Finite Group Databasebeta

← D23×#451 #15682588

Subgroups — D23×#451 #15682588

Order 2944.

740 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C22×Q8×D23C211yesmaximal, characteristic
2C2×Q16×D23C212yesmaximal, complemented
3C22⋊Q8×D23C211yesmaximal, characteristic
4#13092575C211yesmaximal, characteristic, complemented
5#13092574C212yesmaximal, complemented
6#13092573C211yesmaximal, characteristic, complemented
7#13092572C211yesmaximal, characteristic, complemented
8#13092571C212yesmaximal, complemented
9#103×D23C212yesmaximal, complemented
10C22⋊C8×D23C211yesmaximal, characteristic
11C23×#451C211yesmaximal, characteristic, complemented
12C22×Q8×C23C2211yescharacteristic
13C2×Q8×D23—28no—
14C2×Q8×D23—21no—
15C2×Q8×D23—21no—
16C2×Q8×D23C2211yescharacteristic
17C2×Q8×D23C2212yes—
18C2×Q8×D23C2212yes—
19C22×C4×D23—21no—
20C22×C4×D23C2211yescharacteristic
21C22×Dic46—21no—
22C22×Dic46C2211yescharacteristic
23C2×Q16×C23C2212yescomplemented
24C23×C22⋊Q8C2211yescharacteristic
25#10625117C2211yescharacteristic
26C2×#67275C2212yescomplemented
27C2×#67275C2212yescomplemented
28#10625095C2211yescharacteristic
29Q16×D23—28no—
30C2×Dic92C2212yescomplemented
31C2×C8×D23C2212yes—
32#10625068C2211yescharacteristic
33#10625067—21no—
34C4⋊C4×D23—21no—
35C4⋊C4×D23C2211yescharacteristic
36C22⋊C4×D23—21no—
37#10625049—21no—
38C23×#103C2212yescomplemented
39C23×C22⋊C8C2211yescharacteristic
40#10625017C2212yescomplemented
41C22⋊(C23⋊C8)C2211yescharacteristic
42(C2×D23)⋊C8C2212yes—
43#10624998C2212yescomplemented
44#10624992C2212yescomplemented
45C23×D23D411yescharacteristic
46C2×Q8×C23—24no—
47C2×Q8×C23—21no—
48C2×Q8×C23C2311yescharacteristic
49C2×Q8×C23C2312yes—
50C22×C4×C23—21no—
51C22×C4×C23C2311yescharacteristic
52C22×Dic23—21no—
53C22×Dic23D411yescharacteristic
54Q8×D23—28no—
55Q8×D23—28no—
56Q8×D23—24no—
57Q8×D23—24no—
58Q8×D23—24no—
59Q8×D23—24no—
60Q8×D23D418yes—
61C2×C4×D23—28no—
62C2×C4×D23—22no—
63C2×C4×D23—22no—
64C2×C4×D23—24no—
65C2×C4×D23—21no—
66C2×C4×D23—21no—
67C2×C4×D23D411yescharacteristic
68C2×C4×D23D411yescharacteristic
69C2×C4×D23C2311yescharacteristic
70C2×C4×D23C2311yescharacteristic
71C2×Dic46—28no—
72C2×Dic46—21no—
73C2×Dic46—22no—
74C2×Dic46—22no—
75C2×Dic46—24no—
76C2×Dic46—21no—
77C2×Dic46C2311yescharacteristic
78C2×Dic46C2312yes—
79Q16×C23—24no—
80C2×C8×C23C2312yes—
81C4⋊C4×C23—21no—
82C4⋊C4×C23C2311yescharacteristic
83C22⋊C4×C23—21no—
84C22⋊Dic23—21no—
85#67275—24no—
86#67275—24no—
87(C2×D23)⋊C4—21no—
88(C2×D23)⋊C4—21no—
89C4⋊Dic23—21no—
90C4⋊Dic23C2311yescharacteristic
91Dic23⋊C4—21no—
92Dic23⋊C4—21no—
93Dic23⋊C4—21no—
94C2×C23⋊C8C2312yes—
95Dic92—24no—
96C8×D23—24no—
97C23×C23C2×D411yescharacteristic
