Finite Group Databasebeta

← #87522

Subgroups — #87522

Order 384.

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

normal subgroups only

This group has 273 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, complemented
2#5935C212yesmaximal, complemented
3#3021—31nomaximal, Sylow
4#997C2211yescharacteristic, complemented
5#986—42nomaximal
6C4⋊S4—41nomaximal
7#585D311yescharacteristic, complemented
8C2×AGL(1,ℤ/8)—32no—
9#535—31no—
10#454—34no—
11#453—32no—
12#453—32no—
13#421—32no—
14#413—31no—
15C2×S4—42no—
16#41⋊C3—42no—
17C4×A4—41no—
18C4×A4—81no—
19GL(2,𝔽3)—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×D4—31no—
28AGL(1,ℤ/8)—34no—
29AGL(1,ℤ/8)—34no—
30AGL(1,ℤ/8)—62no—
31AGL(1,ℤ/8)—62no—
32C2×SD16—62no—
33C2×D8—62no—
34C2×M16—32no—
35#127—32no—
36#124—32no—
37#121—61no—
38#121—61no—
39#120—61no—
40#120—61no—
41#117—31no—
42#104—34no—
43#104—34no—
44#101—32no—
45#100—32no—
46C2×A4—41no—
47C2×A4—81no—
48S4—82no—
49D12—161no—
50SL(2,𝔽3)—42no—
51C24—61no—
52#41—32no—
53#41—32no—
54#41—32no—
55#41—62no—
56#41—62no—
57#41—62no—
58#41—62no—
59#41—62no—
60#41S412yescomplemented
61C2×Q8—31no—
62C2×Q8—32no—
63C2×Q8—61no—
64C2×D4—32no—
65C2×D4—31no—
66C2×D4—31no—
67C2×D4—31no—
68C2×D4—32no—
69C2×D4—62no—
70C2×D4—62no—
71C2×D4—61no—
72C2×D4—61no—
73C2×D4—62no—
74C2×D4—61no—
75C2×D4—61no—
76C2×D4—62no—
77C2×D4—62no—
78C22×C4—21no—
79C22×C4—31no—
80C22×C4—31no—
81C22×C4—61no—
82C22×C4S411yescharacteristic
83SD16—64no—
84SD16—64no—
85D8—64no—
86D8—64no—
87M16—34no—
88M16—62no—
89C2×C8—62no—
90C4⋊C4—61no—
91C22⋊C4—62no—
92C22⋊C4—62no—
93C22⋊C4—61no—
94C42—32no—
95C2×D3—162no—
96A4—41no—
97A4—81no—
98C3×C4—161no—
99C23—21no—
100C23—31no—
101C23—31no—
102C23—61no—
103C23—62no—
104C23—122no—
105C23—242no—
106C23C2×S411yescharacteristic
107Q8—32no—
108Q8—62no—
109Q8C2×S412yes—
110D4—32no—
111D4—32no—
112D4—62no—
113D4—62no—
114D4—64no—
115D4—62no—
116D4—62no—
117D4—121no—
118D4—121no—
119D4—242no—
120D4—242no—
121C2×C4—31no—
122C2×C4—32no—
123C2×C4—31no—
124C2×C4—32no—
125C2×C4—31no—
126C2×C4—31no—
127C2×C4—61no—
128C2×C4—62no—
129C2×C4—61no—
130C2×C4—61no—
131C2×C4—61no—
132C2×C4—62no—
133C8—64no—
134C2×C3—161no—
135D3—322no—
136C22—31no—
137C22—32no—
138C22—41no—
139C22—61no—
140C22—81no—
141C22—121no—
142C22—121no—
143C22—122no—
144C22—122no—
145C22—122no—
146C22—241no—
147C22—244no—
148C4—32no—
149C4—31no—
150C4—61no—
151C4—122no—
152C4#102011yescharacteristic
153C3—161noSylow
154C2—61no—
155C2—62no—
156C2—121no—
157C2—242no—
158C2C2×#102011yescharacteristic
subgroupG/H
center, Frattini, socleC2C2×#1020
commutator, cosocle, hypocenter#997C22
Fitting#585D3
hypercenterC4#1020
LCSG ▷ #997
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #5950C2semidirect
2. #5935C2semidirect
4. #997C22semidirect
7. #585D3semidirect
60. #41S4semidirect

Maximal subgroups

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

subgroupindex
#59502
#59352
#30213
#9864
C4⋊S44