Finite Group Databasebeta

← C23×#3151 #15670770

Subgroups — C23×#3151 #15670770

Order 2944.

656 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 224 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×C23×AGL(1,ℤ/8)C214yesmaximal, complemented
2C23×#532C211yesmaximal, characteristic, complemented
3C2×C23×#127C211yesmaximal, characteristic, complemented
4C23×#496C214yesmaximal, complemented
5C23×#493C212yesmaximal, complemented
6C23×#492C212yesmaximal, complemented
7C4×M16×C23C211yesmaximal, characteristic, complemented
8C2×#41×C23C2212yes—
9C22×D4×C23—22no—
10C22×D4×C23C2212yes—
11C23×AGL(1,ℤ/8)—216no—
12C2×SD16×C23C2218yescomplemented
13C2×D8×C23C2218yescomplemented
14C2×M16×C23C2212yes—
15C23×C4⋊Q8C2211yescharacteristic, complemented
16C23×#127—22no—
17C23×#127C2212yes—
18C23×#127C2212yes—
19C23×#127C2211yescharacteristic, complemented
20C23×#124C2212yescomplemented
21C23×#121—22no—
22C4×D4×C23—22no—
23C2×C42×C23C2211yescharacteristic
24C23×#97C2212yescomplemented
25C4×C8×C23C2212yescomplemented
26C24×C23—22no—
27#41×C23—28no—
28C2×Q8×C23C2312yes—
29C2×D4×C23—22no—
30C2×D4×C23—28no—
31C2×D4×C23—22no—
32C2×D4×C23—22no—
33C2×D4×C23—24no—
34C2×D4×C23—24no—
35C2×D4×C23—48no—
36C2×D4×C23C2312yes—
37C2×D4×C23C2314yes—
38C22×C4×C23—22no—
39C22×C4×C23D412yes—
40C22×C4×C23C2311yescharacteristic
41SD16×C23—216no—
42D8×C23—216no—
43M16×C23D418yescomplemented
44C2×C8×C23C2314yes—
45C4⋊C4×C23—22no—
46C22⋊C4×C23—24no—
47C42×C23D412yes—
48C42×C23C2311yescharacteristic
49C42×C23C2311yescharacteristic
50C23×C23—22no—
51C23×C23—24no—
52C23×C23—48no—
53C23×C23—44no—
54C23×C23C2×D411yescharacteristic
55Q8×C23—24no—
56D4×C23—24no—
57D4×C23—28no—
58D4×C23—48no—
59D4×C23—44no—
60D4×C23—44no—
61D4×C23—44no—
62D4×C23—48no—
63C2×C4×C23—24no—
64C2×C4×C23—22no—
65C2×C4×C23—44no—
66C2×C4×C23C2×D412yes—
67C2×C4×C23C2×D411yescharacteristic
68C2×C4×C23C2×D414yes—
69C2×C4×C23C2×D412yes—
70C2×C4×C23C2411yescharacteristic
71C8×C23C2×D418yescomplemented
72#3151C2311yesmaximal, characteristic, direct product factor, Sylow
73C22×C23—22no—
74C22×C23—42no—
75C22×C23—44no—
76C22×C23—48no—
77C22×C23—44no—
78C22×C23—88no—
79C22×C23#12712yes—
80C22×C23C22×D411yescharacteristic
81C4×C23—22no—
82C4×C23—42no—
83C4×C23#12712yes—
84C4×C23AGL(1,ℤ/8)14yescomplemented
85C4×C23C22×D412yes—
86C2×AGL(1,ℤ/8)C2×C2314yescomplemented
87#532C2×C2311yescharacteristic, complemented
88C2×#127C2×C2311yescharacteristic, complemented
89#496C2×C2314yescomplemented
90#493C2×C2312yescomplemented
91#492C2×C2312yescomplemented
92C4×M16C2×C2311yescharacteristic, complemented
93C2×C23—22no—
94C2×C23—82no—
95C2×C23—84no—
96C2×C23C2×#12711yescharacteristic
97C2×C23C2×AGL(1,ℤ/8)12yes—
98C2×#41C22×C2312yes—
99C22×D4—22no—
100C22×D4C22×C2312yes—
101AGL(1,ℤ/8)—216no—
102C2×SD16C22×C2318yescomplemented
103C2×D8C22×C2318yescomplemented
