Finite Group Databasebeta

← #16848952

Subgroups — #16848952

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×#11758470C212yesmaximal, complemented
2C2×#11758470C211yesmaximal, characteristic, complemented
3C2×#11758469C211yesmaximal, characteristic, complemented
4#14520975C211yesmaximal, characteristic, complemented
5#14520971C212yesmaximal, complemented
6#14520967C212yesmaximal, complemented
7#14520966C211yesmaximal, characteristic, complemented
8#14520959C211yesmaximal, characteristic
9#14520900C211yesmaximal, characteristic, complemented
10#14520896C212yesmaximal, complemented
11C31×#398C211yesmaximal, characteristic, complemented
12C22×D124C2211yescharacteristic
13C22×D124C2212yes—
14C22×C4×D31C2212yes—
15C22×C4×D31C2211yescharacteristic
16C2×C4⋊C4×C31C2211yescharacteristic
17C2×C4⋊C4×C31C2212yes—
18C2×C4⋊C4×C31C2211yescharacteristic
19C2×(C2×D31)⋊C4C2212yes—
20C2×(C2×D31)⋊C4C2212yes—
21C2×(C2×D31)⋊C4C2211yescharacteristic
22C2×(C2×D31)⋊C4C2212yes—
23C2×(C2×D31)⋊C4C2211yescharacteristic
24C2×(C2×D31)⋊C4C2212yes—
25C2×(C2×D31)⋊C4C2212yes—
26C2×C4⋊Dic31C2211yescharacteristic
27C2×Dic31⋊C4C2212yes—
28C2×C4×Dic31C2211yescharacteristic
29C2×C4×Dic31C2212yes—
30C2×C4×Dic31C2211yescharacteristic
31#11758470—28no—
32#11758470—24no—
33#11758469—24no—
34C31×#95C2212yes—
35C31×#95C2211yescharacteristic
36C31⋊#95C2212yes—
37C31⋊#95C2211yescharacteristic
38C23×D31C2311yescharacteristic
39C23×D31C2312yes—
40C22×C4×C31C2312yes—
41C22×C4×C31C2311yescharacteristic
42C22×C4×C31C2311yescharacteristic
43C22×C4×C31C2311yescharacteristic
44C22×C4×C31C2312yes—
45C22×Dic31C2312yes—
46C22×Dic31C2311yescharacteristic
47C22×Dic31C2311yescharacteristic
48C22×Dic31C2311yescharacteristic
49C2×D124—24no—
50C2×D124—24no—
51C2×D124—24no—
52C2×D124—22no—
53C2×D124—24no—
54C2×C4×D31—22no—
55C2×C4×D31—22no—
56C2×C4×D31—24no—
57C2×C4×D31—24no—
58C2×C4×D31—24no—
59C2×C4×D31D414yescomplemented
60C4⋊C4×C31—22no—
61C4⋊C4×C31—24no—
62C4⋊C4×C31—22no—
63(C2×D31)⋊C4—24no—
64(C2×D31)⋊C4—24no—
65(C2×D31)⋊C4—22no—
66(C2×D31)⋊C4—24no—
67(C2×D31)⋊C4—24no—
68(C2×D31)⋊C4—22no—
69(C2×D31)⋊C4—24no—
70C4⋊Dic31—22no—
71Dic31⋊C4—24no—
72C4×Dic31—22no—
73C4×Dic31—24no—
74C4×Dic31D414yescomplemented
75C23×C31C2411yescharacteristic
76C22×D31—21no—
77C22×D31—22no—
78C22×D31—22no—
79C22×D31—22no—
80C22×D31—21no—
81C22×D31—22no—
82C22×D31—22no—
83C22×D31—22no—
84C22×D31—22no—
85C22×D31—22no—
86C22×D31C2×D412yes—
87C22×D31#4114yes—
88C2×C4×C31—22no—
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×C31—22no—
98C2×C4×C31C2×D412yes—
99C2×C4×C31C2×D412yes—
100C2×C4×C31#4114yes—
101C2×Dic31—22no—
102C2×Dic31—21no—
103C2×Dic31—22no—
104C2×Dic31—22no—
105C2×Dic31—22no—
106C2×Dic31—22no—
107C2×Dic31C2×D412yes—
108C2×Dic31C2×D412yes—
109C2×Dic31C2×D412yes—
110C2×Dic31#4112yes—
111D124—28no—
112D124—216no—
113C4×D31—28no—
114C4×D31—48no—
115#2392—311nomaximal, Sylow
116C22×C31C22×D411yescharacteristic
117C22×C31C22×D411yescharacteristic
118C22×C31C2×#4112yes—
119C22×C31C2×#4111yescharacteristic
120C22×C31C2×#4112yes—
121C2×D31—22no—
122C2×D31—24no—
123C2×D31—24no—
124C2×D31—24no—
125C2×D31—24no—
126C2×D31—42no—
127C2×D31—41no—
128C2×D31—42no—
129C2×D31—42no—
130C2×D31—42no—
131C2×D31—41no—
132C2×D31—42no—
133C4×C31—22no—
134C4×C31—22no—
135C4×C31—24no—
136C4×C31—42no—
137C4×C31—41no—
138Dic31—22no—
139Dic31—42no—
140Dic31—41no—
141Dic31#12114yes—
142C2×C4×D4—311no—
