Finite Group Databasebeta

← C23×#3021 #15670640

Subgroups — C23×#3021 #15670640

Order 2944.

736 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 212 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×#585C211yesmaximal, characteristic, complemented
2C2×C23×AGL(1,ℤ/8)C212yesmaximal, complemented
3C23×#535C211yesmaximal, characteristic, complemented
4C23×#454C214yesmaximal, complemented
5C23×#453C214yesmaximal, complemented
6C23×#421C212yesmaximal, complemented
7C23×#413C211yesmaximal, characteristic, complemented
8C23×2−1+4—22no—
9C23×2−1+4C2212yescomplemented
10C23×H(5,𝔽2)—22no—
11C23×H(5,𝔽2)—22no—
12C23×H(5,𝔽2)C2212yescomplemented
13C2×#41×C23—24no—
14C2×#41×C23—22no—
15C2×#41×C23C2212yes—
16C2×#41×C23C2211yescharacteristic
17C22×D4×C23C2211yescharacteristic, complemented
18C23×AGL(1,ℤ/8)—24no—
19C23×AGL(1,ℤ/8)C2218yescomplemented
20C2×SD16×C23—22no—
21C2×D8×C23—22no—
22C2×M16×C23C2212yescomplemented
23C23×#127C2212yescomplemented
24C23×#124C2212yescomplemented
25C23×#121—22no—
26C23×#120—22no—
27C23×#117C2211yescharacteristic, complemented
28C23×#104C2218yescomplemented
29C23×#101C2212yescomplemented
30C23×#100C2212yescomplemented
31C24×C23—21no—
32#41×C23—24no—
33#41×C23—28no—
34#41×C23—24no—
35#41×C23—24no—
36#41×C23—24no—
37#41×C23—22no—
38#41×C23—22no—
39#41×C23—24no—
40#41×C23—22no—
41#41×C23—22no—
42#41×C23D414yescomplemented
43#41×C23C2314yes—
44C2×Q8×C23—22no—
45C2×Q8×C23—24no—
46C2×Q8×C23D412yescomplemented
47C2×Q8×C23C2311yescharacteristic
48C2×D4×C23—28no—
49C2×D4×C23—22no—
50C2×D4×C23—24no—
51C2×D4×C23—22no—
52C2×D4×C23—24no—
53C2×D4×C23—24no—
54C2×D4×C23—22no—
55C2×D4×C23—22no—
56C2×D4×C23—21no—
57C2×D4×C23D412yes—
58C2×D4×C23D412yescomplemented
59C2×D4×C23C2312yes—
60C2×D4×C23C2311yescharacteristic
61C22×C4×C23—24no—
62C22×C4×C23—22no—
63C22×C4×C23D412yes—
64C22×C4×C23C2311yescharacteristic
65SD16×C23—28no—
66D8×C23—28no—
67M16×C23—22no—
68M16×C23C2314yes—
69C2×C8×C23—22no—
70C4⋊C4×C23—21no—
71C22⋊C4×C23—21no—
72C22⋊C4×C23—24no—
73C42×C23C2312yes—
74C23×C23—24no—
75C23×C23—22no—
76C23×C23—22no—
77C23×C23—42no—
78C23×C23—82no—
79C23×C23C2×D412yes—
80C23×C23C2×D411yescharacteristic
81Q8×C23—24no—
82Q8×C23—22no—
83Q8×C23—22no—
84Q8×C23C2×D414yes—
85D4×C23—28no—
86D4×C23—24no—
87D4×C23—24no—
88D4×C23—24no—
89D4×C23—24no—
90D4×C23—22no—
91D4×C23—22no—
92D4×C23—24no—
93D4×C23—41no—
94D4×C23—41no—
95D4×C23—84no—
96D4×C23C2×D414yes—
97C2×C4×C23—24no—
98C2×C4×C23—28no—
99C2×C4×C23—22no—
100C2×C4×C23—24no—
101C2×C4×C23—22no—
102C2×C4×C23—22no—
103C2×C4×C23—24no—
104C2×C4×C23—22no—
105C2×C4×C23C2×D412yes—
106C2×C4×C23C2×D411yescharacteristic
107C2×C4×C23C2×D412yes—
108C2×C4×C23C2×D412yes—
109C2×C4×C23C2411yescharacteristic
110C8×C23—24no—
111#3021C2311yesmaximal, characteristic, direct product factor, Sylow
112C22×C23—22no—
113C22×C23—24no—
114C22×C23—41no—
115C22×C23—42no—
116C22×C23—44no—
117C22×C23—42no—
118C22×C23—82no—
119C22×C23—84no—
120C22×C23—84no—
121C22×C23#12012yes—
122C22×C23C22×D411yescharacteristic
123C4×C23—24no—
124C4×C23—22no—
125C4×C23—42no—
126C4×C23#12011yescharacteristic
127C4×C23#12011yescharacteristic
128C4×C23C22×D412yes—
129#585C2×C2311yescharacteristic, complemented
130C2×AGL(1,ℤ/8)C2×C2312yescomplemented
131#535C2×C2311yescharacteristic, complemented
132#454C2×C2314yescomplemented
133#453C2×C2314yescomplemented
134#421C2×C2312yescomplemented
135#413C2×C2311yescharacteristic, complemented
136C2×C23—21no—
137C2×C23—22no—
138C2×C23—42no—
139C2×C23—44no—
140C2×C23—82no—
141C2×C23C2×#12011yescharacteristic
1422−1+4—22no—
1432−1+4C22×C2312yescomplemented
144H(5,𝔽2)—22no—
145H(5,𝔽2)—22no—
146H(5,𝔽2)C22×C2312yescomplemented
147C2×#41—24no—
148C2×#41—22no—
149C2×#41C22×C2312yes—
