Finite Group Databasebeta

← C23×#18529 #19166155

Subgroups — C23×#18529 #19166155

Order 5888.

1630 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 1630 conjugacy classes of subgroups — too many to draw. Only its 246 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×D42×C23C211yesmaximal, characteristic
2C23×#3138C211yesmaximal, characteristic, complemented
3C23×#3110C212yesmaximal, complemented
4C23×#1639C218yesmaximal, complemented
5C23×#1555C211yesmaximal, characteristic, complemented
6C23×#1490C212yesmaximal, complemented
7C23×D4×C23—22no—
8D42×C23—22no—
9D42×C23—24no—
10D42×C23C2214yes—
11C23×#533C2211yescharacteristic, complemented
12C2×C23×#127C2211yescharacteristic
13C2×C23×#121—22no—
14C2×C23×#120—22no—
15C2×C4×D4×C23C2212yes—
16C23×#490—22no—
17C23×#486C2214yescomplemented
18C23×#484—22no—
19C23×#474C2214yescomplemented
20C23×#461C2214yescomplemented
21C23×#450—22no—
22C23×#440—24no—
23C23×#432C2212yescomplemented
24C23×#418—22no—
25C4×M16×C23C2211yescharacteristic, complemented
26C23×#331C2214yescomplemented
27C23×D4⋊C8C2218yescomplemented
28C25×C23—22no—
29C22×D4×C23—24no—
30C22×D4×C23—28no—
31C22×D4×C23—28no—
32C22×D4×C23—22no—
33C22×D4×C23—416no—
34C22×D4×C23D412yes—
35C22×D4×C23D412yes—
36C23×C4×C23—22no—
37C2×SD16×C23—24no—
38C2×M16×C23—21no—
39C23×C4⋊Q8C2312yescomplemented
40C23×#127—22no—
41C23×#127D412yes—
42C23×#127C2312yes—
43C23×#125—21no—
44C23×C22⋊Q8—22no—
45C23×#121—28no—
46C23×#121—28no—
47C23×#120—28no—
48C23×#120—28no—
49C4×Q8×C23—22no—
50C4×D4×C23—22no—
51C4×D4×C23—22no—
52C4×D4×C23D414yes—
53C4×D4×C23C2314yes—
54C23×#117—21no—
55C2×C4⋊C4×C23D412yes—
56C2×C22⋊C4×C23—22no—
57C2×C22⋊C4×C23—22no—
58C2×C42×C23C2311yescharacteristic
59C23×#107—24no—
60C23×C4⋊C8C2314yescomplemented
61C23×#103—24no—
62C23×#102—28no—
63C23×#102—24no—
64C23×C22⋊C8—22no—
65C23×#97—21no—
66C23×#97—22no—
67C4×C8×C23C2312yescomplemented
68C24×C23—22no—
69C24×C23—28no—
70C24×C23—22no—
71C24×C23—416no—
72C24×C23—84no—
73C2×Q8×C23—22no—
74C2×D4×C23—28no—
75C2×D4×C23—216no—
76C2×D4×C23—22no—
77C2×D4×C23—24no—
78C2×D4×C23—22no—
79C2×D4×C23—24no—
80C2×D4×C23—48no—
81C2×D4×C23—44no—
82C2×D4×C23—432no—
83C2×D4×C23—416no—
84C2×D4×C23—416no—
85C2×D4×C23—416no—
86C2×D4×C23C2×D414yes—
87C2×D4×C23C2×D414yes—
88C22×C4×C23—22no—
89C22×C4×C23—24no—
90C22×C4×C23—24no—
91C22×C4×C23—48no—
92C22×C4×C23C2×D411yescharacteristic
93C22×C4×C23C2×D412yes—
94SD16×C23—48no—
95M16×C23—42no—
96C2×C8×C23—22no—
97C2×C8×C23—24no—
98C4⋊C4×C23—22no—
99C4⋊C4×C23—22no—
100C4⋊C4×C23—22no—
101C4⋊C4×C23—42no—
102C4⋊C4×C23C2×D414yes—
103C22⋊C4×C23—28no—
104C22⋊C4×C23—24no—
105C22⋊C4×C23—24no—
106C22⋊C4×C23—41no—
107C42×C23—41no—
108C42×C23C2×D411yescharacteristic
109C42×C23C2×D412yes—
110C42×C23C2411yescharacteristic
