Finite Group Databasebeta

← #67772

Subgroups — #67772

Order 384.

112 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 25 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×#455C211yesmaximal, characteristic
2#4542C211yesmaximal, characteristic, complemented
3#4540C211yesmaximal, characteristic, complemented
4#1403—31nomaximal, Sylow
5C3×H(5,𝔽2)C411yescharacteristic
6C3×#137—21no—
7C3×#124C2211yescharacteristic
8C3×#104—21no—
9C3×#100C411yescharacteristic
10#836—21no—
11#803—21no—
12#455D311yescharacteristic
13#355—31no—
14#329—31no—
15C3×#41—21no—
16C3×#41—41no—
17C2×C3×Q8D411yescharacteristic
18C2×C3×D4—41no—
19C2×C3×D4—41no—
20C2×C3×D4C2×C411yescharacteristic
21C3×Q16—41no—
22C3×SD16—41no—
23C3×M16—21no—
24C3×C22⋊C4—41no—
25C3×C42D411yescharacteristic
26#208—41no—
27C2×C3⋊C8—41no—
28H(5,𝔽2)Dic311yescharacteristic
29#137—21no—
30#124C2×D311yescharacteristic
31#104—21no—
32#101—61no—
33#100Dic311yescharacteristic
34#97—61no—
35C23×C3—21no—
36C23×C3—41no—
37C3×Q8—21no—
38C3×Q8—41no—
39C3×D4—21no—
40C3×D4—41no—
41C3×D4—42no—
42C3×D4—41no—
43C2×C3×C4—21no—
44C2×C3×C4—21no—
45C2×C3×C4—41no—
46C2×C3×C4—41no—
47C2×C3×C4C22⋊C411yescharacteristic
48C3×C8—41no—
49C3⋊C8—42no—
50C3⋊C8—81no—
51#41—21no—
52#41—41no—
53C2×Q8#6711yescharacteristic
54C2×D4—41no—
55C2×D4—41no—
56C2×D4C2×Dic311yescharacteristic
57Q16—41no—
58SD16—41no—
59M16—21no—
60M16—121no—
61C2×C8—121no—
62C22⋊C4—41no—
63C42#6711yescharacteristic
64C22×C3—41no—
65C22×C3—41no—
66C22×C3—81no—
67C22×C3—161no—
68C22×C3#9911yescharacteristic
69C3×C4—21no—
70C3×C4—41no—
71C3×C4—41no—
72C3×C4—41no—
73C23—21no—
74C23—41no—
75Q8—21no—
76Q8—41no—
77D4—21no—
78D4—41no—
79D4—42no—
80D4—41no—
81C2×C4—21no—
82C2×C4—21no—
83C2×C4—41no—
84C2×C4—41no—
85C2×C4#21711yescharacteristic
86C8—41no—
87C8—122no—
88C8—241no—
89C2×C3—21no—
90C2×C3—81no—
91C2×C3—81no—
92C2×C3AΣL(1,𝔽16)11yescharacteristic
93C22—41no—
94C22—41no—
95C22—81no—
96C22—161no—
97C22#83411yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—41no—
102C3#140311yescharacteristic, complemented, Sylow
103C2—21no—
104C2—81no—
105C2—81no—
106C2#453611yescharacteristic
subgroupG/H
centerC2#4536
commutatorC2×C3×D4C2×C4
Frattini#124C2×D3
FittingC3×#455C2
socleC2×C3AΣL(1,𝔽16)
prime radicalC3×H(5,𝔽2)C4
cosocleC3×#124C22
hypercenter#455D3
hypocenterC3#1403
LCSG ▷ C2×C3×D4 ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #455

Complements

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

NH ≅ G/Ntype
2. #4542C2semidirect
3. #4540C2semidirect
102. C3#1403semidirect

Maximal subgroups

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

subgroupindex
C3×#4552
#45422
#45402
#14033