Finite Group Databasebeta

← C23×#3291 #15670910

Subgroups — C23×#3291 #15670910

Order 2944.

460 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 184 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×#577C214yesmaximal, complemented
2C23×#492C211yesmaximal, characteristic, complemented
3C23×#471C214yesmaximal, complemented
4C23×#456C212yesmaximal, complemented
5C23×#453C212yesmaximal, complemented
6C23×#443C212yesmaximal, complemented
7C23×2−1+4C2212yes—
8C23×H(5,𝔽2)C2212yescomplemented
9C23×#137—24no—
10C23×#137C2214yescomplemented
11C23×AGL(1,ℤ/8)—24no—
12C23×AGL(1,ℤ/8)C2214yes—
13C23×#135—24no—
14C23×#135C2212yes—
15C2×SD16×C23—24no—
16C2×SD16×C23C2214yescomplemented
17C23×#131C2214yescomplemented
18C23×C4⋊Q8C2211yescharacteristic, complemented
19C23×#127C2211yescharacteristic
20C23×#108C2212yescomplemented
21C23×#104C2214yescomplemented
22C23×#101C2212yescomplemented
23C23×#100C2212yescomplemented
24C4×C8×C23C2211yescharacteristic, complemented
25#41×C23—24no—
26#41×C23—24no—
27#41×C23—42no—
28#41×C23C2314yes—
29C2×Q8×C23—24no—
30C2×Q8×C23C2312yes—
31C2×D4×C23—24no—
32C2×D4×C23—41no—
33C2×D4×C23—42no—
34C2×D4×C23C2312yes—
35Q16×C23—24no—
36Q16×C23D412yescomplemented
37SD16×C23—24no—
38SD16×C23—24no—
39SD16×C23—24no—
40SD16×C23—24no—
41SD16×C23D414yescomplemented
42D8×C23—24no—
43D8×C23D412yescomplemented
44M16×C23—24no—
45M16×C23C2314yes—
46C2×C8×C23—24no—
47C2×C8×C23C2312yes—
48C4⋊C4×C23—41no—
49C42×C23C2311yescharacteristic
50C23×C23—22no—
51C23×C23—42no—
52Q8×C23—24no—
53Q8×C23—42no—
54Q8×C23C2×D414yes—
55D4×C23—24no—
56D4×C23—24no—
57D4×C23—44no—
58D4×C23—42no—
59D4×C23—82no—
60D4×C23C2×D414yes—
61C2×C4×C23—21no—
62C2×C4×C23—24no—
63C2×C4×C23—22no—
64C2×C4×C23—42no—
65C2×C4×C23—44no—
66C2×C4×C23C2411yescharacteristic
67C8×C23—24no—
68C8×C23C2×D412yes—
69C8×C23C2×D412yes—
70#3291C2311yesmaximal, characteristic, direct product factor, Sylow
71C22×C23—24no—
72C22×C23—42no—
73C22×C23—82no—
74C22×C23—162no—
75C22×C23H(5,𝔽2)11yescharacteristic
76C4×C23—22no—
77C4×C23—24no—
78C4×C23—42no—
79C4×C23C22×D412yes—
80#577C2×C2314yescomplemented
81#492C2×C2311yescharacteristic, complemented
82#471C2×C2314yescomplemented
83#456C2×C2312yescomplemented
84#453C2×C2312yescomplemented
85#443C2×C2312yescomplemented
86C2×C23—21no—
87C2×C23—44no—
88C2×C23—82no—
89C2×C23D4211yescharacteristic
902−1+4C22×C2312yes—
91H(5,𝔽2)C22×C2312yescomplemented
92#137—24no—
93#137C22×C2314yescomplemented
94AGL(1,ℤ/8)—24no—
95AGL(1,ℤ/8)C22×C2314yes—
96#135—24no—
97#135C22×C2312yes—
98C2×SD16—24no—
99C2×SD16C22×C2314yescomplemented
100#131C22×C2314yescomplemented
101C4⋊Q8C22×C2311yescharacteristic, complemented
102#127C22×C2311yescharacteristic
103#108C22×C2312yescomplemented
104#104C22×C2314yescomplemented
105#101C22×C2312yescomplemented
106#100C22×C2312yescomplemented
107C4×C8C22×C2311yescharacteristic, complemented
108C23#329111yescharacteristic, direct product factor, Sylow
109#41—24no—
110#41—24no—
111#41—42no—
112#41C23×C2314yes—
113C2×Q8—24no—
114C2×Q8C23×C2312yes—
115C2×D4—24no—
116C2×D4—41no—
117C2×D4—42no—
118C2×D4C23×C2312yes—
119Q16—24no—
120Q16D4×C2312yescomplemented
121SD16—24no—
122SD16—24no—
123SD16—24no—
124SD16—24no—
125SD16D4×C2314yescomplemented
126D8—24no—
127D8D4×C2312yescomplemented
128M16—24no—
129M16C23×C2314yes—
130C2×C8—24no—
131C2×C8C23×C2312yes—
132C4⋊C4—41no—
133C42C23×C2311yescharacteristic
134C23—22no—
135C23—42no—
136Q8—24no—
137Q8—42no—
138Q8C2×D4×C2314yes—
139D4—24no—
140D4—24no—
141D4—44no—
142D4—42no—
143D4—82no—
144D4C2×D4×C2314yes—
145C2×C4—21no—
146C2×C4—24no—
147C2×C4—22no—
148C2×C4—42no—
149C2×C4—44no—
150C2×C4C24×C2311yescharacteristic
151C8—24no—
152C8C2×D4×C2312yes—
153C8C2×D4×C2312yes—
154C22—24no—
155C22—42no—
156C22—82no—
157C22—162no—
158C22C23×H(5,𝔽2)11yescharacteristic
159C4—22no—
160C4—24no—
161C4—42no—
162C4C22×D4×C2312yes—
163C2—21no—
164C2—44no—
165C2—82no—
166C2D42×C2311yescharacteristic
subgroupG/H
center, socleC2×C23D42
commutator, Frattini, cosocleC2×C4C24×C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#577C2semidirect
2. C23×#492C2semidirect
3. C23×#471C2semidirect
4. C23×#456C2semidirect
5. C23×#453C2semidirect
6. C23×#443C2semidirect
8. C23×H(5,𝔽2)C22semidirect
10. C23×#137C22semidirect
16. C2×SD16×C23C22semidirect
17. C23×#131C22semidirect
18. C23×C4⋊Q8C22semidirect
20. C23×#108C22semidirect
21. C23×#104C22semidirect
22. C23×#101C22semidirect
23. C23×#100C22semidirect
24. C4×C8×C23C22semidirect
36. Q16×C23D4semidirect
41. SD16×C23D4semidirect
43. D8×C23D4semidirect
70. #3291C23direct (N × H)
80. #577C2×C23semidirect
81. #492C2×C23semidirect
82. #471C2×C23semidirect
83. #456C2×C23semidirect
84. #453C2×C23semidirect
85. #443C2×C23semidirect
91. H(5,𝔽2)C22×C23semidirect
93. #137C22×C23semidirect
99. C2×SD16C22×C23semidirect
100. #131C22×C23semidirect
101. C4⋊Q8C22×C23semidirect
103. #108C22×C23semidirect
104. #104C22×C23semidirect
105. #101C22×C23semidirect
106. #100C22×C23semidirect
107. C4×C8C22×C23semidirect
108. C23#3291direct (N × H)
120. Q16D4×C23semidirect
125. SD16D4×C23semidirect
127. D8D4×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#5772
C23×#4922
C23×#4712
C23×#4562
C23×#4532
C23×#4432
#329123