Finite Group Databasebeta

← #70105

Subgroups — #70105

Order 384.

314 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 85 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#5816C211yesmaximal, characteristic
2#5812C211yesmaximal, characteristic, complemented
3C2×C3×#101C211yesmaximal, characteristic, complemented
4#5139C212yesmaximal
5#4557C212yesmaximal, complemented
6#1912—31nomaximal, Sylow
7C22×C3×Q8C2211yescharacteristic
8C2×#235C2211yescharacteristic
9C2×C3×M16C2212yes—
10#947C412yescomplemented
11#946—22no—
12Q8×Dic3—24no—
13#944C412yescomplemented
14#929—22no—
15#926—22no—
16#926C2211yescharacteristic
17#926C2212yes—
18C3×#101—22no—
19#825—24no—
20#536—31no—
21#519—31no—
22#421—32no—
23C2×#101D311yescharacteristic, complemented
24H(3,ℤ/4)—32no—
25C2×C3×Q8—22no—
26C2×C3×Q8—22no—
27C2×C3×Q8C2×C412yes—
28C2×C3×Q8D412yes—
29C22×C3×C4D412yes—
30C22×C3×C4C2311yescharacteristic
31C2×#67—21no—
32#235—22no—
33#235—22no—
34C2×D12C2×C411yescharacteristic
35C2×C4×D3—21no—
36C2×Dic6C2×C411yescharacteristic
37C3×M16—22no—
38C3×M16—22no—
39C2×C3×C8—22no—
40#217—22no—
41#217—21no—
42(C2×D3)⋊C4—24no—
43C4⋊Dic3—24no—
44C4⋊Dic3—22no—
45C4⋊Dic3—21no—
46Dic3⋊C4—24no—
47C4×Dic3—24no—
48C4×Dic3C2×C412yes—
49C4×Dic3D414yes—
50C2×#41—31no—
51C22×Q8C2×D311yescharacteristic
52C2×M16C2×D312yes—
53C4⋊Q8—32no—
54#124—32no—
55#123—62no—
56C22⋊Q8—62no—
57C4×Q8—64no—
58#117—32no—
59#117—31no—
60#117—62no—
61#104—64no—
62#101—22no—
63C23×C3C2×D411yescharacteristic
64C22×D3—21no—
65C3×Q8—24no—
66C3×Q8—24no—
67C2×C3×C4—22no—
68C2×C3×C4—22no—
69C2×C3×C4C22⋊C414yes—
70C2×C3×C4C22×C411yescharacteristic
71C2×C3×C4C2×D412yes—
72C2×C3×C4C2×D411yescharacteristic
73C2×C3×C4#4112yes—
74#67—41no—
75#67—41no—
76C2×Dic3—22no—
77C2×Dic3—21no—
78C2×Dic3—24no—
79D12—22no—
80C4×D3—42no—
81Dic6—22no—
82C3×C8—42no—
83#41—62no—
84#41—62no—
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8—31no—
88C2×Q8C4×D312yes—
89C2×Q8#6712yes—
90C2×D4—31no—
91C2×D4—61no—
92C22×C4—61no—
93C22×C4D1212yes—
94C22×C4C22×D311yescharacteristic
95M16—22no—
96M16—22no—
97C2×C8—22no—
98C4⋊C4—64no—
99C4⋊C4—64no—
100C4⋊C4—62no—
101C4⋊C4—61no—
102C22⋊C4—62no—
103C22⋊C4—61no—
104C22⋊C4—64no—
105C42—32no—
106C42—34no—
107C42—64no—
108C22×C3—41no—
109C22×C3C2×C22⋊C411yescharacteristic
110C22×C3#12011yescharacteristic
111C22×C3#12311yescharacteristic
112C2×D3—42no—
113C2×D3—41no—
114C3×C4—24no—
115C3×C4C4×D412yes—
116C3×C4#12112yes—
117Dic3—44no—
118Dic3—42no—
119Dic3—41no—
120C23—61no—
121C23C2×D1211yescharacteristic
122Q8—24no—
123Q8—24no—
124Q8—62no—
125D4—62no—
126D4—121no—
127D4—121no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—64no—
131C2×C4—62no—
132C2×C4—61no—
133C2×C4—122no—
134C2×C4(C2×D3)⋊C414yes—
135C2×C4C2×C4×D311yescharacteristic
136C2×C4D3×D412yes—
137C2×C4#23712yes—
138C2×C4C2×#6711yescharacteristic
139C8—42no—
140C2×C3—21no—
141C2×C3—21no—
142C2×C3—21no—
143C2×C3#38611yescharacteristic
144D3—81no—
145C22—41no—
146C22—122no—
147C22—121no—
148C22#88211yescharacteristic
149C22#88611yescharacteristic
150C22C2×(C2×D3)⋊C411yescharacteristic
151C4—24no—
152C4—124no—
153C4—122no—
154C4—121no—
155C4#67⋊C412yes—
156C4#93912yes—
157C3#191211yescharacteristic, complemented, Sylow
158C2—21no—
159C2—21no—
160C2—21no—
161C2—241no—
162C2#495811yescharacteristic
subgroupG/H
centerC2#4958
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC2×C3×#101C2
socleC2×C3#386
prime radicalC2×#235C22
cosocleC22×C3×C4C23
hypercenterC2×#101D3
hypocenterC3#1912
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. #5812C2semidirect
3. C2×C3×#101C2semidirect
5. #4557C2semidirect
10. #947C4semidirect
13. #944C4semidirect
23. C2×#101D3semidirect
157. C3#1912semidirect

Maximal subgroups

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

subgroupindex
#58162
#58122
C2×C3×#1012
#51392
#45572
#19123