Finite Group Databasebeta

← #37583000

Subgroups — #37583000

Order 24448.

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

normal subgroups only

This group has 596 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
1C2×#20259346C212yesmaximal, complemented
2C2×#20259341C211yesmaximal, characteristic, complemented
3C2×D8×D191C211yesmaximal, characteristic, complemented
4#27133226C212yesmaximal, complemented
5#27133051C212yesmaximal, complemented
6#27133050C212yesmaximal, complemented
7#27133047C211yesmaximal, characteristic, complemented
8#27133046C211yesmaximal, characteristic, complemented
9#27133038C211yesmaximal, characteristic, complemented
10#27132617C211yesmaximal, characteristic, complemented
11#493×C191C211yesmaximal, characteristic, complemented
12C2×#15690286C2212yes—
13C2×D4×D191C2212yes—
14#20259346—28no—
15#20259341—24no—
16C2×Dic764C2211yescharacteristic, complemented
17C2×#15690276C2212yescomplemented
18C2×#15690275C2212yescomplemented
19C2×#15690275C2212yescomplemented
20C2×D1528C2211yescharacteristic
21C2×#15690272C2212yes—
22C2×#15690272C2212yescomplemented
23D8×D191—24no—
24C2×#15690270C2211yescharacteristic
25C2×C8×D191C2211yescharacteristic
26#20259318C2212yescomplemented
27#20259310C2212yescomplemented
28#20259309C2212yescomplemented
29#20259294—24no—
30#20259290—24no—
31#20259279C2211yescharacteristic
32#20259233C2211yescharacteristic, complemented
33#20259232C2211yescharacteristic, complemented
34C8×Dic191C2211yescharacteristic, complemented
35C2×D8×C191C2211yescharacteristic
36C2×D8×C191C2211yescharacteristic, complemented
37C2×D8×C191C2212yescomplemented
38#127×C191C2212yescomplemented
39C4×C8×C191C2211yescharacteristic, complemented
40C23×D191—22no—
41#15690286—24no—
42#15690286—24no—
43C2×Dic382C2312yes—
44C2×D764C2312yes—
45C2×C22⋊D191—22no—
46C2×C22⋊D191—22no—
47C2×C22⋊D191—24no—
48D4×D191—24no—
49D4×D191—24no—
50C22×Dic191—22no—
51C2×C4×D191C2311yescharacteristic
52Dic764—22no—
53#15690276—24no—
54#15690275—24no—
55#15690275—24no—
56D1528—22no—
57#15690272—24no—
58#15690272—24no—
59#15690270D414yescomplemented
60C2×C191⋊C8C2311yescharacteristic
61C2×C191⋊C8C2311yescharacteristic
62C8×D191D414yescomplemented
63Dic191⋊C4D412yes—
64C22⋊Dic191—24no—
65(C2×D191)⋊C4—24no—
66(C2×D191)⋊C4D412yes—
67C4×Dic191C2311yescharacteristic
68C2×D4×C191—24no—
69C2×D4×C191C2312yes—
70C2×D4×C191C2312yes—
71D8×C191—22no—
72D8×C191—22no—
73D8×C191—24no—
74C2×C8×C191C2311yescharacteristic
75C2×C8×C191C2311yescharacteristic
76C42×C191C2311yescharacteristic
77C22×D191—22no—
78C22×D191—44no—
79C22×D191—84no—
80C22×D191C2×D411yescharacteristic
81Dic382—22no—
82Dic382—22no—
83D764—22no—
84D764—22no—
85C22⋊D191—44no—
86C22⋊D191—44no—
87C22⋊D191—84no—
88C2×Dic191—22no—
89C2×Dic191—44no—
90C2×Dic191C2×D412yes—
91C2×Dic191C2×D411yescharacteristic
92C4×D191C2×D412yes—
93C4×D191C2×D412yes—
94C191⋊C8C2×D412yes—
95C191⋊C8C2×D412yes—
96C23×C191—22no—
97C23×C191—22no—
98D4×C191—22no—
99D4×C191—22no—
100D4×C191—22no—
101D4×C191—22no—
102D4×C191—84no—
103C2×C4×C191C2×D412yes—
104C2×C4×C191C2411yescharacteristic
105C8×C191C2×D412yes—
106C8×C191C2×D412yes—
107C22×C191—42no—
108C22×C191—42no—
109C22×C191—44no—
110C22×C191—44no—
111C22×C191C22×D411yescharacteristic
112C2×D191—22no—
113C2×D191—42no—
114C2×D191—44no—
115C2×D191—88no—
116C2×D191#12712yes—
117C4×C191—22no—
118C4×C191C22×D411yescharacteristic
119C4×C191C22×D411yescharacteristic
120Dic191—22no—
121Dic191—42no—
122Dic191#12712yes—
123C2×C191—82no—
124C2×C191—82no—
125C2×C191C2×#12711yescharacteristic
126C2×C191#57612yes—
127D191—22no—
128D191—82no—
129C191#315511yescharacteristic, complemented, Sylow
