Finite Group Databasebeta

← #5923

Subgroups — #5923

Order 192.

144 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 27 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×#41⋊C3C211yesmaximal, characteristic, complemented
2#986C211yesmaximal, characteristic, complemented
3#985C212yesmaximal, complemented
4#984C211yesmaximal, characteristic, complemented
5#983C212yesmaximal, complemented
6#575—31nomaximal, Sylow
7#235—41nomaximal
8#41⋊C3C2211yescharacteristic
9#41⋊C3C2211yescharacteristic, complemented
10C2×SL(2,𝔽3)C2211yescharacteristic, complemented
11GL(2,𝔽3)C2212yescomplemented
12#226C2212yescomplemented
13C2×#41—31no—
14C2×#41D311yescharacteristic, complemented
15#137—31no—
16#137—31no—
17#137—32no—
18AGL(1,ℤ/8)—32no—
19AGL(1,ℤ/8)—31no—
20AGL(1,ℤ/8)—31no—
21#135—32no—
22#135—32no—
23C2×M16—31no—
24C2×C3×C4—41no—
25#67—42no—
26D12—41no—
27C4×D3—42no—
28Dic6—41no—
29SL(2,𝔽3)C2311yescharacteristic
30#41—31no—
31#41—31no—
32#41—32no—
33#41—32no—
34#41—61no—
35#41—61no—
36#41C2×D311yescharacteristic
37#41C2×D311yescharacteristic
38C2×Q8—31no—
39C2×Q8C2×D311yescharacteristic
40C2×D4—31no—
41C2×D4—31no—
42C2×D4—61no—
43C22×C4—31no—
44C22×C4—61no—
45Q16—32no—
46Q16—32no—
47SD16—32no—
48SD16—32no—
49SD16—32no—
50SD16—32no—
51D8—32no—
52D8—32no—
53M16—32no—
54M16—31no—
55M16—31no—
56C2×C8—32no—
57C22×C3—41no—
58C2×D3—42no—
59C3×C4—41no—
60C3×C4—41no—
61Dic3—42no—
62C23—31no—
63C23—61no—
64Q8—31no—
65Q8—32no—
66Q8—61no—
67Q8C22×D311yescharacteristic
68D4—31no—
69D4—31no—
70D4—32no—
71D4—61no—
72D4—62no—
73D4—62no—
74D4—61no—
75C2×C4—31no—
76C2×C4—31no—
77C2×C4—31no—
78C2×C4—31no—
79C2×C4—31no—
80C2×C4—62no—
81C2×C4—61no—
82C2×C4—62no—
83C2×C4S411yescharacteristic
84C8—32no—
85C8—32no—
86C2×C3—41no—
87C2×C3—81no—
88D3—82no—
89C22—31no—
90C22—31no—
91C22—62no—
92C22—121no—
93C22—122no—
94C22C2×S411yescharacteristic
95C4—31no—
96C4—31no—
97C4—62no—
98C4C2×S411yescharacteristic
99C4C2×S411yescharacteristic
100C3—41noSylow
101C2—21no—
102C2—61no—
103C2—61no—
104C2—122no—
105C2C22×S411yescharacteristic
subgroupG/H
centerC4C2×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C23
Frattini, socleC2C22×S4
FittingC2×#41D3
hypercenterC2×C4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#41⋊C3C2semidirect
2. #986C2semidirect
3. #985C2semidirect
4. #984C2semidirect
5. #983C2semidirect
9. #41⋊C3C22semidirect
10. C2×SL(2,𝔽3)C22semidirect
11. GL(2,𝔽3)C22semidirect
12. #226C22semidirect
14. C2×#41D3semidirect

Maximal subgroups

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

subgroupindex
C2×#41⋊C32
#9862
#9852
#9842
#9832
#5753
#2354