Finite Group Databasebeta

← #73000

Subgroups — #73000

Order 384.

126 conjugacy classes of subgroups; only the 12 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 12 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#5931C211yesmaximal, characteristic, complemented
2#5930C211yesmaximal, characteristic, complemented
3#393⋊C3C211yesmaximal, characteristic, complemented
4#1606—31nomaximal, Sylow
5C22⋊SL(2,𝔽3)C2211yescharacteristic
6GL(2,ℤ/4)—41nomaximal
7#538—31no—
8#451—31no—
9#448—31no—
10#393D311yescharacteristic, complemented
11#328—31no—
12#327—31no—
13#323—31no—
14C22×A4—41no—
15C2×S4—41no—
16SL(2,ℤ/4)—41no—
17GL(2,𝔽3)—81no—
18#226—81no—
19C22×Q8C2×D311yescharacteristic
20C2×Q16—61no—
21C2×SD16—61no—
22#126—61no—
23#124—61no—
24#123—61no—
25C22⋊Q8—31no—
26#121—31no—
27#121—61no—
28#120—61no—
29C4×D4—61no—
30C2×C22⋊C4—31no—
31#103—61no—
32#103—61no—
33C22⋊C8—31no—
34C22⋊C8—31no—
35#95—31no—
36C2×A4—41no—
37C2×A4—81no—
38S4—81no—
39#67—161no—
40SL(2,𝔽3)—81no—
41SL(2,𝔽3)—81no—
42C24S411yescharacteristic
43C2×Q8—31no—
44C2×Q8—61no—
45C2×Q8—61no—
46C2×D4—31no—
47C2×D4—61no—
48C2×D4—61no—
49C2×D4—61no—
50C22×C4—31no—
51C22×C4—31no—
52C22×C4—61no—
53Q16—121no—
54Q16—121no—
55SD16—121no—
56SD16—121no—
57C2×C8—61no—
58C2×C8—61no—
59C4⋊C4—31no—
60C4⋊C4—31no—
61C4⋊C4—61no—
62C4⋊C4—61no—
63C22⋊C4—61no—
64C22⋊C4—61no—
65C22⋊C4—61no—
66C22⋊C4—61no—
67C22⋊C4—61no—
68C22⋊C4—61no—
69C22⋊C4—61no—
70C42—61no—
71C22×C3—161no—
72C2×D3—161no—
73A4—41no—
74Dic3—161no—
75C23—21no—
76C23—61no—
77C23—61no—
78C23—61no—
79C23C2×S411yescharacteristic
80Q8—21no—
81Q8—21no—
82Q8—61no—
83Q8—61no—
84Q8—121no—
85D4—121no—
86D4—121no—
87D4—121no—
88D4—241no—
89D4—241no—
90C2×C4—31no—
91C2×C4—31no—
92C2×C4—61no—
93C2×C4—61no—
94C2×C4—61no—
95C2×C4—61no—
96C2×C4—121no—
97C2×C4—121no—
98C2×C4—241no—
99C8—121no—
100C8—121no—
101C2×C3—161no—
102C2×C3—321no—
103D3—321no—
104C22—31no—
105C22—31no—
106C22—41no—
107C22—121no—
108C22—121no—
109C22—121no—
110C22—121no—
111C22—121no—
112C22#98311yescharacteristic
113C4—61no—
114C4—61no—
115C4—121no—
116C4—121no—
117C4—121no—
118C4—121no—
119C3—161noSylow
120C2—31no—
121C2—31no—
122C2—81no—
123C2—241no—
124C2#1020⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2#1020⋊C2
commutator, cosocle, hypocenterC22⋊SL(2,𝔽3)C22
Frattini, socleC23C2×S4
Fitting#393D3
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #5931C2semidirect
2. #5930C2semidirect
3. #393⋊C3C2semidirect
10. #393D3semidirect

Maximal subgroups

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

subgroupindex
#59312
#59302
#393⋊C32
#16063
GL(2,ℤ/4)4