Finite Group Databasebeta

← C23×#3038 #15670657

Subgroups — C23×#3038 #15670657

Order 2944.

584 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 208 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)C212yesmaximal, complemented
2C2×C23×#127C211yesmaximal, characteristic, complemented
3C4×#41×C23C211yesmaximal, characteristic, complemented
4C23×#460C214yesmaximal, complemented
5C23×#459C214yesmaximal, complemented
6C23×#423C211yesmaximal, characteristic, complemented
7C23×#418C212yesmaximal, complemented
8C2×#41×C23C2211yescharacteristic
9C22×D4×C23—22no—
10C22×D4×C23C2212yes—
11C23×AGL(1,ℤ/8)—28no—
12C2×SD16×C23C2214yescomplemented
13C2×D8×C23C2214yescomplemented
14C2×M16×C23C2212yes—
15C23×#127—21no—
16C23×#127—21no—
17C23×#127C2214yes—
18C4×Q8×C23C2212yescomplemented
19C4×D4×C23—22no—
20C4×D4×C23C2212yescomplemented
21C23×#117—21no—
22C23×#117C2211yescharacteristic
23C2×C42×C23—21no—
24C2×C42×C23C2211yescharacteristic
25C23×C4⋊C8C2214yescomplemented
26C23×#102C2218yescomplemented
27C24×C23—22no—
28#41×C23—22no—
29#41×C23D414yescomplemented
30C2×Q8×C23C2311yescharacteristic
31C2×D4×C23—21no—
32C2×D4×C23—24no—
33C2×D4×C23—22no—
34C2×D4×C23—24no—
35C2×D4×C23—24no—
36C2×D4×C23—24no—
37C2×D4×C23—48no—
38C2×D4×C23C2314yes—
39C2×D4×C23C2311yescharacteristic
40C22×C4×C23—21no—
41C22×C4×C23—22no—
42C22×C4×C23D412yes—
43C22×C4×C23C2311yescharacteristic
44SD16×C23—28no—
45D8×C23—28no—
46M16×C23—24no—
47C2×C8×C23C2314yes—
48C4⋊C4×C23—22no—
49C4⋊C4×C23C2312yes—
50C22⋊C4×C23—21no—
51C22⋊C4×C23—22no—
52C42×C23—21no—
53C42×C23—22no—
54C42×C23D411yescharacteristic
55C42×C23D411yescharacteristic
56C42×C23C2312yes—
57C23×C23—21no—
58C23×C23—24no—
59C23×C23—48no—
60C23×C23—44no—
61C23×C23C2×D411yescharacteristic
62Q8×C23—21no—
63Q8×C23C2×D412yescomplemented
64D4×C23—24no—
65D4×C23—24no—
66D4×C23—21no—
67D4×C23—24no—
68D4×C23—48no—
69D4×C23—44no—
70D4×C23—48no—
71D4×C23C2×D412yescomplemented
72C2×C4×C23—21no—
73C2×C4×C23—24no—
74C2×C4×C23—24no—
75C2×C4×C23—22no—
76C2×C4×C23—44no—
77C2×C4×C23C2×D412yes—
78C2×C4×C23C2×D411yescharacteristic
79C2×C4×C23C2×D412yes—
80C2×C4×C23C2×D412yes—
81C2×C4×C23#4112yes—
82C2×C4×C23C2411yescharacteristic
83C8×C23—24no—
84#3038C2311yesmaximal, characteristic, direct product factor, Sylow
85C22×C23—22no—
86C22×C23—22no—
87C22×C23—42no—
88C22×C23—44no—
89C22×C23—48no—
90C22×C23—88no—
91C22×C23#12112yes—
92C22×C23C22×D411yescharacteristic
93C4×C23—22no—
94C4×C23—22no—
95C4×C23—42no—
96C4×C23#12112yes—
97C4×C23AGL(1,ℤ/8)14yescomplemented
98C4×C23C22×D411yescharacteristic
99C4×C23C2×#4111yescharacteristic
100C2×AGL(1,ℤ/8)C2×C2312yescomplemented
101C2×#127C2×C2311yescharacteristic, complemented
102C4×#41C2×C2311yescharacteristic, complemented
103#460C2×C2314yescomplemented
104#459C2×C2314yescomplemented
105#423C2×C2311yescharacteristic, complemented
106#418C2×C2312yescomplemented
107C2×C23—22no—
108C2×C23—42no—
109C2×C23—84no—
110C2×C23C2×#12111yescharacteristic
111C2×C23C2×AGL(1,ℤ/8)12yes—
112C2×#41C22×C2311yescharacteristic
113C22×D4—22no—
114C22×D4C22×C2312yes—
