D31)⋊C4C2214yescomplemented
25C2×(C2×D31)⋊C4C2214yescomplemented
26C2×C4⋊Dic31C2211yescharacteristic, complemented
27C2×Dic31⋊C4C2212yescomplemented
28C2×C4×Dic31C2212yescomplemented
29C2×C4×Dic31C2211yescharacteristic, complemented
30#11758469—24no—
31#11758449—24no—
32C24×C31C2312yes—
33C23×D31—24no—
34C23×D31—24no—
35C23×D31—44no—
36C23×D31—44no—
37C23×D31C2×C418yes—
38C23×D31C2312yes—
39C23×D31C2311yescharacteristic
40C2×D4×C31—22no—
41C2×D4×C31—22no—
42C2×D4×C31C2×C414yes—
43C22×C4×C31C2312yes—
44C22×C4×C31C2312yes—
45C22×C4×C31C2311yescharacteristic
46C2×C22⋊D31—24no—
47C2×C22⋊D31—24no—
48C2×C22⋊D31C2×C418yes—
49C22×Dic31C2312yes—
50C22×Dic31C2312yes—
51C22×Dic31C2311yescharacteristic
52D4×D31—232no—
53C2×D124—22no—
54C2×D124—22no—
55C2×D124C2×C414yes—
56C2×C4×D31C2×C414yes—
57C2×C4×D31D414yes—
58C2×C4×D31D414yes—
59C4⋊C4×C31—22no—
60C22⋊C4×C31—44no—
61C42×C31—22no—
62C22⋊Dic31—44no—
63(C2×D31)⋊C4—24no—
64(C2×D31)⋊C4—24no—
65(C2×D31)⋊C4—44no—
66(C2×D31)⋊C4—44no—
67C4⋊Dic31—22no—
68Dic31⋊C4—24no—
69C4×Dic31—22no—
70C4×Dic31—24no—
71C23×C31—22no—
72C23×C31—22no—
73C23×C31—24no—
74C23×C31—24no—
75C23×C31C22×C414yes—
76C23×C31C2411yescharacteristic
77C22×D31—22no—
78C22×D31—22no—
79C22×D31—22no—
80C22×D31—22no—
81C22×D31—24no—
82C22×D31—24no—
83C22×D31—216no—
84C22×D31—416no—
85C22×D31—48no—
86C22×D31—416no—
87C22×D31—48no—
88C22×D31C22×C414yes—
89C22×D31C22×C412yes—
90C22×D31C2×D412yes—
91C22×D31C2×D412yes—
92D4×C31—28no—
93C2×C4×C31—22no—
94C2×C4×C31—22no—
95C2×C4×C31—22no—
96C2×C4×C31—24no—
97C2×C4×C31—22no—
98C2×C4×C31C22×C412yes—
99C2×C4×C31C2×D412yes—
100C2×C4×C31C2×D412yes—
101C22⋊D31—216no—
102C2×Dic31—22no—
103C2×Dic31—22no—
104C2×Dic31—22no—
105C2×Dic31—24no—
106C2×Dic31—22no—
107C2×Dic31C22×C412yes—
108C2×Dic31C2×D412yes—
109C2×Dic31C2×D412yes—
110D124—28no—
111C4×D31C22⋊C4116yescomplemented
112#2309—311nomaximal, Sylow
113C22×C31—24no—
114C22×C31—28no—
115C22×C31—42no—
116C22×C31—44no—
117C22×C31—42no—
118C22×C31C23×C411yescharacteristic
119C22×C31C22×D411yescharacteristic
120C22×C31C22×D411yescharacteristic
121C22×C31H(5,𝔽2)12yes—
122C22×C31H(5,𝔽2)12yes—
123C2×D31—24no—
124C2×D31—22no—
125C2×D31—24no—
126C2×D31—22no—
127C2×D31—24no—
128C2×D31—28no—
129C2×D31—432no—
130C2×D31—42no—
131C2×D31—44no—
132C2×D31—42no—
133C2×D31C2×C22⋊C414yes—
134C4×C31—42no—
135C4×C31—42no—
136C4×C31C2×C22⋊C414yes—
137Dic31—42no—
138Dic31—42no—
139Dic31C2×C22⋊C414yes—
140C23×D4—311no—
141C2×#117—311no—
142C2×#117—311no—
143#390—311no—
144#390—311no—
145#390—314no—
146#390—311no—
147#390D3111yescharacteristic, complemented
148#379—314no—
149C2×C31—44no—
150C2×C31C22×C22⋊C411yescharacteristic
151C2×C31#51812yes—
152C2×C31#53511yescharacteristic
153C2×C31#53512yes—
154C2×C31#53511yescharacteristic
155D31—24no—
156D31—44no—
157C25—312no—
158C22×D4—312no—
159C22×D4—318no—
160C22×D4—311no—
161C22×D4—624no—
162C22×D4—624no—
163C22×D4C2×D3111yescharacteristic
164C23×C4—311no—
165#117—624no—
166#117—624no—
167C2×C4⋊C4—312no—
168C2×C4⋊C4—311no—
169C2×C4⋊C4C2×D3111yescharacteristic, complemented
170C2×C22⋊C4—312no—
171C2×C22⋊C4—312no—
172C2×C22⋊C4—314no—
173C2×C22⋊C4—314no—
174C2×C22⋊C4—314no—
