Finite Group Databasebeta

← #82788

Subgroups — #82788

Order 384.

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

normal subgroups only

This group has 508 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
1C2×C3×C22⋊Q8C211yesmaximal, characteristic
2C22×Dic3⋊C4C211yesmaximal, characteristic
3#5230C212yesmaximal, complemented
4#5212C211yesmaximal, characteristic
5#4985C212yesmaximal, complemented
6#4982C214yesmaximal, complemented
7#4946C212yesmaximal
8#4943C212yesmaximal
9#2627—31nomaximal, Sylow
10C22×C3×Q8C2211yescharacteristic
11C23×C3×C4C2211yescharacteristic
12C23×Dic3C2211yescharacteristic
13C23×Dic3C2211yescharacteristic
14C3×C22⋊Q8—24no—
15C2×C3×C4⋊C4C2212yes—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2212yes—
18C2×#217C2212yes—
19C2×C4⋊Dic3C2212yes—
20C2×Dic3⋊C4—24no—
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2214yes—
24C3⋊#95C2214yes—
25C3⋊#95C2214yes—
26C3⋊#95C2214yes—
27C3⋊#95C2212yes—
28C2×C22⋊Q8D311yescharacteristic
29C22×C4⋊C4—31no—
30#400—32no—
31#398—34no—
32#395—32no—
33#385—32no—
34#381—32no—
35#380—31no—
36C24×C3C2311yescharacteristic
37C2×C3×Q8—22no—
38C2×C3×Q8—42no—
39C22×C3×C4—42no—
40C22×C3×C4D414yes—
41C22×C3×C4C2312yes—
42C22×C3×C4C2312yes—
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2311yescharacteristic
45C22×Dic3—22no—
46C22×Dic3—22no—
47C22×Dic3—22no—
48C22×Dic3—42no—
49C22×Dic3Q814yescomplemented
50C22×Dic3C2314yes—
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C3×C4⋊C4—22no—
54C3×C4⋊C4—24no—
55C3×C22⋊C4—24no—
56#217—24no—
57C4⋊Dic3—42no—
58Dic3⋊C4—24no—
59Dic3⋊C4—24no—
60Dic3⋊C4—44no—
61C22×Q8C2×D311yescharacteristic
62C23×C4—31no—
63C23×C4—31no—
64C23×C4C2×D311yescharacteristic
65C22⋊Q8—24no—
66C2×C4⋊C4—32no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—34no—
70C2×C4⋊C4—64no—
71C2×C4⋊C4C2×D312yes—
72C2×C4⋊C4C2×D311yescharacteristic
73C2×C22⋊C4—32no—
74C2×C22⋊C4C2×D312yes—
75#95—32no—
76#95—34no—
77#95—34no—
78#95—34no—
79C23×C3—22no—
80C23×C3—22no—
81C23×C3C2×D412yes—
82C23×C3C2×Q812yes—
83C23×C3#4112yes—
84C23×C3C2411yescharacteristic
85C3×Q8—44no—
86C2×C3×C4—22no—
87C2×C3×C4—22no—
88C2×C3×C4—22no—
89C2×C3×C4—22no—
90C2×C3×C4—24no—
91C2×C3×C4—24no—
92C2×C3×C4—44no—
93C2×C3×C4C2×D412yes—
94C2×C3×C4C2×D412yes—
95C2×Dic3—28no—
96C2×Dic3—22no—
97C2×Dic3—22no—
98C2×Dic3—22no—
99C2×Dic3—24no—
100C2×Dic3—24no—
101C2×Dic3—24no—
102C2×Dic3—24no—
103C2×Dic3—44no—
104C2×Dic3C2×Q814yes—
105C24C22×D311yescharacteristic
106C2×Q8—22no—
107C2×Q8—42no—
108C22×C4—34no—
109C22×C4—32no—
110C22×C4—34no—
111C22×C4—32no—
112C22×C4—42no—
113C22×C4—62no—
114C22×C4—62no—
115C22×C4—62no—
116C22×C4—122no—
117C22×C4#6714yes—
118C22×C4C22×D312yes—
119C22×C4C22×D312yes—
120C22×C4C22×D311yescharacteristic
121C22×C4C22×D311yescharacteristic
122C4⋊C4—22no—
123C4⋊C4—24no—
124C4⋊C4—64no—
125C4⋊C4—64no—
126C4⋊C4—124no—
127C4⋊C4—122no—
128C22⋊C4—24no—
129C22⋊C4—64no—
130C22×C3—22no—
131C22×C3—22no—
132C22×C3—24no—
133C22×C3—24no—
134C22×C3C22⋊Q814yes—
135C22×C3C22×D411yescharacteristic
136C22×C3C22×Q811yescharacteristic
137C22×C3C2×#4111yescharacteristic
138C22×C3H(5,𝔽2)12yes—
139C22×C32−1+412yes—
140C3×C4—22no—
141C3×C4—22no—
142C3×C4—42no—
143C3×C4—42no—
144Dic3—24no—
145Dic3—42no—
146Dic3—44no—
147C23—22no—
148C23—22no—
149C23D3×Q812yes—
150C23#23912yes—
151C23C2×#6712yes—
152C23C23×D311yescharacteristic
153Q8—44no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—34no—
161C2×C4—44no—
162C2×C4—68no—
163C2×C4—62no—
164C2×C4—62no—
165C2×C4—64no—
166C2×C4—64no—
167C2×C4—64no—
168C2×C4—64no—
169C2×C4—62no—
170C2×C4—124no—
171C2×C4C2×#6712yes—
172C2×C4C2×#6712yes—
173C2×C3—24no—
174C2×C3C2×C22⋊Q811yescharacteristic
175C2×C3#53411yescharacteristic
176C2×C3#53611yescharacteristic
177C2×C3#53912yes—
178C2×C3#54412yes—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—24no—
183C22#94614yes—
184C22#100412yes—
185C22C2×D3×Q811yescharacteristic
186C22C2×#23911yescharacteristic
187C22#101012yes—
188C22C22×#6711yescharacteristic
189C4—22no—
190C4—22no—
191C4—42no—
192C4—42no—
193C4—64no—
194C4—124no—
195C4—122no—
196C3#262711yescharacteristic, complemented, Sylow
197C2—24no—
198C2#562312yes—
199C2#564512yes—
200C2#580411yescharacteristic
201C2C2×#94611yescharacteristic
202C2#582511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C22⋊Q8C2
prime radicalC24×C3C23
hypercenterC2×C22⋊Q8D3
hypocenterC3#2627
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #5230C2semidirect
5. #4985C2semidirect
6. #4982C2semidirect
49. C22×Dic3Q8semidirect
196. C3#2627semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
C22×Dic3⋊C42
#52302
#52122
#49852
#49822
#49462
#49432
#26273