130#3155—1911nomaximal, Sylow
131C2×AGL(1,ℤ/8)—1912no—
132C2×#135—1911no—
133C22×D8—1911no—
134#535—1912no—
135#496—1912no—
136#496—1912no—
137#495—1911no—
138#495—1911no—
139#493—1911no—
140#493D19111yescharacteristic, complemented
141#405—1911no—
142C2×#41—1912no—
143C22×D4—1912no—
144AGL(1,ℤ/8)—3828no—
145#135—3824no—
146C2×Q16—1911no—
147C2×SD16—1912no—
148C2×SD16—1912no—
149C2×SD16—1912no—
150C2×D8—3824no—
151C2×D8—1912no—
152C2×D8—1912no—
153C2×D8—1911no—
154C2×D8C2×D19111yescharacteristic
155C2×D8C2×D19111yescharacteristic, complemented
156C2×D8C2×D19112yescomplemented
157C2×M16—1911no—
158C22×C8—1911no—
159#127—1912no—
160#127C2×D19112yescomplemented
161#124—1912no—
162#124—1912no—
163#121—3824no—
164#120—3824no—
165#117—1911no—
166#97—1911no—
167#97—1911no—
168C4×C8—1911no—
169C4×C8C2×D19111yescharacteristic, complemented
170C24—3822no—
171#41—3824no—
172#41—3824no—
173C2×Q8—1912no—
174C2×D4—24no—
175C2×D4—3822no—
176C2×D4—3824no—
177C2×D4—3824no—
178C2×D4—3824no—
179C2×D4—3822no—
180C2×D4—1912no—
181C2×D4C22×D19112yes—
182C2×D4C22×D19112yes—
183C22×C4—3822no—
184C22×C4—1911no—
185Q16—3822no—
186SD16—3824no—
187SD16—3824no—
188SD16—3824no—
189D8—22no—
190D8—22no—
191D8—24no—
192D8—3824no—
193D8—3824no—
194D8—3822no—
195M16—1914no—
196C2×C8—1911no—
197C2×C8—1911no—
198C2×C8—1914no—
199C2×C8C22×D19111yescharacteristic
200C2×C8C22×D19111yescharacteristic
201C4⋊C4—1912no—
202C22⋊C4—3824no—
203C22⋊C4—3824no—
204C22⋊C4—1912no—
205C42—1911no—
206C42C22×D19111yescharacteristic
207C23—22no—
208C23—22no—
209C23—3822no—
210C23—1911no—
211C23—15284no—
212C23—7644no—
213Q8—3822no—
214Q8—3822no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—22no—
219D4—84no—
220D4—3822no—
221D4—3822no—
222D4—15284no—
223D4—7644no—
224D4—7644no—
225C2×C4—3822no—
226C2×C4—1912no—
227C2×C4—1911no—
228C2×C4—1912no—
229C2×C4—1912no—
230C2×C4—7644no—
231C2×C4D4×D19112yes—
232C2×C4C23×D19111yescharacteristic
233C8—1912no—
234C8—1912no—
235C8D4×D19112yes—
236C8D4×D19112yes—
237C22—42no—
238C22—42no—
239C22—44no—
240C22—44no—
241C22—3822no—
242C22—1912no—
243C22—15288no—
244C22—7642no—
245C22—7644no—
246C22C2×D4×D19111yescharacteristic
247C4—22no—
248C4—3822no—
249C4—1912no—
250C4—7642no—
251C4C2×D4×D19111yescharacteristic
252C4C2×D4×D19111yescharacteristic
253C2—82no—
254C2—82no—
255C2—3822no—
256C2—15282no—
257C2#127×D19111yescharacteristic
258C2#2713346412yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#493×C191C2
socleC22×C191C22×D4
hypercenter#493D191
hypocenterC191#3155
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#20259346C2semidirect
2. C2×#20259341C2semidirect
3. C2×D8×D191C2semidirect
4. #27133226C2semidirect
5. #27133051C2semidirect
6. #27133050C2semidirect
7. #27133047C2semidirect
8. #27133046C2semidirect
9. #27133038C2semidirect
10. #27132617C2semidirect
11. #493×C191C2semidirect
16. C2×Dic764C22semidirect
17. C2×#15690276C22semidirect
18. C2×#15690275C22semidirect
19. C2×#15690275C22semidirect
22. C2×#15690272C22semidirect
26. #20259318C22semidirect
27. #20259310C22semidirect
28. #20259309C22semidirect
32. #20259233C22semidirect
33. #20259232C22semidirect
34. C8×Dic191C22semidirect
36. C2×D8×C191C22semidirect
37. C2×D8×C191C22semidirect
38. #127×C191C22semidirect
39. C4×C8×C191C22semidirect
59. #15690270D4semidirect
62. C8×D191D4semidirect
129. C191#3155semidirect
140. #493D191semidirect
155. C2×D8C2×D191semidirect
156. C2×D8C2×D191semidirect
160. #127C2×D191semidirect
169. C4×C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593462
C2×#202593412
C2×D8×D1912
#271332262
#271330512
#271330502
#271330472
#271330462
#271330382
#271326172
#493×C1912
#3155191