175C2×C22⋊C4C2×D3114yescomplemented
176C2×C42—312no—
177C2×C42—311no—
178C2×C42C2×D3111yescharacteristic, complemented
179C31#230911yescharacteristic, complemented, Sylow
180C24—311no—
181C24—312no—
182C24—318no—
183C24—624no—
184C24—624no—
185C24—1244no—
186C24—1244no—
187C24C22×D3112yes—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—318no—
191C2×D4—314no—
192C2×D4—6232no—
193C2×D4—624no—
194C2×D4—624no—
195C2×D4—622no—
196C2×D4—622no—
197C2×D4C4×D3114yes—
198C22×C4—311no—
199C22×C4—314no—
200C22×C4—314no—
201C22×C4—314no—
202C22×C4—312no—
203C22×C4—312no—
204C22×C4C22×D3112yes—
205C22×C4C22×D3112yes—
206C22×C4C22×D3111yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—624no—
209C4⋊C4—622no—
210C22⋊C4—44no—
211C22⋊C4—624no—
212C22⋊C4—624no—
213C22⋊C4—1244no—
214C22⋊C4—1244no—
215C22⋊C4—1244no—
216C42—22no—
217C42—624no—
218C42—622no—
219C23—22no—
220C23—22no—
221C23—24no—
222C23—24no—
223C23—312no—
224C23—312no—
225C23—312no—
226C23—314no—
227C23—6216no—
228C23—622no—
229C23—622no—
230C23—622no—
231C23—622no—
232C23—624no—
233C23—624no—
234C23—12416no—
235C23—1248no—
236C23—12416no—
237C23—1248no—
238C23C2×C4×D3114yes—
239C23C23×D3111yescharacteristic
240D4—28no—
241D4—6216no—
242D4—628no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—312no—
249C2×C4—312no—
250C2×C4—312no—
251C2×C4—3116no—
252C2×C4—622no—
253C2×C4—622no—
254C2×C4—622no—
255C2×C4—622no—
256C2×C4—624no—
257C2×C4C2×C4×D3112yes—
258C2×C4D4×D3112yes—
259C2×C4D4×D3112yes—
260C22—24no—
261C22—28no—
262C22—42no—
263C22—44no—
264C22—42no—
265C22—314no—
266C22—622no—
267C22—624no—
268C22—622no—
269C22—624no—
270C22—624no—
271C22—628no—
272C22—12432no—
273C22—1242no—
274C22—1244no—
275C22—1242no—
276C22C22×C4×D3111yescharacteristic
277C22C2×D4×D3111yescharacteristic
278C22C2×D4×D3111yescharacteristic
279C22#1175856112yes—
280C22#1175856612yes—
281C4—42no—
282C4—42no—
283C4—314no—
284C4—1242no—
285C4—1242no—
286C4C22⋊C4×D3114yes—
287C2—44no—
288C2—624no—
289C2—1244no—
290C2C2×C22⋊C4×D3111yescharacteristic
291C2#1452152112yes—
292C2#1452161211yescharacteristic
293C2#1452161512yes—
294C2#1452161711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#390C2
socle, cosocleC23×C31C24
prime radicalC22×D4×D31C2
hypercenter#390D31
hypocenterC31#2309
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#11758469C2semidirect
3. C2×#11758449C2semidirect
4. #14520952C2semidirect
5. #14520951C2semidirect
6. #14520950C2semidirect
7. #14520949C2semidirect
8. #14520904C2semidirect
9. C31×#390C2semidirect
15. C2×D4×D31C4semidirect
18. C2×C4⋊C4×C31C22semidirect
19. C2×C22⋊C4×C31C22semidirect
20. C2×C42×C31C22semidirect
21. C2×C22⋊Dic31C22semidirect
24. C2×(C2×D31)⋊C4C22semidirect
25. C2×(C2×D31)⋊C4C22semidirect
26. C2×C4⋊Dic31C22semidirect
27. C2×Dic31⋊C4C22semidirect
28. C2×C4×Dic31C22semidirect
29. C2×C4×Dic31C22semidirect
111. C4×D31C22⋊C4semidirect
147. #390D31semidirect
169. C2×C4⋊C4C2×D31semidirect
175. C2×C22⋊C4C2×D31semidirect
178. C2×C42C2×D31semidirect
179. C31#2309semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D312
C2×#117584692
C2×#117584492
#145209522
#145209512
#145209502
#145209492
#145209042
C31×#3902
#230931