Finite Group Databasebeta

← #82181

Subgroups — #82181

Order 384.

544 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 173 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×#924C211yesmaximal, characteristic
3#5210C212yesmaximal
4#5210C212yesmaximal
5C4×#217C211yesmaximal, characteristic
6#4981C212yesmaximal, complemented
7#4979C212yesmaximal, complemented
8#4976C214yesmaximal, complemented
9#2583—31nomaximal, Sylow
10C23×C3×C4C2211yescharacteristic
11C23×C3×C4C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C22×Dic6C2211yescharacteristic
14C2×C3×C4⋊C4—24no—
15C2×C3×C4⋊C4C2212yes—
16C2×C3×C4⋊C4C2212yes—
17C2×#217C2212yes—
18C2×#217C2212yes—
19C2×C4⋊Dic3C2211yescharacteristic
20#924—24no—
21C2×Dic3⋊C4C2214yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2214yes—
24C2×C4×Dic3C2212yes—
25C3⋊#95C2212yes—
26C3⋊#95C2214yes—
27C3⋊#95C2214yes—
28C2×C22⋊Q8—31no—
29C22×C4⋊C4D311yescharacteristic
30#398—32no—
31#395—32no—
32#389—34no—
33#385—32no—
34#385—32no—
35C4×C22⋊C4—31no—
36C24×C3C2311yescharacteristic
37C22×C3×C4—22no—
38C22×C3×C4—24no—
39C22×C3×C4—22no—
40C22×C3×C4—42no—
41C22×C3×C4—42no—
42C22×C3×C4D414yes—
43C22×C3×C4C2312yes—
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2312yes—
47C22×Dic3C2312yes—
48C22×Dic3C2312yes—
49C22×Dic3C2314yes—
50C2×Dic6—22no—
51C2×Dic6—42no—
52C3×C4⋊C4—24no—
53C3×C4⋊C4—24no—
54#217—24no—
55#217Q818yescomplemented
56C4⋊Dic3—22no—
57Dic3⋊C4—24no—
58Dic3⋊C4—28no—
59Dic3⋊C4—44no—
60C4×Dic3—42no—
61C22×Q8—31no—
62C23×C4C2×D311yescharacteristic
63C23×C4C2×D311yescharacteristic
64C23×C4C2×D311yescharacteristic
65C22⋊Q8—64no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4—31no—
69C2×C4⋊C4—34no—
70C2×C4⋊C4—34no—
71C2×C4⋊C4C2×D312yes—
72C2×C4⋊C4C2×D312yes—
73C2×C22⋊C4—32no—
74C2×C22⋊C4—32no—
75C2×C42—32no—
76#95—32no—
77#95—34no—
78#95—34no—
79C23×C3—22no—
80C23×C3—22no—
81C23×C3C2×D412yes—
82C23×C3C2×Q814yes—
83C23×C3C2411yescharacteristic
84C2×C3×C4—28no—
85C2×C3×C4—22no—
86C2×C3×C4—22no—
87C2×C3×C4—22no—
88C2×C3×C4—22no—
89C2×C3×C4—22no—
90C2×C3×C4—24no—
91C2×C3×C4—44no—
92C2×C3×C4—44no—
93C2×C3×C4C2×D412yes—
94C2×C3×C4C2×D412yes—
95C2×C3×C4#4114yes—
96C2×Dic3—24no—
97C2×Dic3—24no—
98C2×Dic3—22no—
99C2×Dic3—22no—
100C2×Dic3—24no—
101C2×Dic3—24no—
102C2×Dic3C2×Q818yes—
103Dic6—44no—
104C24C22×D311yescharacteristic
105C2×Q8—62no—
106C2×Q8—122no—
107C22×C4—22no—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—34no—
111C22×C4—32no—
112C22×C4—32no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4#6714yes—
116C22×C4C22×D312yes—
117C22×C4C22×D312yes—
118C22×C4C22×D311yescharacteristic
119C22×C4C22×D311yescharacteristic
120C4⋊C4—24no—
121C4⋊C4—24no—
122C4⋊C4—68no—
123C4⋊C4—64no—
124C4⋊C4—62no—
125C4⋊C4—124no—
126C22⋊C4—38no—
127C22⋊C4—64no—
128C42—122no—
129C22×C3—22no—
130C22×C3—22no—
131C22×C3—28no—
132C22×C3C4⋊Q814yes—
133C22×C3C22×D411yescharacteristic
134C22×C3C22×Q812yes—
135C22×C3C2×#4112yes—
136C22×C3H(5,𝔽2)12yes—
137C3×C4—24no—
138C3×C4—22no—
139C3×C4—22no—
140C3×C4—42no—
141Dic3—28no—
142Dic3—42no—
143Dic3—42no—
144C23—22no—
145C23—22no—
146C23D3×Q814yes—
147C23C2×#6712yes—
148C23C23×D311yescharacteristic
149Q8—124no—
150C2×C4—28no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—22no—
157C2×C4—38no—
158C2×C4—44no—
159C2×C4—44no—
160C2×C4—64no—
161C2×C4—64no—
162C2×C4—64no—
163C2×C4—64no—
164C2×C4—62no—
165C2×C4—62no—
166C2×C4#23514yes—
167C2×C4C2×#6712yes—
168C2×C4C2×#6712yes—
169C2×C3—24no—
170C2×C3#52511yescharacteristic
171C2×C3C2×C4⋊Q811yescharacteristic
172C2×C3#53411yescharacteristic
173C2×C3#55414yes—
174C22—28no—
175C22—22no—
176C22—22no—
177C22#94414yes—
178C22C2×#23512yes—
179C22#100412yes—
180C22C2×D3×Q812yes—
181C22C22×#6711yescharacteristic
182C4—24no—
183C4—22no—
184C4—22no—
185C4—42no—
186C4—68no—
187C4—122no—
188C4—122no—
189C3#258311yescharacteristic, complemented, Sylow
190C2—24no—
191C2#551114yes—
192C2#579111yescharacteristic
193C2#580411yescharacteristic
194C2C2×#94411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×C4⋊C4C2
prime radicalC24×C3C23
hypercenterC22×C4⋊C4D3
hypocenterC3#2583
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #4981C2semidirect
7. #4979C2semidirect
8. #4976C2semidirect
55. #217Q8semidirect
189. C3#2583semidirect

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×#9242
#52102
C4×#2172
#49812
#49792
#49762
#25833