Finite Group Databasebeta

← #69164

Subgroups — #69164

Order 384.

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

normal subgroups only

This group has 230 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#5675C211yesmaximal, characteristic
2D3×#101C212yesmaximal, complemented
3C3×#355C211yesmaximal, characteristic, complemented
4#4542C211yesmaximal, characteristic, complemented
5#4478C212yesmaximal, complemented
6#2133—31nomaximal, Sylow
7C2×D3×Q8C2211yescharacteristic
8C2×#237C411yescharacteristic
9C3×#124C2211yescharacteristic
10#944C2211yescharacteristic
11#936C411yescharacteristic
12D3×M16—22no—
13#883—22no—
14#878—22no—
15#872C411yescharacteristic
16#869C411yescharacteristic
17C3×#101C2212yes—
18#836C2212yes—
19#824—22no—
20#536—31no—
21C2×#101—32no—
22#356—32no—
23#355—31no—
24#355D311yescharacteristic, complemented
25C2×C3×Q8C2311yescharacteristic
26C2×C3×D4C2×C411yescharacteristic
27C2×#67—21no—
28C22×Dic3—21no—
29D3×Q8—42no—
30#237—42no—
31C2×C4×D3C2×C411yescharacteristic
32C2×C4×D3D412yes—
33C2×Dic6C2×C411yescharacteristic
34C2×Dic6C2×C411yescharacteristic
35C2×Dic6D412yes—
36C3×M16—22no—
37C3×C22⋊C4—22no—
38C3×C42C2×C411yescharacteristic
39#217—22no—
40(C2×D3)⋊C4—21no—
41C4⋊Dic3—22no—
42Dic3⋊C4—22no—
43Dic3⋊C4—21no—
44C4×Dic3C2×C411yescharacteristic
45#208—22no—
46ΓL(1,𝔽25)—42no—
47D3×C8—42no—
48C2×#41—31no—
49C22×Q8—31no—
50C2×M16—62no—
51C4⋊Q8—31no—
52C4⋊Q8—31no—
53#124—31no—
54#124C2×D311yescharacteristic
55#123—62no—
56C22⋊Q8—62no—
57#117—31no—
58#101—32no—
59#101—62no—
60#101C2×D312yes—
61C23×C3—21no—
62C22×D3C22⋊C411yescharacteristic
63C3×Q8—41no—
64C3×D4—41no—
65C2×C3×C4—21no—
66C2×C3×C4C22×C411yescharacteristic
67C2×C3×C4C2×D412yes—
68#67—42no—
69C2×Dic3—21no—
70C2×Dic3—21no—
71C2×Dic3—42no—
72C2×Dic3C22⋊C411yescharacteristic
73C2×Dic3C22⋊C412yes—
74C4×D3—22no—
75C4×D3—22no—
76C4×D3—22no—
77Dic6—41no—
78Dic6—42no—
79Dic6—41no—
80C3×C8—42no—
81C3⋊C8—42no—
82#41—122no—
83C2×Q8—31no—
84C2×Q8—32no—
85C2×Q8—31no—
86C2×Q8—122no—
87C2×Q8C22×D311yescharacteristic
88C2×D4—61no—
89C2×D4C4×D311yescharacteristic
90C22×C4—31no—
91C22×C4—32no—
92C22×C4—61no—
93M16—22no—
94M16—62no—
95M16—122no—
96C2×C8—122no—
97C4⋊C4—61no—
98C4⋊C4—62no—
99C4⋊C4—62no—
100C22⋊C4—22no—
101C22⋊C4—61no—
102C22⋊C4—62no—
103C42—31no—
104C42C4×D311yescharacteristic
105C22×C3—42no—
106C22×C3—41no—
107C22×C3C2×C22⋊C411yescharacteristic
108C2×D3—41no—
109C2×D3#9912yes—
110C3×C4—22no—
111C3×C4—21no—
112C3×C4—41no—
113Dic3—21no—
114Dic3—22no—
115Dic3—41no—
116Dic3—41no—
117C23—21no—
118C23—31no—
119Q8—41no—
120Q8—122no—
121Q8—121no—
122Q8—121no—
123D4—41no—
124D4—122no—
125C2×C4—21no—
126C2×C4—32no—
127C2×C4—31no—
128C2×C4—62no—
129C2×C4—62no—
130C2×C4—62no—
131C2×C4—61no—
132C2×C4—61no—
133C2×C4—122no—
134C2×C4C2×C4×D311yescharacteristic
135C2×C4D3×D412yes—
136C8—42no—
137C8—122no—
138C2×C3—21no—
139C2×C3—81no—
140C2×C3C2×#9911yescharacteristic
141D3—22no—
142C22—32no—
143C22—42no—
144C22—41no—
145C22—121no—
146C22D3×C22⋊C411yescharacteristic
147C4—22no—
148C4—21no—
149C4—41no—
150C4—61no—
151C4—62no—
152C4—121no—
153C4—121no—
154C3#213311yescharacteristic, complemented, Sylow
155C2—21no—
156C2—62no—
157C2—81no—
158C2D3×#9911yescharacteristic
subgroupG/H
centerC2D3×#99
commutatorC2×C3×C4C22×C4
FrattiniC2×Q8C22×D3
FittingC3×#355C2
socleC2×C3C2×#99
prime radicalC2×#237C4
cosocleC2×C3×Q8C23
hypercenter#355D3
hypocenterC3#2133
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. D3×#101C2semidirect
3. C3×#355C2semidirect
4. #4542C2semidirect
5. #4478C2semidirect
24. #355D3semidirect
154. C3#2133semidirect

Maximal subgroups

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

subgroupindex
#56752
D3×#1012
C3×#3552
#45422
#44782
#21333