111#18529C2311yesmaximal, characteristic, direct product factor, Sylow
112C23×C23—28no—
113C23×C23—24no—
114C23×C23—24no—
115C23×C23—44no—
116C23×C23—48no—
117C23×C23—432no—
118C23×C23—44no—
119C23×C23—44no—
120C23×C23—88no—
121C23×C23—832no—
122C23×C23#12011yescharacteristic
123Q8×C23—44no—
124D4×C23—28no—
125D4×C23—24no—
126D4×C23—44no—
127D4×C23—48no—
128D4×C23—432no—
129D4×C23—48no—
130D4×C23—82no—
131D4×C23AGL(1,ℤ/8)18yes—
132C2×C4×C23—22no—
133C2×C4×C23—24no—
134C2×C4×C23—24no—
135C2×C4×C23—48no—
136C2×C4×C23—44no—
137C2×C4×C23—48no—
138C2×C4×C23—42no—
139C2×C4×C23#12011yescharacteristic
140C2×C4×C23#12012yes—
141C2×C4×C23C22×D411yescharacteristic
142C2×C4×C23C22×D412yes—
143C8×C23—42no—
144C8×C23—44no—
145C2×D42C2×C2311yescharacteristic
146#3138C2×C2311yescharacteristic, complemented
147#3110C2×C2312yescomplemented
148#1639C2×C2318yescomplemented
149#1555C2×C2311yescharacteristic, complemented
150#1490C2×C2312yescomplemented
151C22×C23—22no—
152C22×C23—28no—
153C22×C23—416no—
154C22×C23—44no—
155C22×C23—44no—
156C22×C23—48no—
157C22×C23—44no—
158C22×C23—88no—
159C22×C23—832no—
160C22×C23#45512yes—
161C22×C23C2×#12011yescharacteristic
162C4×C23—22no—
163C4×C23—21no—
164C4×C23—21no—
165C4×C23—44no—
166C4×C23—82no—
167C4×C23C2×AGL(1,ℤ/8)14yes—
168C23×D4—22no—
169D42—22no—
170D42—24no—
171D42C22×C2314yes—
172#533C22×C2311yescharacteristic, complemented
173C2×#127C22×C2311yescharacteristic
174C2×#121—22no—
175C2×#120—22no—
176C2×C4×D4C22×C2312yes—
177#490—22no—
178#486C22×C2314yescomplemented
179#484—22no—
180#474C22×C2314yescomplemented
181#461C22×C2314yescomplemented
182#450—22no—
183#440—24no—
184#432C22×C2312yescomplemented
185#418—22no—
186C4×M16C22×C2311yescharacteristic, complemented
187#331C22×C2314yescomplemented
188D4⋊C8C22×C2318yescomplemented
189C2×C23—22no—
190C2×C23—48no—
191C2×C23—84no—
192C2×C23#300312yes—
193C2×C23C2×#45511yescharacteristic
194C25—22no—
195C22×D4—24no—
196C22×D4—28no—
197C22×D4—28no—
198C22×D4—22no—
199C22×D4—416no—
200C22×D4D4×C2312yes—
201C22×D4D4×C2312yes—
202C23×C4—22no—
203C2×SD16—24no—
204C2×M16—21no—
205C4⋊Q8C23×C2312yescomplemented
206#127—22no—
207#127D4×C2312yes—
208#127C23×C2312yes—
209#125—21no—
210C22⋊Q8—22no—
211#121—28no—
212#121—28no—
213#120—28no—
214#120—28no—
215C4×Q8—22no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4D4×C2314yes—
219C4×D4C23×C2314yes—
220#117—21no—
221C2×C4⋊C4D4×C2312yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C42C23×C2311yescharacteristic
225#107—24no—
226C4⋊C8C23×C2314yescomplemented
227#103—24no—
228#102—28no—
229#102—24no—
230C22⋊C8—22no—
231#97—21no—
232#97—22no—
233C4×C8C23×C2312yescomplemented
234C23#1852911yescharacteristic, direct product factor, Sylow
235C24—22no—
236C24—28no—
237C24—22no—
238C24—416no—
