Finite Group Databasebeta

← #82808

Subgroups — #82808

Order 384.

444 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 133 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
2C2×#924C211yesmaximal, characteristic
3#5230C212yesmaximal, complemented
4#5211C211yesmaximal, characteristic
5#4986C212yesmaximal, complemented
6#4985C212yesmaximal, complemented
7#4979C212yesmaximal, complemented
8#4952C214yesmaximal
9#2817—31nomaximal, Sylow
10C22×C3×Q8C2211yescharacteristic
11C23×C3×C4C2211yescharacteristic
12C22×Dic6C2211yescharacteristic
13C3×C22⋊Q8—24no—
14C2×C3×C4⋊C4C2212yes—
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C22⋊C4C2212yes—
17C2×#217C2212yes—
18C2×#217C2212yes—
19C2×C4⋊Dic3C2211yescharacteristic
20#924—24no—
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2214yes—
23C3⋊#95C2214yes—
24C3⋊#95C2214yes—
25C3⋊#95C2214yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C2×C22⋊Q8—31no—
29C2×C22⋊Q8D311yescharacteristic
30#400—32no—
31#400—32no—
32#397—34no—
33#395—32no—
34#395—32no—
35#381—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×Dic3C2314yes—
46C22×Dic3C2312yes—
47C22×Dic3C2312yes—
48C2×Dic6—22no—
49C2×Dic6—42no—
50C3×C4⋊C4—22no—
51C3×C4⋊C4—24no—
52C3×C22⋊C4—24no—
53#217—24no—
54#217—42no—
55C4⋊Dic3—22no—
56Dic3⋊C4—24no—
57Dic3⋊C4—44no—
58C22×Q8—31no—
59C22×Q8C2×D311yescharacteristic
60C23×C4C2×D311yescharacteristic
61C22⋊Q8—24no—
62C22⋊Q8—64no—
63C2×C4⋊C4—32no—
64C2×C4⋊C4—31no—
65C2×C4⋊C4—34no—
66C2×C4⋊C4C2×D312yes—
67C2×C4⋊C4C2×D311yescharacteristic
68C2×C22⋊C4—32no—
69C2×C22⋊C4—32no—
70C2×C22⋊C4C2×D312yes—
71#95—32no—
72#95—32no—
73#95—34no—
74#95—34no—
75#95—34no—
76C23×C3—22no—
77C23×C3—22no—
78C23×C3—22no—
79C23×C3C2×D412yes—
80C23×C3C2411yescharacteristic
81C3×Q8—44no—
82C2×C3×C4—22no—
83C2×C3×C4—22no—
84C2×C3×C4—22no—
85C2×C3×C4—22no—
86C2×C3×C4—24no—
87C2×C3×C4—24no—
88C2×C3×C4—44no—
89C2×C3×C4C2×D412yes—
90C2×C3×C4C2×D412yes—
91C2×Dic3—22no—
92C2×Dic3—22no—
93C2×Dic3—24no—
94C2×Dic3—24no—
95C2×Dic3—24no—
96C2×Dic3—24no—
97C2×Dic3—24no—
98Dic6—44no—
99C24C22×D311yescharacteristic
100C2×Q8—22no—
101C2×Q8—42no—
102C2×Q8—62no—
103C2×Q8—122no—
104C22×C4—34no—
105C22×C4—32no—
106C22×C4—32no—
107C22×C4—42no—
108C22×C4#6714yes—
109C22×C4C22×D312yes—
110C22×C4C22×D312yes—
111C22×C4C22×D311yescharacteristic
112C22×C4C22×D311yescharacteristic
113C4⋊C4—22no—
114C4⋊C4—24no—
115C4⋊C4—64no—
116C4⋊C4—62no—
117C4⋊C4—124no—
118C22⋊C4—24no—
119C22⋊C4—64no—
120C22⋊C4—122no—
121C22×C3—22no—
122C22×C3—22no—
123C22×C3—22no—
124C22×C3—44no—
125C22×C3C22×D411yescharacteristic
126C22×C3H(5,𝔽2)12yes—
127C22×C32−1+412yes—
128C22×C32−1+412yes—
129C3×C4—22no—
130C3×C4—22no—
131C3×C4—42no—
132C3×C4—42no—
133Dic3—42no—
134Dic3—42no—
135Dic3—44no—
136C23—22no—
137C23—22no—
138C23—22no—
139C23C2×#6712yes—
140C23C23×D311yescharacteristic
141Q8—44no—
142Q8—124no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—24no—
146C2×C4—24no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—44no—
150C2×C4—64no—
151C2×C4—64no—
152C2×C4—64no—
153C2×C4—64no—
154C2×C4—64no—
155C2×C4—62no—
156C2×C4—62no—
157C2×C4C2×#6712yes—
158C2×C4C2×#6712yes—
159C2×C3—42no—
160C2×C3#53411yescharacteristic
161C2×C3#53611yescharacteristic
162C2×C3#53611yescharacteristic
163C2×C3#56314yes—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22#100412yes—
169C22#100712yes—
170C22#101012yes—
171C22C22×#6711yescharacteristic
172C4—22no—
173C4—22no—
174C4—42no—
175C4—42no—
176C4—124no—
177C4—122no—
178C4—122no—
179C3#281711yescharacteristic, complemented, Sylow
180C2—42no—
181C2#564614yes—
182C2#580411yescharacteristic
183C2#581611yescharacteristic
184C2#582511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C22⋊Q8C2
prime radicalC24×C3C23
hypercenterC2×C22⋊Q8D3
hypocenterC3#2817
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. #4986C2semidirect
6. #4985C2semidirect
7. #4979C2semidirect
179. C3#2817semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
C2×#9242
#52302
#52112
#49862
#49852
#49792
#49522
#28173