Finite Group Databasebeta

← #67461

Subgroups — #67461

Order 384.

112 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 23 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#5061C211yesmaximal, characteristic
2C3×#329C211yesmaximal, characteristic, complemented
3#4542C211yesmaximal, characteristic, complemented
4#1403—31nomaximal, Sylow
5#1004C411yescharacteristic
6C3×#124C2211yescharacteristic
7#942—21no—
8C3×#97—21no—
9#836—21no—
10#833C411yescharacteristic
11#805—21no—
12#455—31no—
13#355—31no—
14#329D311yescharacteristic, complemented
15C2×C3×Q8D411yescharacteristic
16C2×C3×D4C2×C411yescharacteristic
17C2×#67—41no—
18#237—41no—
19D3×D4—41no—
20#235—21no—
21C2×C3×C8—41no—
22C3×C22⋊C4—41no—
23C3×C42D411yescharacteristic
24#216—41no—
25#215—41no—
26#208—21no—
27#208—41no—
28H(5,𝔽2)—31no—
29#137—61no—
30#124C2×D311yescharacteristic
31#104—61no—
32#101—61no—
33#100—31no—
34#97—21no—
35C23×C3—21no—
36C22×D3—41no—
37C3×Q8—41no—
38C3×D4—41no—
39C2×C3×C4—21no—
40C2×C3×C4—21no—
41C2×C3×C4C22⋊C411yescharacteristic
42#67—42no—
43#67—41no—
44C2×Dic3—41no—
45D12—21no—
46C4×D3—41no—
47Dic6—21no—
48C3×C8—42no—
49C3⋊C8—41no—
50C3⋊C8—81no—
51#41—61no—
52#41—121no—
53C2×Q8D1211yescharacteristic
54C2×D4—121no—
55C2×D4—121no—
56C2×D4C4×D311yescharacteristic
57Q16—121no—
58SD16—121no—
59M16—61no—
60M16—121no—
61C2×C8—41no—
62C22⋊C4—41no—
63C42#6711yescharacteristic
64C22×C3—41no—
65C22×C3—81no—
66C22×C3#9911yescharacteristic
67C2×D3—41no—
68C2×D3—161no—
69C3×C4—21no—
70C3×C4—41no—
71C3×C4—41no—
72Dic3—41no—
73C23—21no—
74C23—121no—
75Q8—41no—
76Q8—61no—
77D4—41no—
78D4—61no—
79D4—122no—
80D4—121no—
81C2×C4—21no—
82C2×C4—21no—
83C2×C4—121no—
84C2×C4—121no—
85C2×C4(C2×D3)⋊C411yescharacteristic
86C8—42no—
87C8—121no—
88C8—241no—
89C2×C3—21no—
90C2×C3—81no—
91C2×C3AΣL(1,𝔽16)11yescharacteristic
92D3—81no—
93C22—41no—
94C22—81no—
95C22—121no—
96C22—481no—
97C22#80611yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—121no—
102C3#140311yescharacteristic, complemented, Sylow
103C2—21no—
104C2—81no—
105C2—241no—
106C2#447511yescharacteristic
subgroupG/H
centerC2#4475
commutatorC2×C3×D4C2×C4
Frattini#124C2×D3
FittingC3×#329C2
socleC2×C3AΣL(1,𝔽16)
prime radical#1004C4
cosocleC3×#124C22
hypercenter#329D3
hypocenterC3#1403
LCSG ▷ C2×C3×D4 ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #329

Complements

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

NH ≅ G/Ntype
2. C3×#329C2semidirect
3. #4542C2semidirect
14. #329D3semidirect
102. C3#1403semidirect

Maximal subgroups

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

subgroupindex
#50612
C3×#3292
#45422
#14033