239C24—84no—
240C2×Q8—22no—
241C2×D4—28no—
242C2×D4—216no—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—22no—
246C2×D4—24no—
247C2×D4—48no—
248C2×D4—44no—
249C2×D4—432no—
250C2×D4—416no—
251C2×D4—416no—
252C2×D4—416no—
253C2×D4C2×D4×C2314yes—
254C2×D4C2×D4×C2314yes—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—48no—
259C22×C4C2×D4×C2312yes—
260C22×C4C2×D4×C2311yescharacteristic
261SD16—48no—
262M16—42no—
263C2×C8—22no—
264C2×C8—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—42no—
269C4⋊C4C2×D4×C2314yes—
270C22⋊C4—28no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—41no—
274C42—41no—
275C42C2×D4×C2311yescharacteristic
276C42C2×D4×C2312yes—
277C42C24×C2311yescharacteristic
278C23—28no—
279C23—24no—
280C23—24no—
281C23—44no—
282C23—48no—
283C23—432no—
284C23—44no—
285C23—44no—
286C23—88no—
287C23—832no—
288C23C23×#12011yescharacteristic
289Q8—44no—
290D4—28no—
291D4—24no—
292D4—44no—
293D4—48no—
294D4—432no—
295D4—48no—
296D4—82no—
297D4C23×AGL(1,ℤ/8)18yes—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—48no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—42no—
305C2×C4C23×#12011yescharacteristic
306C2×C4C23×#12012yes—
307C2×C4C22×D4×C2311yescharacteristic
308C2×C4C22×D4×C2312yes—
309C8—42no—
310C8—44no—
311C22—22no—
312C22—28no—
313C22—416no—
314C22—44no—
315C22—44no—
316C22—48no—
317C22—44no—
318C22—88no—
319C22—832no—
320C22C23×#45512yes—
321C22C2×C23×#12011yescharacteristic
322C4—22no—
323C4—21no—
324C4—21no—
325C4—44no—
326C4—82no—
327C4C2×C23×AGL(1,ℤ/8)14yes—
328C2—22no—
329C2—48no—
330C2—84no—
331C2C23×#300312yes—
332C2C2×C23×#45511yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC2×D42×C23C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#3138C2semidirect
3. C23×#3110C2semidirect
4. C23×#1639C2semidirect
5. C23×#1555C2semidirect
6. C23×#1490C2semidirect
11. C23×#533C22semidirect
17. C23×#486C22semidirect
19. C23×#474C22semidirect
20. C23×#461C22semidirect
23. C23×#432C22semidirect
25. C4×M16×C23C22semidirect
26. C23×#331C22semidirect
27. C23×D4⋊C8C22semidirect
39. C23×C4⋊Q8C23semidirect
60. C23×C4⋊C8C23semidirect
67. C4×C8×C23C23semidirect
111. #18529C23direct (N × H)
146. #3138C2×C23semidirect
147. #3110C2×C23semidirect
148. #1639C2×C23semidirect
149. #1555C2×C23semidirect
150. #1490C2×C23semidirect
172. #533C22×C23semidirect
178. #486C22×C23semidirect
180. #474C22×C23semidirect
181. #461C22×C23semidirect
184. #432C22×C23semidirect
186. C4×M16C22×C23semidirect
187. #331C22×C23semidirect
188. D4⋊C8C22×C23semidirect
205. C4⋊Q8C23×C23semidirect
226. C4⋊C8C23×C23semidirect
233. C4×C8C23×C23semidirect
234. C23#18529direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D42×C232
C23×#31382
C23×#31102
C23×#16392
C23×#15552
C23×#14902
#1852923