Finite Group Databasebeta

← #16848950

Subgroups — #16848950

Order 3968.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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×D4×Dic31C211yesmaximal, characteristic, complemented
2C2×(C22⋊D31)⋊C4C212yesmaximal, complemented
3C2×#11758469C211yesmaximal, characteristic, complemented
4#14520979C211yesmaximal, characteristic, complemented
5#14520976C211yesmaximal, characteristic, complemented
6#14520972C212yesmaximal, complemented
7#14520964C212yesmaximal, complemented
8#14520957C211yesmaximal, characteristic
9#14520900C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic31C212yesmaximal, complemented
11C31×#394C211yesmaximal, characteristic, complemented
12C22×D4×C31C2211yescharacteristic
13C22×C22⋊D31C2212yes—
14C23×Dic31C2212yes—
15C22×C4×D31C2211yescharacteristic
16C2×C4⋊C4×C31C2211yescharacteristic
17C2×C22⋊C4×C31C2212yes—
18C2×C22⋊C4×C31C2212yes—
19C2×C22⋊Dic31C2212yes—
20C2×C22⋊Dic31C2212yes—
21D4×Dic31—24no—
22C2×(C2×D31)⋊C4C2211yescharacteristic
23C2×(C2×D31)⋊C4C2211yescharacteristic
24C2×(C2×D31)⋊C4C2212yes—
25C2×C4⋊Dic31C2211yescharacteristic
26C2×C4⋊Dic31C2211yescharacteristic
27C2×Dic31⋊C4C2212yes—
28C2×Dic31⋊C4C2212yes—
29C2×C4×Dic31C2211yescharacteristic
30C2×C4×Dic31C2212yes—
31C2×C4×Dic31C2211yescharacteristic
32#11758469—24no—
33(C22⋊D31)⋊C4—28no—
34C31×#95C2211yescharacteristic
35C31⋊#95C2212yes—
36C31⋊#95C2212yes—
37C31⋊#95C2211yescharacteristic
38C24×C31C2312yes—
39C23×D31C2311yescharacteristic
40C2×D4×C31—24no—
41C2×D4×C31—22no—
42C22×C4×C31C2311yescharacteristic
43C22×C4×C31C2312yes—
44C22×C4×C31C2311yescharacteristic
45C22×C4×C31C2311yescharacteristic
46C2×C22⋊D31—24no—
47C2×C22⋊D31—24no—
48C2×C22⋊D31—24no—
49C22×Dic31—24no—
50C22×Dic31—24no—
51C22×Dic31—24no—
52C22×Dic31C2311yescharacteristic
53C22×Dic31C2312yes—
54C22×Dic31C2311yescharacteristic
55C22×Dic31C2312yes—
56C22×Dic31C2311yescharacteristic
57C2×C4×D31—22no—
58C2×C4×D31—22no—
59C2×C4×D31D414yescomplemented
60C4⋊C4×C31—22no—
61C22⋊C4×C31—24no—
62C22⋊C4×C31—24no—
63C22⋊Dic31—24no—
64C22⋊Dic31—24no—
65(C2×D31)⋊C4—24no—
66(C2×D31)⋊C4—22no—
67(C2×D31)⋊C4—22no—
68C4⋊Dic31—22no—
69C4⋊Dic31—22no—
70Dic31⋊C4—24no—
71Dic31⋊C4—24no—
72C4×Dic31—24no—
73C4×Dic31—22no—
74C4×Dic31D414yescomplemented
75C23×C31—22no—
76C23×C31—22no—
77C23×C31—22no—
78C23×C31—22no—
79C23×C31—22no—
80C23×C31—22no—
81C23×C31#4114yes—
82C23×C31C2411yescharacteristic
83C22×D31—21no—
84C22×D31—21no—
85C22×D31—22no—
86C22×D31—22no—
87C22×D31C2×D412yes—
88D4×C31—28no—
89C2×C4×C31—21no—
90C2×C4×C31—21no—
91C2×C4×C31—22no—
92C2×C4×C31—22no—
93C2×C4×C31—21no—
94C2×C4×C31—22no—
95C2×C4×C31—22no—
96C2×C4×C31—22no—
97C2×C4×C31C2×D412yes—
98C2×C4×C31C2×D412yes—
99C22⋊D31—216no—
100C2×Dic31—22no—
101C2×Dic31—22no—
102C2×Dic31—22no—
103C2×Dic31—21no—
104C2×Dic31—22no—
105C2×Dic31—22no—
106C2×Dic31—22no—
107C2×Dic31—22no—
108C2×Dic31—48no—
109C2×Dic31C2×D412yes—
110C2×Dic31C2×D412yes—
111C2×Dic31C2×D412yes—
112C2×Dic31#4114yes—
113C2×Dic31#4112yes—
114C4×D31—28no—
115#2392—311nomaximal, Sylow
116C22×C31—24no—
117C22×C31—24no—
118C22×C31—24no—
119C22×C31—42no—
120C22×C31—42no—
121C22×C31—42no—
122C22×C31—42no—
123C22×C31C22×D411yescharacteristic
124C22×C31C22×D411yescharacteristic
125C22×C31C2×#4112yes—
126C22×C31C2×#4111yescharacteristic
127C22×C31C2×#4112yes—
128C2×D31—22no—
129C2×D31—24no—
130C2×D31—41no—
131C2×D31—42no—
132C2×D31—41no—
133C4×C31—22no—
134C4×C31—22no—
135C4×C31—41no—
136C4×C31—42no—
137Dic31—24no—
138Dic31—22no—
