Finite Group Databasebeta

← #15683310

Subgroups — #15683310

Order 2944.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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#13092928C211yesmaximal, characteristic
2#13092758C212yesmaximal, complemented
3#13092752C211yesmaximal, characteristic, complemented
4#13092747C211yesmaximal, characteristic, complemented
5#13092741C212yesmaximal, complemented
6#13092735C211yesmaximal, characteristic, complemented
7#13092730C211yesmaximal, characteristic, complemented
8#13092453C212yesmaximal, complemented
9#13092411C212yesmaximal, complemented
10C2×C22⋊(C23⋊C8)C211yesmaximal, characteristic, complemented
11C2×C23×#117C211yesmaximal, characteristic, complemented
12C23×C4×C23C2211yescharacteristic
13C2×#67288C2211yescharacteristic
14C23×#117—21no—
15C23×#117—21no—
16C23×#117C2212yes—
17C23×#117C2212yes—
18C2×C4⋊C4×C23C2211yescharacteristic
19C2×C4⋊C4×C23C2211yescharacteristic
20C2×C22⋊C4×C23—21no—
21C2×C42×C23—21no—
22#10625124—21no—
23#10625101C2211yescharacteristic
24#10625101C2211yescharacteristic
25#10625100—21no—
26C4×C22⋊D23—21no—
27C4×C22⋊D23—21no—
28#10625097C2211yescharacteristic
29#10625095C2211yescharacteristic
30#10625095C2211yescharacteristic
31C22×C23⋊C8C2212yes—
32C22⋊(C23⋊C8)C2211yescharacteristic
33C22⋊(C23⋊C8)C2211yescharacteristic
34C22⋊(C23⋊C8)C2212yes—
35#10624992C2212yescomplemented
36#10624992C2212yescomplemented
37#10624991C2212yescomplemented
38#10624991C2212yescomplemented
39#10624990C2212yescomplemented
40#10624990C2212yescomplemented
41#10624989C2212yescomplemented
42#10624989C2212yescomplemented
43C24×C23D411yescharacteristic
44C22×C4×C23—24no—
45C22×C4×C23—21no—
46C22×C4×C23—21no—
47C22×C4×C23D411yescharacteristic
48C22×C4×C23D411yescharacteristic
49C22×C4×C23D411yescharacteristic
50C22×C4×C23C2311yescharacteristic
51C22×C4×C23C2311yescharacteristic
52C22×C4×C23C2311yescharacteristic
53C2×C22⋊D23—21no—
54C2×C22⋊D23—21no—
55C2×C22⋊D23—21no—
56#67288—24no—
57C2×D92C2311yescharacteristic
58C2×C4×D23—21no—
59C2×Dic46C2311yescharacteristic
60C4⋊C4×C23—21no—
61C4⋊C4×C23—21no—
62C4⋊C4×C23C2312yes—
63C4⋊C4×C23C2312yes—
64C22⋊C4×C23—22no—
65C22⋊C4×C23—22no—
66C42×C23—22no—
67C42×C23—22no—
68C22⋊Dic23—21no—
69C22⋊Dic23—21no—
70C22⋊Dic23—21no—
71(C2×D23)⋊C4—21no—
72(C2×D23)⋊C4—21no—
73C4⋊Dic23C2311yescharacteristic
74C4⋊Dic23C2311yescharacteristic
75Dic23⋊C4—21no—
76Dic23⋊C4—21no—
77C4×Dic23—21no—
78C2×C23⋊C8—24no—
79C2×C23⋊C8C2312yes—
80C2×C23⋊C8C2312yes—
81C23×C23—22no—
82C23×C23—22no—
83C23×C23—41no—
84C23×C23C2×D411yescharacteristic
85C23×C23C2×D411yescharacteristic
86C23×C23C2×D411yescharacteristic
87C22×D23—21no—
88C2×C4×C23—22no—
89C2×C4×C23—22no—
90C2×C4×C23—22no—
91C2×C4×C23—22no—
92C2×C4×C23—22no—
93C2×C4×C23—22no—
94C2×C4×C23—22no—
95C2×C4×C23—42no—
96C2×C4×C23—42no—
97C2×C4×C23C2×D411yescharacteristic
98C2×C4×C23C2×D411yescharacteristic
99C2×C4×C23C2×D411yescharacteristic
100C2×C4×C23#4114yes—
101C2×C4×C23C2411yescharacteristic
102C22⋊D23—42no—
103C22⋊D23—42no—
104C22⋊D23—42no—
105C2×Dic23—21no—
106C2×Dic23—21no—
107C2×Dic23—21no—
108D92—22no—
109C4×D23—42no—
110Dic46—22no—
111C23⋊C8—22no—
112C23⋊C8—22no—
113#3091—231nomaximal, Sylow
114C22×C23—21no—
115C22×C23—21no—
116C22×C23—21no—
117C22×C23—21no—
118C22×C23—21no—
119C22×C23—21no—
120C22×C23—21no—
121C22×C23—44no—
122C22×C23#12312yes—
123C22×C23#13512yes—
124C22×C23C22×D411yescharacteristic
125C2×D23—41no—
126C2×D23—41no—
127C2×D23—41no—
128C4×C23—21no—
129C4×C23—21no—
130C4×C23—42no—
131C4×C23—42no—
132C4×C23#12312yes—
133C4×C23C2×#4112yes—
134Dic23—41no—
135Dic23—41no—
136Dic23—41no—
137#525—231no—
138C2×#117D2311yescharacteristic, complemented
