Finite Group Databasebeta

← #15682679

Subgroups — #15682679

Order 2944.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10625158C211yesmaximal, characteristic, complemented
2C2×#10625103C211yesmaximal, characteristic, complemented
3#13092984C211yesmaximal, characteristic, complemented
4#13092561C214yesmaximal, complemented
5#13092554C214yesmaximal, complemented
6#13092426C211yesmaximal, characteristic, complemented
7#13092414C211yesmaximal, characteristic, complemented
8#13092383C211yesmaximal, characteristic, complemented
9C22×SD16×C23C211yesmaximal, characteristic, complemented
10C22×Q8×C23C2211yescharacteristic
11C22×D4×C23C2211yescharacteristic
12#10625158—24no—
13#10625158—24no—
14C2×#67292—21no—
15C2×#67292C2212yescomplemented
16C2×Q8×D23—21no—
17C2×Q8×D23C2212yescomplemented
18C2×#67288—22no—
19C2×#67288C2211yescharacteristic
20C2×#67288C2211yescharacteristic
21C2×SD16×C23—24no—
22C2×SD16×C23C2214yes—
23C22×C8×C23C2211yescharacteristic
24#10625124—22no—
25#10625111—21no—
26#10625109C2214yes—
27#10625107—21no—
28C2×#67273C2212yes—
29#10625103—24no—
30C2×#67272C2212yes—
31#10625097C2211yescharacteristic
32C2×#67267C2211yescharacteristic
33#10625017C2214yescomplemented
34#10625003C2212yescomplemented
35(C2×D23)⋊C8C2214yescomplemented
36#10624998C2212yescomplemented
37C24×C23—21no—
38C2×Q8×C23—21no—
39C2×Q8×C23—22no—
40C2×Q8×C23D414yescomplemented
41C2×Q8×C23C2312yes—
42C2×D4×C23—22no—
43C2×D4×C23—22no—
44C2×D4×C23—21no—
45C2×D4×C23C2312yes—
46C22×C4×C23—21no—
47C22×C4×C23C2311yescharacteristic
48C2×C22⋊D23—21no—
49C2×C22⋊D23—24no—
50C2×C22⋊D23D412yes—
51#67292—28no—
52#67292—24no—
53#67292—24no—
54Q8×D23—28no—
55Q8×D23—24no—
56Q8×D23—24no—
57#67288—216no—
58#67288—22no—
59#67288—22no—
60#67288—22no—
61#67288D414yescomplemented
62C2×D92—22no—
63C2×D92D411yescharacteristic
64C2×D92C2311yescharacteristic
65C2×C4×D23—22no—
66C2×C4×D23—21no—
67C2×C4×D23—22no—
68C2×C4×D23C2312yes—
69C2×Dic46—22no—
70C2×Dic46D411yescharacteristic
71C2×Dic46C2311yescharacteristic
72SD16×C23—28no—
73C2×C8×C23—22no—
74C2×C8×C23C2312yes—
75C22⋊Dic23—21no—
76C22⋊Dic23—24no—
77#67273—24no—
78#67272—24no—
79C4⋊Dic23C2312yes—
80C4×Dic23—21no—
81#67267—22no—
82C2×C23⋊C8C2312yes—
83C23×C23—22no—
84C23×C23—42no—
85C23×C23—82no—
86C23×C23C2×D411yescharacteristic
87C22×D23—21no—
88C22×D23C2×D412yes—
89Q8×C23—24no—
90Q8×C23—22no—
91Q8×C23C2×D414yes—
92D4×C23—22no—
93D4×C23—22no—
94D4×C23—42no—
95C2×C4×C23—24no—
96C2×C4×C23—22no—
97C2×C4×C23C2×D411yescharacteristic
98C2×C4×C23C2×D412yes—
99C2×C4×C23C2×D412yes—
100C2×C4×C23C2411yescharacteristic
101C22⋊D23—24no—
102C22⋊D23—24no—
103C22⋊D23—84no—
104C2×Dic23—21no—
105C2×Dic23—22no—
106C2×Dic23C2×D412yes—
107D92—28no—
108D92—22no—
109D92—21no—
110D92C2×D412yes—
111C4×D23—28no—
112C4×D23—28no—
113C4×D23—24no—
114C4×D23—22no—
115C4×D23—22no—
116Dic46—28no—
117Dic46—22no—
118Dic46—21no—
119Dic46C2×D412yes—
120C8×C23—22no—
121C23⋊C8—22no—
122#3010—231nomaximal, Sylow
123C22×C23—22no—
124C22×C23—42no—
125C22×C23—44no—
126C22×C23—84no—
127C22×C23#12012yes—
128C22×C23C22×D411yescharacteristic
129C2×D23—22no—
130C2×D23—22no—
131C2×D23—24no—
132C2×D23—42no—
133C4×C23—24no—
134C4×C23#12012yes—
135C4×C23C22×D411yescharacteristic
136C4×C23C22×D411yescharacteristic
137Dic23—22no—
138Dic23—22no—
139Dic23—42no—
140C2×2−1+4—231no—
141C2×AGL(1,ℤ/8)—231no—
142C22×SD16D2311yescharacteristic, complemented
143#535—231no—
144#449—234no—
145#448—234no—
146#419—231no—
