Finite Group Databasebeta

← #68362

Subgroups — #68362

Order 384.

226 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 63 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
1#5688C211yesmaximal, characteristic
2#4707C212yesmaximal
3C3×#330C211yesmaximal, characteristic, complemented
4#4543C211yesmaximal, characteristic, complemented
5#4465C212yesmaximal, complemented
6#1526—31nomaximal, Sylow
7C3×#125C2211yescharacteristic
8C2×ΓL(1,𝔽25)—22no—
9#896—22no—
10#892C412yes—
11#888C412yes—
12#887—22no—
13C42×D3C2211yescharacteristic
14#870C2211yescharacteristic
15C3×#97C2212yes—
16C8×Dic3—22no—
17#803C2212yes—
18#533—31no—
19C4×M16—32no—
20#330—31no—
21#330—32no—
22#330D311yescharacteristic, complemented
23C2×C4×D3C2×C411yescharacteristic
24C2×C4×D3D412yes—
25C2×Dic6—22no—
26C2×C3×C8—22no—
27C3×C4⋊C4—22no—
28C3×C4⋊C4C2×C412yes—
29C3×C42C2311yescharacteristic
30(C2×D3)⋊C4—22no—
31C4⋊Dic3C2×C412yes—
32Dic3⋊C4—22no—
33Dic3⋊C4—22no—
34C4×Dic3—22no—
35C4×Dic3C2×C411yescharacteristic
36C4×Dic3D412yes—
37C2×C3⋊C8—22no—
38ΓL(1,𝔽25)—28no—
39C2×M16—62no—
40C4⋊Q8—32no—
41#125—31no—
42#125C2×D311yescharacteristic
43C22⋊Q8—62no—
44C4×Q8—62no—
45#117—32no—
46C2×C42—31no—
47#97—32no—
48#97C2×D312yes—
49C4×C8—62no—
50C22×D3C22⋊C411yescharacteristic
51C2×C3×C4—22no—
52C2×C3×C4C22×C411yescharacteristic
53C2×C3×C4C2×D412yes—
54C2×Dic3—22no—
55C2×Dic3C22⋊C411yescharacteristic
56C2×Dic3C22⋊C412yes—
57C4×D3—22no—
58C4×D3—24no—
59Dic6—44no—
60C3×C8—24no—
61C3⋊C8—24no—
62C2×Q8—62no—
63C22×C4—31no—
64C22×C4—32no—
65M16—68no—
66C2×C8—22no—
67C2×C8—62no—
68C4⋊C4—22no—
69C4⋊C4—32no—
70C4⋊C4—62no—
71C4⋊C4—62no—
72C4⋊C4C4×D312yes—
73C22⋊C4—62no—
74C42—31no—
75C42—32no—
76C42—62no—
77C42C22×D311yescharacteristic
78C22×C3C2×C22⋊C411yescharacteristic
79C2×D3—21no—
80C2×D3—22no—
81C3×C4—21no—
82C3×C4—22no—
83C3×C4—42no—
84Dic3—42no—
85Dic3#10112yes—
86Dic3#10414yes—
87C23—31no—
88Q8—124no—
89C2×C4—22no—
90C2×C4—31no—
91C2×C4—32no—
92C2×C4—64no—
93C2×C4—62no—
94C2×C4—62no—
95C2×C4C2×C4×D311yescharacteristic
96C2×C4D3×D412yes—
97C8—24no—
98C8—64no—
99C2×C3C2×#10111yescharacteristic
100C2×C3C2×#10412yes—
101D3—41no—
102C22—61no—
103C22—62no—
104C22D3×C22⋊C411yescharacteristic
105C4—21no—
106C4—22no—
107C4—32no—
108C4—34no—
109C4—42no—
110C4—122no—
111C3#152611yescharacteristic, complemented, Sylow
112C2—121no—
113C2D3×#10111yescharacteristic
114C2D3×#10412yes—
subgroupG/H
centerC22D3×C22⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC42C22×D3
FittingC3×#330C2
socleC22×C3C2×C22⋊C4
prime radicalC22×D3C22⋊C4
cosocleC3×C42C23
hypercenter#330D3
hypocenterC3#1526
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
3. C3×#330C2semidirect
4. #4543C2semidirect
5. #4465C2semidirect
22. #330D3semidirect
111. C3#1526semidirect

Maximal subgroups

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

subgroupindex
#56882
#47072
C3×#3302
#45432
#44652
#15263