Finite Group Databasebeta

← #69152

Subgroups — #69152

Order 384.

270 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 55 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#5715C211yesmaximal, characteristic
2#4740C212yesmaximal, complemented
3C3×#353C211yesmaximal, characteristic, complemented
4#4545C211yesmaximal, characteristic, complemented
5#4474C212yesmaximal, complemented
6#2128—31nomaximal, Sylow
7C2×#237C2211yescharacteristic
8C2×D3×D4C411yescharacteristic, complemented
9C3×#127C2211yescharacteristic
10#940C2211yescharacteristic
11#939—22no—
12#937—22no—
13#936C411yescharacteristic, complemented
14#877—22no—
15#875C411yescharacteristic, complemented
16#872C411yescharacteristic, complemented
17C3×#99C2212yescomplemented
18#834C2212yescomplemented
19#806—22no—
20#535—31no—
21#410—32no—
22#353—31no—
23#353—32no—
24#353D311yescharacteristic, complemented
25C23×D3—21no—
26C2×C3×D4—22no—
27C2×C3×D4C2×C411yescharacteristic
28C2×C3×D4C2311yescharacteristic
29C2×#67—22no—
30C2×#67—21no—
31C2×#67D412yes—
32C22×Dic3D412yes—
33#237—42no—
34D3×D4—42no—
35C2×D12C2×C411yescharacteristic
36C2×C4×D3C2×C411yescharacteristic
37C2×Dic6C2×C411yescharacteristic
38C3×C22⋊C4—22no—
39C3×C42C2×C411yescharacteristic
40#217—22no—
41#217—22no—
42(C2×D3)⋊C4—21no—
43(C2×D3)⋊C4—22no—
44Dic3⋊C4—21no—
45Dic3⋊C4—42no—
46C4×Dic3—42no—
47C4×Dic3C2×C411yescharacteristic
48C2×#41—31no—
49C22×D4—31no—
50#127—31no—
51#127C2×D311yescharacteristic
52#124—31no—
53#124—31no—
54#121—62no—
55#120—62no—
56#117—31no—
57#117—62no—
58#99—32no—
59#99—62no—
60#99C2×D312yescomplemented
61C23×C3—21no—
62C23×C3C2×D412yes—
63C22×D3—21no—
64C22×D3—42no—
65C22×D3—82no—
66C22×D3C22⋊C411yescharacteristic
67C3×D4—41no—
68C3×D4—41no—
69C3×D4—82no—
70C2×C3×C4—21no—
71C2×C3×C4—42no—
72C2×C3×C4C22×C411yescharacteristic
73#67—42no—
74#67—42no—
75#67—82no—
76C2×Dic3—22no—
77C2×Dic3—22no—
78C2×Dic3—21no—
79C2×Dic3—42no—
80C2×Dic3C22⋊C412yes—
81C2×Dic3C22⋊C411yescharacteristic
82D12—41no—
83C4×D3—22no—
84Dic6—41no—
85C24—61no—
86#41—122no—
87C2×Q8—31no—
88C2×D4—22no—
89C2×D4—32no—
90C2×D4—31no—
91C2×D4—61no—
92C2×D4—62no—
93C2×D4—122no—
94C2×D4C4×D311yescharacteristic
95C2×D4C22×D311yescharacteristic
96C22×C4—32no—
97C22×C4—31no—
98C4⋊C4—61no—
99C4⋊C4—122no—
100C22⋊C4—22no—
101C22⋊C4—62no—
102C22⋊C4—61no—
103C22⋊C4—62no—
104C22⋊C4—62no—
105C42—31no—
106C42—122no—
107C42C4×D311yescharacteristic
108C22×C3—22no—
109C22×C3—42no—
110C22×C3—42no—
111C22×C3—41no—
112C22×C3C2×C22⋊C411yescharacteristic
113C2×D3—21no—
114C2×D3—41no—
115C2×D3—42no—
116C2×D3—41no—
117C2×D3—84no—
118C3×C4—21no—
119C3×C4—41no—
120C3×C4—82no—
121Dic3—22no—
122Dic3—41no—
123Dic3—82no—
124Dic3#9912yes—
125C23—21no—
126C23—31no—
127C23—61no—
128C23—122no—
129C23—242no—
130C23D3×D412yes—
131Q8—121no—
132D4—41no—
133D4—41no—
134D4—82no—
135D4—121no—
136D4—122no—
137D4—122no—
138D4—242no—
139C2×C4—21no—
140C2×C4—31no—
141C2×C4—32no—
142C2×C4—42no—
143C2×C4—62no—
144C2×C4—62no—
145C2×C4—62no—
146C2×C4—61no—
147C2×C4—122no—
148C2×C4C2×C4×D311yescharacteristic
149C2×C3—21no—
150C2×C3—42no—
151C2×C3—81no—
152C2×C3C2×#9911yescharacteristic
153D3—41no—
154D3—81no—
155C22—22no—
156C22—42no—
157C22—42no—
158C22—41no—
159C22—61no—
160C22—121no—
161C22—122no—
162C22—121no—
163C22—244no—
164C22D3×C22⋊C411yescharacteristic
165C4—21no—
166C4—32no—
167C4—41no—
168C4—62no—
169C4—82no—
170C4—121no—
171C4—242no—
172C3#212811yescharacteristic, complemented, Sylow
173C2—21no—
174C2—42no—
175C2—81no—
176C2—121no—
177C2—241no—
178C2D3×#9911yescharacteristic
subgroupG/H
centerC2D3×#99
commutatorC2×C3×C4C22×C4
FrattiniC2×D4C22×D3
FittingC3×#353C2
socleC2×C3C2×#99
prime radical#5715C2
cosocleC2×C3×D4C23
hypercenter#353D3
hypocenterC3#2128
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #4740C2semidirect
3. C3×#353C2semidirect
4. #4545C2semidirect
5. #4474C2semidirect
8. C2×D3×D4C4semidirect
13. #936C4semidirect
15. #875C4semidirect
16. #872C4semidirect
17. C3×#99C22semidirect
18. #834C22semidirect
24. #353D3semidirect
60. #99C2×D3semidirect
172. C3#2128semidirect

Maximal subgroups

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

subgroupindex
#57152
#47402
C3×#3532
#45452
#44742
#21283