Finite Group Databasebeta

← #85527

Subgroups — #85527

Order 384.

299 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 38 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
1C2×#994C211yesmaximal, characteristic
2C2×#984C211yesmaximal, characteristic, complemented
3#5425C214yesmaximal, complemented
4#5424C211yesmaximal, characteristic, complemented
5#3009—31nomaximal, Sylow
6#994—22no—
7C2×#41⋊C3C2212yes—
8C2×#41⋊C3C2211yescharacteristic
9#984—22no—
10C2×#226C2212yescomplemented
11#944—41nomaximal
12#859C2212yescomplemented
13C2×H(5,𝔽2)D311yescharacteristic
14C2×#137—31no—
15C22×SD16—31no—
16#536—31no—
17#452—34no—
18#450—34no—
19#418—31no—
20#416—31no—
21#408—31no—
22C2×C3×Q8—41no—
23C2×Dic6—41no—
24#41⋊C3—22no—
25#41⋊C3D412yes—
26C2×SL(2,𝔽3)C2311yescharacteristic
27#226—22no—
28Dic3⋊C4—44no—
29C4×Dic3—41no—
30H(5,𝔽2)—22no—
31H(5,𝔽2)—62no—
32C2×#41—31no—
33C2×#41C2×D312yes—
34C2×#41C2×D311yescharacteristic
35C22×Q8—31no—
36C22×D4—31no—
37C22×D4—32no—
38#137—62no—
39#137—62no—
40C2×Q16—32no—
41C2×SD16—34no—
42C2×SD16—32no—
43C2×SD16—64no—
44C2×M16—31no—
45C22×C8—31no—
46C4⋊Q8—61no—
47#124—61no—
48#123—62no—
49C22⋊Q8—34no—
50#117—31no—
51#103—32no—
52#102—32no—
53#102—34no—
54C22⋊C8—34no—
55C3×Q8—82no—
56C2×C3×C4—41no—
57C2×C3×C4—42no—
58C2×Dic3—42no—
59C2×Dic3—42no—
60Dic6—82no—
61SL(2,𝔽3)C2×D411yescharacteristic
62C24—61no—
63#41—22no—
64#41—32no—
65#41—62no—
66#41—62no—
67#41—62no—
68#41#6712yes—
69C2×Q8—32no—
70C2×Q8—62no—
71C2×Q8—62no—
72C2×Q8—61no—
73C2×Q8S411yescharacteristic
74C2×Q8C22×D311yescharacteristic
75C2×D4—31no—
76C2×D4—32no—
77C2×D4—34no—
78C2×D4—32no—
79C2×D4—31no—
80C2×D4—62no—
81C2×D4—64no—
82C2×D4—64no—
83C22×C4—31no—
84C22×C4—32no—
85C22×C4—61no—
86Q16—62no—
87Q16—62no—
88SD16—68no—
89SD16—64no—
90M16—62no—
91C2×C8—32no—
92C2×C8—32no—
93C2×C8—62no—
94C4⋊C4—32no—
95C4⋊C4—64no—
96C22⋊C4—61no—
97C22⋊C4—64no—
98C42—61no—
99C22×C3—41no—
100C3×C4—42no—
101C3×C4—82no—
102Dic3—44no—
103Dic3—82no—
104C23—31no—
105C23—32no—
106C23—64no—
107C23—242no—
108Q8—22no—
109Q8—31no—
110Q8—62no—
111Q8—62no—
112Q8—122no—
113Q8C2×#6711yescharacteristic
114D4—32no—
115D4—34no—
116D4—64no—
117D4—61no—
118D4—62no—
119D4—62no—
120C2×C4—31no—
121C2×C4—32no—
122C2×C4—32no—
123C2×C4—62no—
124C2×C4—62no—
125C2×C4—62no—
126C2×C4—62no—
127C2×C4—122no—
128C2×C4C2×S412yes—
129C2×C4C2×S411yescharacteristic
130C8—62no—
131C8—62no—
132C2×C3—42no—
133C2×C3—41no—
134C22—32no—
135C22—62no—
136C22—64no—
137C22—62no—
138C22—244no—
139C22C22×S411yescharacteristic
140C4—22no—
141C4—31no—
142C4—31no—
143C4—122no—
144C4—122no—
145C4GL(2,ℤ/4)12yes—
146C3—41noSylow
147C2—62no—
148C2—122no—
149C2#592412yes—
150C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
FittingC2×H(5,𝔽2)D3
prime radicalC2×#994C2
hypercenterC2×Q8S4
hypocenterSL(2,𝔽3)C2×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#984C2semidirect
3. #5425C2semidirect
4. #5424C2semidirect
10. C2×#226C22semidirect
12. #859C22semidirect

Maximal subgroups

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

subgroupindex
C2×#9942
C2×#9842
#54252
#54242
#30093
#9444