Finite Group Databasebeta

← #73002

Subgroups — #73002

Order 384.

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

normal subgroups only

This group has 120 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
2#5930C211yesmaximal, characteristic, complemented
3#395⋊C3C211yesmaximal, characteristic, complemented
4#1603—31nomaximal, Sylow
5C22⋊SL(2,𝔽3)C2211yescharacteristic
6C4×S4—41nomaximal
7#539—31no—
8#451—31no—
9#448—31no—
10#395D311yescharacteristic, complemented
11#328—31no—
12#327—31no—
13#324—31no—
14C2×S4—41no—
15C4×A4—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#125—61no—
24#123—61no—
25#123—61no—
26C22⋊Q8—31no—
27C22⋊Q8—61no—
28#121—31no—
29C4×D4—61no—
30C2×C4⋊C4—31no—
31#103—61no—
32#103—61no—
33C22⋊C8—31no—
34C22⋊C8—31no—
35#95—31no—
36C2×A4—41no—
37S4—42no—
38C4×D3—161no—
39SL(2,𝔽3)—81no—
40SL(2,𝔽3)—81no—
41C2×Q8—31no—
42C2×Q8—61no—
43C2×Q8—61no—
44C2×D4—31no—
45C2×D4—61no—
46C22×C4—31no—
47C22×C4—31no—
48C22×C4—61no—
49C22×C4S411yescharacteristic
50Q16—121no—
51Q16—121no—
52SD16—121no—
53SD16—121no—
54C2×C8—61no—
55C2×C8—61no—
56C4⋊C4—31no—
57C4⋊C4—31no—
58C4⋊C4—61no—
59C4⋊C4—61no—
60C4⋊C4—61no—
61C4⋊C4—61no—
62C4⋊C4—61no—
63C4⋊C4—61no—
64C22⋊C4—61no—
65C22⋊C4—61no—
66C22⋊C4—61no—
67C22⋊C4—61no—
68C22⋊C4—61no—
69C42—61no—
70C2×D3—161no—
71A4—41no—
72C3×C4—161no—
73Dic3—161no—
74C23—61no—
75C23C2×S411yescharacteristic
76Q8—21no—
77Q8—21no—
78Q8—61no—
79Q8—61no—
80Q8—121no—
81D4—121no—
82D4—121no—
83D4—121no—
84C2×C4—31no—
85C2×C4—31no—
86C2×C4—61no—
87C2×C4—61no—
88C2×C4—61no—
89C2×C4—61no—
90C2×C4—61no—
91C2×C4—61no—
92C2×C4—61no—
93C2×C4—121no—
94C2×C4—241no—
95C8—121no—
96C8—121no—
97C2×C3—161no—
98D3—162no—
99C22—31no—
100C22—31no—
101C22—121no—
102C22—121no—
103C22—121no—
104C22#98511yescharacteristic
105C4—41no—
106C4—61no—
107C4—61no—
108C4—121no—
109C4—121no—
110C4—121no—
111C4—121no—
112C3—161noSylow
113C2—31no—
114C2—31no—
115C2—241no—
116C2#1020⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2#1020⋊C2
commutator, cosocle, hypocenterC22⋊SL(2,𝔽3)C22
Frattini, socleC23C2×S4
Fitting#395D3
prime radical#5931C2
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. #5930C2semidirect
3. #395⋊C3C2semidirect
10. #395D3semidirect

Maximal subgroups

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

subgroupindex
#59312
#59302
#395⋊C32
#16033
C4×S44