Finite Group Databasebeta

← #16848774

Subgroups — #16848774

Order 3968.

832 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 832 conjugacy classes of subgroups — too many to draw. Only its 209 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
2C2×C4×C22⋊D31C212yesmaximal, complemented
3C2×(C22⋊D31)⋊C4C212yesmaximal, complemented
4#14520949C212yesmaximal, complemented
5#14520949C212yesmaximal, complemented
6#14520929C211yesmaximal, characteristic, complemented
7#14520927C212yesmaximal, complemented
8#14520900C212yesmaximal, complemented
9C31×#386C211yesmaximal, characteristic, complemented
10C22×D4×C31C2211yescharacteristic
11C23×C4×C31C2211yescharacteristic
12C22×C22⋊D31C2212yes—
13C22×C22⋊D31C2212yes—
14C23×Dic31C2211yescharacteristic
15C22×D124C2211yescharacteristic
16C22×C4×D31C2212yes—
17C2×C22⋊C4×C31C2212yes—
18C2×C22⋊C4×C31C2211yescharacteristic
19C2×C22⋊Dic31C2212yes—
20C2×C22⋊Dic31C2211yescharacteristic
21#11758517C418yescomplemented
22C4×C22⋊D31—28no—
23C2×(C2×D31)⋊C4C2212yes—
24C2×(C2×D31)⋊C4C2212yes—
25C2×(C2×D31)⋊C4C2212yes—
26C2×Dic31⋊C4C2212yes—
27C2×Dic31⋊C4C2212yes—
28C2×C4×Dic31C2212yes—
29C2×C4×Dic31C2212yes—
30C2×C4×Dic31C2211yescharacteristic
31(C22⋊D31)⋊C4—28no—
32C31×#95C2212yes—
33C31⋊#95C2212yes—
34C24×C31C2311yescharacteristic
35C24×C31C2311yescharacteristic
36C23×D31C2312yes—
37C2×D4×C31—41no—
38C2×D4×C31—41no—
39C2×D4×C31C2×C414yescomplemented
40C22×C4×C31—24no—
41C22×C4×C31—41no—
42C22×C4×C31C2312yes—
43C22×C4×C31C2311yescharacteristic
44C22×C4×C31C2312yes—
45C2×C22⋊D31—24no—
46C2×C22⋊D31—24no—
47C2×C22⋊D31—24no—
48C2×C22⋊D31—24no—
49C2×C22⋊D31C2×C418yescomplemented
50C2×C22⋊D31C2×C418yescomplemented
51C22×Dic31—24no—
52C22×Dic31—41no—
53C22×Dic31C2312yes—
54C22×Dic31C2312yes—
55C22×Dic31C2312yes—
56C2×D124—41no—
57C2×D124—41no—
58C2×D124C2×C414yescomplemented
59C2×C4×D31—24no—
60C2×C4×D31—24no—
61C2×C4×D31—42no—
62C22⋊C4×C31—24no—
63C22⋊C4×C31—41no—
64C22⋊Dic31—24no—
65C22⋊Dic31—41no—
66(C2×D31)⋊C4—24no—
67(C2×D31)⋊C4—24no—
68(C2×D31)⋊C4—42no—
69Dic31⋊C4—24no—
70Dic31⋊C4D418yescomplemented
71C4×Dic31—24no—
72C4×Dic31—24no—
73C4×Dic31C2×C414yescomplemented
74C23×C31—22no—
75C23×C31—22no—
76C23×C31—22no—
77C23×C31—22no—
78C23×C31—21no—
79C23×C31—21no—
80C23×C31—21no—
81C23×C31—21no—
82C23×C31C22×C412yes—
83C23×C31C22×C412yes—
84C23×C31C2411yescharacteristic
85C22×D31—22no—
86C22×D31—22no—
87C22×D31—22no—
88C22×D31—22no—
89C22×D31—22no—
90C22×D31—22no—
91C22×D31C22×C414yes—
92D4×C31—44no—
93C2×C4×C31—22no—
94C2×C4×C31—22no—
95C2×C4×C31—21no—
96C2×C4×C31—21no—
97C2×C4×C31—22no—
98C2×C4×C31—22no—
99C2×C4×C31—22no—
100C2×C4×C31—44no—
101C2×C4×C31C22×C412yes—
102C2×C4×C31C2×D414yes—
103C22⋊D31—216no—
104C22⋊D31—216no—
105C2×Dic31—22no—
106C2×Dic31—22no—
107C2×Dic31—22no—
108C2×Dic31—22no—
109C2×Dic31—22no—
110C2×Dic31—44no—
111C2×Dic31C22×C414yes—
112C2×Dic31C2×D414yes—
113C2×Dic31C2×D414yes—
114C2×Dic31#4114yes—
115D124—44no—
116C4×D31—48no—
117#2366—311nomaximal, Sylow
118C22×C31—24no—
119C22×C31—24no—
120C22×C31—22no—
121C22×C31—22no—
122C22×C31—41no—
123C22×C31—41no—
124C22×C31—42no—
125C22×C31—42no—
126C22×C31—41no—
127C22×C31—41no—
128C22×C31C23×C411yescharacteristic
129C22×C31C22×D412yes—
130C22×C31C2×#4112yes—
131C22×C31H(5,𝔽2)11yescharacteristic
132C22×C31H(5,𝔽2)11yescharacteristic
133C2×D31—24no—
134C2×D31—24no—
135C2×D31—24no—
136C2×D31—42no—
137C2×D31—42no—
138C2×D31—42no—
139C2×D31—42no—
140C4×C31—22no—
141C4×C31—24no—
142C4×C31—42no—
143Dic31—24no—
144Dic31—42no—
145Dic31C4×D418yescomplemented
146C2×#127—311no—
147C2×C4×D4—312no—
148C2×C4×D4—312no—
149#390—312no—
150#390—312no—
