Finite Group Databasebeta

← #69166

Subgroups — #69166

Order 384.

262 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 55 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#5674C211yesmaximal, characteristic
2#4751C212yesmaximal, complemented
3C3×#355C211yesmaximal, characteristic, complemented
4#4542C211yesmaximal, characteristic, complemented
5#4477C212yesmaximal, complemented
6#2134—31nomaximal, Sylow
7C2×#239C2211yescharacteristic
8C2×D3×D4C411yescharacteristic
9C3×#124C2211yescharacteristic
10#947C2211yescharacteristic
11#940C411yescharacteristic
12D3×M16—22no—
13#884—22no—
14#882—22no—
15#874C411yescharacteristic
16#872C411yescharacteristic
17C3×#101C2212yescomplemented
18#836C2212yescomplemented
19#823—22no—
20#535—31no—
21#413—32no—
22#355—31no—
23#355—32no—
24#355D311yescharacteristic, complemented
25C23×D3—21no—
26C2×C3×Q8C2311yescharacteristic
27C2×C3×D4C2×C411yescharacteristic
28C2×#67—22no—
29C2×#67—21no—
30#239—42no—
31D3×D4—42no—
32C2×D12—22no—
33C2×D12C2×C411yescharacteristic
34C2×D12C2×C411yescharacteristic
35C2×D12D412yes—
36C2×C4×D3C2×C411yescharacteristic
37C2×C4×D3D412yes—
38C3×M16—22no—
39C3×C22⋊C4—22no—
40C3×C42C2×C411yescharacteristic
41(C2×D3)⋊C4—21no—
42(C2×D3)⋊C4—22no—
43Dic3⋊C4—21no—
44C4×Dic3C2×C411yescharacteristic
45#208—22no—
46ΓL(1,𝔽25)—42no—
47D3×C8—42no—
48C2×#41—31no—
49C22×D4—31no—
50C2×M16—62no—
51#127—31no—
52#127—31no—
53#124—31no—
54#124C2×D311yescharacteristic
55#121—62no—
56#120—62no—
57#117—31no—
58#101—32no—
59#101C2×D312yescomplemented
60#100—62no—
61C23×C3—21no—
62C22×D3—21no—
63C22×D3—42no—
64C22×D3—82no—
65C22×D3C22⋊C411yescharacteristic
66C22×D3C22⋊C412yes—
67C3×Q8—41no—
68C3×D4—41no—
69C2×C3×C4—21no—
70C2×C3×C4C22×C411yescharacteristic
71C2×C3×C4C2×D412yes—
72#67—42no—
73#67—82no—
74C2×Dic3—21no—
75C2×Dic3C22⋊C411yescharacteristic
76D12—41no—
77D12—42no—
78D12—41no—
79D12—82no—
80C4×D3—22no—
81C4×D3—22no—
82C4×D3—22no—
83C3×C8—42no—
84C3⋊C8—42no—
85C24—61no—
86#41—122no—
87C2×Q8C22×D311yescharacteristic
88C2×D4—31no—
89C2×D4—31no—
90C2×D4—32no—
91C2×D4—62no—
92C2×D4—61no—
93C2×D4—62no—
94C2×D4—122no—
95C2×D4C4×D311yescharacteristic
96C22×C4—31no—
97C22×C4—32no—
98M16—22no—
99M16—62no—
100M16—122no—
101C2×C8—122no—
102C4⋊C4—61no—
103C22⋊C4—22no—
104C22⋊C4—61no—
105C22⋊C4—62no—
106C42—31no—
107C42C4×D311yescharacteristic
108C22×C3—42no—
109C22×C3—41no—
110C22×C3C2×C22⋊C411yescharacteristic
111C2×D3—21no—
112C2×D3—22no—
113C2×D3—42no—
114C2×D3—41no—
115C2×D3—42no—
116C2×D3—41no—
117C2×D3—84no—
118C3×C4—22no—
119C3×C4—21no—
120C3×C4—41no—
121Dic3—41no—
122Dic3#9912yes—
123C23—21no—
124C23—31no—
125C23—32no—
126C23—61no—
127C23—122no—
128C23—242no—
129Q8—41no—
130D4—41no—
131D4—122no—
132D4—121no—
133D4—121no—
134D4—122no—
135D4—242no—
136D4—242no—
137C2×C4—21no—
138C2×C4—31no—
139C2×C4—62no—
140C2×C4—62no—
141C2×C4—62no—
142C2×C4—61no—
143C2×C4C2×C4×D311yescharacteristic
144C2×C4D3×D412yes—
145C8—42no—
146C8—122no—
147C2×C3—21no—
148C2×C3—81no—
149C2×C3C2×#9911yescharacteristic
150D3—42no—
151D3—41no—
152D3—81no—
153C22—42no—
154C22—41no—
155C22—61no—
156C22—62no—
157C22—122no—
158C22—121no—
159C22—122no—
160C22—121no—
161C22—244no—
162C22D3×C22⋊C411yescharacteristic
163C4—22no—
164C4—21no—
165C4—32no—
166C4—41no—
167C4—121no—
168C3#213411yescharacteristic, complemented, Sylow
169C2—21no—
170C2—81no—
171C2—121no—
172C2—122no—
173C2—241no—
174C2D3×#9911yescharacteristic
subgroupG/H
centerC2D3×#99
commutatorC2×C3×C4C22×C4
FrattiniC2×Q8C22×D3
FittingC3×#355C2
socleC2×C3C2×#99
prime radical#5674C2
cosocleC2×C3×Q8C23
hypercenter#355D3
hypocenterC3#2134
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #4751C2semidirect
3. C3×#355C2semidirect
4. #4542C2semidirect
5. #4477C2semidirect
17. C3×#101C22semidirect
18. #836C22semidirect
24. #355D3semidirect
59. #101C2×D3semidirect
168. C3#2134semidirect

Maximal subgroups

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

subgroupindex
#56742
#47512
C3×#3552
#45422
#44772
#21343