Finite Group Databasebeta

← #68128

Subgroups — #68128

Order 384.

204 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 63 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×#889C211yesmaximal, characteristic
2#4719C211yesmaximal, characteristic
3#4718C211yesmaximal, characteristic
4C3×#324C211yesmaximal, characteristic
5#4528C211yesmaximal, characteristic
6#4469C211yesmaximal, characteristic
7#4469C211yesmaximal, characteristic
8#1470—31nomaximal, Sylow
9C2×C3×C4⋊C4C2211yescharacteristic
10C2×C4⋊Dic3C411yescharacteristic
11C2×Dic3⋊C4C411yescharacteristic
12C2×Dic3⋊C4C411yescharacteristic
13C2×Dic3⋊C4C411yescharacteristic
14C2×Dic3⋊C4C2211yescharacteristic
15C2×C4×Dic3C2211yescharacteristic
16#889—21no—
17#889—21no—
18#889—21no—
19#889—21no—
20C3×C22⋊C8C2211yescharacteristic
21C3×C22⋊C8C2211yescharacteristic
22#830C2211yescharacteristic
23#830C2211yescharacteristic
24#815—21no—
25Dic3⋊C8—21no—
26Dic3⋊C8—21no—
27C8×Dic3—21no—
28C2×#125—31no—
29#432—31no—
30#431—31no—
31#324—31no—
32#324—31no—
33#324—31no—
34#324D311yescharacteristic
35C22×C3×C4C2×C411yescharacteristic
36C22×C3×C4C2×C411yescharacteristic
37C22×C3×C4C2311yescharacteristic
38C22×Dic3C2×C411yescharacteristic
39C22×Dic3C2×C411yescharacteristic
40C22×Dic3C2×C411yescharacteristic
41C22×Dic3C2×C411yescharacteristic
42C2×C3×C8—21no—
43C2×C3×C8—21no—
44C3×C4⋊C4—21no—
45C3×C4⋊C4—21no—
46C4⋊Dic3—21no—
47C4⋊Dic3—21no—
48Dic3⋊C4—21no—
49Dic3⋊C4—21no—
50Dic3⋊C4—21no—
51Dic3⋊C4—21no—
52Dic3⋊C4—21no—
53Dic3⋊C4—21no—
54Dic3⋊C4—21no—
55Dic3⋊C4—21no—
56C4×Dic3D411yescharacteristic
57C4×Dic3D411yescharacteristic
58C4×Dic3D411yescharacteristic
59C4×Dic3D411yescharacteristic
60C2×C3⋊C8—21no—
61C2×C3⋊C8—21no—
62#125—61no—
63#125—61no—
64#125—61no—
65#125—61no—
66C2×C4⋊C4—31no—
67C2×C4⋊C4—31no—
68C2×C4⋊C4—31no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4—31no—
71C2×C4⋊C4C2×D311yescharacteristic
72C2×C42—31no—
73C4⋊C8—61no—
74C4⋊C8—61no—
75C22⋊C8—31no—
76C22⋊C8—31no—
77C22⋊C8C2×D311yescharacteristic
78C22⋊C8C2×D311yescharacteristic
79#97—61no—
80C4×C8—61no—
81C23×C3C22×C411yescharacteristic
82C2×C3×C4—21no—
83C2×C3×C4—21no—
84C2×C3×C4—21no—
85C2×C3×C4—21no—
86C2×C3×C4—41no—
87C2×C3×C4—41no—
88C2×C3×C4C2×D411yescharacteristic
89C2×C3×C4C2×D411yescharacteristic
90C2×Dic3—21no—
91C2×Dic3—21no—
92C2×Dic3—22no—
93C2×Dic3—21no—
94C2×Dic3—21no—
95C2×Dic3—21no—
96C2×Dic3—21no—
97C2×Dic3—41no—
98C2×Dic3C22⋊C411yescharacteristic
99C2×Dic3C22⋊C411yescharacteristic
100C2×Dic3C22⋊C411yescharacteristic
101C2×Dic3C22⋊C411yescharacteristic
102C3×C8—41no—
103C3×C8—41no—
104C3⋊C8—41no—
105C3⋊C8—41no—
106C22×C4—31no—
107C22×C4—31no—
108C22×C4—31no—
109C22×C4—31no—
110C22×C4C4×D311yescharacteristic
111C22×C4C4×D311yescharacteristic
112C22×C4C22×D311yescharacteristic
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—61no—
116C2×C8—61no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—61no—
120C4⋊C4—61no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C4⋊C4—61no—
127C4⋊C4—61no—
128C4⋊C4—61no—
129C42—31no—
130C42—31no—
131C42—31no—
132C42—31no—
133C22×C3—21no—
134C22×C3—21no—
135C22×C3C2×C22⋊C411yescharacteristic
136C22×C3#13111yescharacteristic
137C22×C3#13111yescharacteristic
138C3×C4—21no—
139C3×C4—21no—
140C3×C4—41no—
141C3×C4—41no—
142Dic3—21no—
143Dic3—21no—
144Dic3—21no—
145Dic3—41no—
146Dic3—41no—
147Dic3#10112yes—
148C23C2×C4×D311yescharacteristic
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—31no—
154C2×C4—31no—
155C2×C4—31no—
156C2×C4—31no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—62no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—61no—
163C2×C4—61no—
164C2×C4—61no—
165C2×C4—61no—
166C2×C4—121no—
167C2×C4D3×D411yescharacteristic
168C2×C4D3×D411yescharacteristic
169C8—41no—
170C8—41no—
171C8—121no—
172C8—121no—
173C2×C3—21no—
174C2×C3—21no—
175C2×C3#40811yescharacteristic
176C2×C3#41011yescharacteristic
177C2×C3C2×#10111yescharacteristic
178C22—21no—
179C22—21no—
180C22D3×C22⋊C411yescharacteristic
181C22#90111yescharacteristic
182C22#90711yescharacteristic
183C4—21no—
184C4—21no—
185C4—32no—
186C4—41no—
187C4—41no—
188C4—61no—
189C4—61no—
190C4—61no—
191C4—121no—
192C4—121no—
193C3#147011yescharacteristic, complemented, Sylow
194C2—21no—
195C2—21no—
196C2#472711yescharacteristic
197C2#474011yescharacteristic
198C2D3×#10111yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutator, prime radicalC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC3×#324C2
socleC22×C3C2×C22⋊C4
cosocleC22×C3×C4C23
hypercenter#324D3
hypocenterC3#1470
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
193. C3#1470semidirect

Maximal subgroups

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

subgroupindex
C2×#8892
#47192
#47182
C3×#3242
#45282
#44692
#14703