Finite Group Databasebeta

← #15683028

Subgroups — #15683028

Order 2944.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×#10625080C211yesmaximal, characteristic, complemented
2C2×M16×D23C211yesmaximal, characteristic, complemented
3#13092929C211yesmaximal, characteristic
4C2×#10625068C211yesmaximal, characteristic, complemented
5#13092676C212yesmaximal, complemented
6#13092664C212yesmaximal, complemented
7#13092659C212yesmaximal, complemented
8#13092647C212yesmaximal, complemented
9#13092425C211yesmaximal, characteristic, complemented
10#13092416C211yesmaximal, characteristic, complemented
11C23×#428C211yesmaximal, characteristic, complemented
12C2×#67288C2211yescharacteristic
13C22×C4×D23C2211yescharacteristic
14C22×Dic46C2211yescharacteristic
15C2×M16×C23C2211yescharacteristic
16C23×#117C2211yescharacteristic
17C2×C4⋊C4×C23C2211yescharacteristic
18C2×(C2×D23)⋊C4—21no—
19C2×C4⋊Dic23—21no—
20C2×#67267C2211yescharacteristic
21#10625080—24no—
22M16×D23—24no—
23C2×Dic92C2212yescomplemented
24C2×#67263C2212yescomplemented
25C2×#67262C2212yes—
26C2×C8×D23C2212yes—
27#10625068—21no—
28#10625068—21no—
29#10625068C2212yes—
30#10625068C2211yescharacteristic
31#10625068C2212yes—
32#10625068C2211yescharacteristic
33#10625066C2211yescharacteristic
34#10625066C2211yescharacteristic
35#10625057—21no—
36#10625053—21no—
37C4×D92—21no—
38C4×D92—21no—
39C23×#107C2212yescomplemented
40C23×#106C2212yescomplemented
41#10624992C2212yescomplemented
42#10624992C2212yescomplemented
43#10624992C2212yescomplemented
44#10624991C2212yescomplemented
45C23×D23D411yescharacteristic
46C22×C4×C23—21no—
47C22×C4×C23C2311yescharacteristic
48C2×C22⋊D23—21no—
49C22×Dic23—21no—
50C22×Dic23D411yescharacteristic
51#67288—22no—
52#67288—22no—
53C2×D92—21no—
54C2×D92—21no—
55C2×D92C2311yescharacteristic
56C2×C4×D23—22no—
57C2×C4×D23—22no—
58C2×C4×D23—21no—
59C2×C4×D23D411yescharacteristic
60C2×C4×D23D411yescharacteristic
61C2×C4×D23C2311yescharacteristic
62C2×C4×D23C2311yescharacteristic
63C2×Dic46—22no—
64C2×Dic46—22no—
65C2×Dic46—21no—
66C2×Dic46C2312yes—
67C2×Dic46C2311yescharacteristic
68M16×C23D414yescomplemented
69C2×C8×C23C2312yes—
70C4⋊C4×C23—21no—
71C4⋊C4×C23C2311yescharacteristic
72C4⋊C4×C23C2312yes—
73C4⋊C4×C23C2311yescharacteristic
74C22⋊C4×C23—21no—
75C42×C23—21no—
76(C2×D23)⋊C4—21no—
77(C2×D23)⋊C4—21no—
78(C2×D23)⋊C4—22no—
79(C2×D23)⋊C4—22no—
80(C2×D23)⋊C4—21no—
81(C2×D23)⋊C4—21no—
82C4⋊Dic23—21no—
83C4⋊Dic23—21no—
84C4⋊Dic23—22no—
85C4⋊Dic23—22no—
86#67267—22no—
87C2×C23⋊C8C2312yes—
88Dic92—24no—
89#67263—24no—
90#67262—24no—
91C8×D23—24no—
92C23×C23C2×D411yescharacteristic
93C22×D23—22no—
94C22×D23—22no—
95C22×D23—21no—