98C22×D23—22no—
99C22×D23—22no—
100C22×D23Q1614yescomplemented
101C22×D23C2×D411yescharacteristic
102C22×D23C2×D411yescharacteristic
103Q8×C23—24no—
104Q8×C23—22no—
105Q8×C23—22no—
106Q8×C23C2×D414yes—
107C2×C4×C23—24no—
108C2×C4×C23—22no—
109C2×C4×C23—22no—
110C2×C4×C23—21no—
111C2×C4×C23—21no—
112C2×C4×C23C2×D411yescharacteristic
113C2×C4×C23C2411yescharacteristic
114C2×Dic23—24no—
115C2×Dic23—22no—
116C2×Dic23—22no—
117C2×Dic23—21no—
118C2×Dic23—21no—
119C2×Dic23C2×D411yescharacteristic
120C2×Dic23C2×D411yescharacteristic
121C4×D23—28no—
122C4×D23—28no—
123C4×D23—22no—
124C4×D23—22no—
125C4×D23—22no—
126C4×D23—42no—
127C4×D23—42no—
128C4×D23C2×D414yes—
129Dic46—28no—
130Dic46—24no—
131Dic46—22no—
132Dic46—28no—
133Dic46—22no—
134Dic46—42no—
135Dic46C2×D414yes—
136C8×C23—22no—
137C23⋊C8—22no—
138C2×#451—231nomaximal, Sylow
139C22×C23—21no—
140C22×C23—21no—
141C22×C23C2×Q1612yescomplemented
142C22×C23C22×D411yescharacteristic
143C2×D23—21no—
144C2×D23—21no—
145C2×D23—28no—
146C2×D23#12012yes—
147C2×D23C2×Q1612yes—
148C2×D23C2×Q1612yescomplemented
149C2×D23#13712yes—
150C4×C23—24no—
151C4×C23—21no—
152C4×C23—41no—
153C4×C23—41no—
154C4×C23C22×D412yes—
155Dic23—21no—
156Dic23—24no—
157Dic23—41no—
158Dic23—41no—
159Dic23#12012yes—
160C23×Q8—231no—
161C22×Q16—232no—
162C2×C22⋊Q8—231no—
163#451—231no—
164#451—232no—
165#451—231no—
166#451—232no—
167#451—231no—
168#451D2311yescharacteristic, direct product factor
169C2×#103—232no—
170C2×C22⋊C8—231no—
171C2×C23—22no—
172C2×C23C2×#12011yescharacteristic
173C2×C23C22×Q1611yescharacteristic
174C2×C23C2×#13711yescharacteristic
175D23—22no—
176D23#45114yesdirect product factor
177C22×Q8—232no—
178C22×Q8—231no—
179C22×Q8—232no—
180C22×Q8—231no—
181C22×Q8—461no—
182C22×Q8—461no—
183C22×Q8—461no—
184C22×Q8—468no—
185C22×Q8C2×D2311yescharacteristic
186C23×C4—231no—
187C23×C4—461no—
188C2×Q16—232no—
189C2×Q16—232no—
190C2×Q16—232no—
191C2×Q16—468no—
192C2×Q16C2×D2312yescomplemented
193C22×C8—232no—
194C22⋊Q8—231no—
195C22⋊Q8—231no—
196C22⋊Q8—231no—
197C22⋊Q8—461no—
198C22⋊Q8—461no—
199C22⋊Q8C2×D2311yescharacteristic
200C2×C4⋊C4—231no—
201C2×C4⋊C4—461no—
202C2×C22⋊C4—461no—
203#103—232no—
204#103—232no—
205#103—232no—
206#103C2×D2312yescomplemented
207C22⋊C8—231no—
208C22⋊C8—232no—
209C22⋊C8C2×D2311yescharacteristic
210C23C2×#45111yescharacteristic, complemented, Sylow
211C24—231no—
212C2×Q8—24no—
213C2×Q8—21no—
214C2×Q8—238no—
215C2×Q8—232no—
216C2×Q8—231no—
217C2×Q8—462no—
218C2×Q8—462no—
219C2×Q8—461no—
220C2×Q8—464no—
221C2×Q8—468no—
222C2×Q8—464no—
223C2×Q8—464no—
224C2×Q8—464no—
225C2×Q8—468no—
226C2×Q8—468no—
227C2×Q8—464no—
228C2×Q8—461no—
229C2×Q8C22×D2311yescharacteristic
230C2×Q8C22×D2312yes—