139Dic31—42no—
140Dic31—41no—
141Dic31#12114yes—
142C2×C4×D4—311no—
143C2×C4×D4—312no—
144C2×#117—311no—
145#398—311no—
146#397—311no—
147#396—312no—
148#394—312no—
149#394D3111yescharacteristic, complemented
150#392—311no—
151C4×C4⋊C4—311no—
152C4×C22⋊C4—312no—
153C2×C31—44no—
154C2×C31C2×#12111yescharacteristic
155C2×C31#52511yescharacteristic
156C2×C31#52512yes—
157C2×C31#52912yes—
158C2×C31#53211yescharacteristic
159D31—42no—
160C22×D4—312no—
161C22×D4C2×D3111yescharacteristic
162C23×C4—311no—
163C23×C4—312no—
164C4×D4—624no—
165C4×D4—628no—
166#117—624no—
167C2×C4⋊C4—311no—
168C2×C4⋊C4—312no—
169C2×C4⋊C4—311no—
170C2×C4⋊C4—312no—
171C2×C4⋊C4C2×D3111yescharacteristic
172C2×C22⋊C4—312no—
173C2×C22⋊C4—312no—
174C2×C22⋊C4—311no—
175C2×C22⋊C4—311no—
176C2×C22⋊C4—312no—
177C2×C22⋊C4C2×D3112yes—
178C2×C22⋊C4C2×D3112yes—
179C2×C42—311no—
180C2×C42—311no—
181C2×C42—312no—
182#95—312no—
183#95—312no—
184#95—311no—
185#95C2×D3111yescharacteristic
186C31#239211yescharacteristic, complemented, Sylow
187C24—311no—
188C24C22×D3112yes—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—624no—
192C2×D4—624no—
193C2×D4—624no—
194C22×C4—311no—
195C22×C4—312no—
196C22×C4—311no—
197C22×C4—314no—
198C22×C4—311no—
199C22×C4—312no—
200C22×C4—622no—
201C22×C4—622no—
202C22×C4—624no—
203C22×C4—624no—
204C22×C4—624no—
205C22×C4C22×D3111yescharacteristic
206C22×C4C22×D3112yes—
207C22×C4C22×D3111yescharacteristic
208C22×C4C22×D3111yescharacteristic
209C4⋊C4—22no—
210C4⋊C4—624no—
211C4⋊C4—622no—
212C4⋊C4—624no—
213C4⋊C4—622no—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—622no—
217C22⋊C4—624no—
218C22⋊C4—624no—
219C22⋊C4—624no—
220C22⋊C4—622no—
221C42—314no—
222C42—622no—
223C42—624no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—312no—
231C23—621no—
232C23—622no—
233C23—621no—
234C23—622no—
235C23#9249714yes—
236C23C23×D3111yescharacteristic
237D4—28no—
238D4—6216no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—312no—
248C2×C4—312no—
249C2×C4—312no—
250C2×C4—312no—
251C2×C4—314no—
252C2×C4—628no—
253C2×C4—622no—
254C2×C4—622no—
255C2×C4—622no—
256C2×C4—621no—
257C2×C4—622no—
258C2×C4—622no—
259C2×C4—622no—
260C2×C4—622no—
261C2×C4—1248no—
262C2×C4D4×D3112yes—
263C2×C4D4×D3112yes—
264C22—24no—
265C22—24no—
266C22—24no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—622no—
272C22—624no—
273C22—1241no—
274C22—1242no—
275C22—1241no—
276C22C2×D4×D3111yescharacteristic
277C22C2×D4×D3111yescharacteristic
278C22C2×#9249712yes—
279C22#41×D3111yescharacteristic
280C22#41×D3112yes—
281C4—22no—
282C4—22no—
283C4—41no—
284C4—42no—
285C4—314no—
286C4—624no—
287C4—622no—
288C4—1242no—
289C4—1241no—
290C2—44no—
291C2—1242no—
292C2#121×D3111yescharacteristic
293C2#1452155511yescharacteristic
294C2#1452155912yes—
295C2#1452157612yes—
296C2#1452159811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#394C2
prime radical#14520957C2
hypercenter#394D31
hypocenterC31#2392
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic31C2semidirect
2. C2×(C22⋊D31)⋊C4C2semidirect
3. C2×#11758469C2semidirect
4. #14520979C2semidirect
5. #14520976C2semidirect
6. #14520972C2semidirect
7. #14520964C2semidirect
9. #14520900C2semidirect
10. C22⋊C4×Dic31C2semidirect
11. C31×#394C2semidirect
59. C2×C4×D31D4semidirect
74. C4×Dic31D4semidirect
149. #394D31semidirect
186. C31#2392semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic312
C2×(C22⋊D31)⋊C42
C2×#117584692
#145209792
#145209762
#145209722
#145209642
#145209572
#145209002
C22⋊C4×Dic312
C31×#3942
#239231