Finite Group Databasebeta

← #16854060

Subgroups — #16854060

Order 3968.

644 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 189 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×#11758521C211yesmaximal, characteristic, complemented
2C2×#11758474C211yesmaximal, characteristic
3C2×#11758523C211yesmaximal, characteristic
4C2×C42×D31C211yesmaximal, characteristic
5#14520955C211yesmaximal, characteristic, complemented
6#14520946C214yesmaximal, complemented
7#14520918C211yesmaximal, characteristic, complemented
8#14520908C214yesmaximal
9C31×#391C211yesmaximal, characteristic, complemented
10C22×Q8×C31C2211yescharacteristic
11C22×C4×D31C2211yescharacteristic
12C22×C4×D31C2212yes—
13C22×Dic62C2211yescharacteristic
14C2×C4⋊C4×C31C2211yescharacteristic
15C2×C42×C31C2211yescharacteristic
16#11758523—24no—
17#11758521—24no—
18C2×(C2×D31)⋊C4C2212yes—
19C2×(C2×D31)⋊C4C2212yes—
20C2×C4⋊Dic31C2211yescharacteristic
21C2×C4⋊Dic31C2212yes—
22C2×Dic31⋊C4C2214yes—
23C2×Dic31⋊C4C2214yes—
24C2×Dic31⋊C4C2212yes—
25C2×C4×Dic31C2212yes—
26C2×C4×Dic31C2211yescharacteristic
27#11758474—24no—
28C42×D31—24no—
29C31×#95C2214yes—
30C31⋊#95C2214yes—
31C23×D31C2311yescharacteristic
32C2×Q8×C31—22no—
33C2×Q8×C31—22no—
34C2×Q8×C31—22no—
35C22×C4×C31C2312yes—
36C22×C4×C31C2312yes—
37C22×C4×C31C2312yes—
38C22×C4×C31C2311yescharacteristic
39C22×Dic31C2312yes—
40C22×Dic31C2312yes—
41C22×Dic31C2311yescharacteristic
42C22×Dic31C2312yes—
43C2×C4×D31—22no—
44C2×C4×D31—22no—
45C2×C4×D31—24no—
46C2×C4×D31—24no—
47C2×C4×D31D414yes—
48C2×C4×D31Q818yescomplemented
49C2×Dic62—22no—
50C2×Dic62—22no—
51C2×Dic62—22no—
52C4⋊C4×C31—22no—
53C42×C31—22no—
54(C2×D31)⋊C4—24no—
55(C2×D31)⋊C4—24no—
56C4⋊Dic31—22no—
57C4⋊Dic31—24no—
58Dic31⋊C4—28no—
59Dic31⋊C4—24no—
60Dic31⋊C4—28no—
61C4×Dic31—24no—
62C4×Dic31D414yes—
63C23×C31C2411yescharacteristic
64C22×D31—21no—
65C22×D31—21no—
66C22×D31—22no—
67C22×D31C2×D412yes—
68C22×D31C2×Q814yes—
69Q8×C31—44no—
70C2×C4×C31—22no—
71C2×C4×C31—22no—
72C2×C4×C31—22no—
73C2×C4×C31—22no—
74C2×C4×C31—22no—
75C2×C4×C31—24no—
76C2×C4×C31—22no—
77C2×C4×C31C2×D412yes—
78C2×C4×C31C2×D412yes—
79C2×C4×C31C2×Q814yes—
80C2×C4×C31#4112yes—
81C2×Dic31—21no—
82C2×Dic31—21no—
83C2×Dic31—22no—
84C2×Dic31—22no—
85C2×Dic31—22no—
86C2×Dic31—24no—
87C2×Dic31—22no—
88C2×Dic31C2×D412yes—
89C2×Dic31C2×D414yes—
90C2×Dic31C2×Q814yes—
91C2×Dic31#4114yes—
92C4×D31—28no—
93C4×D31—216no—
94Dic62—44no—
95#2857—311nomaximal, Sylow
96C22×C31C22×D411yescharacteristic
97C22×C31C22×D411yescharacteristic
98C22×C31C22×Q812yes—
99C22×C31C2×#4111yescharacteristic
100C22×C31C2×#4112yes—
101C2×D31—22no—
102C2×D31—24no—
103C2×D31—22no—
104C2×D31—24no—
105C2×D31C4⋊Q814yescomplemented
106C4×C31—24no—
107C4×C31—42no—
108C4×C31—42no—
109C4×C31#12114yes—
110Dic31—22no—
111Dic31—42no—
112Dic31—42no—
113Dic31C22⋊Q818yescomplemented
114C2×C4⋊Q8—311no—
115C2×C22⋊Q8—311no—
116C2×C22⋊Q8—311no—
117C22×C42—311no—