231C22×C4—21no—
232C22×C4—231no—
233C22×C4—231no—
234C22×C4—231no—
235C22×C4—231no—
236C22×C4—231no—
237C22×C4—464no—
238C22×C4—462no—
239C22×C4—462no—
240C22×C4—468no—
241C22×C4—461no—
242C22×C4—461no—
243C22×C4—461no—
244C22×C4C22×D2311yescharacteristic
245Q16—24no—
246Q16—464no—
247Q16—464no—
248Q16—464no—
249C2×C8—232no—
250C2×C8—464no—
251C2×C8C22×D2312yes—
252C4⋊C4—21no—
253C4⋊C4—231no—
254C4⋊C4—461no—
255C4⋊C4—461no—
256C4⋊C4—461no—
257C4⋊C4—461no—
258C4⋊C4C22×D2311yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—461no—
261C22⋊C4—461no—
262C22⋊C4—461no—
263C23—231no—
264C23—234no—
265C23—231no—
266C23—462no—
267C23—462no—
268C23D4×D2311yescharacteristic
269Q8—24no—
270Q8—22no—
271Q8—22no—
272Q8—234no—
273Q8—468no—
274Q8—468no—
275Q8—464no—
276Q8—462no—
277Q8—462no—
278Q8—922no—
279Q8D4×D2314yes—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—231no—
286C2×C4—234no—
287C2×C4—231no—
288C2×C4—462no—
289C2×C4—462no—
290C2×C4—468no—
291C2×C4—468no—
292C2×C4—462no—
293C2×C4—464no—
294C2×C4—462no—
295C2×C4—462no—
296C2×C4—461no—
297C2×C4—461no—
298C2×C4—922no—
299C2×C4—922no—
300C2×C4D4×D2311yescharacteristic
301C2×C4C23×D2311yescharacteristic
302C8—22no—
303C8—462no—
304C22—21no—
305C22—21no—
306C22—232no—
307C22—232no—
308C22—232no—
309C22—232no—
310C22—468no—
311C22—461no—
312C22—461no—
313C22Q16×D2312yescomplemented
314C22C2×D4×D2311yescharacteristic
315C4—24no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—232no—
320C4—461no—
321C4—464no—
322C4—921no—
323C4—921no—
324C4C2×D4×D2312yes—
325C2—22no—
326C2—234no—
327C2—462no—
328C2#120×D2311yescharacteristic
329C2C2×Q16×D2311yescharacteristic
330C2#137×D2311yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#451C2
socleC22×C23C22×D4
prime radicalC23×D23D4
hypercenter#451D23
hypocenterC23C2×#451
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×Q16×D23C2semidirect
4. #13092575C2semidirect
5. #13092574C2semidirect
6. #13092573C2semidirect
7. #13092572C2semidirect
8. #13092571C2semidirect
9. #103×D23C2semidirect
11. C23×#451C2semidirect
23. C2×Q16×C23C22semidirect
26. C2×#67275C22semidirect
27. C2×#67275C22semidirect
30. C2×Dic92C22semidirect
38. C23×#103C22semidirect
40. #10625017C22semidirect
43. #10624998C22semidirect
44. #10624992C22semidirect
100. C22×D23Q16semidirect
141. C22×C23C2×Q16semidirect
148. C2×D23C2×Q16semidirect
168. #451D23direct (N × H)
176. D23#451direct (N × H)
192. C2×Q16C2×D23semidirect
206. #103C2×D23semidirect
210. C23C2×#451semidirect
313. C22Q16×D23semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D232
C2×Q16×D232
C22⋊Q8×D232
#130925752
#130925742
#130925732
#130925722
#130925712
#103×D232
C22⋊C8×D232
C23×#4512
C2×#45123