Finite Group Databasebeta

← D23×#480 #15683265

Subgroups — D23×#480 #15683265

Order 2944.

512 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 135 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
1#121×D23C211yesmaximal, characteristic
2C2×C4⋊C4×D23C211yesmaximal, characteristic, complemented
3#13092732C212yesmaximal, complemented
4#13092731C211yesmaximal, characteristic, complemented
5#13092730C211yesmaximal, characteristic, complemented
6#13092729C212yesmaximal, complemented
7#13092728C211yesmaximal, characteristic, complemented
8#107×D23C212yesmaximal, complemented
9#102×D23C212yesmaximal, complemented
10C22⋊C8×D23C211yesmaximal, characteristic, complemented
11C23×#480C211yesmaximal, characteristic, complemented
12C2×D4×D23—21no—
13C2×D4×D23C2211yescharacteristic
14C22×C4×D23—21no—
15C22×C4×D23C2211yescharacteristic
16C23×#121C2211yescharacteristic
17C2×C4⋊C4×C23C2211yescharacteristic, complemented
18#10625110—21no—
19#10625109C2211yescharacteristic
20#10625101C2211yescharacteristic
21C2×C4⋊Dic23C2211yescharacteristic, complemented
22C2×Dic23⋊C4—21no—
23C2×C8×D23C2212yes—
24#10625066C2211yescharacteristic
25C4⋊C4×D23—21no—
26C4⋊C4×D23—21no—
27C4⋊C4×D23C2212yes—
28C4⋊C4×D23C2211yescharacteristic
29C4⋊C4×D23C2212yescomplemented
30#10625055—21no—
31C22⋊C4×D23—21no—
32C23×#107C2212yescomplemented
33C23×#102C2212yescomplemented
34C23×C22⋊C8C2211yescharacteristic, complemented
35#10625014C2212yescomplemented
36C22⋊(C23⋊C8)C2211yescharacteristic, complemented
37#10625003C2212yescomplemented
38(C2×D23)⋊C8C2212yescomplemented
39#10625000C2212yescomplemented
40#10624991C2212yescomplemented
41#10624989C2212yescomplemented
42#10624989C2212yescomplemented
43C23×D23—21no—
44C23×D23D411yescharacteristic
45C2×D4×C23—21no—
46C2×D4×C23C2311yescharacteristic
47C22×C4×C23—21no—
48C22×C4×C23C2311yescharacteristic
49C2×C22⋊D23—21no—
50C2×C22⋊D23—21no—
51C2×C22⋊D23—21no—
52C22×Dic23—21no—
53C22×Dic23D411yescharacteristic
54D4×D23—24no—
55D4×D23—44no—
56C2×D92—21no—
57C2×D92C2311yescharacteristic
58C2×C4×D23—24no—
59C2×C4×D23—21no—
60C2×C4×D23—22no—
61C2×C4×D23—22no—
62C2×C4×D23—44no—
63C2×C4×D23D411yescharacteristic
64C2×C4×D23D411yescharacteristic
65C2×C4×D23C2311yescharacteristic
66C2×C4×D23C2311yescharacteristic
67C2×C8×C23C2312yes—
68C4⋊C4×C23—21no—
69C4⋊C4×C23C2312yes—
70C4⋊C4×C23C2311yescharacteristic
71C22⋊C4×C23—21no—
72C22⋊Dic23—21no—
73(C2×D23)⋊C4—21no—
74(C2×D23)⋊C4—21no—
75C4⋊Dic23—21no—
76C4⋊Dic23C2312yes—
77C4⋊Dic23C2311yescharacteristic
78Dic23⋊C4—21no—
79Dic23⋊C4—22no—
80Dic23⋊C4—22no—
81C2×C23⋊C8C2312yes—
82C8×D23—24no—
83C23×C23—21no—
84C23×C23C2×D411yescharacteristic
85C22×D23—22no—
86C22×D23—22no—
87C22×D23—21no—
88C22×D23—42no—
