Finite Group Databasebeta

← #15683845

Subgroups — #15683845

Order 2944.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#10625079C211yesmaximal, characteristic, complemented
2C2×#10625103C211yesmaximal, characteristic, complemented
3C2×#10625120C211yesmaximal, characteristic, complemented
4#13092963C211yesmaximal, characteristic, complemented
5C2×#10625111C211yesmaximal, characteristic, complemented
6#13092805C212yesmaximal, complemented
7#13092803C212yesmaximal, complemented
8#13092792C212yesmaximal, complemented
9#13092788C212yesmaximal, complemented
10M16×Dic23C211yesmaximal, characteristic, complemented
11C2×C23×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
12C2×#41×C23C2211yescharacteristic
13C22×D4×C23C2211yescharacteristic
14C22×C22⋊D23—22no—
15C2×#67288C2211yescharacteristic
16C22×D92C2211yescharacteristic
17C23×AGL(1,ℤ/8)—24no—
18C2×SD16×C23C2212yescomplemented
19C2×D8×C23C2212yescomplemented
20C2×M16×C23C2211yescharacteristic
21#10625120—24no—
22#10625118C2211yescharacteristic, complemented
23#10625115C2211yescharacteristic, complemented
24C2×#67274C2212yescomplemented
25#10625111—22no—
26#10625111C2212yes—
27#10625111C2212yes—
28#10625111C2211yescharacteristic, complemented
29#10625110—22no—
30#10625107C2211yescharacteristic, complemented
31C2×#67273C2212yescomplemented
32#10625103—24no—
33C2×#67272C2212yescomplemented
34C2×#67272C2212yescomplemented
35C4×C22⋊D23—22no—
36C2×C4×Dic23C2211yescharacteristic
37C2×#67267C2211yescharacteristic
38#10625079—24no—
39C2×D184C2212yescomplemented
40C2×#67263C2212yescomplemented
41#10624997C2212yescomplemented
42C8×Dic23C2212yescomplemented
43C24×C23—21no—
44C23×D23—21no—
45#41×C23—22no—
46#41×C23—22no—
47C2×Q8×C23C2311yescharacteristic
48C2×D4×C23—21no—
49C2×D4×C23—22no—
50C2×D4×C23—22no—
51C2×D4×C23—21no—
52C2×D4×C23C2311yescharacteristic
53C2×D4×C23C2312yes—
54C22×C4×C23—21no—
55C22×C4×C23C2311yescharacteristic
56C2×C22⋊D23—21no—
57C2×C22⋊D23—22no—
58C2×C22⋊D23—24no—
59C2×C22⋊D23—24no—
60C2×C22⋊D23—48no—
61C22×Dic23D412yes—
62#67288—22no—
63#67288—22no—
64C2×D92—22no—
65C2×D92—22no—
66C2×D92—21no—
67C2×D92C2312yes—
68C2×D92C2311yescharacteristic
69C2×C4×D23—21no—
70C2×Dic46C2311yescharacteristic
71SD16×C23—24no—
72D8×C23—24no—
73M16×C23D414yescomplemented
74C2×C8×C23C2312yes—
75C22⋊Dic23—22no—
76#67274—24no—
77#67273—24no—
78#67272—24no—
79#67272—24no—
80(C2×D23)⋊C4—22no—
81Dic23⋊C4—22no—
82C4×Dic23D412yes—
83C4×Dic23C2311yescharacteristic
84C4×Dic23C2311yescharacteristic
85#67267D414yescomplemented
86C2×C23⋊C8C2312yes—
87D184—24no—
88#67263—24no—
89C23×C23—21no—
90C23×C23—22no—
91C23×C23—44no—
92C23×C23—42no—
93C23×C23C2×D411yescharacteristic
94C22×D23—21no—
95C22×D23—22no—
96C22×D23—44no—
97C22×D23—42no—
98Q8×C23—21no—
99Q8×C23—21no—
100D4×C23—21no—
101D4×C23—22no—
102D4×C23—21no—
103D4×C23—22no—
104D4×C23—42no—
105D4×C23—42no—
106C2×C4×C23—21no—
107C2×C4×C23—42no—
108C2×C4×C23C2×D411yescharacteristic
109C2×C4×C23C2×D412yes—
110C2×C4×C23C2×D412yes—
111C2×C4×C23C2411yescharacteristic
112C22⋊D23—48no—
113C22⋊D23—42no—
114C22⋊D23—44no—
115C22⋊D23—48no—
116C2×Dic23—24no—
117C2×Dic23—21no—
118C2×Dic23C2×D412yes—
119C2×Dic23C2×D412yes—
120D92—22no—
121D92—21no—
122D92—21no—
123D92—22no—
124D92—42no—
125C4×D23—42no—
126Dic46—21no—
127Dic46—21no—
128C8×C23C2×D414yescomplemented
129C23⋊C8C2×D414yescomplemented
130#3151—231nomaximal, Sylow
131C22×C23—22no—
132C22×C23—41no—
133C22×C23—42no—
134C22×C23—44no—
135C22×C23—42no—
136C22×C23—84no—
137C22×C23#12712yes—
138C22×C23C22×D411yescharacteristic
139C2×D23—41no—
140C2×D23—42no—
141C2×D23—44no—
142C2×D23—42no—
143C2×D23—84no—
144C4×C23—41no—
145C4×C23#12712yes—
146C4×C23C22×D411yescharacteristic
147C4×C23C22×D411yescharacteristic
148Dic23—22no—
149Dic23—41no—
150Dic23AGL(1,ℤ/8)14yescomplemented
151C2×AGL(1,ℤ/8)—231no—
152C2×AGL(1,ℤ/8)—231no—
153C2×AGL(1,ℤ/8)—231no—
154C2×AGL(1,ℤ/8)D2311yescharacteristic, complemented
155#532—231no—
156C2×#127—231no—
157#496—232no—
158#496—232no—
159#493—232no—
160#492—232no—
161C4×M16—231no—
