Finite Group Databasebeta

← #76543

Subgroups — #76543

Order 384.

500 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 185 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×Q8×Dic3C211yesmaximal, characteristic
2C2×#888C211yesmaximal, characteristic
3C3×#391C211yesmaximal, characteristic
4#5229C211yesmaximal, characteristic
5#4975C211yesmaximal, characteristic
6#4935C211yesmaximal, characteristic
7C4×Dic3⋊C4C211yesmaximal, characteristic
8#4656C214yesmaximal
9#4647C214yesmaximal
10#2567—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×Dic6C2212yes—
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C42C2211yescharacteristic
15Q8×Dic3—24no—
16C2×C4⋊Dic3C2212yes—
17C2×C4⋊Dic3C2211yescharacteristic
18C2×C4⋊Dic3C2211yescharacteristic
19C2×C4⋊Dic3C2214yes—
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2214yes—
22C2×Dic3⋊C4C2212yes—
23C2×C4×Dic3C2212yes—
24C2×C4×Dic3C2211yescharacteristic
25C2×C4×Dic3C2211yescharacteristic
26#888—24no—
27C3×#95C2214yes—
28C3⋊#95C2214yes—
29C3⋊#95C2212yes—
30C2×C4⋊Q8—31no—
31C2×C4×Q8—31no—
32#395—34no—
33#391—31no—
34#391D311yescharacteristic
35#389—34no—
36#384—31no—
37#384—31no—
38C4×C4⋊C4—31no—
39C2×C3×Q8—22no—
40C2×C3×Q8—22no—
41C2×C3×Q8Q814yescomplemented
42C22×C3×C4C2312yes—
43C22×C3×C4C2312yes—
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×Dic3C2312yes—
47C22×Dic3C2312yes—
48C22×Dic3C2311yescharacteristic
49C22×Dic3C2312yes—
50C22×Dic3C2311yescharacteristic
51C2×Dic6—24no—
52C2×Dic6—44no—
53C3×C4⋊C4—22no—
54C3×C42—22no—
55C4⋊Dic3—22no—
56C4⋊Dic3—22no—
57C4⋊Dic3—24no—
58C4⋊Dic3—44no—
59Dic3⋊C4—24no—
60Dic3⋊C4—44no—
61Dic3⋊C4Q818yescomplemented
62C4×Dic3—22no—
63C4×Dic3—24no—
64C4×Dic3D414yes—
65C22×Q8—32no—
66C22×Q8C2×D311yescharacteristic
67C4⋊Q8—64no—
68C4×Q8—64no—
69C2×C4⋊C4—32no—
70C2×C4⋊C4—31no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—31no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4—34no—
75C2×C4⋊C4—34no—
76C2×C4⋊C4C2×D311yescharacteristic
77C2×C42—31no—
78C2×C42—32no—
79C2×C42—31no—
80C2×C42C2×D311yescharacteristic
81#95—32no—
82#95—34no—
83#95C2×D314yes—
84C23×C3C2411yescharacteristic
85C3×Q8—28no—
86C2×C3×C4—22no—
87C2×C3×C4—22no—
88C2×C3×C4—22no—
89C2×C3×C4—22no—
90C2×C3×C4—22no—
91C2×C3×C4—24no—
92C2×C3×C4C2×D412yes—
93C2×C3×C4C2×Q814yes—
94C2×C3×C4C2×Q812yes—
95C2×C3×C4C2×Q814yes—
96C2×C3×C4#4112yes—
97C2×Dic3—22no—
98C2×Dic3—22no—
99C2×Dic3—22no—
100C2×Dic3—22no—
101C2×Dic3—22no—
102C2×Dic3—24no—
103C2×Dic3—24no—
104C2×Dic3C2×D412yes—
105C2×Dic3C2×D412yes—
106C2×Dic3C2×Q814yes—
107C2×Dic3C2×Q814yes—
108Dic6—48no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—64no—
112C2×Q8—124no—
113C2×Q8Dic614yescomplemented
114C22×C4—31no—
115C22×C4—32no—
116C22×C4—31no—
117C22×C4—32no—
118C22×C4—32no—
119C22×C4C22×D312yes—
120C22×C4C22×D312yes—
121C22×C4C22×D312yes—
122C22×C4C22×D311yescharacteristic
123C4⋊C4—22no—
124C4⋊C4—38no—
125C4⋊C4—64no—
126C4⋊C4—64no—
127C4⋊C4—62no—
128C4⋊C4—62no—
129C4⋊C4—124no—
130C4⋊C4—124no—
131C42—22no—
132C42—34no—
133C42—64no—
134C42—62no—
135C22×C3C22×D411yescharacteristic
136C22×C3C22×Q811yescharacteristic
137C22×C3C22×Q812yes—
138C22×C3C2×#4111yescharacteristic
139C22×C3H(5,𝔽2)12yes—
140C3×C4—24no—
141C3×C4—24no—
142C3×C4—42no—
143C3×C4C22⋊Q814yes—
144Dic3—22no—
145Dic3—24no—
146Dic3—42no—
147Dic3—42no—
148Dic3C4⋊Q814yes—
149C23C23×D311yescharacteristic
150Q8—28no—
151Q8—128no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—32no—
159C2×C4—34no—
160C2×C4—32no—
161C2×C4—34no—
162C2×C4—62no—
163C2×C4—62no—
164C2×C4—62no—
165C2×C4—62no—
166C2×C4—62no—
167C2×C4—64no—
168C2×C4—64no—
169C2×C4C2×Dic614yes—
170C2×C4C2×Dic612yes—
171C2×C4D3×D412yes—
172C2×C4#23712yes—
173C2×C4D3×Q814yes—
174C2×C3C2×C22⋊Q811yescharacteristic
175C2×C3C2×C4⋊Q811yescharacteristic
176C2×C3#53511yescharacteristic
177C2×C3#55412yes—
178C2×C3Q8212yes—
179C22C22×Dic611yescharacteristic
180C22C2×D3×D411yescharacteristic
181C22C2×#23711yescharacteristic
182C22C2×D3×Q812yes—
183C22#100912yes—
184C4—24no—
185C4—24no—
186C4—34no—
187C4—42no—
188C4—64no—
189C4—62no—
190C4—122no—
191C4—122no—
192C4#87814yes—
193C3#256711yescharacteristic, complemented, Sylow
194C2C2×#87811yescharacteristic
195C2Q8×Dic612yes—
196C2#563512yes—
197C2#567411yescharacteristic
198C2D3×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#391C2
hypercenter#391D3
hypocenterC3#2567
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
41. C2×C3×Q8Q8semidirect
61. Dic3⋊C4Q8semidirect
113. C2×Q8Dic6semidirect
193. C3#2567semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic32
C2×#8882
C3×#3912
#52292
#49752
#49352
C4×Dic3⋊C42
#46562
#46472
#25673