Finite Group Databasebeta

← #69643

Subgroups — #69643

Order 384.

282 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 69 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#5500C211yesmaximal, characteristic, complemented
2C3×#390C211yesmaximal, characteristic
3#5215C212yesmaximal
4#4530C211yesmaximal, characteristic, complemented
5#4523C212yesmaximal, complemented
6#2027—31nomaximal, Sylow
7C22×C3×D4C2211yescharacteristic
8C2×C3×C4⋊C4C2211yescharacteristic
9C2×C3×C22⋊C4—22no—
10C2×C3×C42C2211yescharacteristic
11#926C2212yes—
12#924—22no—
13C2×#208C2212yes—
14#890—22no—
15#888—22no—
16#832—22no—
17#832—22no—
18#544—31no—
19#418—32no—
20#390D311yescharacteristic
21#343—31no—
22#339—32no—
23C24×C3—21no—
24C2×C3×D4—22no—
25C2×C3×D4—22no—
26C2×C3×D4—21no—
27C2×C3×D4Q812yescomplemented
28C22×C3×C4—21no—
29C22×C3×C4D412yes—
30C22×C3×C4C2311yescharacteristic
31C2×Dic6—22no—
32C3×C4⋊C4—22no—
33C3×C22⋊C4—44no—
34C3×C42—22no—
35#217—22no—
36C4⋊Dic3—22no—
37C4⋊Dic3D412yes—
38C4⋊Dic3D412yes—
39Dic3⋊C4—24no—
40C4×Dic3—22no—
41#208—22no—
42#208—22no—
43C2×C3⋊C8—22no—
44C22×D4C2×D311yescharacteristic
45C2×M16—32no—
46C4⋊Q8—62no—
47#125—62no—
48C22⋊Q8—62no—
49#117—32no—
50C2×C4⋊C4C2×D311yescharacteristic
51C2×C22⋊C4—22no—
52C2×C42C2×D311yescharacteristic
53#102—62no—
54#102—62no—
55C23×C3—22no—
56C23×C3—42no—
57C23×C3—42no—
58C23×C3—41no—
59C23×C3—41no—
60C23×C3C2×D411yescharacteristic
61C3×D4—24no—
62C3×D4—42no—
63C2×C3×C4—22no—
64C2×C3×C4—22no—
65C2×C3×C4—41no—
66C2×C3×C4—41no—
67C2×C3×C4—42no—
68C2×C3×C4C2×D412yes—
69C2×C3×C4C2×Q811yescharacteristic
70C2×C3×C4#4111yescharacteristic
71C2×C3×C4#4112yes—
72C2×Dic3—22no—
73C2×Dic3—22no—
74Dic6—82no—
75C3⋊C8—42no—
76C24—21no—
77C2×Q8—62no—
78C2×D4—21no—
79C2×D4—22no—
80C2×D4—22no—
81C2×D4Dic612yescomplemented
82C22×C4—21no—
83C22×C4#6712yes—
84C22×C4C22×D311yescharacteristic
85M16—62no—
86M16—62no—
87C2×C8—62no—
88C4⋊C4—22no—
89C4⋊C4—32no—
90C4⋊C4—32no—
91C4⋊C4—62no—
92C4⋊C4—64no—
93C22⋊C4—44no—
94C22⋊C4—62no—
95C42—22no—
96C42—62no—
97C22×C3—21no—
98C22×C3—21no—
99C22×C3—44no—
100C22×C3—42no—
101C22×C3—82no—
102C22×C3—81no—
103C22×C3—81no—
104C22×C3#12011yescharacteristic
105C22×C3C22⋊Q811yescharacteristic
106C22×C3#12411yescharacteristic
107C3×C4—42no—
108C3×C4—42no—
109C3×C4C22⋊Q812yes—
110C3×C4#12412yes—
111Dic3—42no—
112Dic3—42no—
113C23—22no—
114C23—42no—
115C23—42no—
116C23—41no—
117C23—41no—
118C23C2×#6711yescharacteristic
119Q8—242no—
120D4—24no—
121D4—42no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—42no—
127C2×C4—62no—
128C2×C4—62no—
129C2×C4C2×Dic611yescharacteristic
130C2×C4#23511yescharacteristic
131C2×C4D3×D412yes—
132C2×C4#23712yes—
133C8—122no—
134C2×C3—21no—
135C2×C3—21no—
136C2×C3—82no—
137C2×C3#39311yescharacteristic
138C2×C3#45311yescharacteristic
139C2×C3#45411yescharacteristic
140C22—21no—
141C22—21no—
142C22—44no—
143C22—42no—
144C22—82no—
145C22—81no—
146C22—81no—
147C22#92411yescharacteristic
148C22#93611yescharacteristic
149C22#93711yescharacteristic
150C4—42no—
151C4—42no—
152C4—122no—
153C4—122no—
154C4#87812yes—
155C4#88512yes—
156C3#202711yescharacteristic, complemented, Sylow
157C2—21no—
158C2—21no—
159C2—82no—
160C2#494711yescharacteristic
161C2#506211yescharacteristic
162C2#507711yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#390C2
socleC22×C3#120
prime radicalC22×C3×D4C22
hypercenter#390D3
hypocenterC3#2027
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #5500C2semidirect
4. #4530C2semidirect
5. #4523C2semidirect
27. C2×C3×D4Q8semidirect
81. C2×D4Dic6semidirect
156. C3#2027semidirect

Maximal subgroups

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

subgroupindex
#55002
C3×#3902
#52152
#45302
#45232
#20273