Finite Group Databasebeta

← #16856099

Subgroups — #16856099

Order 3968.

400 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#14521602C211yesmaximal, characteristic, complemented
2C2×#11758524C211yesmaximal, characteristic
3#14521410C214yesmaximal, complemented
4#14521405C214yesmaximal, complemented
5#14521058C211yesmaximal, characteristic, complemented
6#14521055C211yesmaximal, characteristic
7#14521053C211yesmaximal, characteristic
8M16×Dic31C211yesmaximal, characteristic, complemented
9C31×#419C211yesmaximal, characteristic, complemented
10C22×Q8×C31C2211yescharacteristic
11C22×D124C2211yescharacteristic
12C2×M16×C31C2211yescharacteristic
13C31×#117C2211yescharacteristic
14#11758524—22no—
15#11758524C2212yes—
16#11758524C2212yes—
17C2×(C2×D31)⋊C4—22no—
18#11758503C2211yescharacteristic
19C2×C4×Dic31C2211yescharacteristic
20C2×#92474C2211yescharacteristic
21#11758467C2212yescomplemented
22#11758465C2212yescomplemented
23#11758455—22no—
24C4×Dic62—22no—
25C31×#103C2214yescomplemented
26#11758423C2214yescomplemented
27#11758409C2214yes—
28#11758403C2212yescomplemented
29C8×Dic31C2212yescomplemented
30#11758397C2214yes—
31C23×D31—21no—
32C2×Q8×C31—21no—
33C2×Q8×C31—22no—
34C2×Q8×C31—22no—
35C2×Q8×C31C2312yes—
36C22×C4×C31—21no—
37C22×C4×C31C2311yescharacteristic
38C22×Dic31D412yes—
39C2×D124—22no—
40C2×D124—22no—
41C2×D124—21no—
42C2×D124C2312yes—
43C2×Dic62—22no—
44M16×C31—22no—
45C2×C8×C31C2312yes—
46C4⋊C4×C31C2312yes—
47C22⋊C4×C31—21no—
48C42×C31—21no—
49C22⋊Dic31—21no—
50(C2×D31)⋊C4—24no—
51(C2×D31)⋊C4—24no—
52C4⋊Dic31—22no—
53C4⋊Dic31C2312yes—
54Dic31⋊C4—22no—
55Dic31⋊C4—22no—
56C4×Dic31—21no—
57C4×Dic31D412yes—
58C4×Dic31C2311yescharacteristic
59C4×Dic31C2311yescharacteristic
60#92474—22no—
61C2×C31⋊C8C2312yes—
62C23×C31C2×D411yescharacteristic
63C22×D31—22no—
64C22×D31—44no—
65C22×D31—42no—
66Q8×C31—22no—
67Q8×C31—22no—
68Q8×C31—42no—
69C2×C4×C31—22no—
70C2×C4×C31—22no—
71C2×C4×C31—42no—
72C2×C4×C31C2×D411yescharacteristic
73C2×C4×C31#4112yes—
74C2×C4×C31#4112yes—
75C2×C4×C31C2411yescharacteristic
76C2×Dic31—22no—
77C2×Dic31—42no—
78C2×Dic31C2×D412yes—
79C2×Dic31C2×D412yes—
80D124—22no—
81D124—22no—
82D124—42no—
83Dic62—44no—
84C8×C31—22no—
85C31⋊C8—22no—
86#3140—311nomaximal, Sylow
87C22×C31—22no—
88C22×C31#12412yes—
89C22×C31C22×D411yescharacteristic
90C2×D31—42no—
91C2×D31—44no—
92C2×D31—84no—
93C4×C31—42no—
94C4×C31—42no—
95C4×C31#12412yes—
96C4×C31C2×#4111yescharacteristic
97C4×C31C2×#4111yescharacteristic
98Dic31—22no—
99Dic31—22no—
100Dic31—42no—
101#533—311no—
102C2×#124—311no—
103#489—314no—
104#488—314no—
105#419—311no—
106#419D3111yescharacteristic, complemented
107#418—311no—
108#418—311no—
109C4×M16—311no—
110C2×C31—22no—
111C2×C31C2×#12411yescharacteristic
112C2×C31#57512yes—
113D31—82no—
114C22×Q8C2×D3111yescharacteristic
115C22×D4—311no—
116C2×M16—311no—
117C2×M16C2×D3111yescharacteristic
118C4⋊Q8—312no—
119#125—312no—
120#124—312no—
121#124—312no—
122#124—622no—
123C22⋊Q8—622no—
124C4×Q8—622no—
125#117—311no—
126#117C2×D3111yescharacteristic
127C2×C22⋊C4—622no—
128C2×C42—311no—
129#103—314no—
130#103C2×D3114yescomplemented
131#102—314no—
132#102—314no—
133#97—312no—
134C4×C8—312no—
135C31#314011yescharacteristic, complemented, Sylow
136C24—621no—
137C2×Q8—21no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—622no—
141C2×Q8C22×D3112yes—
142C2×D4—312no—
143C2×D4—622no—
144C2×D4—621no—
145C2×D4—622no—
146C22×C4—21no—
147C22×C4—312no—
148C22×C4C22×D3111yescharacteristic
149M16—22no—
150M16—622no—
151C2×C8—312no—
152C2×C8C22×D3112yes—
153C4⋊C4—312no—
154C4⋊C4—622no—
155C4⋊C4—622no—
156C4⋊C4—622no—
157C4⋊C4C22×D3112yes—
158C22⋊C4—21no—
159C22⋊C4—624no—
160C22⋊C4—624no—
161C22⋊C4—621no—
162C42—21no—
163C42—312no—
164C42—311no—
165C42—311no—
166C42—621no—
167C23—622no—
168C23—1244no—
169C23—1242no—
170C23D4×D3111yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—42no—
174Q8—1244no—
175D4—622no—
176D4—622no—
177D4—1242no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—312no—
182C2×C4—312no—
183C2×C4—622no—
184C2×C4—1242no—
185C2×C4#9249512yes—
186C2×C4D4×D3111yescharacteristic
187C2×C4#9249712yes—
188C2×C4C23×D3111yescharacteristic
189C8—22no—
190C8—622no—
191C22—22no—
192C22—1242no—
193C22—1244no—
194C22—2484no—
195C22#1175846212yes—
196C22C2×D4×D3111yescharacteristic
197C4—42no—
198C4—42no—
199C4—622no—
200C4—622no—
201C4—1242no—
202C4#1175846212yes—
203C4C2×#9249511yescharacteristic
204C4C2×#9249711yescharacteristic
205C2—22no—
206C2—2482no—
207C2C2×#1175846211yescharacteristic
208C2#1452184812yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#419C2
socleC22×C31C22×D4
prime radicalC22×D124C22
hypercenter#419D31
hypocenterC31#3140
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. #14521602C2semidirect
3. #14521410C2semidirect
4. #14521405C2semidirect
5. #14521058C2semidirect
8. M16×Dic31C2semidirect
9. C31×#419C2semidirect
21. #11758467C22semidirect
22. #11758465C22semidirect
25. C31×#103C22semidirect
26. #11758423C22semidirect
28. #11758403C22semidirect
29. C8×Dic31C22semidirect
106. #419D31semidirect
130. #103C2×D31semidirect
135. C31#3140semidirect

Maximal subgroups

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

subgroupindex
#145216022
C2×#117585242
#145214102
#145214052
#145210582
#145210552
#145210532
M16×Dic312
C31×#4192
#314031