Finite Group Databasebeta

← #87510

Subgroups — #87510

Order 384.

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

normal subgroups only

This group has 293 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, complemented
2#5931C212yesmaximal, complemented
3#3006—31nomaximal, Sylow
4C22⋊SL(2,𝔽3)C2211yescharacteristic, complemented
5#986—42nomaximal
6C4⋊S4—41nomaximal
7C22×#41D311yescharacteristic, complemented
8C2×AGL(1,ℤ/8)—32no—
9C2×#121—31no—
10#449—34no—
11#448—32no—
12#447—32no—
13#417—32no—
14#407—31no—
15C2×S4—42no—
16#41⋊C3—42no—
17#41⋊C3—81no—
18C4×A4—41no—
19GL(2,𝔽3)—44no—
20C2×#41—32no—
21C2×#41—32no—
22C2×#41—61no—
23C2×#41—61no—
24C22×Q8C2×D311yescharacteristic
25C22×D4—31no—
26C22×D4—31no—
27C22×D4—61no—
28C23×C4—31no—
29AGL(1,ℤ/8)—62no—
30AGL(1,ℤ/8)—62no—
31AGL(1,ℤ/8)—62no—
32AGL(1,ℤ/8)—62no—
33C2×SD16—34no—
34C2×D8—34no—
35C2×M16—32no—
36#121—32no—
37#121—32no—
38#121—61no—
39#121—61no—
40C2×C4⋊C4—31no—
41C2×C22⋊C4—61no—
42#103—34no—
43#102—34no—
44C22⋊C8—32no—
45C22⋊C8—32no—
46C2×A4—41no—
47S4—82no—
48D12—161no—
49SL(2,𝔽3)—42no—
50SL(2,𝔽3)—81no—
51C24—31no—
52C24—61no—
53#41—21no—
54#41—32no—
55#41—32no—
56#41—32no—
57#41—61no—
58#41—62no—
59#41—61no—
60#41—61no—
61#41—62no—
62#41S412yescomplemented
63C2×Q8—32no—
64C2×Q8—61no—
65C2×D4—32no—
66C2×D4—32no—
67C2×D4—61no—
68C2×D4—62no—
69C2×D4—62no—
70C2×D4—61no—
71C2×D4—62no—
72C2×D4—62no—
73C2×D4—62no—
74C2×D4—62no—
75C2×D4—122no—
76C2×D4—122no—
77C22×C4—31no—
78C22×C4—31no—
79C22×C4—31no—
80C22×C4—31no—
81C22×C4—31no—
82C22×C4—62no—
83C22×C4—62no—
84C22×C4—61no—
85C22×C4S411yescharacteristic
86SD16—64no—
87SD16—64no—
88D8—64no—
89D8—64no—
90M16—62no—
91M16—62no—
92C2×C8—34no—
93C4⋊C4—32no—
94C4⋊C4—61no—
95C22⋊C4—62no—
96C22⋊C4—62no—
97C2×D3—162no—
98A4—41no—
99C3×C4—161no—
100C23—31no—
101C23—31no—
102C23—62no—
103C23—61no—
104C23—62no—
105C23—61no—
106C23—121no—
107C23—122no—
108C23—122no—
109C23—122no—
110C23C2×S411yescharacteristic
111Q8—21no—
112Q8—32no—
113Q8—61no—
114Q8C2×S412yes—
115D4—32no—
116D4—32no—
117D4—62no—
118D4—64no—
119D4—61no—
120D4—62no—
121D4—61no—
122D4—121no—
123D4—122no—
124D4—124no—
125D4—122no—
126D4—121no—
127C2×C4—31no—
128C2×C4—32no—
129C2×C4—31no—
130C2×C4—32no—
131C2×C4—31no—
132C2×C4—31no—
133C2×C4—62no—
134C2×C4—62no—
135C2×C4—62no—
136C2×C4—61no—
137C2×C4—61no—
138C2×C4—62no—
139C2×C4—122no—
140C8—64no—
141C2×C3—161no—
142D3—322no—
143C22—31no—
144C22—31no—
145C22—32no—
146C22—61no—
147C22—61no—
148C22—61no—
149C22—61no—
150C22—61no—
151C22—122no—
152C22—122no—
153C22—122no—
154C22—122no—
155C22—122no—
156C22—244no—
157C22#98611yescharacteristic, complemented
158C4—32no—
159C4—31no—
160C4—61no—
161C4—122no—
162C4#102011yescharacteristic
163C3—161noSylow
164C2—31no—
165C2—31no—
166C2—62no—
167C2—121no—
168C2—242no—
169C2C2×#102011yescharacteristic
subgroupG/H
center, FrattiniC2C2×#1020
commutator, cosocle, hypocenterC22⋊SL(2,𝔽3)C22
FittingC22×#41D3
socleC23C2×S4
hypercenterC4#1020
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #5948C2semidirect
2. #5931C2semidirect
4. C22⋊SL(2,𝔽3)C22semidirect
7. C22×#41D3semidirect
62. #41S4semidirect
157. C22#986semidirect

Maximal subgroups

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

subgroupindex
#59482
#59312
#30063
#9864
C4⋊S44