Finite Group Databasebeta

← #87508

Subgroups — #87508

Order 384.

263 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 17 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#5948C211yesmaximal, characteristic
2#5930C212yesmaximal, complemented
3#3007—31nomaximal, Sylow
4C22⋊SL(2,𝔽3)C2211yescharacteristic
5#984—42nomaximal
6A4⋊Q8—41nomaximal
7C22×#41D311yescharacteristic
8C2×#137—32no—
9C2×C22⋊Q8—31no—
10#452—34no—
11#451—32no—
12#450—32no—
13#417—32no—
14#407—31no—
15#41⋊C3—42no—
16#41⋊C3—81no—
17C4×A4—41no—
18SL(2,ℤ/4)—42no—
19#226—44no—
20C2×#41—32no—
21C2×#41—32no—
22C2×#41—61no—
23C2×#41—61no—
24C22×Q8—31no—
25C22×Q8C2×D311yescharacteristic
26C22×D4—31no—
27C23×C4—31no—
28#137—62no—
29#137—62no—
30#137—62no—
31#137—62no—
32C2×Q16—34no—
33C2×SD16—34no—
34C2×M16—32no—
35C22⋊Q8—32no—
36C22⋊Q8—32no—
37C22⋊Q8—61no—
38C22⋊Q8—61no—
39C2×C4⋊C4—31no—
40C2×C4⋊C4—61no—
41C2×C22⋊C4—61no—
42#103—34no—
43#102—34no—
44C22⋊C8—32no—
45C22⋊C8—32no—
46C2×A4—41no—
47Dic6—161no—
48SL(2,𝔽3)—42no—
49SL(2,𝔽3)—81no—
50C24—31no—
51#41—21no—
52#41—32no—
53#41—32no—
54#41—32no—
55#41—61no—
56#41—62no—
57#41—61no—
58#41—61no—
59#41—62no—
60#41S412yes—
61C2×Q8—32no—
62C2×Q8—32no—
63C2×Q8—61no—
64C2×Q8—61no—
65C2×Q8—62no—
66C2×Q8—62no—
67C2×D4—32no—
68C2×D4—61no—
69C2×D4—62no—
70C2×D4—62no—
71C22×C4—31no—
72C22×C4—31no—
73C22×C4—31no—
74C22×C4—31no—
75C22×C4—31no—
76C22×C4—62no—
77C22×C4—62no—
78C22×C4—61no—
79C22×C4—61no—
80C22×C4S411yescharacteristic
81Q16—64no—
82Q16—64no—
83SD16—64no—
84SD16—64no—
85M16—62no—
86M16—62no—
87C2×C8—34no—
88C4⋊C4—32no—
89C4⋊C4—61no—
90C4⋊C4—62no—
91C4⋊C4—62no—
92C22⋊C4—62no—
93C22⋊C4—62no—
94A4—41no—
95C3×C4—161no—
96Dic3—162no—
97C23—31no—
98C23—31no—
99C23—62no—
100C23—61no—
101C23—61no—
102C23—121no—
103C23C2×S411yescharacteristic
104Q8—21no—
105Q8—32no—
106Q8—61no—
107Q8—64no—
108Q8—121no—
109Q8—121no—
110Q8C2×S412yes—
111D4—32no—
112D4—32no—
113D4—62no—
114D4—61no—
115D4—62no—
116D4—61no—
117C2×C4—31no—
118C2×C4—32no—
119C2×C4—31no—
120C2×C4—32no—
121C2×C4—31no—
122C2×C4—31no—
123C2×C4—62no—
124C2×C4—62no—
125C2×C4—62no—
126C2×C4—61no—
127C2×C4—61no—
128C2×C4—62no—
129C2×C4—62no—
130C2×C4—122no—
131C2×C4—122no—
132C8—64no—
133C2×C3—161no—
134C22—31no—
135C22—31no—
136C22—32no—
137C22—61no—
138C22—61no—
139C22—61no—
140C22—61no—
141C22—61no—
142C22—122no—
143C22—122no—
144C22#98411yescharacteristic, complemented
145C4—32no—
146C4—31no—
147C4—61no—
148C4—122no—
149C4—122no—
150C4#102011yescharacteristic
151C3—161noSylow
152C2—31no—
153C2—31no—
154C2—62no—
155C2—121no—
156C2C2×#102011yescharacteristic
subgroupG/H
center, FrattiniC2C2×#1020
commutator, cosocle, hypocenterC22⋊SL(2,𝔽3)C22
FittingC22×#41D3
socleC23C2×S4
prime radical#5948C2
hypercenterC4#1020
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #5930C2semidirect
144. C22#984semidirect

Maximal subgroups

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

subgroupindex
#59482
#59302
#30073
#9844
A4⋊Q84