Finite Group Databasebeta

← C2×#72246847 #72344129

Subgroups — C2×#72246847 #72344129

Order 6144.

At least 6986 conjugacy classes of subgroups; only the 563 normal subgroups are shown.

The subgroup lattice has 144 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#72245692C211yesmaximal, characteristic, complemented
2C2×#72245579C211yesmaximal, characteristic, complemented
3C2×#72245678C214yesmaximal, complemented
4C2×#32899⋊C3C2212yescomplemented
5C2×#72245595C212yesmaximal, complemented
6C2×#72245603C212yesmaximal, complemented
7C2×#11711594C2214yescomplemented
8C2×#32976⋊C3C2211yescharacteristic, complemented
9C2×#72245747C211yesmaximal, characteristic, complemented
10C2×#11715717C2212yescomplemented
11C2×#11715748C2211yescharacteristic, complemented
12C2×C4×#5950C2211yescharacteristic, complemented
13C2×#72245330C212yesmaximal, complemented
14C2×#72245288C212yesmaximal, complemented
15C2×C4⋊#5935C2212yescomplemented
16C2×C4⋊#5935C2212yescomplemented
17C2×#11711630C2214yescomplemented
18C2×#11711631C2214yescomplemented
19C2×C4⋊(B(4,𝔽2)⋊C3)C2211yescharacteristic, complemented
20C22×#87522C2212yescomplemented
21C2×C4×B(4,𝔽2)⋊C3C2212yescomplemented
22C2×#6629390D311yescharacteristic, complemented
23C2×C4⋊#5934C2211yescharacteristic, complemented
24#72246847C2116yesmaximal, direct product factor
25#72245747C2218yescomplemented
26#72245692C2218yescomplemented
27#72245678C22132yescomplemented
28#72245603C22116yescomplemented
29#72245595C22116yescomplemented
30#72245579C2218yescomplemented
31#72245330C22116yescomplemented
32#72245288C22116yescomplemented
33C2×C4×#5934C2212yescomplemented
34C22×#73026C2214yescomplemented
35C22×#5934C2312yes—
36C2×#87522C2318yescomplemented
37C22×#5935C2314yes—
38#11715748C2314yescomplemented
39C4⋊#5934C2314yescomplemented
40C4⋊#5935C2318yescomplemented
41C4⋊#5935C2318yescomplemented
42#11715717C2318yescomplemented
43C4×#5934C2318yescomplemented
44C2×#85564C2312yes—
45C2×#73026C23116yescomplemented
46#11711631C23116yescomplemented
47#11711630C23116yescomplemented
48#11711594C23116yescomplemented
49C22×#5950C2311yescharacteristic
50C4×#5950C2314yescomplemented
51C2×C4×#997C2311yescharacteristic
52C2×C4×#997C2311yescharacteristic
53C22×B(4,𝔽2)⋊C3C2312yes—
54#32976⋊C3C2314yescomplemented
55C4⋊(B(4,𝔽2)⋊C3)C2314yescomplemented
56#32899⋊C3C2318yescomplemented
57C4×B(4,𝔽2)⋊C3C2318yescomplemented
58C2×#1790⋊C3C2312yes—
59C2×C4×#585C2×D311yescharacteristic, complemented
60#6629390C2×D318yescomplemented
61C2×#32976C2×D311yescharacteristic, complemented
62C2×#32926C2×D311yescharacteristic
63C2×#32899C2×D312yescomplemented
64C2×C4×B(4,𝔽2)C2×D312yescomplemented
65C2×#5950C2412yes—
66C22×#997C2411yescharacteristic
67C2×#5935C2418yes—
68C2×#5934C2414yes—
69C4×#997C2412yes—
70C4×#997C2412yes—
71#85564C2414yes—
72C2×B(4,𝔽2)⋊C3C2414yes—
73#1790⋊C3C2414yes—
74C22×#585C22×D311yescharacteristic
75C4×#585C22×D314yescomplemented
76C2×C4×H(5,𝔽2)C22×D311yescharacteristic
77C2×C4×H(5,𝔽2)C22×D311yescharacteristic
78C23×C4×D4S411yescharacteristic, complemented
79C22×B(4,𝔽2)C22×D312yes—
80#32976C22×D314yescomplemented
81#32926C22×D314yes—
82#32899C22×D318yescomplemented
83C4×B(4,𝔽2)C22×D318yescomplemented
84C2×#1790C22×D312yes—
85C2×#997H(5,𝔽2)12yes—
86C2×#997C2511yescharacteristic
87C2×#585C23×D312yes—
88C22×H(5,𝔽2)C23×D311yescharacteristic
89C24×D4C2×S411yescharacteristic
90C25×C4C2×S412yes—
91C4×H(5,𝔽2)C23×D312yes—
92C4×H(5,𝔽2)C23×D312yes—
93C22×C4×D4C2×S418yescomplemented
94C23×C4⋊C4C2×S411yescharacteristic
95C23×C22⋊C4C2×S412yes—
96C23×C42C2×S411yescharacteristic