150C2×#41C22×C2311yescharacteristic
151C22×D4C22×C2311yescharacteristic, complemented
152AGL(1,ℤ/8)—24no—
153AGL(1,ℤ/8)C22×C2318yescomplemented
154C2×SD16—22no—
155C2×D8—22no—
156C2×M16C22×C2312yescomplemented
157#127C22×C2312yescomplemented
158#124C22×C2312yescomplemented
159#121—22no—
160#120—22no—
161#117C22×C2311yescharacteristic, complemented
162#104C22×C2318yescomplemented
163#101C22×C2312yescomplemented
164#100C22×C2312yescomplemented
165C23#302111yescharacteristic, direct product factor, Sylow
166C24—21no—
167#41—24no—
168#41—28no—
169#41—24no—
170#41—24no—
171#41—24no—
172#41—22no—
173#41—22no—
174#41—24no—
175#41—22no—
176#41—22no—
177#41D4×C2314yescomplemented
178#41C23×C2314yes—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8D4×C2312yescomplemented
182C2×Q8C23×C2311yescharacteristic
183C2×D4—28no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4—22no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—21no—
192C2×D4D4×C2312yes—
193C2×D4D4×C2312yescomplemented
194C2×D4C23×C2312yes—
195C2×D4C23×C2311yescharacteristic
196C22×C4—24no—
197C22×C4—22no—
198C22×C4D4×C2312yes—
199C22×C4C23×C2311yescharacteristic
200SD16—28no—
201D8—28no—
202M16—22no—
203M16C23×C2314yes—
204C2×C8—22no—
205C4⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—24no—
208C42C23×C2312yes—
209C23—24no—
210C23—22no—
211C23—22no—
212C23—42no—
213C23—82no—
214C23C2×D4×C2312yes—
215C23C2×D4×C2311yescharacteristic
216Q8—24no—
217Q8—22no—
218Q8—22no—
219Q8C2×D4×C2314yes—
220D4—28no—
221D4—24no—
222D4—24no—
223D4—24no—
224D4—24no—
225D4—22no—
226D4—22no—
227D4—24no—
228D4—41no—
229D4—41no—
230D4—84no—
231D4C2×D4×C2314yes—
232C2×C4—24no—
233C2×C4—28no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4C2×D4×C2312yes—
241C2×C4C2×D4×C2311yescharacteristic
242C2×C4C2×D4×C2312yes—
243C2×C4C2×D4×C2312yes—
244C2×C4C24×C2311yescharacteristic
245C8—24no—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—42no—
250C22—44no—
251C22—42no—
252C22—82no—
253C22—84no—
254C22—84no—
255C22C23×#12012yes—
256C22C22×D4×C2311yescharacteristic
257C4—24no—
258C4—22no—
259C4—42no—
260C4C23×#12011yescharacteristic
261C4C23×#12011yescharacteristic
262C4C22×D4×C2312yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—44no—
267C2—82no—
268C2C2×C23×#12011yescharacteristic
subgroupG/H
center, socleC2×C23C2×#120
commutator, Frattini, cosocleC2×C4C24×C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#585C2semidirect
2. C2×C23×AGL(1,ℤ/8)C2semidirect
3. C23×#535C2semidirect
4. C23×#454C2semidirect
5. C23×#453C2semidirect
6. C23×#421C2semidirect
7. C23×#413C2semidirect
9. C23×2−1+4C22semidirect
12. C23×H(5,𝔽2)C22semidirect
17. C22×D4×C23C22semidirect
19. C23×AGL(1,ℤ/8)C22semidirect
22. C2×M16×C23C22semidirect
23. C23×#127C22semidirect
24. C23×#124C22semidirect
27. C23×#117C22semidirect
28. C23×#104C22semidirect
29. C23×#101C22semidirect
30. C23×#100C22semidirect
42. #41×C23D4semidirect
46. C2×Q8×C23D4semidirect
58. C2×D4×C23D4semidirect
111. #3021C23direct (N × H)
129. #585C2×C23semidirect
130. C2×AGL(1,ℤ/8)C2×C23semidirect
131. #535C2×C23semidirect
132. #454C2×C23semidirect
133. #453C2×C23semidirect
134. #421C2×C23semidirect
135. #413C2×C23semidirect
143. 2−1+4C22×C23semidirect
146. H(5,𝔽2)C22×C23semidirect
151. C22×D4C22×C23semidirect
153. AGL(1,ℤ/8)C22×C23semidirect
156. C2×M16C22×C23semidirect
157. #127C22×C23semidirect
158. #124C22×C23semidirect
161. #117C22×C23semidirect
162. #104C22×C23semidirect
163. #101C22×C23semidirect
164. #100C22×C23semidirect
165. C23#3021direct (N × H)
177. #41D4×C23semidirect
181. C2×Q8D4×C23semidirect
193. C2×D4D4×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#5852
C2×C23×AGL(1,ℤ/8)2
C23×#5352
C23×#4542
C23×#4532
C23×#4212
C23×#4132
#302123