Finite Group Databasebeta

← #82176

Subgroups — #82176

Order 384.

512 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C22×C3×C4⋊C4C211yesmaximal, characteristic
2C2×#926C211yesmaximal, characteristic
3#5211C211yesmaximal, characteristic
4#5211C212yesmaximal
5#5210C212yesmaximal
6#4986C212yesmaximal, complemented
7#4981C212yesmaximal, complemented
8#4978C214yesmaximal, complemented
9#2484—31nomaximal, Sylow
10C23×C3×C4C2211yescharacteristic
11C23×C3×C4C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C2×C3×C4⋊C4—24no—
14C2×C3×C4⋊C4C2212yes—
15C2×C3×C4⋊C4C2212yes—
16C2×#217C2212yes—
17C2×#217C2212yes—
18#926—24no—
19C2×C4⋊Dic3C2211yescharacteristic
20C2×C4⋊Dic3C2211yescharacteristic
21C2×Dic3⋊C4C2214yes—
22C2×C4×Dic3C2212yes—
23C3⋊#95C2214yes—
24C3⋊#95C2212yes—
25C3⋊#95C2214yes—
26C3⋊#95C2214yes—
27C3⋊#95C2212yes—
28C2×#117—31no—
29C22×C4⋊C4D311yescharacteristic
30#400—32no—
31#398—32no—
32#387—34no—
33#385—32no—
34#381—31no—
35#381—32no—
36C24×C3C2311yescharacteristic
37C22×C3×C4—22no—
38C22×C3×C4—24no—
39C22×C3×C4—24no—
40C22×C3×C4—22no—
41C22×C3×C4—24no—
42C22×C3×C4D414yes—
43C22×C3×C4C2312yes—
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2312yes—
47C22×Dic3C2314yes—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C3×C4⋊C4—24no—
51C3×C4⋊C4—24no—
52#217—24no—
53#217—24no—
54C4⋊Dic3—22no—
55C4⋊Dic3—22no—
56Dic3⋊C4—28no—
57C4×Dic3—24no—
58C23×C4C2×D311yescharacteristic
59C23×C4C2×D311yescharacteristic
60C23×C4C2×D311yescharacteristic
61#117—64no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—31no—
64C2×C4⋊C4—31no—
65C2×C4⋊C4—34no—
66C2×C4⋊C4C2×D312yes—
67C2×C4⋊C4C2×D312yes—
68C2×C22⋊C4—32no—
69C2×C22⋊C4—32no—
70C2×C42—32no—
71#95—32no—
72#95—34no—
73#95—34no—
74#95—32no—
75#95—34no—
76C23×C3—22no—
77C23×C3—22no—
78C23×C3C2×D412yes—
79C23×C3#4114yes—
80C23×C3C2411yescharacteristic
81C2×C3×C4—28no—
82C2×C3×C4—28no—
83C2×C3×C4—22no—
84C2×C3×C4—22no—
85C2×C3×C4—22no—
86C2×C3×C4—22no—
87C2×C3×C4—22no—
88C2×C3×C4—24no—
89C2×C3×C4—44no—
90C2×C3×C4C2×D412yes—
91C2×C3×C4C2×D412yes—
92C2×C3×C4#4114yes—
93C2×Dic3—22no—
94C2×Dic3—22no—
95C2×Dic3—24no—
96C2×Dic3—24no—
97C2×Dic3—24no—
98C2×Dic3—24no—
99C2×Dic3—24no—
100C24C22×D311yescharacteristic
101C22×C4—24no—
102C22×C4—22no—
103C22×C4—24no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—34no—
107C22×C4—32no—
108C22×C4—32no—
109C22×C4#6714yes—
110C22×C4C22×D312yes—
111C22×C4C22×D312yes—
112C22×C4C22×D311yescharacteristic
113C22×C4C22×D311yescharacteristic
114C4⋊C4—24no—
115C4⋊C4—24no—
116C4⋊C4—68no—
117C4⋊C4—62no—
118C4⋊C4—62no—
119C22⋊C4—64no—
120C22⋊C4—64no—
121C42—64no—
122C22×C3—22no—
123C22×C3—22no—
124C22×C3—28no—
125C22×C3#12314yes—
126C22×C3C22×D411yescharacteristic
127C22×C3C2×#4112yes—
128C22×C3C2×#4112yes—
129C22×C32−1+412yes—
130C3×C4—22no—
131C3×C4—22no—
132C3×C4—24no—
133C3×C4—42no—
134Dic3—42no—
135Dic3—42no—
136Dic3—44no—
137C23—22no—
138C23—22no—
139C23#23714yes—
140C23C2×#6712yes—
141C23C23×D311yescharacteristic
142C2×C4—28no—
143C2×C4—28no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—24no—
149C2×C4—22no—
150C2×C4—44no—
151C2×C4—64no—
152C2×C4—64no—
153C2×C4—64no—
154C2×C4—64no—
155C2×C4—64no—
156C2×C4—62no—
157C2×C4—62no—
158C2×C4#23514yes—
159C2×C4C2×#6712yes—
160C2×C4C2×#6712yes—
161C2×C3—24no—
162C2×C3C2×#12311yescharacteristic
163C2×C3#52511yescharacteristic
164C2×C3#53611yescharacteristic
165C2×C3#55214yes—
166C22—28no—
167C22—22no—
168C22—22no—
169C22#93514yes—
170C22C2×#23512yes—
171C22C2×#23712yes—
172C22#100712yes—
173C22C22×#6711yescharacteristic
174C4—24no—
175C4—22no—
176C4—22no—
177C4—42no—
178C4—124no—
179C4—122no—
180C4—122no—
181C3#248411yescharacteristic, complemented, Sylow
182C2—24no—
183C2#551314yes—
184C2#579111yescharacteristic
185C2C2×#93511yescharacteristic
186C2#581611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×C4⋊C4C2
prime radicalC24×C3C23
hypercenterC22×C4⋊C4D3
hypocenterC3#2484
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #4986C2semidirect
7. #4981C2semidirect
8. #4978C2semidirect
181. C3#2484semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊C42
C2×#9262
#52112
#52102
#49862
#49812
#49782
#24843