Finite Group Databasebeta

← #67463

Subgroups — #67463

Order 384.

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

normal subgroups only

This group has 106 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#5095C211yesmaximal, characteristic
2C3×#333C211yesmaximal, characteristic, complemented
3#4548C211yesmaximal, characteristic, complemented
4#1405—31nomaximal, Sylow
5#1007C411yescharacteristic
6C3×C4⋊Q8C2211yescharacteristic
7#942—21no—
8C3×C4⋊C8—21no—
9#836—21no—
10#836C411yescharacteristic
11#805—21no—
12#456—31no—
13#356—31no—
14#333D311yescharacteristic, complemented
15C2×C3×Q8C2×C411yescharacteristic
16C2×C3×Q8D411yescharacteristic
17#239—41no—
18D3×Q8—41no—
19#235—21no—
20#235—41no—
21C2×C3×C8—41no—
22C3×C4⋊C4—41no—
23C3×C42D411yescharacteristic
24#216—41no—
25#215—41no—
26#208—21no—
27#208—41no—
282−1+4—31no—
29#137—61no—
30C4⋊Q8C2×D311yescharacteristic
31C4⋊C8—21no—
32#104—61no—
33#101—31no—
34#101—61no—
35C3×Q8—41no—
36C3×Q8—41no—
37C2×C3×C4—21no—
38C2×C3×C4—21no—
39C2×C3×C4—21no—
40C2×C3×C4C22⋊C411yescharacteristic
41#67—41no—
42D12—21no—
43D12—41no—
44C4×D3—41no—
45C4×D3—42no—
46Dic6—21no—
47Dic6—41no—
48C3×C8—81no—
49C3⋊C8—41no—
50C3⋊C8—81no—
51#41—61no—
52#41—121no—
53#41—121no—
54C2×Q8—121no—
55C2×Q8C4×D311yescharacteristic
56C2×Q8D1211yescharacteristic
57Q16—121no—
58SD16—121no—
59M16—61no—
60M16—121no—
61C2×C8—41no—
62C4⋊C4—41no—
63C42#6711yescharacteristic
64C22×C3#9911yescharacteristic
65C2×D3—41no—
66C3×C4—21no—
67C3×C4—21no—
68C3×C4—21no—
69C3×C4—41no—
70C3×C4—41no—
71Dic3—41no—
72Q8—41no—
73Q8—41no—
74Q8—61no—
75Q8—121no—
76D4—61no—
77D4—121no—
78D4—121no—
79C2×C4—21no—
80C2×C4—21no—
81C2×C4—21no—
82C2×C4—121no—
83C2×C4—122no—
84C2×C4(C2×D3)⋊C411yescharacteristic
85C8—81no—
86C8—121no—
87C8—241no—
88C2×C3—21no—
89C2×C3AΣL(1,𝔽16)11yescharacteristic
90D3—81no—
91C22—121no—
92C22#80611yescharacteristic
93C4—21no—
94C4—21no—
95C4—21no—
96C4—41no—
97C4—41no—
98C4—121no—
99C3#140511yescharacteristic, complemented, Sylow
100C2—21no—
101C2—241no—
102C2#447511yescharacteristic
subgroupG/H
centerC2#4475
commutatorC2×C3×Q8C2×C4
FrattiniC4⋊Q8C2×D3
FittingC3×#333C2
socleC2×C3AΣL(1,𝔽16)
prime radical#1007C4
cosocleC3×C4⋊Q8C22
hypercenter#333D3
hypocenterC3#1405
LCSG ▷ C2×C3×Q8 ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #333

Complements

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

NH ≅ G/Ntype
2. C3×#333C2semidirect
3. #4548C2semidirect
14. #333D3semidirect
99. C3#1405semidirect

Maximal subgroups

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

subgroupindex
#50952
C3×#3332
#45482
#14053