147#416—231no—
148#408—231no—
149C2×C23—22no—
150C2×C23—82no—
151C2×C23C2×#12011yescharacteristic
152C2×C23#57712yes—
153D23—42no—
154D23—42no—
1552−1+4—464no—
1562−1+4—464no—
157C2×#41—231no—
158C2×#41—232no—
159C2×#41—231no—
160C2×#41—462no—
161C2×#41—461no—
162C22×Q8—232no—
163C22×Q8—461no—
164C22×Q8C2×D2311yescharacteristic
165C22×D4C2×D2311yescharacteristic
166AGL(1,ℤ/8)—464no—
167C2×SD16—24no—
168C2×SD16—232no—
169C2×SD16C2×D2314yes—
170C2×D8—232no—
171C2×M16—231no—
172C22×C8C2×D2311yescharacteristic
173#127—461no—
174#124—461no—
175#121—234no—
176#120—462no—
177#117—231no—
178#103—234no—
179#103—232no—
180#102—232no—
181C22⋊C8—234no—
182C23#301011yescharacteristic, complemented, Sylow
183C24—21no—
184#41—234no—
185#41—4616no—
186#41—464no—
187#41—468no—
188#41—464no—
189#41—462no—
190#41—462no—
191#41—462no—
192C2×Q8—21no—
193C2×Q8—22no—
194C2×Q8—231no—
195C2×Q8—231no—
196C2×Q8—462no—
197C2×Q8—464no—
198C2×Q8—468no—
199C2×Q8—464no—
200C2×Q8C22⋊D2314yescomplemented
201C2×Q8C22×D2312yes—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—232no—
206C2×D4—231no—
207C2×D4—231no—
208C2×D4—462no—
209C2×D4—461no—
210C2×D4—464no—
211C2×D4C22×D2312yes—
212C22×C4—21no—
213C22×C4—232no—
214C22×C4—462no—
215C22×C4—462no—
216C22×C4—461no—
217C22×C4C22×D2311yescharacteristic
218SD16—28no—
219SD16—464no—
220D8—464no—
221M16—462no—
222C2×C8—22no—
223C2×C8—232no—
224C2×C8C22×D2312yes—
225C4⋊C4—232no—
226C22⋊C4—464no—
227C22⋊C4—461no—
228C42—461no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—232no—
233C23—461no—
234C23D4×D2311yescharacteristic
235Q8—24no—
236Q8—22no—
237Q8—232no—
238Q8—468no—
239Q8—462no—
240Q8—461no—
241Q8C2×C22⋊D2314yes—
242D4—22no—
243D4—22no—
244D4—42no—
245D4—232no—
246D4—468no—
247D4—464no—
248D4—464no—
249D4—462no—
250D4—461no—
251D4—1844no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—232no—
255C2×C4—468no—
256C2×C4—468no—
257C2×C4—461no—
258C2×C4—464no—
259C2×C4—462no—
260C2×C4—462no—
261C2×C4—462no—
262C2×C4D4×D2311yescharacteristic
263C2×C4D4×D2312yes—
264C2×C4C2×C22⋊D2312yes—
265C2×C4C23×D2311yescharacteristic
266C8—22no—
267C8—462no—
268C22—22no—
269C22—42no—
270C22—44no—
271C22—84no—
272C22—462no—
273C22—462no—
274C22—464no—
275C22—922no—
276C22#1062510812yes—
277C22C2×D4×D2311yescharacteristic
278C4—24no—
279C4—462no—
280C4—462no—
281C4—922no—
282C4#1062510812yes—
283C4C2×D4×D2311yescharacteristic
284C4C22×C22⋊D2311yescharacteristic
285C2—22no—
286C2—82no—
287C2—922no—
288C2—922no—
289C2C2×#1062510811yescharacteristic
290C2#1309322312yes—
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC22×SD16×C23C2
socleC22×C23C22×D4
hypercenterC22×SD16D23
hypocenterC23#3010
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#10625158C2semidirect
2. C2×#10625103C2semidirect
3. #13092984C2semidirect
4. #13092561C2semidirect
5. #13092554C2semidirect
6. #13092426C2semidirect
7. #13092414C2semidirect
8. #13092383C2semidirect
9. C22×SD16×C23C2semidirect
15. C2×#67292C22semidirect
17. C2×Q8×D23C22semidirect
33. #10625017C22semidirect
34. #10625003C22semidirect
35. (C2×D23)⋊C8C22semidirect
36. #10624998C22semidirect
40. C2×Q8×C23D4semidirect
61. #67288D4semidirect
142. C22×SD16D23semidirect
182. C23#3010semidirect
200. C2×Q8C22⋊D23semidirect

Maximal subgroups

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

subgroupindex
C2×#106251582
C2×#106251032
#130929842
#130925612
#130925542
#130924262
#130924142
#130923832
C22×SD16×C232
#301023