Finite Group Databasebeta

← C32×#2679 #11769728

Subgroups — C32×#2679 #11769728

Order 1152.

3000 conjugacy classes of subgroups; only the 672 normal subgroups are shown.

normal subgroups only

This group has 3000 conjugacy classes of subgroups — too many to draw. Only its 672 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×C32×#120C216yesmaximal, complemented
2C32×#394C216yesmaximal, complemented
3C32×#390C213yesmaximal, complemented
4C3×#2679C314yesmaximal, direct product factor
5C25×C32C2212yescomplemented
6C22×C32×D4C2216yes—
7C22×C32×D4C2213yes—
8C32×#120—224no—
9C2×C32×C4⋊C4C2213yescomplemented
10C2×C32×C22⋊C4C22112yescomplemented
11C2×C32×C22⋊C4C2216yescomplemented
12C2×C32×C42C2211yescharacteristic, complemented
13C32×#95C2212yescomplemented
14C2×C3×#120C2×C3124yescomplemented
15C3×#394C2×C3124yescomplemented
16C3×#390C2×C3112yescomplemented
17C24×C32—212no—
18C24×C32—46no—
19C24×C32—42no—
20C24×C32C2316yes—
21C2×C32×D4—224no—
22C2×C32×D4—43no—
23C2×C32×D4—43no—
24C2×C32×D4—46no—
25C2×C32×D4D4112yescomplemented
26C22×C32×C4C2316yes—
27C22×C32×C4C2313yes—
28C32×C4⋊C4—43no—
29C32×C22⋊C4—212no—
30C32×C22⋊C4—224no—
31C32×C42—41no—
32#2679C3211yescharacteristic, direct product factor, Sylow
33C25×C3C22×C318yescomplemented
34C22×C3×D4C22×C3124yes—
35C22×C3×D4C22×C3112yes—
36C3×#120—296no—
37C2×C3×C4⋊C4C22×C3112yescomplemented
38C2×C3×C22⋊C4C22×C3148yescomplemented
39C2×C3×C22⋊C4C22×C3124yescomplemented
40C2×C3×C42C22×C314yescomplemented
41C3×#95C22×C318yescomplemented
42C23×C32—212no—
43C23×C32—26no—
44C23×C32—26no—
45C23×C32—212no—
46C23×C32—424no—
47C23×C32—424no—
48C23×C32—48no—
49C23×C32C2×D4112yes—
50C23×C32C2411yescharacteristic
51C32×D4—412no—
52C32×D4—424no—
53C2×C32×C4—23no—
54C2×C32×C4—23no—
55C2×C32×C4—212no—
56C2×C32×C4—26no—
57C2×C32×C4C2×D416yes—
58C2×#120C2×C3216yescomplemented
59#394C2×C3216yescomplemented
60#390C2×C3213yescomplemented
61C24×C3—248no—
62C24×C3—424no—
63C24×C3—48no—
64C24×C3C23×C3124yes—
65C2×C3×D4—296no—
66C2×C3×D4—412no—
67C2×C3×D4—412no—
68C2×C3×D4—424no—
69C2×C3×D4C3×D4148yescomplemented
70C22×C3×C4C23×C3124yes—
71C22×C3×C4C23×C3112yes—
72C3×C4⋊C4—412no—
73C3×C22⋊C4—248no—
74C3×C22⋊C4—296no—
75C3×C42—44no—
76C22×C32—212no—
77C22×C32—224no—
78C22×C32—432no—
79C22×C32—412no—
80C22×C32—46no—
81C22×C32—46no—
82C22×C32C22×D413yes—
83C22×C32H(5,𝔽2)11yescharacteristic
84C22×C32H(5,𝔽2)13yes—
85C32×C4—26no—
86C32×C4—46no—
87C25C22×C3212yescomplemented
88C22×D4C22×C3216yes—
89C22×D4C22×C3213yes—
90#120—224no—
91C2×C4⋊C4C22×C3213yescomplemented
92C2×C22⋊C4C22×C32112yescomplemented
93C2×C22⋊C4C22×C3216yescomplemented
94C2×C42C22×C3211yescharacteristic, complemented
95#95C22×C3212yescomplemented
96C23×C3—248no—
97C23×C3—224no—
98C23×C3—224no—
99C23×C3—248no—
