Finite Group Databasebeta

← #71944

Subgroups — #71944

Order 384.

218 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C3×#559C211yesmaximal, characteristic
2#5076C211yesmaximal, characteristic
3#5021C212yesmaximal, complemented
4#4544C211yesmaximal, characteristic
5#4535C212yesmaximal, complemented
6#1630—31nomaximal, Sylow
7C3×#127C2211yescharacteristic
8C3×#124—22no—
9C3×#123—22no—
10C3×C4×Q8C2212yes—
11C3×C4×D4—22no—
12C2×#214—22no—
13#869C2211yescharacteristic
14#835—22no—
15#807—22no—
16C4⋊(C3⋊C8)C2212yes—
17C4×C3⋊C8C2211yescharacteristic
18#559D311yescharacteristic
19#492—31no—
20#477—32no—
21#331—31no—
22Q8⋊C8—32no—
23C2×C3×Q8D412yes—
24C2×C3×D4—22no—
25C2×C3×D4D412yes—
26C22×C3×C4—22no—
27C2×Dic6—21no—
28C3×C4⋊C4—22no—
29C3×C4⋊C4D412yes—
30C3×C22⋊C4—24no—
31C3×C22⋊C4—22no—
32C3×C42—22no—
33C3×C42C2311yescharacteristic
34#214—44no—
35C4⋊Dic3—22no—
36C2×C3⋊C8—21no—
37C2×C3⋊C8—22no—
38C2×SD16—62no—
39C4⋊Q8—31no—
40#127C2×D311yescharacteristic
41#124—22no—
42#123—22no—
43C4×Q8C2×D312yes—
44C4×D4—22no—
45#107—62no—
46C4⋊C8—32no—
47#103—62no—
48C4×C8—31no—
49C23×C3—22no—
50C3×Q8—22no—
51C3×Q8—22no—
52C3×D4—42no—
53C3×D4—42no—
54C3×D4—42no—
55C2×C3×C4—22no—
56C2×C3×C4—22no—
57C2×C3×C4—82no—
58C2×C3×C4C2×D411yescharacteristic
59C2×C3×C4C2×D412yes—
60C2×Dic3—41no—
61Dic6—42no—
62C3⋊C8—22no—
63C3⋊C8—42no—
64C2×Q8—61no—
65C2×Q8#6712yes—
66C2×D4—22no—
67C2×D4#6712yes—
68C22×C4—22no—
69SD16—124no—
70C2×C8—61no—
71C2×C8—62no—
72C4⋊C4—22no—
73C4⋊C4—62no—
74C4⋊C4#6712yes—
75C22⋊C4—24no—
76C22⋊C4—22no—
77C42—22no—
78C42C22×D311yescharacteristic
79C22×C3—44no—
80C22×C3—42no—
81C22×C3#12011yescharacteristic
82C3×C4—21no—
83C3×C4—42no—
84C3×C4—42no—
85C3×C4#13512yes—
86C3×C4#13712yes—
87Dic3—81no—
88C23—22no—
89Q8—22no—
90Q8—22no—
91Q8—122no—
92D4—42no—
93D4—42no—
94D4—42no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—82no—
98C2×C4—121no—
99C2×C4C2×#6711yescharacteristic
100C2×C4C2×#6712yes—
101C8—62no—
102C8—122no—
103C2×C3—82no—
104C2×C3#45212yes—
105C2×C3#45311yescharacteristic
106C22—44no—
107C22—42no—
108C22#95311yescharacteristic
109C4—21no—
110C4—42no—
111C4—42no—
112C4—241no—
113C4#95012yes—
114C4#95112yes—
115C3#163011yescharacteristic, complemented, Sylow
116C2—82no—
117C2#523912yes—
118C2#524111yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#559C2
socleC22×C3#120
prime radicalC3×#127C22
hypercenter#559D3
hypocenterC3#1630
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #559

Complements

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

NH ≅ G/Ntype
3. #5021C2semidirect
5. #4535C2semidirect
115. C3#1630semidirect

Maximal subgroups

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

subgroupindex
C3×#5592
#50762
#50212
#45442
#45352
#16303