143C2×C4×D4—312no—
144C2×#117—311no—
145#398D3111yescharacteristic, complemented
146#397—311no—
147#396—312no—
148#394—311no—
149#394—312no—
150#392—311no—
151C4×C4⋊C4—311no—
152C4×C22⋊C4—312no—
153C2×C31C2×#12111yescharacteristic
154C2×C31#52511yescharacteristic
155C2×C31#52512yes—
156C2×C31#52912yes—
157C2×C31#53211yescharacteristic
158D31—42no—
159D31—44no—
160C22×D4—311no—
161C22×D4—312no—
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⋊C4C2×D3111yescharacteristic
170C2×C4⋊C4C2×D3112yes—
171C2×C4⋊C4C2×D3111yescharacteristic
172C2×C22⋊C4—311no—
173C2×C22⋊C4—311no—
174C2×C22⋊C4—312no—
175C2×C22⋊C4—312no—
176C2×C22⋊C4—312no—
177C2×C22⋊C4—312no—
178C2×C22⋊C4—312no—
179C2×C42—311no—
180C2×C42—311no—
181C2×C42—312no—
182#95—312no—
183#95—311no—
184#95C2×D3112yes—
185#95C2×D3111yescharacteristic
186C31#239211yescharacteristic, complemented, Sylow
187C24—311no—
188C24—312no—
189C2×D4—624no—
190C2×D4—624no—
191C2×D4—624no—
192C2×D4—622no—
193C2×D4—624no—
194C22×C4—311no—
195C22×C4—311no—
196C22×C4—314no—
197C22×C4—312no—
198C22×C4—311no—
199C22×C4—622no—
200C22×C4—622no—
201C22×C4—624no—
202C22×C4—624no—
203C22×C4—624no—
204C22×C4C22×D3112yes—
205C22×C4C22×D3111yescharacteristic
206C22×C4C22×D3111yescharacteristic
207C22×C4C22×D3111yescharacteristic
208C22×C4C22×D3112yes—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—622no—
213C4⋊C4—624no—
214C22⋊C4—622no—
215C22⋊C4—624no—
216C22⋊C4—624no—
217C22⋊C4—624no—
218C22⋊C4—622no—
219C22⋊C4—624no—
220C22⋊C4—624no—
221C42—314no—
222C42—622no—
223C42—624no—
224C23—312no—
225C23—314no—
226C23—621no—
227C23—622no—
228C23—622no—
229C23—622no—
230C23—622no—
231C23—622no—
232C23—621no—
233C23—622no—
234C23—622no—
235C23—622no—
236C23C23×D3111yescharacteristic
237D4—628no—
238D4—6216no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—312no—
250C2×C4—312no—
251C2×C4—312no—
252C2×C4—312no—
253C2×C4—628no—
254C2×C4—622no—
255C2×C4—622no—
256C2×C4—621no—
257C2×C4—622no—
258C2×C4—622no—
259C2×C4—622no—
260C2×C4—1248no—
261C2×C4D4×D3112yes—
262C2×C4D4×D3112yes—
263C2×C4#9249914yes—
264C22—622no—
265C22—624no—
266C22—624no—
267C22—624no—
268C22—624no—
269C22—1241no—
270C22—1242no—
271C22—1242no—
272C22—1241no—
273C22—1242no—
274C22—1242no—
275C22—1242no—
276C22C2×D4×D3111yescharacteristic
277C22C2×D4×D3111yescharacteristic
278C22C2×#9249912yes—
279C22#41×D3111yescharacteristic
280C22#41×D3112yes—
281C4—22no—
282C4—22no—
283C4—24no—
284C4—42no—
285C4—41no—
286C4—314no—
287C4—622no—
288C4—1241no—
289C4—1242no—
290C2—1242no—
291C2—1244no—
292C2#121×D3111yescharacteristic
293C2#1452155711yescharacteristic
294C2#1452156212yes—
295C2#1452158412yes—
296C2#1452160011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#398C2
prime radical#14520959C2
hypercenter#398D31
hypocenterC31#2392
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#11758470C2semidirect
2. C2×#11758470C2semidirect
3. C2×#11758469C2semidirect
4. #14520975C2semidirect
5. #14520971C2semidirect
6. #14520967C2semidirect
7. #14520966C2semidirect
9. #14520900C2semidirect
10. #14520896C2semidirect
11. C31×#398C2semidirect
59. C2×C4×D31D4semidirect
74. C4×Dic31D4semidirect
145. #398D31semidirect
186. C31#2392semidirect

Maximal subgroups

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

subgroupindex
C2×#117584702
C2×#117584692
#145209752
#145209712
#145209672
#145209662
#145209592
#145209002
#145208962
C31×#3982
#239231