Finite Group Databasebeta

← C23×C3×GO+(4,𝔽2) #13181847

Subgroups — C23×C3×GO+(4,𝔽2) #13181847

Order 1728.

4488 conjugacy classes of subgroups; only the 882 normal subgroups are shown.

normal subgroups only

This group has 4488 conjugacy classes of subgroups — too many to draw. Only its 882 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
1C23×C3×D32C212yesmaximal, complemented
2C23×C3×#157C211yesmaximal, characteristic, complemented
3C22×C3×GO+(4,𝔽2)C2128yesmaximal, direct product factor
4C23×GO+(4,𝔽2)C311yesmaximal, characteristic, direct product factor
5C23×C3×C3⋊D3C2211yescharacteristic
6C23×C32×D3—22no—
7C22×C3×D32—214no—
8C22×C3×D32C22128yescomplemented
9C22×C3×#157C22114yescomplemented
10C2×C3×GO+(4,𝔽2)C221112yesdirect product factor
11C24×C3×D3—62no—
12C23×D32C2×C312yescomplemented
13C23×#157C2×C311yescharacteristic, complemented
14C22×GO+(4,𝔽2)C2×C3128yesdirect product factor
15C23×C33D411yescharacteristic, complemented
16C22×C3×C3⋊D3D417yescomplemented
17C22×C3×C3⋊D3C2317yes—
18C22×C32×D3—228no—
19C2×C3×D32—284no—
20C2×C3×D32C23156yescomplemented
21C2×C3×#157C23128yescomplemented
22C3×GO+(4,𝔽2)C23164yesdirect product factor
23C23×C3×D4—91nomaximal
24C24×C32—62no—
25C23×C3⋊D3C22×C311yescharacteristic
26C23×C3×D3—22no—
27C23×C3×D3—22no—
28C23×C3×D3—42no—
29C23×C3×D3—628no—
30C23×C3×D3—628no—
31C23×C3×D3—62no—
32C22×D32—214no—
33C22×D32C22×C3128yescomplemented
34C22×#157C22×C3114yescomplemented
35C2×GO+(4,𝔽2)C22×C31112yesdirect product factor
36C22×C33C2×D417yescomplemented
37C2×C3×C3⋊D3C2×D4121yescomplemented
38C2×C3×C3⋊D3C2417yes—
39C2×C32×D3—256no—
40C3×D32—256no—
41C3×D32C24116yescomplemented
42C3×#157C2418yescomplemented
43C25×C3—92no—
44C24×D3—62no—
45C22×C3×D4—928no—
46C23×C3×C4—91no—
47C23×C32—22no—
48C23×C32—42no—
49C23×C32—628no—
50C23×C32C3×D411yescharacteristic, complemented
51C22×C3⋊D3C3×D417yescomplemented
52C22×C3⋊D3C23×C317yes—
53C22×C3×D3—228no—
54C22×C3×D3—228no—
55C22×C3×D3—428no—
56C22×C3×D3—656no—
57C22×C3×D3—614no—
58C22×C3×D3—614no—
59C22×C3×D3—6168no—
60C2×D32—284no—
61C2×D32C23×C3156yescomplemented
62C2×#157C23×C3128yescomplemented
63GO+(4,𝔽2)C23×C3164yesdirect product factor
64C23×D4—91noSylow
65C2×C33C22×D417yescomplemented
66C3×C3⋊D3C22×D417yescomplemented
67C3×C3⋊D3C2511yescharacteristic
68C32×D3—216no—
69C24×C3—62no—
70C24×C3—62no—
71C24×C3—928no—
72C24×C3—91no—
73C24×C3—122no—
74C24×C3—1814no—
75C23×D3—22no—
76C23×D3—628no—
77C23×D3—62no—
78C23×D3—628no—
79C2×C3×D4—9112no—
80C22×C3×C4—914no—
81C22×C32—214no—
82C22×C32—414no—
83C22×C32—656no—
84C22×C32C2×C3×D417yescomplemented
85C2×C3⋊D3C2×C3×D4121yescomplemented
86C2×C3⋊D3C24×C317yes—
87C2×C3×D3—256no—
88C2×C3×D3—256no—
89C2×C3×D3—456no—
90C2×C3×D3—616no—
91C2×C3×D3—642no—
92C2×C3×D3—614no—
93C2×C3×D3—6112no—
94D32—256no—
95D32C24×C3116yescomplemented
96#157C24×C318yescomplemented
97C25—92no—
98C22×D4—928no—
99C23×C4—91no—
100C33C23×D411yescharacteristic, complemented, Sylow
101C23×C3—22no—
102C23×C3—42no—
103C23×C3—628no—
104C23×C3—628no—
105C23×C3—956no—
106C23×C3—97no—
107C23×C3—97no—
108C23×C3—1228no—
109C23×C3—1884no—
110C23×C3GO+(4,𝔽2)11yescharacteristic, direct product factor
