Finite Group Databasebeta

← #68990

Subgroups — #68990

Order 384.

316 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 81 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×D3×C4⋊C4C211yesmaximal, characteristic
2#5126C212yesmaximal
3C3×#343C211yesmaximal, characteristic, complemented
4#4554C212yesmaximal, complemented
5#4530C211yesmaximal, characteristic, complemented
6#1821—31nomaximal, Sylow
7C22×C4×D3C412yes—
8C22×C4×D3C2211yescharacteristic
9C2×C3×M16C2212yes—
10C2×C3×C4⋊C4C2211yescharacteristic
11C2×C4⋊Dic3C2211yescharacteristic
12C2×Dic3⋊C4C412yes—
13C2×#208C2212yes—
14D3×C4⋊C4—22no—
15D3×C4⋊C4—22no—
16(C2×D3)⋊C8—22no—
17(C2×D3)⋊C8—22no—
18C22×C4⋊C4—31no—
19#407—32no—
20#343—32no—
21#343—31no—
22#343D311yescharacteristic, complemented
23C23×D3C2×C411yescharacteristic
24C22×C3×C4C2×C412yes—
25C22×C3×C4C2311yescharacteristic
26C22×Dic3C2×C411yescharacteristic
27C22×Dic3C2×C412yes—
28C2×C4×D3—22no—
29C2×C4×D3—21no—
30C2×C4×D3—21no—
31C2×C4×D3—22no—
32C2×C4×D3—22no—
33C2×C4×D3—44no—
34C2×C4×D3D412yes—
35C2×C4×D3D412yes—
36C3×M16—22no—
37C3×M16—22no—
38C2×C3×C8—22no—
39C3×C4⋊C4—22no—
40C4⋊Dic3—22no—
41Dic3⋊C4—24no—
42#208—22no—
43#208—22no—
44C2×C3⋊C8—22no—
45C23×C4—31no—
46C23×C4—32no—
47C2×M16—32no—
48C2×M16C2×D312yes—
49C2×C4⋊C4—32no—
50C2×C4⋊C4—31no—
51C2×C4⋊C4—62no—
52C2×C4⋊C4—62no—
53C2×C4⋊C4C2×D311yescharacteristic
54C22⋊C8—62no—
55C22⋊C8—62no—
56C23×C3C22×C411yescharacteristic
57C22×D3—21no—
58C22×D3—21no—
59C22×D3—22no—
60C22×D3—22no—
61C22×D3C22⋊C411yescharacteristic
62C22×D3C22⋊C411yescharacteristic
63C2×C3×C4—22no—
64C2×C3×C4—42no—
65C2×C3×C4C2×D412yes—
66C2×C3×C4#4112yes—
67C2×C3×C4#4111yescharacteristic
68C2×C3×C4#4111yescharacteristic
69C2×Dic3—21no—
70C2×Dic3—21no—
71C2×Dic3—22no—
72C2×Dic3—42no—
73C2×Dic3C22⋊C412yes—
74C4×D3—22no—
75C4×D3—22no—
76C4×D3—22no—
77C4×D3—22no—
78C4×D3—44no—
79C4×D3—44no—
80C3×C8—42no—
81C3⋊C8—42no—
82C24—31no—
83C22×C4—31no—
84C22×C4—32no—
85C22×C4—32no—
86C22×C4—32no—
87C22×C4—62no—
88C22×C4—62no—
89C22×C4—62no—
90C22×C4—61no—
91C22×C4—61no—
92C22×C4—124no—
93C22×C4C4×D312yes—
94C22×C4C22×D311yescharacteristic
95M16—22no—
96M16—22no—
97M16—62no—
98M16—62no—
99C2×C8—22no—
100C2×C8—62no—
101C4⋊C4—22no—
102C4⋊C4—64no—
103C4⋊C4—62no—
104C22×C3—21no—
105C22×C3—21no—
106C22×C3C2×C22⋊C411yescharacteristic
107C22×C3#11711yescharacteristic
108C22×C3#11711yescharacteristic
109C2×D3—24no—
110C2×D3—24no—
111C2×D3—21no—
112C2×D3—21no—
113C2×D3—21no—
114C2×D3—21no—
115C2×D3#10012yes—
116C2×D3#10112yes—
117C3×C4—42no—
118C3×C4#12312yes—
119C3×C4#12312yes—
120Dic3—22no—
121Dic3—42no—
122C23—31no—
123C23—31no—
124C23—62no—
125C23—62no—
126C23—61no—
127C23—61no—
128C23C2×C4×D311yescharacteristic
129C2×C4—22no—
130C2×C4—32no—
131C2×C4—42no—
132C2×C4—61no—
133C2×C4—61no—
134C2×C4—62no—
135C2×C4—62no—
136C2×C4—62no—
137C2×C4—62no—
138C2×C4—62no—
139C2×C4—124no—
140C2×C4—124no—
141C2×C4—122no—
142C2×C4#23512yes—
143C2×C4D3×D412yes—
144C2×C4#23711yescharacteristic
145C2×C4#23911yescharacteristic
146C8—42no—
147C8—122no—
148C2×C3—21no—
149C2×C3—21no—
150C2×C3#38111yescharacteristic
151C2×C3C2×#10011yescharacteristic
152C2×C3C2×#10111yescharacteristic
153D3—22no—
154D3—22no—
155C22—21no—
156C22—21no—
157C22—32no—
158C22—32no—
159C22—64no—
160C22—64no—
161C22—61no—
162C22—61no—
163C22—61no—
164C22—61no—
165C22#87211yescharacteristic
166C22D3×C22⋊C411yescharacteristic
167C22#89211yescharacteristic
168C4—42no—
169C4—62no—
170C4—122no—
171C4#88312yes—
172C4#89412yes—
173C3#182111yescharacteristic, complemented, Sylow
174C2—21no—
175C2—21no—
176C2—62no—
177C2—62no—
178C2#466411yescharacteristic
179C2D3×#10011yescharacteristic
180C2D3×#10111yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC3×#343C2
socleC22×C3C2×C22⋊C4
prime radicalC23×D3C2×C4
cosocleC22×C3×C4C23
hypercenter#343D3
hypocenterC3#1821
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
3. C3×#343C2semidirect
4. #4554C2semidirect
5. #4530C2semidirect
22. #343D3semidirect
173. C3#1821semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊C42
#51262
C3×#3432
#45542
#45302
#18213