Finite Group Databasebeta

← #69169

Subgroups — #69169

Order 384.

224 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 41 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#5722C211yesmaximal, characteristic, complemented
2#5095C211yesmaximal, characteristic
3#4752C212yesmaximal
4C3×#356C211yesmaximal, characteristic, complemented
5#4478C212yesmaximal, complemented
6#2205—31nomaximal, Sylow
7#1007C2211yescharacteristic
8C3×C4⋊Q8C2211yescharacteristic
9#944—21no—
10#944C2212yes—
11#942—21no—
12#909—22no—
13#888—22no—
14#887—22no—
15C4×Dic6—21no—
16C3×#101C2212yes—
17#836C2211yescharacteristic
18#825—22no—
19#805—21no—
20Q82—31no—
21#456—31no—
22#456—32no—
23#356—32no—
24#356D311yescharacteristic, complemented
25C2×C3×Q8D411yescharacteristic
26C2×C3×Q8C2311yescharacteristic
27#239—41no—
28D3×Q8—41no—
29#235—21no—
30#235—22no—
31C2×Dic6—21no—
32C2×Dic6—21no—
33C2×Dic6D412yes—
34C3×M16—22no—
35C3×C4⋊C4—22no—
36C3×C42D411yescharacteristic
37#216—41no—
38#215—41no—
39C4⋊Dic3—21no—
40Dic3⋊C4—22no—
41Dic3⋊C4—22no—
42Dic3⋊C4—22no—
43C4×Dic3—22no—
44C4×Dic3—21no—
45C4×Dic3D412yes—
46#208—21no—
47Dic12—42no—
48#204—42no—
492−1+4—31no—
50#137—62no—
51#137—61no—
52C4⋊Q8—32no—
53C4⋊Q8—62no—
54C4⋊Q8—61no—
55C4⋊Q8C2×D311yescharacteristic
56C4×Q8—62no—
57C4×Q8—61no—
58#104—62no—
59#104—61no—
60#101—31no—
61#101C2×D312yes—
62C3×Q8—41no—
63C3×Q8—41no—
64C2×C3×C4—21no—
65C2×C3×C4—21no—
66C2×C3×C4C2×D411yescharacteristic
67C2×C3×C4C2×D412yes—
68#67—41no—
69C2×Dic3—22no—
70C2×Dic3—22no—
71D12—22no—
72D12—21no—
73C4×D3—41no—
74C4×D3—42no—
75Dic6—24no—
76Dic6—22no—
77Dic6—21no—
78Dic6—42no—
79Dic6—81no—
80C3×C8—42no—
81C3⋊C8—41no—
82#41—61no—
83#41—62no—
84#41—121no—
85C2×Q8—32no—
86C2×Q8—61no—
87C2×Q8—61no—
88C2×Q8—121no—
89C2×Q8D1211yescharacteristic
90C2×Q8C22×D311yescharacteristic
91Q16—122no—
92Q16—121no—
93SD16—122no—
94SD16—121no—
95M16—22no—
96M16—61no—
97C4⋊C4—22no—
98C4⋊C4—62no—
99C4⋊C4—61no—
100C4⋊C4—62no—
101C4⋊C4—62no—
102C42—32no—
103C42—62no—
104C42—61no—
105C42D1211yescharacteristic
106C22×C3#12011yescharacteristic
107C2×D3—41no—
108C3×C4—22no—
109C3×C4—21no—
110C3×C4—22no—
111C3×C4—41no—
112Dic3—24no—
113Dic3—41no—
114Dic3—42no—
115Q8—41no—
116Q8—41no—
117Q8—64no—
118Q8—62no—
119Q8—61no—
120Q8—122no—
121Q8—241no—
122D4—62no—
123D4—61no—
124D4—121no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—62no—
128C2×C4—62no—
129C2×C4—122no—
130C2×C4—121no—
131C2×C4C2×D1211yescharacteristic
132C2×C4D3×D412yes—
133C8—42no—
134C8—121no—
135C2×C3—21no—
136C2×C3B(4,𝔽2)11yescharacteristic
137D3—81no—
138C22—121no—
139C22#88211yescharacteristic
140C4—22no—
141C4—22no—
142C4—21no—
143C4—41no—
144C4—64no—
145C4—122no—
146C4—121no—
147C3#220511yescharacteristic, complemented, Sylow
148C2—21no—
149C2—241no—
150C2#474111yescharacteristic
subgroupG/H
centerC2#4741
commutator, cosocleC2×C3×Q8C23
FrattiniC2×Q8C22×D3
FittingC3×#356C2
socleC2×C3B(4,𝔽2)
prime radical#1007C22
hypercenter#356D3
hypocenterC3#2205
LCSG ▷ C2×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #5722C2semidirect
4. C3×#356C2semidirect
5. #4478C2semidirect
24. #356D3semidirect
147. C3#2205semidirect

Maximal subgroups

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

subgroupindex
#57222
#50952
#47522
C3×#3562
#44782
#22053