Finite Group Databasebeta

← #87518

Subgroups — #87518

Order 384.

253 conjugacy classes of subgroups; only the 16 normal subgroups are shown.

normal subgroups only

This group has 253 conjugacy classes of subgroups — too many to draw. Only its 16 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#5950C211yesmaximal, characteristic
2#5933C212yesmaximal, complemented
3#3022—31nomaximal, Sylow
4#997C2211yescharacteristic
5#984—42nomaximal
6A4⋊Q8—41nomaximal
7#585D311yescharacteristic
8C2×#137—32no—
9#536—31no—
10#456—34no—
11#455—32no—
12#455—32no—
13#421—32no—
14#413—31no—
15#41⋊C3—42no—
16C4×A4—41no—
17C4×A4—81no—
18SL(2,ℤ/4)—42no—
19#226—44no—
202−1+4—32no—
21H(5,𝔽2)—31no—
22H(5,𝔽2)—61no—
23H(5,𝔽2)C2×D311yescharacteristic
24C2×#41—31no—
25C2×#41—32no—
26C2×#41—61no—
27C22×Q8—31no—
28#137—34no—
29#137—34no—
30#137—62no—
31#137—62no—
32C2×Q16—62no—
33C2×SD16—62no—
34C2×M16—32no—
35C4⋊Q8—32no—
36#124—32no—
37#123—61no—
38#123—61no—
39C22⋊Q8—61no—
40C22⋊Q8—61no—
41#117—31no—
42#104—34no—
43#104—34no—
44#101—32no—
45#100—32no—
46C2×A4—41no—
47C2×A4—81no—
48Dic6—161no—
49SL(2,𝔽3)—42no—
50#41—32no—
51#41—32no—
52#41—32no—
53#41—62no—
54#41—62no—
55#41—62no—
56#41—62no—
57#41—62no—
58#41S412yes—
59C2×Q8—31no—
60C2×Q8—32no—
61C2×Q8—32no—
62C2×Q8—61no—
63C2×Q8—62no—
64C2×Q8—62no—
65C2×Q8—61no—
66C2×D4—32no—
67C2×D4—31no—
68C2×D4—31no—
69C2×D4—31no—
70C2×D4—61no—
71C2×D4—62no—
72C2×D4—61no—
73C2×D4—61no—
74C22×C4—21no—
75C22×C4—31no—
76C22×C4—31no—
77C22×C4—61no—
78C22×C4—61no—
79C22×C4S411yescharacteristic
80Q16—64no—
81Q16—64no—
82SD16—64no—
83SD16—64no—
84M16—34no—
85M16—62no—
86C2×C8—62no—
87C4⋊C4—61no—
88C4⋊C4—62no—
89C4⋊C4—62no—
90C22⋊C4—61no—
91C22⋊C4—62no—
92C22⋊C4—62no—
93C42—32no—
94A4—41no—
95A4—81no—
96C3×C4—161no—
97Dic3—162no—
98C23—21no—
99C23—31no—
100C23—31no—
101C23—61no—
102C23C2×S411yescharacteristic
103Q8—32no—
104Q8—62no—
105Q8—64no—
106Q8—121no—
107Q8—121no—
108Q8C2×S412yes—
109D4—32no—
110D4—32no—
111D4—62no—
112D4—62no—
113D4—62no—
114D4—62no—
115C2×C4—31no—
116C2×C4—32no—
117C2×C4—31no—
118C2×C4—32no—
119C2×C4—31no—
120C2×C4—31no—
121C2×C4—61no—
122C2×C4—62no—
123C2×C4—61no—
124C2×C4—61no—
125C2×C4—61no—
126C2×C4—62no—
127C2×C4—62no—
128C2×C4—122no—
129C8—64no—
130C2×C3—161no—
131C22—31no—
132C22—32no—
133C22—41no—
134C22—61no—
135C22—81no—
136C22—121no—
137C22—121no—
138C22—241no—
139C4—32no—
140C4—31no—
141C4—61no—
142C4—122no—
143C4—122no—
144C4#102011yescharacteristic
145C3—161noSylow
146C2—61no—
147C2—62no—
148C2—121no—
149C2C2×#102011yescharacteristic
subgroupG/H
center, Frattini, socleC2C2×#1020
commutator, cosocle, hypocenter#997C22
Fitting#585D3
prime radical#5950C2
hypercenterC4#1020
LCSG ▷ #997
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #5933C2semidirect

Maximal subgroups

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

subgroupindex
#59502
#59332
#30223
#9844
A4⋊Q84