Finite Group Databasebeta

← #16854085

Subgroups — #16854085

Order 3968.

524 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 153 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×#11758467C211yesmaximal, characteristic, complemented
2C2×#11758524C211yesmaximal, characteristic
3#14520956C211yesmaximal, characteristic, complemented
4#14520951C211yesmaximal, characteristic
5#14520950C211yesmaximal, characteristic
6#14520941C214yesmaximal, complemented
7#14520938C214yesmaximal, complemented
8C42×Dic31C211yesmaximal, characteristic, complemented
9C31×#391C211yesmaximal, characteristic, complemented
10C22×Q8×C31C2211yescharacteristic
11C22×D124C2211yescharacteristic
12C2×C4⋊C4×C31C2211yescharacteristic
13C2×C42×C31C2211yescharacteristic
14#11758524—24no—
15C2×(C2×D31)⋊C4C2212yes—
16C2×(C2×D31)⋊C4C2214yes—
17C2×(C2×D31)⋊C4C2214yes—
18C2×(C2×D31)⋊C4C2212yes—
19C2×C4⋊Dic31C2212yes—
20C2×C4⋊Dic31C2211yescharacteristic
21C2×Dic31⋊C4C2212yes—
22C2×C4×Dic31C2212yes—
23C2×C4×Dic31C2212yes—
24C2×C4×Dic31C2211yescharacteristic
25C2×C4×Dic31C2211yescharacteristic
26#11758467—24no—
27C31×#95C2214yes—
28C31⋊#95C2214yes—
29C23×D31C2312yes—
30C2×Q8×C31—22no—
31C2×Q8×C31—22no—
32C2×Q8×C31—22no—
33C22×C4×C31C2312yes—
34C22×C4×C31C2312yes—
35C22×C4×C31C2312yes—
36C22×C4×C31C2311yescharacteristic
37C22×Dic31C2312yes—
38C22×Dic31C2311yescharacteristic
39C22×Dic31C2311yescharacteristic
40C22×Dic31C2312yes—
41C2×D124—22no—
42C2×D124—22no—
43C2×D124—22no—
44C4⋊C4×C31—22no—
45C42×C31—22no—
46(C2×D31)⋊C4—28no—
47(C2×D31)⋊C4—24no—
48(C2×D31)⋊C4—24no—
49(C2×D31)⋊C4—28no—
50C4⋊Dic31—22no—
51C4⋊Dic31—24no—
52Dic31⋊C4—24no—
53C4×Dic31—22no—
54C4×Dic31—24no—
55C4×Dic31—24no—
56C4×Dic31D414yes—
57C23×C31C2411yescharacteristic
58C22×D31—22no—
59C22×D31—24no—
60C22×D31—24no—
61C22×D31—22no—
62C22×D31—22no—
63Q8×C31—44no—
64C2×C4×C31—22no—
65C2×C4×C31—22no—
66C2×C4×C31—22no—
67C2×C4×C31—22no—
68C2×C4×C31—24no—
69C2×C4×C31—22no—
70C2×C4×C31—22no—
71C2×C4×C31C2×D412yes—
72C2×C4×C31#4112yes—
73C2×C4×C31#4112yes—
74C2×C4×C31#4114yes—
75C2×Dic31—22no—
76C2×Dic31—22no—
77C2×Dic31—22no—
78C2×Dic31—22no—
79C2×Dic31—22no—
80C2×Dic31—24no—
81C2×Dic31C2×D412yes—
82C2×Dic31C2×D412yes—
83C2×Dic31#4114yes—
84D124—44no—
85#2861—311nomaximal, Sylow
86C22×C31C22×D411yescharacteristic
87C22×C31C2×#4111yescharacteristic
88C22×C31C2×#4111yescharacteristic
89C22×C31C2×#4112yes—
90C22×C31C2×#4112yes—
91C2×D31—42no—
92C2×D31—44no—
93C2×D31—44no—
94C2×D31—42no—
95C2×D31—42no—
96C4×C31—24no—
97C4×C31—42no—
98C4×C31—42no—
99C4×C31#12414yes—
100Dic31—24no—
101Dic31—22no—
102Dic31—22no—
103Dic31—42no—
104C2×#125—311no—
105C2×#124—311no—
106#391—311no—