97C2×B(4,𝔽2)C23×D314yes—
98#1790C23×D314yes—
99#997C2×H(5,𝔽2)11yescharacteristic
100C26C22×S412yes—
101C2×H(5,𝔽2)#100912yes—
102C2×H(5,𝔽2)C24×D311yescharacteristic
103C23×D4C22×S414yes—
104C24×C4C22×S411yescharacteristic
105C24×C4C22×S411yescharacteristic
106C24×C4C22×S411yescharacteristic
107C24×C4C22×S418yes—
108C24×C4C22×S412yes—
109C22×C4⋊C4C22×S414yes—
110C22×C22⋊C4C22×S418yes—
111C22×C42C22×S414yes—
112C25C23×S414yes—
113C25C23×S411yescharacteristic
114H(5,𝔽2)C2×#100911yescharacteristic
115C23×C4C23×S412yes—
116C23×C4C23×S412yes—
117C23×C4C23×S412yes—
118C23×C4C23×S414yes—
119C2×C42#1020⋊C211yescharacteristic
120C24#8748912yes—
121C24C24×S411yescharacteristic
122C22×C4C2×#1020⋊C211yescharacteristic
123C22×C4C2×#1020⋊C211yescharacteristic
124C22×C4C2×#1020⋊C211yescharacteristic
125C42C2×#1020⋊C214yes—
126C23C22×#1020⋊C211yescharacteristic
127C23C2×#8748911yescharacteristic
128C2×C4#1171417414yes—
129C2×C4#1171417514yes—
130C2×C4C22×#1020⋊C212yes—
131C2×C4C22×#1020⋊C212yes—
132C2×C4C22×#1020⋊C212yes—
133C22#7222061812yes—
134C22C23×#1020⋊C211yescharacteristic
135C22C2×#1171417512yes—
136C22C2×#1171417412yes—
137C4C2×#1171417514yes—
138C4C2×#1171417414yes—
139C2#7224684714yesdirect product factor
140C2C22×#1171417411yescharacteristic
141C2C22×#1171417511yescharacteristic
142C2C2×#7222061811yescharacteristic
subgroupG/H
center, socleC23C22×#1020⋊C2
commutator, cosocleC2×#997C25
FrattiniC24C24×S4
FittingC2×#6629390D3
hypercenterC2×C42#1020⋊C2
hypocenter#997C2×H(5,𝔽2)
LCSG ▷ C2×#997 ▷ #997
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C2×#72245692C2semidirect
2. C2×#72245579C2semidirect
3. C2×#72245678C2semidirect
4. C2×#32899⋊C3C22semidirect
5. C2×#72245595C2semidirect
6. C2×#72245603C2semidirect
7. C2×#11711594C22semidirect
8. C2×#32976⋊C3C22semidirect
9. C2×#72245747C2semidirect
10. C2×#11715717C22semidirect
11. C2×#11715748C22semidirect
12. C2×C4×#5950C22semidirect
13. C2×#72245330C2semidirect
14. C2×#72245288C2semidirect
15. C2×C4⋊#5935C22semidirect
16. C2×C4⋊#5935C22semidirect
17. C2×#11711630C22semidirect
18. C2×#11711631C22semidirect
19. C2×C4⋊(B(4,𝔽2)⋊C3)C22semidirect
20. C22×#87522C22semidirect
21. C2×C4×B(4,𝔽2)⋊C3C22semidirect
22. C2×#6629390D3semidirect
23. C2×C4⋊#5934C22semidirect
24. #72246847C2direct (N × H)
25. #72245747C22semidirect
26. #72245692C22semidirect
27. #72245678C22semidirect
28. #72245603C22semidirect
29. #72245595C22semidirect
30. #72245579C22semidirect
31. #72245330C22semidirect
32. #72245288C22semidirect
33. C2×C4×#5934C22semidirect
34. C22×#73026C22semidirect
36. C2×#87522C23semidirect
38. #11715748C23semidirect
39. C4⋊#5934C23semidirect
40. C4⋊#5935C23semidirect
41. C4⋊#5935C23semidirect
42. #11715717C23semidirect
43. C4×#5934C23semidirect
45. C2×#73026C23semidirect
46. #11711631C23semidirect
47. #11711630C23semidirect
48. #11711594C23semidirect
50. C4×#5950C23semidirect
54. #32976⋊C3C23semidirect
55. C4⋊(B(4,𝔽2)⋊C3)C23semidirect
56. #32899⋊C3C23semidirect
57. C4×B(4,𝔽2)⋊C3C23semidirect
59. C2×C4×#585C2×D3semidirect
60. #6629390C2×D3semidirect
61. C2×#32976C2×D3semidirect
63. C2×#32899C2×D3semidirect
64. C2×C4×B(4,𝔽2)C2×D3semidirect
75. C4×#585C22×D3semidirect
78. C23×C4×D4S4semidirect
80. #32976C22×D3semidirect
82. #32899C22×D3semidirect
83. C4×B(4,𝔽2)C22×D3semidirect
93. C22×C4×D4C2×S4semidirect
139. C2#72246847direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#722456922
C2×#722455792
C2×#722456782
C2×#722455952
C2×#722456032
C2×#722457472
C2×D4×C4⋊S44
C2×#722453302
C2×#722452882
C2×#722365963
#722468472