162C2×C23—22no—
163C2×C23—81no—
164C2×C23—82no—
165C2×C23C2×#12711yescharacteristic
166C2×C23C2×AGL(1,ℤ/8)12yes—
167D23—81no—
168D23—82no—
169C2×#41—231no—
170C2×#41C2×D2311yescharacteristic
171C22×D4—231no—
172C22×D4—462no—
173C22×D4C2×D2311yescharacteristic
174AGL(1,ℤ/8)—24no—
175AGL(1,ℤ/8)—464no—
176AGL(1,ℤ/8)—464no—
177AGL(1,ℤ/8)—464no—
178C2×SD16—232no—
179C2×SD16—232no—
180C2×SD16—232no—
181C2×SD16C2×D2312yescomplemented
182C2×D8—232no—
183C2×D8—232no—
184C2×D8—232no—
185C2×D8C2×D2312yescomplemented
186C2×M16—231no—
187C2×M16C2×D2311yescharacteristic
188C4⋊Q8—231no—
189#127—232no—
190#127—232no—
191#127—231no—
192#127—462no—
193#124—231no—
194#124—231no—
195#121—462no—
196C4×D4—462no—
197C2×C42—231no—
198#97—232no—
199C4×C8—232no—
200C23#315111yescharacteristic, complemented, Sylow
201C24—21no—
202C24—461no—
203#41—22no—
204#41—22no—
205#41—462no—
206#41—462no—
207C2×Q8—231no—
208C2×Q8C22×D2311yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—231no—
214C2×D4—232no—
215C2×D4—462no—
216C2×D4—464no—
217C2×D4—464no—
218C2×D4—461no—
219C2×D4—462no—
220C2×D4—461no—
221C2×D4—462no—
222C2×D4—928no—
223C2×D4C22×D2311yescharacteristic
224C2×D4C22×D2312yes—
225C22×C4—21no—
226C22×C4—232no—
227C22×C4—461no—
228C22×C4C22×D2311yescharacteristic
229SD16—24no—
230SD16—464no—
231SD16—464no—
232SD16—464no—
233D8—24no—
234D8—464no—
235D8—464no—
236D8—464no—
237M16—234no—
238M16C22⋊D2314yescomplemented
239C2×C8—232no—
240C2×C8C22×D2312yes—
241C4⋊C4—462no—
242C22⋊C4—462no—
243C22⋊C4—462no—
244C42—232no—
245C42—231no—
246C42—231no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—461no—
252C23—462no—
253C23—924no—
254C23—922no—
255C23D4×D2311yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—461no—
259Q8—461no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—461no—
267D4—462no—
268D4—461no—
269D4—462no—
270D4—928no—
271D4—924no—
272D4—928no—
273D4—922no—
274D4—922no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—232no—
278C2×C4—232no—
279C2×C4—464no—
280C2×C4—461no—
281C2×C4—922no—
282C2×C4D4×D2311yescharacteristic
283C2×C4D4×D2312yes—
284C2×C4C2×C22⋊D2312yes—
285C2×C4C23×D2311yescharacteristic
286C8—234no—
287C8C2×C22⋊D2314yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—924no—
295C22—922no—
296C22—921no—
297C22—922no—
298C22—1844no—
299C22#1062511112yes—
300C22C2×D4×D2311yescharacteristic
301C4—41no—
302C4—234no—
303C4—462no—
304C4—921no—
305C4#1062511112yes—
306C4C2×D4×D2311yescharacteristic
307C4C22×C22⋊D2311yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—1842no—
312C2—1841no—
313C2C2×#1062511111yescharacteristic
314C2AGL(1,ℤ/8)×D2312yes—
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC2×C23×AGL(1,ℤ/8)C2
socleC22×C23C22×D4
hypercenterC2×AGL(1,ℤ/8)D23
hypocenterC23#3151
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#10625079C2semidirect
2. C2×#10625103C2semidirect
3. C2×#10625120C2semidirect
4. #13092963C2semidirect
5. C2×#10625111C2semidirect
6. #13092805C2semidirect
7. #13092803C2semidirect
8. #13092792C2semidirect
9. #13092788C2semidirect
10. M16×Dic23C2semidirect
11. C2×C23×AGL(1,ℤ/8)C2semidirect
18. C2×SD16×C23C22semidirect
19. C2×D8×C23C22semidirect
22. #10625118C22semidirect
23. #10625115C22semidirect
24. C2×#67274C22semidirect
28. #10625111C22semidirect
30. #10625107C22semidirect
31. C2×#67273C22semidirect
33. C2×#67272C22semidirect
34. C2×#67272C22semidirect
39. C2×D184C22semidirect
40. C2×#67263C22semidirect
41. #10624997C22semidirect
42. C8×Dic23C22semidirect
73. M16×C23D4semidirect
85. #67267D4semidirect
128. C8×C23C2×D4semidirect
129. C23⋊C8C2×D4semidirect
150. Dic23AGL(1,ℤ/8)semidirect
154. C2×AGL(1,ℤ/8)D23semidirect
181. C2×SD16C2×D23semidirect
185. C2×D8C2×D23semidirect
200. C23#3151semidirect
238. M16C22⋊D23semidirect
287. C8C2×C22⋊D23semidirect

Maximal subgroups

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

subgroupindex
C2×#106250792
C2×#106251032
C2×#106251202
#130929632
C2×#106251112
#130928052
#130928032
#130927922
#130927882
M16×Dic232
C2×C23×AGL(1,ℤ/8)2
#315123