96C22×D23—41no—
97C22×D23C2×D411yescharacteristic
98C22×D23C2×D411yescharacteristic
99C2×C4×C23—22no—
100C2×C4×C23—21no—
101C2×C4×C23—21no—
102C2×C4×C23—42no—
103C2×C4×C23C2×D412yes—
104C2×C4×C23C2×D411yescharacteristic
105C2×C4×C23#4112yes—
106C2×C4×C23C2411yescharacteristic
107C22⋊D23—42no—
108C2×Dic23—22no—
109C2×Dic23—22no—
110C2×Dic23—21no—
111C2×Dic23—42no—
112C2×Dic23C2×D411yescharacteristic
113C2×Dic23C2×D411yescharacteristic
114D92—21no—
115D92—21no—
116D92—42no—
117D92—42no—
118C4×D23—22no—
119C4×D23—22no—
120C4×D23—21no—
121C4×D23—21no—
122C4×D23—21no—
123C4×D23—21no—
124C4×D23—42no—
125Dic46—22no—
126Dic46—21no—
127Dic46—22no—
128Dic46—21no—
129Dic46—42no—
130C8×C23C2×D414yes—
131C23⋊C8—22no—
132#3056—231nomaximal, Sylow
133C22×C23—21no—
134C22×C23—21no—
135C22×C23#12112yes—
136C22×C23C22×D411yescharacteristic
137C2×D23—21no—
138C2×D23—21no—
139C2×D23—21no—
140C2×D23—21no—
141C2×D23—21no—
142C2×D23—44no—
143C2×D23—41no—
144C2×D23—41no—
145C2×D23—41no—
146C2×D23#13712yescomplemented
147C4×C23—41no—
148C4×C23—41no—
149C4×C23—42no—
150C4×C23#12112yes—
151C4×C23C22×D411yescharacteristic
152C4×C23C2×#4111yescharacteristic
153Dic23—21no—
154Dic23—21no—
155Dic23—42no—
156Dic23—41no—
157C2×#137—231no—
158C22×M16—231no—
159#525—231no—
160C2×C22⋊Q8—231no—
161#470—232no—
162#468—232no—
163#467—232no—
164#465—232no—
165#428D2311yescharacteristic, complemented
166#419—231no—
167#417—231no—
168C2×C23—22no—
169C2×C23C2×#12111yescharacteristic
170C2×C23C2×#13711yescharacteristic
171C2×C23#57511yescharacteristic
172D23—22no—
173D23—41no—
174D23—81no—
175C2×#41—231no—
176C22×Q8—231no—
177C23×C4—231no—
178#137—464no—
179C2×Q16—232no—
180C2×SD16—232no—
181C2×M16—231no—
182C2×M16—232no—
183C2×M16—464no—
184C2×M16C2×D2311yescharacteristic
185C22×C8—232no—
186#123—461no—
187C22⋊Q8—232no—
188C22⋊Q8—232no—
189C22⋊Q8—231no—
190C22⋊Q8—231no—
191C22⋊Q8—461no—
192C22⋊Q8—461no—
193#121—231no—
194#121—231no—
195#120—461no—
196C4×D4—461no—
197C4×D4—461no—
198#117C2×D2311yescharacteristic
199C2×C4⋊C4—461no—
200C2×C4⋊C4C2×D2311yescharacteristic
201C2×C22⋊C4—461no—
202#107C2×D2312yescomplemented
203#106C2×D2312yescomplemented
204#103—232no—
205#103—232no—
206#103—232no—
207#102—232no—
208C23#305611yescharacteristic, complemented, Sylow
209C24—231no—
210#41—462no—
211#41—462no—
212C2×Q8—232no—
213C2×Q8—231no—
214C2×Q8—461no—
215C2×Q8—462no—
216C2×Q8—462no—
217C2×D4—231no—
218C2×D4—461no—
219C2×D4—461no—
220C2×D4—461no—
221C22×C4—21no—
222C22×C4—231no—
223C22×C4—231no—
224C22×C4—231no—
225C22×C4—231no—
226C22×C4—231no—
227C22×C4—462no—
228C22×C4—462no—
229C22×C4—461no—
230C22×C4—461no—
231C22×C4C22×D2311yescharacteristic
232Q16—464no—
