Finite Group Databasebeta

← C23×#3003 #15670622

Subgroups — C23×#3003 #15670622

Order 2944.

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

normal subgroups only

This group has 994 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
1C23×D4×C23C211yesmaximal, characteristic, complemented
2C2×C23×AGL(1,ℤ/8)C212yesmaximal, complemented
3C23×#525C211yesmaximal, characteristic, complemented
4C23×#450C214yesmaximal, complemented
5C23×#447C214yesmaximal, complemented
6C23×#418C212yesmaximal, complemented
7C23×#407C211yesmaximal, characteristic, complemented
8C25×C23—21no—
9C2×#41×C23C2211yescharacteristic, complemented
10C22×D4×C23—28no—
11C22×D4×C23—22no—
12C22×D4×C23—21no—
13C22×D4×C23C2214yes—
14C22×D4×C23C2212yes—
15C23×C4×C23C2211yescharacteristic
16C23×AGL(1,ℤ/8)—28no—
17C2×SD16×C23C2214yescomplemented
18C2×D8×C23C2214yescomplemented
19C2×M16×C23C2212yescomplemented
20C23×#123—21no—
21C23×C22⋊Q8C2212yescomplemented
22C23×#121C2212yescomplemented
23C23×#120—21no—
24C4×D4×C23—22no—
25C23×#117C2211yescharacteristic, complemented
26C23×#102C2218yescomplemented
27C23×C22⋊C8C2214yescomplemented
28C24×C23—28no—
29C24×C23—24no—
30C24×C23—22no—
31C24×C23—48no—
32C24×C23D411yescharacteristic
33#41×C23—24no—
34C2×Q8×C23C2311yescharacteristic, complemented
35C2×D4×C23—28no—
36C2×D4×C23—216no—
37C2×D4×C23—216no—
38C2×D4×C23—22no—
39C2×D4×C23—21no—
40C2×D4×C23—24no—
41C2×D4×C23—22no—
42C2×D4×C23—24no—
43C2×D4×C23D418yescomplemented
44C2×D4×C23C2314yes—
45C2×D4×C23C2311yescharacteristic, complemented
46C22×C4×C23—24no—
47C22×C4×C23—21no—
48C22×C4×C23D412yes—
49C22×C4×C23D411yescharacteristic
50C22×C4×C23C2312yes—
51C22×C4×C23C2311yescharacteristic
52SD16×C23—28no—
53D8×C23—28no—
54M16×C23—24no—
55C2×C8×C23C2314yescomplemented
56C4⋊C4×C23—22no—
57C4⋊C4×C23C2312yescomplemented
58C22⋊C4×C23—21no—
59C22⋊C4×C23—22no—
60C22⋊C4×C23—22no—
61C42×C23—21no—
62C23×C23—22no—
63C23×C23—24no—
64C23×C23—216no—
65C23×C23—28no—
66C23×C23—24no—
67C23×C23—21no—
68C23×C23—432no—
69C23×C23—48no—
70C23×C23—48no—
71C23×C23—48no—
72C23×C23C2×D411yescharacteristic
73C23×C23C2×D412yes—
74Q8×C23—22no—
75D4×C23—216no—
76D4×C23—28no—
77D4×C23—24no—
78D4×C23—22no—
79D4×C23—42no—
80D4×C23—44no—
81D4×C23C2×D418yes—
82C2×C4×C23—22no—
83C2×C4×C23—24no—
84C2×C4×C23—24no—
85C2×C4×C23—22no—
86C2×C4×C23—21no—
87C2×C4×C23—42no—
88C2×C4×C23C2×D412yes—
89C2×C4×C23C2×D411yescharacteristic
90C2×C4×C23C2×D414yes—
91C2×C4×C23C2411yescharacteristic
92C8×C23—24no—
93#3003C2311yesmaximal, characteristic, direct product factor, Sylow
94C22×C23—28no—
95C22×C23—22no—
96C22×C23—22no—
97C22×C23—22no—
98C22×C23—28no—
99C22×C23—416no—
100C22×C23—432no—
101C22×C23—48no—
102C22×C23—41no—
103C22×C23—42no—
104C22×C23#12012yes—
105C22×C23AGL(1,ℤ/8)14yescomplemented
106C22×C23C22×D411yescharacteristic
107C4×C23—22no—
108C4×C23—41no—
109C4×C23—42no—
110C4×C23#12012yes—
111C4×C23C22×D412yes—
112C23×D4C2×C2311yescharacteristic, complemented
113C2×AGL(1,ℤ/8)C2×C2312yescomplemented
114#525C2×C2311yescharacteristic, complemented
115#450C2×C2314yescomplemented
116#447C2×C2314yescomplemented
117#418C2×C2312yescomplemented
118#407C2×C2311yescharacteristic, complemented
119C2×C23—22no—
120C2×C23—24no—
121C2×C23—48no—
122C2×C23—81no—
123C2×C23C2×#12011yescharacteristic
124C2×C23C2×AGL(1,ℤ/8)12yes—
125C25—21no—
126C2×#41C22×C2311yescharacteristic, complemented
127C22×D4—28no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4C22×C2314yes—
131C22×D4C22×C2312yes—
