Finite Group Databasebeta

← #68365

Subgroups — #68365

Order 384.

212 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 61 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#5688C211yesmaximal, characteristic
2#4705C211yesmaximal, characteristic
3D3×#97C211yesmaximal, characteristic
4C3×#330C211yesmaximal, characteristic, complemented
5#4543C211yesmaximal, characteristic, complemented
6#4465C211yesmaximal, characteristic, complemented
7#4465C211yesmaximal, characteristic, complemented
8#1531—31nomaximal, Sylow
9C3×#125C2211yescharacteristic
10C2×D3×C8—21no—
11#896—21no—
12#896—21no—
13#892C411yescharacteristic
14#892C411yescharacteristic
15#888C411yescharacteristic
16#888C411yescharacteristic
17#887—21no—
18#887—21no—
19C42×D3C2211yescharacteristic
20#870C2211yescharacteristic
21C3×#97C2211yescharacteristic
22C3×#97C2211yescharacteristic
23(C2×D3)⋊C8—21no—
24#815—21no—
25Dic3⋊C8—21no—
26#803C2211yescharacteristic
27#803C2211yescharacteristic
28#533—31no—
29#432—31no—
30C2×#97—31no—
31#330—31no—
32#330—31no—
33#330—31no—
34#330D311yescharacteristic, complemented
35C2×C4×D3C2×C411yescharacteristic
36C2×C4×D3D411yescharacteristic
37C2×C4×D3D411yescharacteristic
38C2×Dic6—21no—
39C2×Dic6—21no—
40C2×C3×C8—21no—
41C2×C3×C8—21no—
42C3×C4⋊C4—21no—
43C3×C4⋊C4—21no—
44C3×C4⋊C4C2×C411yescharacteristic
45C3×C4⋊C4C2×C411yescharacteristic
46C3×C42C2311yescharacteristic
47(C2×D3)⋊C4—21no—
48(C2×D3)⋊C4—21no—
49C4⋊Dic3C2×C411yescharacteristic
50C4⋊Dic3C2×C411yescharacteristic
51Dic3⋊C4—21no—
52Dic3⋊C4—21no—
53Dic3⋊C4—21no—
54Dic3⋊C4—21no—
55C4×Dic3—21no—
56C4×Dic3—21no—
57C4×Dic3C2×C411yescharacteristic
58C4×Dic3D411yescharacteristic
59C4×Dic3D411yescharacteristic
60C2×C3⋊C8—21no—
61C2×C3⋊C8—21no—
62D3×C8—24no—
63C22×C8—61no—
64C4⋊Q8—31no—
65C4⋊Q8—31no—
66#125—31no—
67#125C2×D311yescharacteristic
68C22⋊Q8—61no—
69C22⋊Q8—61no—
70C4×Q8—61no—
71C4×Q8—61no—
72#117—31no—
73#117—31no—
74C2×C42—31no—
75C4⋊C8—61no—
76C22⋊C8—61no—
77#97—31no—
78#97—31no—
79#97—61no—
80#97C2×D311yescharacteristic
81#97C2×D311yescharacteristic
82C22×D3C22⋊C411yescharacteristic
83C2×C3×C4—21no—
84C2×C3×C4—21no—
85C2×C3×C4C22×C411yescharacteristic
86C2×C3×C4C2×D411yescharacteristic
87C2×C3×C4C2×D411yescharacteristic
88C2×Dic3—21no—
89C2×Dic3—21no—
90C2×Dic3C22⋊C411yescharacteristic
91C2×Dic3C22⋊C411yescharacteristic
92C2×Dic3C22⋊C411yescharacteristic
93C4×D3—22no—
94C4×D3—22no—
95C4×D3—41no—
96Dic6—42no—
97Dic6—42no—
98C3×C8—22no—
99C3×C8—41no—
100C3⋊C8—22no—
101C3⋊C8—41no—
102C2×Q8—61no—
103C2×Q8—61no—
104C22×C4—31no—
105C22×C4—31no—
106C22×C4—31no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—64no—
110C2×C8—61no—
111C2×C8—61no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—31no—
115C4⋊C4—31no—
116C4⋊C4—61no—
117C4⋊C4—61no—
118C4⋊C4—61no—
119C4⋊C4—61no—
120C4⋊C4C4×D311yescharacteristic
121C4⋊C4C4×D311yescharacteristic
122C22⋊C4—61no—
123C22⋊C4—61no—
124C42—31no—
125C42—31no—
126C42—31no—
127C42—61no—
128C42—61no—
129C42C22×D311yescharacteristic
130C22×C3C2×C22⋊C411yescharacteristic
131C2×D3—21no—
132C2×D3—21no—
133C2×D3#10112yes—
134C3×C4—21no—
135C3×C4—21no—
136C3×C4—21no—
137C3×C4—41no—
138C3×C4—41no—
139Dic3—21no—
140Dic3—21no—
141Dic3—41no—
142Dic3—41no—
143Dic3#10412yes—
144C23—31no—
145Q8—122no—
146Q8—122no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—31no—
150C2×C4—31no—
151C2×C4—31no—
152C2×C4—62no—
153C2×C4—62no—
154C2×C4—61no—
155C2×C4—61no—
156C2×C4—121no—
157C2×C4C2×C4×D311yescharacteristic
158C2×C4D3×D411yescharacteristic
159C2×C4D3×D411yescharacteristic
160C8—22no—
161C8—41no—
162C8—62no—
163C8—121no—
164C2×C3C2×#10111yescharacteristic
165C2×C3C2×#10411yescharacteristic
166C2×C3#42111yescharacteristic
167D3—22no—
168C22—32no—
169C22—61no—
170C22—61no—
171C22D3×C22⋊C411yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—32no—
176C4—41no—
177C4—41no—
178C4—61no—
179C4—61no—
180C4—121no—
181C4—121no—
182C3#153111yescharacteristic, complemented, Sylow
183C2—62no—
184C2D3×#10111yescharacteristic
185C2D3×#10411yescharacteristic
186C2#482111yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC42C22×D3
FittingC3×#330C2
socleC22×C3C2×C22⋊C4
prime radicalC22×D3C22⋊C4
cosocleC3×C42C23
hypercenter#330D3
hypocenterC3#1531
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
4. C3×#330C2semidirect
5. #4543C2semidirect
6. #4465C2semidirect
7. #4465C2semidirect
34. #330D3semidirect
182. C3#1531semidirect

Maximal subgroups

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

subgroupindex
#56882
#47052
D3×#972
C3×#3302
#45432
#44652
#15313