151#386—311no—
152#386—312no—
153#386D3111yescharacteristic, complemented
154C4×C4⋊C4—312no—
155C2×C31—42no—
156C2×C31—42no—
157C2×C31C2×C4×D412yes—
158C2×C31#51811yescharacteristic
159C2×C31D4211yescharacteristic
160C2×C31#55012yes—
161C2×C31#55911yescharacteristic
162D31—44no—
163C22×D4—312no—
164C22×D4—312no—
165C22×D4—311no—
166C22×D4C2×D3111yescharacteristic
167C23×C4—311no—
168C23×C4—312no—
169C23×C4C2×D3111yescharacteristic
170#127—318no—
171C4×D4—628no—
172C4×D4—628no—
173C2×C4⋊C4—312no—
174C2×C4⋊C4—312no—
175C2×C22⋊C4—312no—
176C2×C22⋊C4—312no—
177C2×C22⋊C4—312no—
178C2×C22⋊C4—311no—
179C2×C22⋊C4—312no—
180C2×C22⋊C4C2×D3112yes—
181C2×C22⋊C4C2×D3111yescharacteristic
182C2×C42—311no—
183C2×C42—312no—
184C2×C42—312no—
185#95—312no—
186#95C2×D3112yes—
187C31#236611yescharacteristic, complemented, Sylow
188C24—312no—
189C24C22×D3111yescharacteristic
190C24C22×D3111yescharacteristic
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—318no—
194C2×D4—318no—
195C2×D4—314no—
196C2×D4—624no—
197C2×D4—624no—
198C2×D4—624no—
199C2×D4—624no—
200C2×D4—1241no—
201C2×D4—1241no—
202C2×D4C4×D3114yescomplemented
203C22×C4—24no—
204C22×C4—41no—
205C22×C4—312no—
206C22×C4—312no—
207C22×C4—312no—
208C22×C4—624no—
209C22×C4—624no—
210C22×C4—624no—
211C22×C4—1242no—
212C22×C4—1241no—
213C22×C4C22×D3112yes—
214C22×C4C22×D3111yescharacteristic
215C22×C4C22×D3112yes—
216C4⋊C4—318no—
217C4⋊C4—624no—
218C22⋊C4—24no—
219C22⋊C4—41no—
220C22⋊C4—624no—
221C22⋊C4—624no—
222C22⋊C4—624no—
223C22⋊C4—1242no—
224C22⋊C4—1241no—
225C42—314no—
226C42—624no—
227C42—624no—
228C23—22no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—314no—
237C23—622no—
238C23—622no—
239C23—622no—
240C23—622no—
241C23—622no—
242C23—622no—
243C23C2×C4×D3112yes—
244C23C2×C4×D3112yes—
245C23C23×D3111yescharacteristic
246D4—44no—
247D4—6216no—
248D4—6216no—
249D4—1244no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—314no—
259C2×C4—314no—
260C2×C4—314no—
261C2×C4—314no—
262C2×C4—622no—
263C2×C4—622no—
264C2×C4—622no—
265C2×C4—622no—
266C2×C4—622no—
267C2×C4—1248no—
268C2×C4—1244no—
269C2×C4C2×C4×D3112yes—
270C2×C4D4×D3114yes—
271C22—24no—
272C22—24no—
273C22—22no—
274C22—22no—
275C22—41no—
276C22—41no—
277C22—42no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—624no—
282C22—624no—
283C22—624no—
284C22—1242no—
285C22—1242no—
286C22—1242no—
287C22—1242no—
288C22C22×C4×D3111yescharacteristic
289C22C2×D4×D3112yes—
290C22#1175856111yescharacteristic
291C22#1175856111yescharacteristic
292C22#41×D3112yes—
293C4—24no—
294C4—22no—
295C4—42no—
296C4—318no—
297C4—624no—
298C4—1242no—
299C2—42no—
300C2—42no—
301C2—1244no—
302C2C4×D4×D3112yes—
303C2#1452152211yescharacteristic
304C2#1452168111yescharacteristic
305C2#1452171712yes—
306C2#1452178311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#386C2
socle, cosocleC23×C31C24
prime radicalC2×#11758517C2
hypercenter#386D31
hypocenterC31#2366
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D31C2semidirect
3. C2×(C22⋊D31)⋊C4C2semidirect
4. #14520949C2semidirect
5. #14520949C2semidirect
6. #14520929C2semidirect
7. #14520927C2semidirect
8. #14520900C2semidirect
9. C31×#386C2semidirect
21. #11758517C4semidirect
39. C2×D4×C31C2×C4semidirect
49. C2×C22⋊D31C2×C4semidirect
50. C2×C22⋊D31C2×C4semidirect
58. C2×D124C2×C4semidirect
70. Dic31⋊C4D4semidirect
73. C4×Dic31C2×C4semidirect
145. Dic31C4×D4semidirect
153. #386D31semidirect
187. C31#2366semidirect
202. C2×D4C4×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585172
C2×C4×C22⋊D312
C2×(C22⋊D31)⋊C42
#145209492
#145209292
#145209272
#145209002
C31×#3862
#236631