Finite Group Databasebeta

← #5964

Subgroups — #5964

Order 192.

810 conjugacy classes of subgroups; only the 443 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 443 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×C3×#41C211yesmaximal, characteristic, complemented
2#1010C214yesmaximal, complemented
3#1009C214yesmaximal, complemented
4D3×#41C218yesmaximal, complemented
5#1007C212yesmaximal, complemented
6#1004C216yesmaximal, complemented
7C2×#235C216yesmaximal, complemented
8#585—31nomaximal, Sylow
9C3×#41C2214yescomplemented
10C3×#41C2214yescomplemented
11C2×C3×Q8C2211yescharacteristic, complemented
12C2×C3×D4C2213yescomplemented
13C22×C3×C4C2213yescomplemented
14C2×#67C22112yescomplemented
15#239C2218yescomplemented
16D3×Q8C2218yescomplemented
17#237C22124yescomplemented
18D3×D4C22124yescomplemented
19#235C2214yescomplemented
20#235C22112yescomplemented
21#235C22112yescomplemented
22#235C22112yescomplemented
23C2×D12C2216yescomplemented
24C2×C4×D3C22112yescomplemented
25C2×Dic6C2216yescomplemented
262−1+4—34no—
272−1+4—32no—
28H(5,𝔽2)—34no—
29H(5,𝔽2)—36no—
30C2×#41—38no—
31C2×#41—36no—
32C2×#41D311yescharacteristic, complemented
33C23×C3C2313yes—
34C22×D3C23112yes—
35C3×Q8C2314yes—
36C3×D4C23112yes—
37C2×C3×C4C2316yes—
38C2×C3×C4C2316yes—
39C2×C3×C4C2311yescharacteristic
40C2×C3×C4C2313yes—
41#67C2318yes—
42#67C23124yes—
43C2×Dic3C23112yes—
44D12C2314yes—
45D12C23112yes—
46C4×D3C2318yes—
47C4×D3C23124yes—
48Dic6C2314yes—
49Dic6C23112yes—
50#41—38no—
51#41—324no—
52#41—312no—
53#41—34no—
54#41—312no—
55#41—312no—
56#41C2×D314yescomplemented
57#41C2×D314yescomplemented
58C2×Q8—38no—
59C2×Q8—36no—
60C2×Q8C2×D311yescharacteristic, complemented
61C2×D4—312no—
62C2×D4—324no—
63C2×D4—36no—
64C2×D4C2×D313yescomplemented
65C22×C4—312no—
66C22×C4C2×D313yescomplemented
67C22×C3—43no—
68C22×C3C2416yes—
69C22×C3C2411yescharacteristic
70C2×D3—412no—
71C2×D3C2418yes—
72C3×C4C2411yescharacteristic
73C3×C4C2416yes—
74C3×C4C2411yescharacteristic
75Dic3C2418yes—
76C23—312no—
77C23C22×D313yes—
78Q8—312no—
79Q8—34no—
80Q8C22×D314yes—
81D4—324no—
82D4—38no—
83D4—312no—
84D4—34no—
85D4C22×D3112yes—
86C2×C4—312no—
87C2×C4—324no—
88C2×C4—38no—
89C2×C4C22×D316yes—
90C2×C4C22×D313yes—
91C2×C4C22×D316yes—
92C2×C4C22×D311yescharacteristic
93C2×C3—26no—
94C2×C3—21no—
95C2×C3C2511yescharacteristic
96D3—28no—
97C22—38no—
98C22—43no—
99C22—1212no—
100C22C23×D316yes—
101C22C23×D311yescharacteristic
102C4—38no—
103C4C23×D316yes—
104C4C23×D311yescharacteristic
105C4C23×D311yescharacteristic
106C3#58511yescharacteristic, complemented, Sylow
107C2—21no—
108C2—26no—
109C2—68no—
110C2C24×D311yescharacteristic
subgroupG/H
centerC4C23×D3
commutator, socle, cosocleC2×C3C25
FrattiniC2C24×D3
FittingC2×C3×#41C2
hypercenterC2×#41D3
hypocenterC3#585
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×C3×#41C2semidirect
2. #1010C2semidirect
3. #1009C2semidirect
4. D3×#41C2semidirect
5. #1007C2semidirect
6. #1004C2semidirect
7. C2×#235C2semidirect
9. C3×#41C22semidirect
10. C3×#41C22semidirect
11. C2×C3×Q8C22semidirect
12. C2×C3×D4C22semidirect
13. C22×C3×C4C22semidirect
14. C2×#67C22semidirect
15. #239C22semidirect
16. D3×Q8C22semidirect
17. #237C22semidirect
18. D3×D4C22semidirect
19. #235C22semidirect
20. #235C22semidirect
21. #235C22semidirect
22. #235C22semidirect
23. C2×D12C22semidirect
24. C2×C4×D3C22semidirect
25. C2×Dic6C22semidirect
32. C2×#41D3semidirect
56. #41C2×D3semidirect
57. #41C2×D3semidirect
60. C2×Q8C2×D3semidirect
64. C2×D4C2×D3semidirect
66. C22×C4C2×D3semidirect
106. C3#585semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#412
#10102
#10092
D3×#412
#10072
#10042
C2×#2352
#5853