118#391—311no—
119#391D3111yescharacteristic, complemented
120#389—314no—
121#384—311no—
122#380—314no—
123C2×C31C2×#12111yescharacteristic
124C2×C31C2×C22⋊Q812yes—
125C2×C31C2×C4⋊Q811yescharacteristic
126C2×C31#53211yescharacteristic
127C2×C31#53312yes—
128D31—24no—
129C22×Q8—311no—
130C22×Q8C2×D3111yescharacteristic
131C23×C4—311no—
132C23×C4—312no—
133C4⋊Q8—624no—
134C22⋊Q8—624no—
135C22⋊Q8—624no—
136C2×C4⋊C4—311no—
137C2×C4⋊C4—312no—
138C2×C4⋊C4—312no—
139C2×C4⋊C4—314no—
140C2×C4⋊C4—314no—
141C2×C4⋊C4C2×D3111yescharacteristic
142C2×C22⋊C4—312no—
143C2×C22⋊C4—312no—
144C2×C42—311no—
145C2×C42—312no—
146C2×C42—624no—
147C2×C42C2×D3111yescharacteristic
148#95—314no—
149#95C2×D3114yes—
150C31#285711yescharacteristic, complemented, Sylow
151C24—311no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—622no—
156C2×Q8—622no—
157C2×Q8—622no—
158C22×C4—312no—
159C22×C4—311no—
160C22×C4—314no—
161C22×C4—318no—
162C22×C4—312no—
163C22×C4—312no—
164C22×C4—624no—
165C22×C4—624no—
166C22×C4—622no—
167C22×C4—622no—
168C22×C4C22×D3112yes—
169C22×C4C22×D3112yes—
170C22×C4C22×D3112yes—
171C22×C4C22×D3111yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—628no—
174C4⋊C4—624no—
175C4⋊C4—628no—
176C4⋊C4—624no—
177C4⋊C4—622no—
178C22⋊C4—624no—
179C22⋊C4—624no—
180C42—22no—
181C42—314no—
182C42—624no—
183C23—312no—
184C23—314no—
185C23—621no—
186C23—621no—
187C23—622no—
188C23C23×D3111yescharacteristic
189Q8—44no—
190Q8—1244no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—314no—
199C2×C4—314no—
200C2×C4—314no—
201C2×C4—312no—
202C2×C4—6216no—
203C2×C4—621no—
204C2×C4—621no—
205C2×C4—628no—
206C2×C4—622no—
207C2×C4—622no—
208C2×C4—622no—
209C2×C4—622no—
210C2×C4—624no—
211C2×C4#9249512yes—
212C2×C4D4×D3112yes—
213C2×C4D4×D3112yes—
214C2×C4Q8×D3114yes—
215C22—314no—
216C22—624no—
217C22—622no—
218C22—624no—
219C22—622no—
220C22C2×#9249511yescharacteristic
221C22C2×D4×D3111yescharacteristic
222C22C2×D4×D3111yescharacteristic
223C22C2×Q8×D3112yes—
224C22#41×D3112yes—
225C4—24no—
226C4—42no—
227C4—42no—
228C4—318no—
229C4—622no—
230C4—1242no—
231C4—1242no—
232C4#1175846114yes—
233C2—624no—
234C2C2×#1175846111yescharacteristic
235C2C22⋊Q8×D3112yes—
236C2C4⋊Q8×D3111yescharacteristic
237C2#1452159811yescharacteristic
238C2#1452160312yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#391C2
prime radicalC23×D31C23
hypercenter#391D31
hypocenterC31#2857
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#11758521C2semidirect
5. #14520955C2semidirect
6. #14520946C2semidirect
7. #14520918C2semidirect
9. C31×#391C2semidirect
48. C2×C4×D31Q8semidirect
105. C2×D31C4⋊Q8semidirect
113. Dic31C22⋊Q8semidirect
119. #391D31semidirect
150. C31#2857semidirect

Maximal subgroups

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

subgroupindex
C2×#117585212
C2×#117584742
C2×#117585232
C2×C42×D312
#145209552
#145209462
#145209182
#145209082
C31×#3912
#285731