Finite Group Databasebeta

← C23×#1007 #11709077

Subgroups — C23×#1007 #11709077

Order 768.

59806 conjugacy classes of subgroups; only the 32139 normal subgroups are shown.

normal subgroups only

This group has 59806 conjugacy classes of subgroups — too many to draw. Only its 32139 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
1C24×C3×Q8C211yesmaximal, characteristic, complemented
2C22×#1007C21112yesmaximal, direct product factor
3C23×D3×Q8C214yesmaximal, complemented
4C23×#239C214yesmaximal, complemented
5C23×#235C216yesmaximal, complemented
6C23×2−1+4—31nomaximal, Sylow
7C23×C3×Q8C2214yes—
8C23×C3×Q8C22156yescomplemented
9C24×C3×C4C2213yes—
10C23×#67C2214yes—
11C2×#1007C2211792yesdirect product factor
12C22×#239C221224yescomplemented
13C22×D3×Q8C221224yescomplemented
14C22×#235C221336yescomplemented
15C23×D12C2216yes—
16C23×C4×D3C22112yes—
17C23×Dic6C2216yes—
18C22×2−1+4—3112no—
19C23×#41—34no—
20C23×#41—36no—
21C24×Q8—34no—
22C24×Q8D311yescharacteristic, complemented
23C25×C3C2311yescharacteristic
24C24×D3C2314yes—
25C22×C3×Q8C231112yes—
26C22×C3×Q8C231448yescomplemented
27C23×C3×C4C2316yes—
28C23×C3×C4C23184yes—
29C22×#67C231112yes—
30C23×Dic3C2314yes—
31#1007C2314096yesdirect product factor
32C2×#239C2311792yescomplemented
33C2×D3×Q8C2311792yescomplemented
34C2×#235C2312688yescomplemented
35C22×D12C231168yes—
36C22×C4×D3C231336yes—
37C22×Dic6C231168yes—
38C2×2−1+4—31792no—
39C22×#41—3224no—
40C22×#41—3336no—
41C23×Q8—3224no—
42C23×Q8—36no—
43C23×Q8C2×D314yes—
44C23×Q8C2×D3156yescomplemented
45C23×D4—34no—
46C23×D4—36no—
47C24×C4—312no—
48C24×C4C2×D313yes—
49C24×C3—28no—
50C24×C3C24114yes—
51C24×C3C2411yescharacteristic
52C23×D3—232no—
53C23×D3C24156yes—
54C2×C3×Q8C241448yes—
55C2×C3×Q8C241512yescomplemented
56C22×C3×C4C24184yes—
57C22×C3×C4C241336yes—
58C2×#67C241448yes—
59C22×Dic3C24156yes—
60#239C2412048yescomplemented
61D3×Q8C2412048yescomplemented
62#235C2413072yescomplemented
63C2×D12C241672yes—
64C2×C4×D3C2411344yes—
65C2×Dic6C241672yes—
66C25—34no—
67C25C22×D311yescharacteristic
682−1+4—34096no—
69C2×#41—31792no—
70C2×#41—32688no—
71C22×Q8—31792no—
72C22×Q8—3168no—
73C22×Q8C22×D31112yes—
74C22×Q8C22×D31448yescomplemented
75C22×D4—3112no—
76C22×D4—3168no—
77C23×C4—34no—
78C23×C4—3336no—
79C23×C4C22×D3184yes—
80C23×C4C22×D316yes—
81C23×C3—256no—
82C23×C32−1+418yescomplemented
83C23×C3C25128yes—
84C23×C3C2517yes—
85C22×D3—2224no—
86C22×D3C251112yes—
87C3×Q8C251256yes—
88C2×C3×C4C251168yes—
89C2×C3×C4C251192yes—
90#67C251256yes—
91C2×Dic3C251112yes—
92D12C251384yes—
93C4×D3C251768yes—
94Dic6C251384yes—
95C24—28no—
96C24—356no—
97C24—632no—
98C24C23×D3114yes—
99C24C23×D311yescharacteristic
100#41—32048no—
101#41—33072no—
102C2×Q8—32048no—
103C2×Q8—3672no—
104C2×Q8C23×D31448yes—
105C2×Q8C23×D31512yescomplemented
106C2×D4—3448no—
107C2×D4—3672no—
108C22×C4—356no—
109C22×C4—31344no—
110C22×C4C23×D3184yes—
111C22×C4C23×D31336yes—
112C22×C3—256no—
113C22×C3C2×2−1+4128yescomplemented
114C22×C3C2618yes—
115C22×C3C2617yes—
116C2×D3—2224no—
117C2×D3C26132yes—
118C3×C4C26148yes—
119Dic3C26132yes—
120C23—256no—
121C23—3112no—
122C23—6224no—
123C23#100718yesdirect product factor
124C23C24×D3128yes—
125C23C24×D317yes—
126Q8—3384no—
127Q8C24×D31256yes—
128D4—3256no—
129D4—3384no—
130C2×C4—3112no—
131C2×C4—3768no—
132C2×C4C24×D31168yes—
133C2×C4C24×D31192yes—
134C2×C3—28no—
135C2×C3C22×2−1+4114yescomplemented
136C2×C3C2711yescharacteristic
137D3—232no—
138C22—256no—
139C22—332no—
140C22—6224no—
141C22C2×#1007128yesdirect product factor
142C22C25×D318yes—
143C22C25×D317yes—
144C4—332no—
145C4C25×D3148yes—
146C3C23×2−1+411yescharacteristic, complemented, Sylow
147C2—28no—
148C2—632no—
149C2C22×#1007114yesdirect product factor
150C2C26×D311yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC2×C3C27
FrattiniC2C26×D3
FittingC24×C3×Q8C2
socleC24×C3C24
hypercenterC24×Q8D3
hypocenterC3C23×2−1+4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C24 ◁ C24×Q8

Complements

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

NH ≅ G/Ntype
1. C24×C3×Q8C2semidirect
2. C22×#1007C2direct (N × H)
3. C23×D3×Q8C2semidirect
4. C23×#239C2semidirect
5. C23×#235C2semidirect
8. C23×C3×Q8C22semidirect
11. C2×#1007C22direct (N × H)
12. C22×#239C22semidirect
13. C22×D3×Q8C22semidirect
14. C22×#235C22semidirect
22. C24×Q8D3semidirect
26. C22×C3×Q8C23semidirect
31. #1007C23direct (N × H)
32. C2×#239C23semidirect
33. C2×D3×Q8C23semidirect
34. C2×#235C23semidirect
44. C23×Q8C2×D3semidirect
55. C2×C3×Q8C24semidirect
60. #239C24semidirect
61. D3×Q8C24semidirect
62. #235C24semidirect
74. C22×Q8C22×D3semidirect
82. C23×C32−1+4semidirect
105. C2×Q8C23×D3semidirect
113. C22×C3C2×2−1+4semidirect
123. C23#1007direct (N × H)
135. C2×C3C22×2−1+4semidirect
141. C22C2×#1007direct (N × H)
146. C3C23×2−1+4semidirect
149. C2C22×#1007direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3×Q82
C22×#10072
C23×D3×Q82
C23×#2392
C23×#2352
C23×2−1+43