89C22×D23—42no—
90C22×D23—42no—
91C22×D23D814yes—
92C22×D23C2×D411yescharacteristic
93C22×D23C2×D411yescharacteristic
94D4×C23—22no—
95D4×C23—42no—
96C2×C4×C23—22no—
97C2×C4×C23—22no—
98C2×C4×C23—21no—
99C2×C4×C23—42no—
100C2×C4×C23C2×D411yescharacteristic
101C2×C4×C23C2411yescharacteristic
102C22⋊D23—42no—
103C22⋊D23—42no—
104C22⋊D23—42no—
105C2×Dic23—22no—
106C2×Dic23—22no—
107C2×Dic23—21no—
108C2×Dic23—42no—
109C2×Dic23C2×D411yescharacteristic
110C2×Dic23C2×D411yescharacteristic
111D92—22no—
112D92—42no—
113C4×D23—22no—
114C4×D23—22no—
115C4×D23—22no—
116C4×D23—44no—
117C4×D23—44no—
118C4×D23—42no—
119C4×D23#4114yes—
120C8×C23—22no—
121C23⋊C8—22no—
122C2×#480—231nomaximal, Sylow
123C22×C23—21no—
124C22×C23—21no—
125C22×C23—41no—
126C22×C23—41no—
127C22×C23—41no—
128C22×C23C2×D812yes—
129C22×C23C22×D411yescharacteristic
130C2×D23—21no—
131C2×D23—21no—
132C2×D23—28no—
133C2×D23—41no—
134C2×D23—41no—
135C2×D23—41no—
136C2×D23—84no—
137C2×D23#12312yes—
138C2×D23C2×D812yes—
139C2×D23C2×D812yes—
140C2×D23#13712yes—
141C4×C23—21no—
142C4×C23—42no—
143C4×C23—41no—
144C4×C23C2×#4112yes—
145Dic23—21no—
146Dic23—42no—
147Dic23—41no—
148Dic23#12312yes—
149C2×#121—231no—
150C22×C4⋊C4—231no—
151#480—231no—
152#480—231no—
153#480—232no—
154#480—232no—
155#480—231no—
156#480D2311yescharacteristic, direct product factor
157C2×#107—232no—
158C2×#102—232no—
159C2×C22⋊C8—231no—
160C2×C23—22no—
161C2×C23—81no—
162C2×C23C2×#12311yescharacteristic
163C2×C23C22×D811yescharacteristic
164C2×C23C2×#13711yescharacteristic
165D23—22no—
166D23—81no—
167D23#48014yesdirect product factor
168C22×D4—231no—
169C22×D4—461no—
170C23×C4—231no—
171C23×C4—461no—
172C22×C8—232no—
173#121—231no—
174#121—231no—
175#121—231no—
176#121—461no—
177#121—461no—
178#121C2×D2311yescharacteristic
179C2×C4⋊C4—231no—
180C2×C4⋊C4—232no—
181C2×C4⋊C4—232no—
182C2×C4⋊C4—231no—
183C2×C4⋊C4—461no—
184C2×C4⋊C4—461no—
185C2×C4⋊C4—461no—
186C2×C4⋊C4C2×D2311yescharacteristic, complemented
187C2×C22⋊C4—461no—
188#107—232no—
189#107—232no—
190#107—232no—
191#107C2×D2312yescomplemented
192#102—232no—
193#102—232no—
194#102—232no—
195#102C2×D2312yescomplemented
196C22⋊C8—231no—
197C22⋊C8—232no—
198C22⋊C8C2×D2311yescharacteristic, complemented
199C23C2×#48011yescharacteristic, complemented, Sylow
200C24—231no—
201C24—461no—
202C2×D4—21no—
203C2×D4—231no—
204C2×D4—461no—
205C2×D4—464no—
206C2×D4—461no—
207C2×D4—461no—
208C2×D4—461no—
209C2×D4—924no—
210C2×D4C22×D2311yescharacteristic
211C22×C4—21no—
212C22×C4—231no—
213C22×C4—231no—
214C22×C4—231no—
215C22×C4—231no—
216C22×C4—231no—
217C22×C4—464no—
218C22×C4—461no—
219C22×C4—462no—
220C22×C4—462no—
221C22×C4—461no—
222C22×C4—924no—
223C22×C4C22×D2311yescharacteristic