139#485—232no—
140#484—231no—
141#483—231no—
142#482—232no—
143#481—231no—
144#480—231no—
145#427—232no—
146#416—232no—
147C2×C22⋊C8—231no—
148C2×C23—22no—
149C2×C23—22no—
150C2×C23—41no—
151C2×C23C2×#12311yescharacteristic
152C2×C23C2×#13511yescharacteristic
153C2×C23#57511yescharacteristic
154D23—81no—
155C2×#41—231no—
156C23×C4C2×D2311yescharacteristic
157C22×C8—232no—
158#123—461no—
159C22⋊Q8—231no—
160C22⋊Q8—231no—
161#121—231no—
162#121—231no—
163#120—461no—
164C4×D4—461no—
165C4×D4—461no—
166#117—21no—
167#117—21no—
168#117—231no—
169#117C2×D2312yes—
170#117C2×D2312yes—
171C2×C4⋊C4C2×D2311yescharacteristic
172C2×C4⋊C4C2×D2311yescharacteristic
173C2×C22⋊C4—21no—
174C2×C42—21no—
175#107—232no—
176#107—232no—
177#106—232no—
178#106—232no—
179#103—232no—
180#103—232no—
181#102—232no—
182#102—232no—
183C22⋊C8—231no—
184C22⋊C8—231no—
185C22⋊C8—232no—
186C23#309111yescharacteristic, complemented, Sylow
187C24C22⋊D2311yescharacteristic
188#41—464no—
189C2×Q8—231no—
190C2×D4—231no—
191C2×D4—461no—
192C2×D4—461no—
193C2×D4—461no—
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—461no—
198C22×C4C22⋊D2311yescharacteristic
199C22×C4C22⋊D2311yescharacteristic
200C22×C4C22⋊D2311yescharacteristic
201C22×C4C22×D2311yescharacteristic
202C22×C4C22×D2311yescharacteristic
203C22×C4C22×D2311yescharacteristic
204C2×C8—232no—
205C2×C8—232no—
206C2×C8—464no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—231no—
210C4⋊C4—231no—
211C4⋊C4—461no—
212C4⋊C4—461no—
213C4⋊C4C22×D2312yes—
214C4⋊C4C22×D2312yes—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—461no—
218C22⋊C4—461no—
219C22⋊C4—461no—
220C22⋊C4—461no—
221C22⋊C4—461no—
222C42—22no—
223C42—22no—
224C42—461no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—461no—
229C23C2×C22⋊D2311yescharacteristic
230C23C2×C22⋊D2311yescharacteristic
231C23C2×C22⋊D2311yescharacteristic
232Q8—462no—
233D4—462no—
234D4—922no—
235D4—922no—
236D4—922no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—461no—
247C2×C4—461no—
248C2×C4—461no—
249C2×C4—922no—
250C2×C4#6728814yes—
251C2×C4C2×C22⋊D2311yescharacteristic
252C2×C4C2×C22⋊D2311yescharacteristic
253C2×C4C2×C22⋊D2311yescharacteristic
254C2×C4C23×D2311yescharacteristic
255C8—462no—
256C8—462no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—921no—
266C22—921no—
267C22—921no—
268C22#1062510012yes—
269C22#1062512112yes—
270C22C22×C22⋊D2311yescharacteristic
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—921no—
276C4—921no—
277C4—921no—
278C4#1062510012yes—
279C4C2×#6728812yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—1841no—
284C2C2×#1062510011yescharacteristic
285C2C2×#1062512111yescharacteristic
286C2#1309321511yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC2×C23×#117C2
socleC22×C23C22×D4
prime radical#13092928C2
hypercenterC2×#117D23
hypocenterC23#3091
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #13092758C2semidirect
3. #13092752C2semidirect
4. #13092747C2semidirect
5. #13092741C2semidirect
6. #13092735C2semidirect
7. #13092730C2semidirect
8. #13092453C2semidirect
9. #13092411C2semidirect
10. C2×C22⋊(C23⋊C8)C2semidirect
11. C2×C23×#117C2semidirect
35. #10624992C22semidirect
36. #10624992C22semidirect
37. #10624991C22semidirect
38. #10624991C22semidirect
39. #10624990C22semidirect
40. #10624990C22semidirect
41. #10624989C22semidirect
42. #10624989C22semidirect
138. C2×#117D23semidirect
186. C23#3091semidirect

Maximal subgroups

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

subgroupindex
#130929282
#130927582
#130927522
#130927472
#130927412
#130927352
#130927302
#130924532
#130924112
C2×C22⋊(C23⋊C8)2
C2×C23×#1172
#309123