Finite Group Databasebeta

← #15685204

Subgroups — #15685204

Order 2944.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D92C211yesmaximal, characteristic, complemented
2#13092765C211yesmaximal, characteristic, complemented
3#13092730C212yesmaximal, complemented
4#13092694C211yesmaximal, characteristic, complemented
5#13092651C212yesmaximal, complemented
6#13092612C211yesmaximal, characteristic, complemented
7#13092550C212yesmaximal, complemented
8C4×#67272C211yesmaximal, characteristic, complemented
9D4×C23⋊C8C211yesmaximal, characteristic, complemented
10C2×#10624991C212yesmaximal, complemented
11C23×#547C211yesmaximal, characteristic, complemented
12C2×#41×C23—21no—
13C2×D4×D23—21no—
14C22×D92C2212yes—
15C23×C4⋊Q8C2211yescharacteristic, complemented
16C23×#123—22no—
17C23×C22⋊Q8—21no—
18C23×#121C2212yescomplemented
19C4×D4×C23C2211yescharacteristic
20C4×D4×C23C2211yescharacteristic, complemented
21C2×C4⋊C4×C23C2212yescomplemented
22C2×#67272C2211yescharacteristic
23C2×#67272C2212yescomplemented
24#10625101C2212yes—
25C22×C23⋊C8C2212yescomplemented
26#10625066—21no—
27#10625053—22no—
28#10625045C2211yescharacteristic
29C4×D92C2211yescharacteristic
30#10625014C2211yescharacteristic, complemented
31C22⋊(C23⋊C8)C2212yescomplemented
32#10624991C2211yescharacteristic
33#10624991C2212yes—
34#10624991C2212yescomplemented
35#10624991C2212yescomplemented
36#10624991C2212yescomplemented
37#10624989C2212yescomplemented
38#10624989C2211yescharacteristic, complemented
39C4⋊(C23⋊C8)C2211yescharacteristic, complemented
40C4×C23⋊C8C2211yescharacteristic, complemented
41C23×D23—22no—
42#41×C23—44no—
43C2×Q8×C23—21no—
44C2×D4×C23—22no—
45C2×D4×C23D411yescharacteristic
46C2×D4×C23C2311yescharacteristic
47C22×C4×C23—22no—
48C22×C4×C23—21no—
49C22×C4×C23C2312yes—
50C2×C22⋊D23—22no—
51D4×D23—44no—
52C2×D92—24no—
53C2×D92—21no—
54C2×D92—22no—
55C2×D92—21no—
56C2×D92—42no—
57C2×D92C2312yes—
58C2×D92C2311yescharacteristic
59C2×C4×D23—21no—
60C4⋊C4×C23—22no—
61C4⋊C4×C23—21no—
62C4⋊C4×C23D411yescharacteristic
63C4⋊C4×C23C2311yescharacteristic
64C4⋊C4×C23C2312yes—
65C22⋊C4×C23—21no—
66C22⋊C4×C23—22no—
67C22⋊C4×C23D412yes—
68C42×C23C2311yescharacteristic
69#67272—22no—
70#67272—42no—
71(C2×D23)⋊C4—21no—
72(C2×D23)⋊C4—22no—
73C4⋊Dic23C2311yescharacteristic
74C2×C23⋊C8—42no—
75C2×C23⋊C8C2311yescharacteristic
76C2×C23⋊C8C2312yes—
77C2×C23⋊C8C2311yescharacteristic
78C23×C23—21no—
79C23×C23C2×D412yes—
80C22×D23—21no—
81C22×D23—22no—
82C22×D23—44no—
83C22×D23—44no—
84C22×D23—82no—
85Q8×C23—42no—
86D4×C23—21no—
87D4×C23—21no—
88D4×C23—44no—
89D4×C23D814yescomplemented
90C2×C4×C23—22no—
91C2×C4×C23—22no—
92C2×C4×C23—21no—
93C2×C4×C23—22no—
94C2×C4×C23—44no—
95C2×C4×C23—42no—
96C2×C4×C23C2×D411yescharacteristic
97C2×C4×C23C2×D412yes—
98C2×C4×C23C2×D411yescharacteristic
99C2×C4×C23C2411yescharacteristic
100C22⋊D23—44no—
101C2×Dic23—21no—
102D92—22no—
103D92—21no—
104D92—22no—
105D92—44no—
106D92—42no—
107D92—81no—
108D92#4112yescomplemented
109C4×D23—42no—
110C23⋊C8—21no—
111C23⋊C8—21no—
112C23⋊C8—22no—
113#3294—231nomaximal, Sylow
114C22×C23—22no—
115C22×C23—22no—
116C22×C23—41no—
117C22×C23—41no—
118C22×C23—41no—
119C22×C23C2×D814yes—
120C22×C23C22×D411yescharacteristic
121C2×D23—22no—
122C2×D23—41no—
123C2×D23—41no—
124C2×D23—42no—
125C2×D23—42no—
126C2×D23—42no—
127C2×D23—88no—
128C4×C23—22no—
129C4×C23—21no—
130C4×C23—41no—
131C4×C23—42no—
132C4×C23C2×D812yes—
133C4×C23C2×#4111yescharacteristic
134C4×C23H(5,𝔽2)11yescharacteristic
135Dic23—41no—
136#547D2311yescharacteristic, complemented
137D42—231no—
138#486—231no—
139#480—232no—
140#474—231no—
141#466—232no—
142#459—231no—
143#447—232no—
144C4×D8—231no—
145C8×D4—231no—
146C2×#102—232no—
147C2×C23—24no—
148C2×C23—81no—
149C2×C23#54611yescharacteristic