132C23×C4C22×C2311yescharacteristic
133AGL(1,ℤ/8)—28no—
134C2×SD16C22×C2314yescomplemented
135C2×D8C22×C2314yescomplemented
136C2×M16C22×C2312yescomplemented
137#123—21no—
138C22⋊Q8C22×C2312yescomplemented
139#121C22×C2312yescomplemented
140#120—21no—
141C4×D4—22no—
142#117C22×C2311yescharacteristic, complemented
143#102C22×C2318yescomplemented
144C22⋊C8C22×C2314yescomplemented
145C23#300311yescharacteristic, direct product factor, Sylow
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24D4×C2311yescharacteristic
151#41—24no—
152C2×Q8C23×C2311yescharacteristic, complemented
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—22no—
157C2×D4—21no—
158C2×D4—24no—
159C2×D4—22no—
160C2×D4—24no—
161C2×D4D4×C2318yescomplemented
162C2×D4C23×C2314yes—
163C2×D4C23×C2311yescharacteristic, complemented
164C22×C4—24no—
165C22×C4—21no—
166C22×C4D4×C2312yes—
167C22×C4D4×C2311yescharacteristic
168C22×C4C23×C2312yes—
169C22×C4C23×C2311yescharacteristic
170SD16—28no—
171D8—28no—
172M16—24no—
173C2×C8C23×C2314yescomplemented
174C4⋊C4—22no—
175C4⋊C4C23×C2312yescomplemented
176C22⋊C4—21no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C42—21no—
180C23—22no—
181C23—24no—
182C23—216no—
183C23—28no—
184C23—24no—
185C23—21no—
186C23—432no—
187C23—48no—
188C23—48no—
189C23—48no—
190C23C2×D4×C2311yescharacteristic
191C23C2×D4×C2312yes—
192Q8—22no—
193D4—216no—
194D4—28no—
195D4—24no—
196D4—22no—
197D4—42no—
198D4—44no—
199D4C2×D4×C2318yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4C2×D4×C2312yes—
207C2×C4C2×D4×C2311yescharacteristic
208C2×C4C2×D4×C2314yes—
209C2×C4C24×C2311yescharacteristic
210C8—24no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—416no—
217C22—432no—
218C22—48no—
219C22—42no—
220C22—41no—
221C22C23×#12012yes—
222C22C23×AGL(1,ℤ/8)14yescomplemented
223C22C22×D4×C2311yescharacteristic
224C4—22no—
225C4—42no—
226C4—41no—
227C4C23×#12012yes—
228C4C22×D4×C2312yes—
229C2—22no—
230C2—24no—
231C2—48no—
232C2—81no—
233C2C2×C23×#12011yescharacteristic
234C2C2×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 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×D4×C23C2semidirect
2. C2×C23×AGL(1,ℤ/8)C2semidirect
3. C23×#525C2semidirect
4. C23×#450C2semidirect
5. C23×#447C2semidirect
6. C23×#418C2semidirect
7. C23×#407C2semidirect
9. C2×#41×C23C22semidirect
17. C2×SD16×C23C22semidirect
18. C2×D8×C23C22semidirect
19. C2×M16×C23C22semidirect
21. C23×C22⋊Q8C22semidirect
22. C23×#121C22semidirect
25. C23×#117C22semidirect
26. C23×#102C22semidirect
27. C23×C22⋊C8C22semidirect
34. C2×Q8×C23C23semidirect
43. C2×D4×C23D4semidirect
45. C2×D4×C23C23semidirect
55. C2×C8×C23C23semidirect
57. C4⋊C4×C23C23semidirect
93. #3003C23direct (N × H)
105. C22×C23AGL(1,ℤ/8)semidirect
112. C23×D4C2×C23semidirect
113. C2×AGL(1,ℤ/8)C2×C23semidirect
114. #525C2×C23semidirect
115. #450C2×C23semidirect
116. #447C2×C23semidirect
117. #418C2×C23semidirect
118. #407C2×C23semidirect
126. C2×#41C22×C23semidirect
134. C2×SD16C22×C23semidirect
135. C2×D8C22×C23semidirect
136. C2×M16C22×C23semidirect
138. C22⋊Q8C22×C23semidirect
139. #121C22×C23semidirect
142. #117C22×C23semidirect
143. #102C22×C23semidirect
144. C22⋊C8C22×C23semidirect
145. C23#3003direct (N × H)
152. C2×Q8C23×C23semidirect
161. C2×D4D4×C23semidirect
163. C2×D4C23×C23semidirect
173. C2×C8C23×C23semidirect
175. C4⋊C4C23×C23semidirect
222. C22C23×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
C23×D4×C232
C2×C23×AGL(1,ℤ/8)2
C23×#5252
C23×#4502
C23×#4472
C23×#4182
C23×#4072
#300323