115AGL(1,ℤ/8)—28no—
116C2×SD16C22×C2314yescomplemented
117C2×D8C22×C2314yescomplemented
118C2×M16C22×C2312yes—
119#127—21no—
120#127—21no—
121#127C22×C2314yes—
122C4×Q8C22×C2312yescomplemented
123C4×D4—22no—
124C4×D4C22×C2312yescomplemented
125#117—21no—
126#117C22×C2311yescharacteristic
127C2×C42—21no—
128C2×C42C22×C2311yescharacteristic
129C4⋊C8C22×C2314yescomplemented
130#102C22×C2318yescomplemented
131C23#303811yescharacteristic, direct product factor, Sylow
132C24—22no—
133#41—22no—
134#41D4×C2314yescomplemented
135C2×Q8C23×C2311yescharacteristic
136C2×D4—21no—
137C2×D4—24no—
138C2×D4—22no—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—24no—
142C2×D4—48no—
143C2×D4C23×C2314yes—
144C2×D4C23×C2311yescharacteristic
145C22×C4—21no—
146C22×C4—22no—
147C22×C4D4×C2312yes—
148C22×C4C23×C2311yescharacteristic
149SD16—28no—
150D8—28no—
151M16—24no—
152C2×C8C23×C2314yes—
153C4⋊C4—22no—
154C4⋊C4C23×C2312yes—
155C22⋊C4—21no—
156C22⋊C4—22no—
157C42—21no—
158C42—22no—
159C42D4×C2311yescharacteristic
160C42D4×C2311yescharacteristic
161C42C23×C2312yes—
162C23—21no—
163C23—24no—
164C23—48no—
165C23—44no—
166C23C2×D4×C2311yescharacteristic
167Q8—21no—
168Q8C2×D4×C2312yescomplemented
169D4—24no—
170D4—24no—
171D4—21no—
172D4—24no—
173D4—48no—
174D4—44no—
175D4—48no—
176D4C2×D4×C2312yescomplemented
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—44no—
182C2×C4C2×D4×C2312yes—
183C2×C4C2×D4×C2311yescharacteristic
184C2×C4C2×D4×C2312yes—
185C2×C4C2×D4×C2312yes—
186C2×C4#41×C2312yes—
187C2×C4C24×C2311yescharacteristic
188C8—24no—
189C22—22no—
190C22—22no—
191C22—42no—
192C22—48no—
193C22—44no—
194C22—88no—
195C22C23×#12112yes—
196C22C22×D4×C2311yescharacteristic
197C4—22no—
198C4—22no—
199C4—42no—
200C4C23×#12112yes—
201C4C23×AGL(1,ℤ/8)14yescomplemented
202C4C22×D4×C2311yescharacteristic
203C4C2×#41×C2311yescharacteristic
204C2—22no—
205C2—42no—
206C2—84no—
207C2C2×C23×#12111yescharacteristic
208C2C2×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. C2×C23×#127C2semidirect
3. C4×#41×C23C2semidirect
4. C23×#460C2semidirect
5. C23×#459C2semidirect
6. C23×#423C2semidirect
7. C23×#418C2semidirect
12. C2×SD16×C23C22semidirect
13. C2×D8×C23C22semidirect
18. C4×Q8×C23C22semidirect
20. C4×D4×C23C22semidirect
25. C23×C4⋊C8C22semidirect
26. C23×#102C22semidirect
29. #41×C23D4semidirect
63. Q8×C23C2×D4semidirect
71. D4×C23C2×D4semidirect
84. #3038C23direct (N × H)
97. C4×C23AGL(1,ℤ/8)semidirect
100. C2×AGL(1,ℤ/8)C2×C23semidirect
101. C2×#127C2×C23semidirect
102. C4×#41C2×C23semidirect
103. #460C2×C23semidirect
104. #459C2×C23semidirect
105. #423C2×C23semidirect
106. #418C2×C23semidirect
116. C2×SD16C22×C23semidirect
117. C2×D8C22×C23semidirect
122. C4×Q8C22×C23semidirect
124. C4×D4C22×C23semidirect
129. C4⋊C8C22×C23semidirect
130. #102C22×C23semidirect
131. C23#3038direct (N × H)
134. #41D4×C23semidirect
168. Q8C2×D4×C23semidirect
176. D4C2×D4×C23semidirect
201. 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
C2×C23×#1272
C4×#41×C232
C23×#4602
C23×#4592
C23×#4232
C23×#4182
#303823