111C22×D3—228no—
112C22×D3—656no—
113C22×D3—614no—
114C22×D3—614no—
115C22×D3—6168no—
116C3×D4—964no—
117C2×C3×C4—928no—
118C2×C32—214no—
119C2×C32—414no—
120C2×C32—616no—
121C2×C32C22×C3×D417yescomplemented
122C3⋊D3C22×C3×D417yescomplemented
123C3⋊D3C25×C311yescharacteristic
124C3×D3—216no—
125C3×D3—216no—
126C3×D3—416no—
127C3×D3—614no—
128C3×D3—62no—
129C24—62no—
130C24—928no—
131C24—91no—
132C24—1814no—
133C2×D4—9112no—
134C22×C4—914no—
135C22×C3—214no—
136C22×C3—414no—
137C22×C3—656no—
138C22×C3—656no—
139C22×C3—916no—
140C22×C3—97no—
141C22×C3—921no—
142C22×C3—1256no—
143C22×C3—1856no—
144C22×C3C2×GO+(4,𝔽2)17yesdirect product factor
145C2×D3—256no—
146C2×D3—616no—
147C2×D3—642no—
148C2×D3—614no—
149C2×D3—6112no—
150C3×C4—98no—
151C32—22no—
152C32—42no—
153C32C23×C3×D411yescharacteristic, complemented
154C23—628no—
155C23—956no—
156C23—97no—
157C23—97no—
158C23—1884no—
159C23C3×GO+(4,𝔽2)11yescharacteristic, direct product factor
160D4—964no—
161C2×C4—928no—
162C2×C3—214no—
163C2×C3—414no—
164C2×C3—616no—
165C2×C3—616no—
166C2×C3—91no—
167C2×C3—97no—
168C2×C3—1216no—
169C2×C3C22×GO+(4,𝔽2)17yesdirect product factor
170D3—216no—
171D3—62no—
172D3—614no—
173C22—656no—
174C22—916no—
175C22—97no—
176C22—921no—
177C22—1856no—
178C22C2×C3×GO+(4,𝔽2)17yesdirect product factor
179C4—98no—
180C3—22no—
181C3—42no—
182C3C23×GO+(4,𝔽2)11yescharacteristic, direct product factor
183C2—616no—
184C2—91no—
185C2—97no—
186C2C22×C3×GO+(4,𝔽2)17yesdirect product factor
subgroupG/H
center, hypercenterC23×C3GO+(4,𝔽2)
commutator, cosocleC3⋊D3C25×C3
FrattiniC1G
Fitting, socleC23×C33D4
hypocenterC32C23×C3×D4
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C23×C3×D32C2semidirect
2. C23×C3×#157C2semidirect
3. C22×C3×GO+(4,𝔽2)C2direct (N × H)
4. C23×GO+(4,𝔽2)C3direct (N × H)
8. C22×C3×D32C22semidirect
9. C22×C3×#157C22semidirect
10. C2×C3×GO+(4,𝔽2)C22direct (N × H)
12. C23×D32C2×C3semidirect
13. C23×#157C2×C3semidirect
14. C22×GO+(4,𝔽2)C2×C3direct (N × H)
15. C23×C33D4semidirect
16. C22×C3×C3⋊D3D4semidirect
20. C2×C3×D32C23semidirect
21. C2×C3×#157C23semidirect
22. C3×GO+(4,𝔽2)C23direct (N × H)
33. C22×D32C22×C3semidirect
34. C22×#157C22×C3semidirect
35. C2×GO+(4,𝔽2)C22×C3direct (N × H)
36. C22×C33C2×D4semidirect
37. C2×C3×C3⋊D3C2×D4semidirect
41. C3×D32C24semidirect
42. C3×#157C24semidirect
50. C23×C32C3×D4semidirect
51. C22×C3⋊D3C3×D4semidirect
61. C2×D32C23×C3semidirect
62. C2×#157C23×C3semidirect
63. GO+(4,𝔽2)C23×C3direct (N × H)
65. C2×C33C22×D4semidirect
66. C3×C3⋊D3C22×D4semidirect
84. C22×C32C2×C3×D4semidirect
85. C2×C3⋊D3C2×C3×D4semidirect
95. D32C24×C3semidirect
96. #157C24×C3semidirect
100. C33C23×D4semidirect
110. C23×C3GO+(4,𝔽2)direct (N × H)
121. C2×C32C22×C3×D4semidirect
122. C3⋊D3C22×C3×D4semidirect
144. C22×C3C2×GO+(4,𝔽2)direct (N × H)
153. C32C23×C3×D4semidirect
159. C23C3×GO+(4,𝔽2)direct (N × H)
169. C2×C3C22×GO+(4,𝔽2)direct (N × H)
178. C22C2×C3×GO+(4,𝔽2)direct (N × H)
182. C3C23×GO+(4,𝔽2)direct (N × H)
186. C2C22×C3×GO+(4,𝔽2)direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×D322
C23×C3×#1572
C22×C3×GO+(4,𝔽2)2
C23×GO+(4,𝔽2)3
C23×C3×D49