233SD16—464no—
234M16—462no—
235M16—464no—
236M16D9214yescomplemented
237C2×C8—232no—
238C2×C8—464no—
239C2×C8C22×D2312yes—
240C4⋊C4—21no—
241C4⋊C4—462no—
242C4⋊C4—462no—
243C4⋊C4—461no—
244C4⋊C4—461no—
245C4⋊C4C22×D2312yes—
246C4⋊C4C22×D2311yescharacteristic
247C4⋊C4C22×D2311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—461no—
250C22⋊C4—461no—
251C22⋊C4—462no—
252C22⋊C4—462no—
253C22⋊C4—461no—
254C22⋊C4—461no—
255C42—21no—
256C23—231no—
257C23—231no—
258C23—462no—
259C23—462no—
260C23—461no—
261C23—921no—
262C23D4×D2311yescharacteristic
263Q8—462no—
264Q8—461no—
265Q8—462no—
266Q8—461no—
267Q8—922no—
268D4—461no—
269D4—461no—
270D4—922no—
271D4—922no—
272D4—922no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—231no—
278C2×C4—231no—
279C2×C4—462no—
280C2×C4—462no—
281C2×C4—462no—
282C2×C4—461no—
283C2×C4—461no—
284C2×C4—461no—
285C2×C4—461no—
286C2×C4—461no—
287C2×C4—462no—
288C2×C4—922no—
289C2×C4—922no—
290C2×C4C2×D9212yes—
291C2×C4D4×D2311yescharacteristic
292C2×C4#6729212yes—
293C2×C4C23×D2311yescharacteristic
294C8—462no—
295C8C2×D9214yes—
296C22—21no—
297C22—21no—
298C22—232no—
299C22—461no—
300C22—461no—
301C22—461no—
302C22—461no—
303C22—461no—
304C22—924no—
305C22—921no—
306C22—921no—
307C22—921no—
308C22#1062506612yes—
309C22C2×D4×D2311yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—461no—
314C4—461no—
315C4—921no—
316C4—922no—
317C4#1062506612yes—
318C4C22×D9211yescharacteristic
319C4C2×#6729211yescharacteristic
320C2—22no—
321C2—462no—
322C2—921no—
323C2—1841no—
324C2C2×#1062506611yescharacteristic
325C2#137×D2311yescharacteristic
326C2#1309321311yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#428C2
socleC22×C23C22×D4
prime radical#13092929C2
hypercenter#428D23
hypocenterC23#3056
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#10625080C2semidirect
2. C2×M16×D23C2semidirect
4. C2×#10625068C2semidirect
5. #13092676C2semidirect
6. #13092664C2semidirect
7. #13092659C2semidirect
8. #13092647C2semidirect
9. #13092425C2semidirect
10. #13092416C2semidirect
11. C23×#428C2semidirect
23. C2×Dic92C22semidirect
24. C2×#67263C22semidirect
39. C23×#107C22semidirect
40. C23×#106C22semidirect
41. #10624992C22semidirect
42. #10624992C22semidirect
43. #10624992C22semidirect
44. #10624991C22semidirect
68. M16×C23D4semidirect
146. C2×D23#137semidirect
165. #428D23semidirect
202. #107C2×D23semidirect
203. #106C2×D23semidirect
208. C23#3056semidirect
236. M16D92semidirect

Maximal subgroups

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

subgroupindex
C2×#106250802
C2×M16×D232
#130929292
C2×#106250682
#130926762
#130926642
#130926592
#130926472
#130924252
#130924162
C23×#4282
#305623