107#391D3111yescharacteristic, complemented
108#390—311no—
109#390—311no—
110#388—314no—
111#388—314no—
112C43—311no—
113C2×C31C2×#12411yescharacteristic
114C2×C31#52912yes—
115C2×C31#52912yes—
116C2×C31#53211yescharacteristic
117C2×C31#53211yescharacteristic
118D31—82no—
119C22×Q8C2×D3111yescharacteristic
120C22×D4—311no—
121#125—624no—
122#124—624no—
123C2×C4⋊C4—312no—
124C2×C4⋊C4—312no—
125C2×C4⋊C4—311no—
126C2×C4⋊C4C2×D3111yescharacteristic
127C2×C22⋊C4—312no—
128C2×C22⋊C4—312no—
129C2×C22⋊C4—314no—
130C2×C22⋊C4—314no—
131C2×C42—311no—
132C2×C42—312no—
133C2×C42—312no—
134C2×C42—311no—
135C2×C42C2×D3111yescharacteristic
136#95—314no—
137#95C2×D3114yes—
138C31#286111yescharacteristic, complemented, Sylow
139C24—312no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—22no—
143C2×D4—622no—
144C2×D4—622no—
145C2×D4—622no—
146C22×C4—312no—
147C22×C4—311no—
148C22×C4—311no—
149C22×C4—312no—
150C22×C4C22×D3112yes—
151C22×C4C22×D3112yes—
152C22×C4C22×D3112yes—
153C22×C4C22×D3111yescharacteristic
154C4⋊C4—22no—
155C4⋊C4—624no—
156C4⋊C4—624no—
157C4⋊C4—622no—
158C22⋊C4—628no—
159C22⋊C4—624no—
160C22⋊C4—624no—
161C22⋊C4—628no—
162C42—22no—
163C42—314no—
164C42—624no—
165C42—622no—
166C42—624no—
167C23—622no—
168C23—622no—
169C23—622no—
170C23—624no—
171C23—624no—
172C23C23×D3111yescharacteristic
173Q8—44no—
174D4—1244no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—312no—
183C2×C4—312no—
184C2×C4—314no—
185C2×C4—622no—
186C2×C4—622no—
187C2×C4—622no—
188C2×C4—622no—
189C2×C4—622no—
190C2×C4—624no—
191C2×C4#9249512yes—
192C2×C4D4×D3112yes—
193C2×C4#9249712yes—
194C2×C4#9249914yes—
195C22—1242no—
196C22—1242no—
197C22—1244no—
198C22—1244no—
199C22—1242no—
200C22C2×#9249511yescharacteristic
201C22C2×D4×D3111yescharacteristic
202C22C2×#9249711yescharacteristic
203C22C2×#9249912yes—
204C22#41×D3112yes—
205C4—24no—
206C4—42no—
207C4—42no—
208C4—622no—
209C4—624no—
210C4—622no—
211C4—1242no—
212C4#1175846214yes—
213C2—2482no—
214C2C2×#1175846211yescharacteristic
215C2#1452158512yes—
216C2#1452158612yes—
217C2#1452159811yescharacteristic
218C2#1452160011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#391C2
prime radicalC22×D124C22
hypercenter#391D31
hypocenterC31#2861
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#11758467C2semidirect
3. #14520956C2semidirect
6. #14520941C2semidirect
7. #14520938C2semidirect
8. C42×Dic31C2semidirect
9. C31×#391C2semidirect
107. #391D31semidirect
138. C31#2861semidirect

Maximal subgroups

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

subgroupindex
C2×#117584672
C2×#117585242
#145209562
#145209512
#145209502
#145209412
#145209382
C42×Dic312
C31×#3912
#286131