150C2×C23C22×D811yescharacteristic
151C2×C23#57711yescharacteristic
152D23—42no—
153D23—82no—
154C2×#41—21no—
155C22×D4—232no—
156C22×D4—461no—
157C2×D8—231no—
158C2×D8—232no—
159C22×C8—232no—
160C4⋊Q8C2×D2311yescharacteristic, complemented
161#127—231no—
162#123—22no—
163C22⋊Q8—21no—
164#121—232no—
165#121—461no—
166#121C2×D2312yescomplemented
167#120—462no—
168C4×D4—231no—
169C4×D4C2×D2311yescharacteristic
170C4×D4C2×D2311yescharacteristic, complemented
171C2×C4⋊C4C2×D2312yescomplemented
172#107—231no—
173#107—232no—
174C4⋊C8—231no—
175#102—231no—
176#102—231no—
177#102—232no—
178#102—232no—
179#102—232no—
180#102—232no—
181C22⋊C8—232no—
182C4×C8—231no—
183C23#329411yescharacteristic, complemented, Sylow
184C24—462no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—231no—
189C2×D4—232no—
190C2×D4—464no—
191C2×D4—462no—
192C2×D4—462no—
193C2×D4—461no—
194C2×D4—461no—
195C2×D4—922no—
196C2×D4—924no—
197C2×D4C22⋊D2311yescharacteristic
198C2×D4C22×D2311yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—461no—
202C22×C4C22×D2312yes—
203D8—462no—
204D8—922no—
205C2×C8—231no—
206C2×C8—231no—
207C2×C8—232no—
208C2×C8—922no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—231no—
212C4⋊C4C22⋊D2311yescharacteristic
213C4⋊C4C22×D2311yescharacteristic
214C4⋊C4C22×D2312yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—461no—
218C22⋊C4—462no—
219C22⋊C4C22⋊D2312yes—
220C42C22×D2311yescharacteristic
221C23—21no—
222C23—461no—
223C23—462no—
224C23—924no—
225C23—924no—
226C23—1842no—
227C23C2×C22⋊D2312yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—232no—
233D4—462no—
234D4—461no—
235D4—462no—
236D4—924no—
237D4—924no—
238D4—922no—
239D4—1841no—
240D4#6727214yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—461no—
248C2×C4—922no—
249C2×C4C2×C22⋊D2311yescharacteristic
250C2×C4C2×C22⋊D2312yes—
251C2×C4C2×C22⋊D2311yescharacteristic
252C2×C4C23×D2311yescharacteristic
253C8—461no—
254C8—461no—
255C8—462no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—462no—
262C22—921no—
263C22—921no—
264C22—922no—
265C22—922no—
266C22—922no—
267C22—1848no—
268C22C2×#6727214yes—
269C22C22×C22⋊D2311yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—42no—
274C4—921no—
275C4C2×#6727212yes—
276C4#41×D2311yescharacteristic
277C4#1062516011yescharacteristic
278C2—24no—
279C2—81no—
280C2—922no—
281C2—1842no—
282C2#1309305511yescharacteristic
283C2C22×#6727211yescharacteristic
284C2#1309322511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#547C2
socleC22×C23C22×D4
hypercenter#547D23
hypocenterC23#3294
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D92C2semidirect
2. #13092765C2semidirect
3. #13092730C2semidirect
4. #13092694C2semidirect
5. #13092651C2semidirect
6. #13092612C2semidirect
7. #13092550C2semidirect
8. C4×#67272C2semidirect
9. D4×C23⋊C8C2semidirect
10. C2×#10624991C2semidirect
11. C23×#547C2semidirect
15. C23×C4⋊Q8C22semidirect
18. C23×#121C22semidirect
20. C4×D4×C23C22semidirect
21. C2×C4⋊C4×C23C22semidirect
23. C2×#67272C22semidirect
25. C22×C23⋊C8C22semidirect
30. #10625014C22semidirect
31. C22⋊(C23⋊C8)C22semidirect
34. #10624991C22semidirect
35. #10624991C22semidirect
36. #10624991C22semidirect
37. #10624989C22semidirect
38. #10624989C22semidirect
39. C4⋊(C23⋊C8)C22semidirect
40. C4×C23⋊C8C22semidirect
89. D4×C23D8semidirect
108. D92#41semidirect
136. #547D23semidirect
160. C4⋊Q8C2×D23semidirect
166. #121C2×D23semidirect
170. C4×D4C2×D23semidirect
171. C2×C4⋊C4C2×D23semidirect
183. C23#3294semidirect
240. D4#67272semidirect

Maximal subgroups

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

subgroupindex
D4×D922
#130927652
#130927302
#130926942
#130926512
#130926122
#130925502
C4×#672722
D4×C23⋊C82
C2×#106249912
C23×#5472
#329423