100C23×C3—496no—
101C23×C3—496no—
102C23×C3—432no—
103C23×C3C2×C3×D4148yes—
104C23×C3C24×C314yes—
105C3×D4—448no—
106C3×D4—496no—
107C2×C3×C4—212no—
108C2×C3×C4—212no—
109C2×C3×C4—248no—
110C2×C3×C4—224no—
111C2×C3×C4C2×C3×D4124yes—
112C2×C32—412no—
113C2×C32#53413yes—
114C2×C32D4213yes—
115C2×C32H(3,𝔽4)11yescharacteristic
116C24—212no—
117C24—46no—
118C24—42no—
119C24C23×C3216yes—
120C2×D4—224no—
121C2×D4—43no—
122C2×D4—43no—
123C2×D4—46no—
124C2×D4C32×D4112yescomplemented
125C22×C4C23×C3216yes—
126C22×C4C23×C3213yes—
127C4⋊C4—43no—
128C22⋊C4—212no—
129C22⋊C4—224no—
130C42—41no—
131C22×C3—248no—
132C22×C3—296no—
133C22×C3—4128no—
134C22×C3—424no—
135C22×C3—448no—
136C22×C3—424no—
137C22×C3C22×C3×D4112yes—
138C22×C3C3×H(5,𝔽2)112yes—
139C22×C3C3×H(5,𝔽2)14yes—
140C3×C4—224no—
141C3×C4—424no—
142C32#267911yescharacteristic, direct product factor, Sylow
143C23—212no—
144C23—26no—
145C23—26no—
146C23—212no—
147C23—424no—
148C23—424no—
149C23—48no—
150C23C2×C32×D4112yes—
151C23C24×C3211yescharacteristic
152D4—412no—
153D4—424no—
154C2×C4—23no—
155C2×C4—23no—
156C2×C4—212no—
157C2×C4—26no—
158C2×C4C2×C32×D416yes—
159C2×C3—448no—
160C2×C3C3×#534112yes—
161C2×C3C3×D42112yes—
162C2×C3C3×H(3,𝔽4)14yes—
163C22—212no—
164C22—224no—
165C22—432no—
166C22—46no—
167C22—412no—
168C22—46no—
169C22C22×C32×D413yes—
170C22C32×H(5,𝔽2)13yes—
171C22C32×H(5,𝔽2)11yescharacteristic
172C4—26no—
173C4—46no—
174C3C3×#267914yesdirect product factor
175C2—412no—
176C2C32×#53413yes—
177C2C32×D4213yes—
178C2C32×H(3,𝔽4)11yescharacteristic
subgroupG/H
center, socleC23×C32C24
commutator, Frattini, cosocleC23C24×C32
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C32 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C32×#120C2semidirect
2. C32×#394C2semidirect
3. C32×#390C2semidirect
4. C3×#2679C3direct (N × H)
5. C25×C32C22semidirect
9. C2×C32×C4⋊C4C22semidirect
10. C2×C32×C22⋊C4C22semidirect
11. C2×C32×C22⋊C4C22semidirect
12. C2×C32×C42C22semidirect
13. C32×#95C22semidirect
14. C2×C3×#120C2×C3semidirect
15. C3×#394C2×C3semidirect
16. C3×#390C2×C3semidirect
25. C2×C32×D4D4semidirect
32. #2679C32direct (N × H)
33. C25×C3C22×C3semidirect
37. C2×C3×C4⋊C4C22×C3semidirect
38. C2×C3×C22⋊C4C22×C3semidirect
39. C2×C3×C22⋊C4C22×C3semidirect
40. C2×C3×C42C22×C3semidirect
41. C3×#95C22×C3semidirect
58. C2×#120C2×C32semidirect
59. #394C2×C32semidirect
60. #390C2×C32semidirect
69. C2×C3×D4C3×D4semidirect
87. C25C22×C32semidirect
91. C2×C4⋊C4C22×C32semidirect
92. C2×C22⋊C4C22×C32semidirect
93. C2×C22⋊C4C22×C32semidirect
94. C2×C42C22×C32semidirect
95. #95C22×C32semidirect
124. C2×D4C32×D4semidirect
142. C32#2679direct (N × H)
174. C3C3×#2679direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1202
C32×#3942
C32×#3902
C3×#26793