Finite Group Databasebeta

← #67542

Subgroups — #67542

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#5062C211yesmaximal, characteristic
2C3×#354C211yesmaximal, characteristic, complemented
3#4540C211yesmaximal, characteristic, complemented
4#1404—31nomaximal, Sylow
5#1007C411yescharacteristic, complemented
6C3×#124C2211yescharacteristic
7#932—21no—
8C3×#99—21no—
9#836C411yescharacteristic, complemented
10#805—21no—
11#803—21no—
12#454—31no—
13#354D311yescharacteristic, complemented
14#329—31no—
15C2×C3×Q8C2×C411yescharacteristic
16C2×C3×D4D411yescharacteristic
17#239—41no—
18D3×Q8—41no—
19#235—21no—
20#235—41no—
21C3×C22⋊C4—41no—
22C3×C22⋊C4—41no—
23C3×C42D411yescharacteristic
24#214—41no—
25#213—41no—
26#208—21no—
27C2×C3⋊C8—41no—
282−1+4—31no—
29AGL(1,ℤ/8)—61no—
30#124C2×D311yescharacteristic
31#104—61no—
32#101—31no—
33#99—21no—
34#97—61no—
35C23×C3—21no—
36C3×Q8—41no—
37C3×D4—41no—
38C2×C3×C4—21no—
39C2×C3×C4—21no—
40C2×C3×C4—81no—
41C2×C3×C4C22⋊C411yescharacteristic
42#67—41no—
43D12—21no—
44D12—41no—
45C4×D3—41no—
46C4×D3—42no—
47Dic6—21no—
48Dic6—41no—
49C3⋊C8—41no—
50C3⋊C8—42no—
51#41—61no—
52#41—121no—
53#41—121no—
54C2×Q8—121no—
55C2×Q8C4×D311yescharacteristic
56C2×D4#6711yescharacteristic
57SD16—121no—
58D8—121no—
59M16—61no—
60C2×C8—121no—
61C22⋊C4—41no—
62C22⋊C4—41no—
63C42D1211yescharacteristic
64C22×C3—41no—
65C22×C3—81no—
66C22×C3#9911yescharacteristic
67C2×D3—41no—
68C3×C4—21no—
69C3×C4—41no—
70C3×C4—41no—
71C3×C4—161no—
72Dic3—41no—
73C23—21no—
74Q8—41no—
75Q8—61no—
76Q8—121no—
77D4—41no—
78D4—61no—
79D4—121no—
80D4—121no—
81C2×C4—21no—
82C2×C4—21no—
83C2×C4—81no—
84C2×C4—121no—
85C2×C4—122no—
86C2×C4(C2×D3)⋊C411yescharacteristic
87C8—121no—
88C8—122no—
89C2×C3—21no—
90C2×C3—81no—
91C2×C3AΣL(1,𝔽16)11yescharacteristic
92D3—81no—
93C22—41no—
94C22—81no—
95C22—121no—
96C22#80611yescharacteristic
97C4—21no—
98C4—41no—
99C4—41no—
100C4—121no—
101C4—161no—
102C3#140411yescharacteristic, complemented, Sylow
103C2—21no—
104C2—81no—
105C2—241no—
106C2#447111yescharacteristic
subgroupG/H
centerC2#4471
commutatorC2×C3×Q8C2×C4
Frattini#124C2×D3
FittingC3×#354C2
socleC2×C3AΣL(1,𝔽16)
prime radical#5062C2
cosocleC3×#124C22
hypercenter#354D3
hypocenterC3#1404
LCSG ▷ C2×C3×Q8 ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #354

Complements

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

NH ≅ G/Ntype
2. C3×#354C2semidirect
3. #4540C2semidirect
5. #1007C4semidirect
9. #836C4semidirect
13. #354D3semidirect
102. C3#1404semidirect

Maximal subgroups

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

subgroupindex
#50622
C3×#3542
#45402
#14043