Finite Group Databasebeta

← #85519

Subgroups — #85519

Order 384.

317 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C22×#41⋊C3C211yesmaximal, characteristic
2#5426C212yesmaximal, complemented
3#5424C212yesmaximal, complemented
4#5420C212yesmaximal, complemented
5#2896—31nomaximal, Sylow
6C2×#41⋊C3C414yescomplemented
7C2×#41⋊C3C2211yescharacteristic
8C2×#41⋊C3C2211yescharacteristic
9C22×SL(2,𝔽3)C2211yescharacteristic
10#926—41nomaximal
11#859C2214yescomplemented
12C22×#41D311yescharacteristic
13C22×M16—31no—
14C2×#117—31no—
15#419—32no—
16#418—32no—
17#417—32no—
18#417—32no—
19#416—32no—
20#416—32no—
21C22×C3×C4—41no—
22#41⋊C3C2×C412yes—
23#41⋊C3C2×C412yescomplemented
24C2×SL(2,𝔽3)C2×C412yescomplemented
25C2×SL(2,𝔽3)C2311yescharacteristic
26#217—42no—
27C4⋊Dic3—42no—
28C4×Dic3—42no—
29C2×#41—34no—
30C2×#41—31no—
31C2×#41—31no—
32C2×#41Dic314yescomplemented
33C2×#41C2×D311yescharacteristic
34C2×#41C2×D311yescharacteristic
35C22×Q8C2×D311yescharacteristic
36C22×D4—31no—
37C23×C4—31no—
38C2×M16—32no—
39C2×M16—32no—
40C2×M16—64no—
41C22×C8—32no—
42#117—32no—
43#117—32no—
44#117—61no—
45#117—61no—
46C2×C4⋊C4—32no—
47C2×C22⋊C4—61no—
48C2×C42—61no—
49#103—34no—
50#103—34no—
51#102—34no—
52#102—34no—
53C23×C3—41no—
54C2×C3×C4—44no—
55C2×C3×C4—41no—
56C2×C3×C4—41no—
57C2×Dic3—44no—
58SL(2,𝔽3)C22×C411yescharacteristic
59C24—31no—
60#41—34no—
61#41—34no—
62#41—32no—
63#41—32no—
64#41—62no—
65#41—62no—
66#41C2×Dic312yes—
67#41C2×Dic312yes—
68C2×Q8—31no—
69C2×Q8—32no—
70C2×Q8C2×Dic312yes—
71C2×Q8C22×D311yescharacteristic
72C2×D4—31no—
73C2×D4—31no—
74C2×D4—34no—
75C2×D4—62no—
76C2×D4—61no—
77C22×C4—31no—
78C22×C4—34no—
79C22×C4—34no—
80C22×C4—31no—
81C22×C4—31no—
82C22×C4—31no—
83C22×C4—31no—
84C22×C4—62no—
85C22×C4S411yescharacteristic
86M16—64no—
87M16—64no—
88C2×C8—34no—
89C2×C8—64no—
90C4⋊C4—34no—
91C4⋊C4—62no—
92C22⋊C4—62no—
93C22⋊C4—62no—
94C42—62no—
95C42—62no—
96C22×C3—42no—
97C22×C3—41no—
98C22×C3—82no—
99C3×C4—42no—
100C3×C4—42no—
101Dic3—84no—
102C23—31no—
103C23—31no—
104C23—34no—
105C23—122no—
106C23C2×S411yescharacteristic
107Q8—32no—
108Q8—31no—
109Q8—61no—
110Q8C22×Dic311yescharacteristic
111D4—32no—
112D4—32no—
113D4—61no—
114D4—61no—
115D4—64no—
116C2×C4—32no—
117C2×C4—32no—
118C2×C4—31no—
119C2×C4—31no—
120C2×C4—32no—
121C2×C4—32no—
122C2×C4—32no—
123C2×C4—32no—
124C2×C4—34no—
125C2×C4—34no—
126C2×C4—64no—
127C2×C4—124no—
128C2×C4SL(2,ℤ/4)14yes—
129C2×C4C2×S411yescharacteristic
130C2×C4C2×S411yescharacteristic
131C8—64no—
132C2×C3—42no—
133C2×C3—41no—
134C2×C3—82no—
135C22—22no—
136C22—32no—
137C22—32no—
138C22—62no—
139C22—62no—
140C22—124no—
141C22C2×SL(2,ℤ/4)12yes—
142C22C22×S411yescharacteristic
143C4—31no—
144C4—31no—
145C4—32no—
146C4—124no—
147C4C2×SL(2,ℤ/4)12yes—
148C4C2×SL(2,ℤ/4)12yes—
149C3—41noSylow
150C2—22no—
151C2—62no—
152C2—62no—
153C2#592312yes—
154C2C22×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC2×C4C2×S4
commutator, hypocenterSL(2,𝔽3)C22×C4
Frattini, socleC22C22×S4
FittingC22×#41D3
prime radicalC22×#41⋊C3C2
cosocleC2×SL(2,𝔽3)C23
hypercenterC22×C4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #5426C2semidirect
3. #5424C2semidirect
4. #5420C2semidirect
6. C2×#41⋊C3C4semidirect
11. #859C22semidirect
23. #41⋊C3C2×C4semidirect
24. C2×SL(2,𝔽3)C2×C4semidirect
32. C2×#41Dic3semidirect

Maximal subgroups

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

subgroupindex
C22×#41⋊C32
#54262
#54242
#54202
#28963
#9264