104C2×M16C22×C2312yes—
105C4⋊Q8C22×C2311yescharacteristic, complemented
106#127—22no—
107#127C22×C2312yes—
108#127C22×C2312yes—
109#127C22×C2311yescharacteristic, complemented
110#124C22×C2312yescomplemented
111#121—22no—
112C4×D4—22no—
113C2×C42C22×C2311yescharacteristic
114#97C22×C2312yescomplemented
115C4×C8C22×C2312yescomplemented
116C23#315111yescharacteristic, direct product factor, Sylow
117C24—22no—
118#41—28no—
119C2×Q8C23×C2312yes—
120C2×D4—22no—
121C2×D4—28no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—24no—
125C2×D4—24no—
126C2×D4—48no—
127C2×D4C23×C2312yes—
128C2×D4C23×C2314yes—
129C22×C4—22no—
130C22×C4D4×C2312yes—
131C22×C4C23×C2311yescharacteristic
132SD16—216no—
133D8—216no—
134M16D4×C2318yescomplemented
135C2×C8C23×C2314yes—
136C4⋊C4—22no—
137C22⋊C4—24no—
138C42D4×C2312yes—
139C42C23×C2311yescharacteristic
140C42C23×C2311yescharacteristic
141C23—22no—
142C23—24no—
143C23—48no—
144C23—44no—
145C23C2×D4×C2311yescharacteristic
146Q8—24no—
147D4—24no—
148D4—28no—
149D4—48no—
150D4—44no—
151D4—44no—
152D4—48no—
153D4—44no—
154C2×C4—24no—
155C2×C4—22no—
156C2×C4—44no—
157C2×C4C2×D4×C2312yes—
158C2×C4C2×D4×C2311yescharacteristic
159C2×C4C2×D4×C2314yes—
160C2×C4C2×D4×C2312yes—
161C2×C4C24×C2311yescharacteristic
162C8C2×D4×C2318yescomplemented
163C22—22no—
164C22—48no—
165C22—44no—
166C22—42no—
167C22—44no—
168C22—88no—
169C22C23×#12712yes—
170C22C22×D4×C2311yescharacteristic
171C4—22no—
172C4—42no—
173C4C23×#12712yes—
174C4C23×AGL(1,ℤ/8)14yescomplemented
175C4C22×D4×C2312yes—
176C2—22no—
177C2—84no—
178C2—82no—
179C2C2×C23×#12711yescharacteristic
180C2C2×C23×AGL(1,ℤ/8)12yes—
subgroupG/H
center, socleC22×C23C22×D4
commutator, Frattini, cosocleC2×C4C24×C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C23×AGL(1,ℤ/8)C2semidirect
2. C23×#532C2semidirect
3. C2×C23×#127C2semidirect
4. C23×#496C2semidirect
5. C23×#493C2semidirect
6. C23×#492C2semidirect
7. C4×M16×C23C2semidirect
12. C2×SD16×C23C22semidirect
13. C2×D8×C23C22semidirect
15. C23×C4⋊Q8C22semidirect
19. C23×#127C22semidirect
20. C23×#124C22semidirect
24. C23×#97C22semidirect
25. C4×C8×C23C22semidirect
43. M16×C23D4semidirect
71. C8×C23C2×D4semidirect
72. #3151C23direct (N × H)
84. C4×C23AGL(1,ℤ/8)semidirect
86. C2×AGL(1,ℤ/8)C2×C23semidirect
87. #532C2×C23semidirect
88. C2×#127C2×C23semidirect
89. #496C2×C23semidirect
90. #493C2×C23semidirect
91. #492C2×C23semidirect
92. C4×M16C2×C23semidirect
102. C2×SD16C22×C23semidirect
103. C2×D8C22×C23semidirect
105. C4⋊Q8C22×C23semidirect
109. #127C22×C23semidirect
110. #124C22×C23semidirect
114. #97C22×C23semidirect
115. C4×C8C22×C23semidirect
116. C23#3151direct (N × H)
134. M16D4×C23semidirect
162. C8C2×D4×C23semidirect
174. C4C23×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C2×C23×AGL(1,ℤ/8)2
C23×#5322
C2×C23×#1272
C23×#4962
C23×#4932
C23×#4922
C4×M16×C232
#315123