Finite Group Databasebeta

← #82379

Subgroups — #82379

Order 384.

676 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 197 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×C3×C4×D4C211yesmaximal, characteristic
2C2×#924C214yesmaximal, complemented
3C2×#869C211yesmaximal, characteristic, complemented
4#5221C212yesmaximal
5#4975C212yesmaximal, complemented
6#4949C214yesmaximal, complemented
7C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
8#2542—31nomaximal, Sylow
9C22×C3×D4C2211yescharacteristic
10C23×C3×C4C2212yes—
11C22×Dic6C2212yes—
12C3×C4×D4—24no—
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C22⋊C4C2212yes—
15C2×C3×C42C2211yescharacteristic
16C2×#217C2218yes—
17C2×C4⋊Dic3C2212yes—
18C2×C4⋊Dic3C2212yes—
19C2×C4⋊Dic3C2212yes—
20#924—216no—
21C2×Dic3⋊C4C2218yes—
22C2×C4×Dic3C2212yes—
23#869—24no—
24C3⋊#95C2212yes—
25C2×C4⋊Q8—31no—
26C2×C22⋊Q8—34no—
27C2×C4×D4D311yescharacteristic
28#399—34no—
29#390—32no—
30#384—32no—
31C4×C4⋊C4—31no—
32C24×C3C2312yes—
33C2×C3×D4—22no—
34C2×C3×D4—22no—
35C2×C3×D4Q814yescomplemented
36C22×C3×C4—28no—
37C22×C3×C4—24no—
38C22×C3×C4C2312yes—
39C22×C3×C4C2311yescharacteristic
40C22×C3×C4C2311yescharacteristic
41C22×C3×C4C2311yescharacteristic
42C22×Dic3C2314yes—
43C22×Dic3C2314yes—
44C2×Dic6—24no—
45C2×Dic6—28no—
46C3×C4⋊C4—22no—
47C3×C22⋊C4—24no—
48C3×C42D414yes—
49#217—216no—
50C4⋊Dic3—24no—
51C4⋊Dic3—24no—
52C4⋊Dic3D418yescomplemented
53Dic3⋊C4—216no—
54C4×Dic3—24no—
55C22×Q8—32no—
56C22×D4C2×D311yescharacteristic
57C23×C4C2×D312yes—
58C4⋊Q8—64no—
59C22⋊Q8—616no—
60C4×D4—24no—
61C2×C4⋊C4—32no—
62C2×C4⋊C4—38no—
63C2×C4⋊C4—32no—
64C2×C4⋊C4—32no—
65C2×C4⋊C4C2×D311yescharacteristic
66C2×C22⋊C4—38no—
67C2×C22⋊C4C2×D312yes—
68C2×C42—32no—
69C2×C42C2×D311yescharacteristic
70#95—32no—
71C23×C3—24no—
72C23×C3—22no—
73C23×C3—22no—
74C23×C3—24no—
75C23×C3C2×Q814yes—
76C23×C3C2411yescharacteristic
77C3×D4—28no—
78C2×C3×C4—22no—
79C2×C3×C4—24no—
80C2×C3×C4—22no—
81C2×C3×C4—48no—
82C2×C3×C4C2×D414yes—
83C2×C3×C4C2×D412yes—
84C2×C3×C4C2×D412yes—
85C2×C3×C4C2×D412yes—
86C2×C3×C4C2×Q812yes—
87C2×C3×C4#4112yes—
88C2×Dic3—24no—
89C2×Dic3—24no—
90C2×Dic3—28no—
91C2×Dic3—24no—
92C2×Dic3C2×D418yes—
93Dic6—48no—
94C24C22×D312yes—
95C2×Q8—68no—
96C2×Q8—64no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4Dic614yescomplemented
100C22×C4—28no—
101C22×C4—24no—
102C22×C4—34no—
103C22×C4—34no—
104C22×C4C22×D312yes—
105C22×C4C22×D311yescharacteristic
106C22×C4C22×D311yescharacteristic
107C22×C4C22×D311yescharacteristic
108C4⋊C4—22no—
109C4⋊C4—38no—
110C4⋊C4—616no—
111C4⋊C4—64no—
112C4⋊C4—64no—
113C22⋊C4—24no—
114C22⋊C4—616no—
115C42—64no—
116C42#6714yes—
117C22×C3—24no—
118C22×C3—28no—
119C22×C3—42no—
120C22×C3—42no—
121C22×C3—44no—
122C22×C3C22×D412yes—
123C22×C3C22×D411yescharacteristic
124C22×C3C22×Q811yescharacteristic
125C22×C3C2×#4111yescharacteristic
126C22×C32−1+412yes—
127C3×C4—22no—
128C3×C4—42no—
129C3×C4C22⋊Q814yescomplemented
130C3×C4#12714yes—
131Dic3—28no—
132Dic3—44no—
133C23—22no—
134C23—24no—
135C23—24no—
136C23—22no—
137C23C2×Dic614yes—
138C23C23×D311yescharacteristic
139Q8—128no—
140D4—28no—
141C2×C4—22no—
142C2×C4—24no—
143C2×C4—22no—
144C2×C4—38no—
145C2×C4—48no—
146C2×C4—64no—
147C2×C4—64no—
148C2×C4—68no—
149C2×C4—64no—
150C2×C4C2×Dic612yes—
151C2×C4#23512yes—
152C2×C4D3×D414yes—
153C2×C4C2×#6712yes—
154C2×C4C2×#6712yes—
155C2×C4C2×#6712yes—
156C2×C3—44no—
157C2×C3C2×C22⋊Q811yescharacteristic
158C2×C3C2×#12711yescharacteristic
159C2×C3#53611yescharacteristic
160C2×C3#54712yes—
161C2×C3D4×Q812yes—
162C22—24no—
163C22—28no—
164C22—42no—
165C22—44no—
166C22—42no—
167C22C22×Dic611yescharacteristic
168C22C2×#23511yescharacteristic
169C22C2×D3×D412yes—
170C22#101012yes—
171C22C22×#6711yescharacteristic
172C4—22no—
173C4—42no—
174C4—68no—
175C4—124no—
176C4#92414yescomplemented
177C4#94014yes—
178C3#254211yescharacteristic, complemented, Sylow
179C2—44no—
180C2D4×Dic612yes—
181C2#555112yes—
182C2C2×#92411yescharacteristic
183C2C2×#94011yescharacteristic
184C2#582511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×D4C2
prime radicalC22×C3×D4C22
hypercenterC2×C4×D4D3
hypocenterC3#2542
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#924C2semidirect
3. C2×#869C2semidirect
5. #4975C2semidirect
6. #4949C2semidirect
7. C4×C4⋊Dic3C2semidirect
35. C2×C3×D4Q8semidirect
52. C4⋊Dic3D4semidirect
99. C2×D4Dic6semidirect
129. C3×C4C22⋊Q8semidirect
176. C4#924semidirect
178. C3#2542semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C2×#9242
C2×#8692
#52212
#49752
#49492
C4×C4⋊Dic32
#25423