224C2×C8—232no—
225C2×C8—464no—
226C2×C8C22×D2312yes—
227C4⋊C4—21no—
228C4⋊C4—232no—
229C4⋊C4—231no—
230C4⋊C4—462no—
231C4⋊C4—462no—
232C4⋊C4—461no—
233C4⋊C4—461no—
234C4⋊C4C22×D2312yes—
235C4⋊C4C22×D2311yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—461no—
238C22⋊C4—461no—
239C22⋊C4—461no—
240C23—21no—
241C23—231no—
242C23—234no—
243C23—231no—
244C23—462no—
245C23—462no—
246C23—461no—
247C23—922no—
248C23—922no—
249C23—922no—
250C23D4×D2311yescharacteristic
251D4—22no—
252D4—42no—
253D4—462no—
254D4—922no—
255D4—922no—
256D4—922no—
257D4—922no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—231no—
263C2×C4—234no—
264C2×C4—231no—
265C2×C4—462no—
266C2×C4—462no—
267C2×C4—462no—
268C2×C4—462no—
269C2×C4—461no—
270C2×C4—462no—
271C2×C4—924no—
272C2×C4—924no—
273C2×C4—922no—
274C2×C4—922no—
275C2×C4D4×D2311yescharacteristic
276C2×C4C23×D2311yescharacteristic
277C8—22no—
278C8—462no—
279C22—21no—
280C22—21no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—232no—
285C22—232no—
286C22—232no—
287C22—232no—
288C22—468no—
289C22—461no—
290C22—461no—
291C22—921no—
292C22—921no—
293C22—921no—
294C22—1844no—
295C22D8×D2312yes—
296C22C2×D4×D2311yescharacteristic
297C4—21no—
298C4—42no—
299C4—41no—
300C4—232no—
301C4—461no—
302C4—922no—
303C4—921no—
304C4#41×D2312yes—
305C2—22no—
306C2—81no—
307C2—234no—
308C2—462no—
309C2—1841no—
310C2#123×D2311yescharacteristic
311C2C2×D8×D2311yescharacteristic
312C2#137×D2311yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#480C2
socleC22×C23C22×D4
prime radical#121×D23C2
hypercenter#480D23
hypocenterC23C2×#480
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D23C2semidirect
3. #13092732C2semidirect
4. #13092731C2semidirect
5. #13092730C2semidirect
6. #13092729C2semidirect
7. #13092728C2semidirect
8. #107×D23C2semidirect
9. #102×D23C2semidirect
10. C22⋊C8×D23C2semidirect
11. C23×#480C2semidirect
17. C2×C4⋊C4×C23C22semidirect
21. C2×C4⋊Dic23C22semidirect
29. C4⋊C4×D23C22semidirect
32. C23×#107C22semidirect
33. C23×#102C22semidirect
34. C23×C22⋊C8C22semidirect
35. #10625014C22semidirect
36. C22⋊(C23⋊C8)C22semidirect
37. #10625003C22semidirect
38. (C2×D23)⋊C8C22semidirect
39. #10625000C22semidirect
40. #10624991C22semidirect
41. #10624989C22semidirect
42. #10624989C22semidirect
156. #480D23direct (N × H)
167. D23#480direct (N × H)
186. C2×C4⋊C4C2×D23semidirect
191. #107C2×D23semidirect
195. #102C2×D23semidirect
198. C22⋊C8C2×D23semidirect
199. C23C2×#480semidirect

Maximal subgroups

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

subgroupindex
#121×D232
C2×C4⋊C4×D232
#130927322
#130927312
#130927302
#130927292
#130927282
#107×D232
#102×D232
C22⋊C8×D232
C23×#4802
C2×#48023