Finite Group Databasebeta

← #37582659

Subgroups — #37582659

Order 24448.

450 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 121 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#27133199C211yesmaximal, characteristic, complemented
2C2×D4×Dic191C211yesmaximal, characteristic, complemented
3#27132991C211yesmaximal, characteristic, complemented
4#27132977C211yesmaximal, characteristic, complemented
5#27132961C212yesmaximal, complemented
6#27132880C212yesmaximal, complemented
7#27132797C211yesmaximal, characteristic
8#27132772C212yesmaximal, complemented
9#27132764C212yesmaximal, complemented
10#27132717C211yesmaximal, characteristic, complemented
11#450×C191C211yesmaximal, characteristic, complemented
12C23×Dic191—21no—
13C2×#15690272C2212yes—
14#20259317—21no—
15#20259313C2211yescharacteristic, complemented
16#20259311C2211yescharacteristic, complemented
17#20259306—21no—
18#20259296C2211yescharacteristic
19Q8×Dic191C2211yescharacteristic, complemented
20D4×Dic191—21no—
21D4×Dic191—21no—
22D4×Dic191C2212yes—
23D4×Dic191C2212yes—
24#20259284C2211yescharacteristic, complemented
25(C22⋊D191)⋊C4—21no—
26#20259280—21no—
27C2×C4⋊Dic191C2211yescharacteristic
28C2×C22⋊Dic191—21no—
29C2×C4×Dic191C2211yescharacteristic
30#20259256C2212yescomplemented
31#20259253C2212yescomplemented
32Dic191⋊C8C2212yescomplemented
33#20259246C2212yescomplemented
34#20259244C2212yescomplemented
35#20259243C2212yes—
36C22⋊(C191⋊C8)C2211yescharacteristic
37#20259232C2212yescomplemented
38C22×D4×C191C2211yescharacteristic
39C2×SD16×C191C2212yes—
40C22⋊Q8×C191C2211yescharacteristic
41#102×C191C2212yes—
42C22⋊C8×C191C2211yescharacteristic
43C2×D764C2311yescharacteristic
44C2×C22⋊D191—21no—
45C22×Dic191—22no—
46C22×Dic191—22no—
47C22×Dic191—21no—
48C22×Dic191—44no—
49C22×Dic191D411yescharacteristic
50C22×Dic191D411yescharacteristic
51C2×C4×D191—21no—
52#15690272—24no—
53C2×C191⋊C8C2312yes—
54C4⋊Dic191—21no—
55C4⋊Dic191—21no—
56C4⋊Dic191—21no—
57C4⋊Dic191C2311yescharacteristic
58C4⋊Dic191C2312yes—
59Dic191⋊C4—21no—
60C22⋊Dic191—22no—
61C22⋊Dic191—22no—
62(C2×D191)⋊C4—21no—
63(C2×D191)⋊C4—21no—
64(C2×D191)⋊C4—21no—
65(C2×D191)⋊C4—21no—
66C4×Dic191—21no—
67C4×Dic191D411yescharacteristic
68C4×Dic191D411yescharacteristic
69C4×Dic191C2311yescharacteristic
70C4×Dic191C2311yescharacteristic
71C24×C191—21no—
72C2×Q8×C191C2311yescharacteristic
73C2×D4×C191—24no—
74C2×D4×C191—21no—
75C2×D4×C191C2312yes—
76C22×C4×C191C2311yescharacteristic
77SD16×C191—24no—
78C2×C8×C191C2312yes—
79C4⋊C4×C191—21no—
80C4⋊C4×C191C2311yescharacteristic
81C22⋊C4×C191—21no—
82C22×D191—21no—
83D764—22no—
84C22⋊D191—42no—
85C2×Dic191—22no—
86C2×Dic191—21no—
87C2×Dic191—22no—
88C2×Dic191—41no—
89C2×Dic191—44no—
90C2×Dic191—44no—
91C2×Dic191—42no—
92C2×Dic191C2×D411yescharacteristic
93C2×Dic191C2×D411yescharacteristic
94C2×Dic191C2×D411yescharacteristic
95C2×Dic191C2×D411yescharacteristic
96C4×D191—42no—
97C191⋊C8—22no—
98C23×C191—24no—
99C23×C191—22no—
100C23×C191—42no—
101C23×C191—42no—
102C23×C191C2×D411yescharacteristic
103Q8×C191—22no—
104D4×C191—24no—
105D4×C191—22no—
106D4×C191#4114yes—
107C2×C4×C191—22no—
108C2×C4×C191—21no—
109C2×C4×C191—21no—
110C2×C4×C191C2×D411yescharacteristic
111C2×C4×C191C2411yescharacteristic
112C8×C191—22no—
113C22×C191—21no—
114C22×C191—21no—
115C22×C191—24no—
116C22×C191—48no—
117C22×C191—42no—
118C22×C191—42no—
119C22×C191AGL(1,ℤ/8)12yes—
120C22×C191C22×D411yescharacteristic
121C2×D191—41no—
122C2×D191—41no—
123C2×D191—41no—
124C4×C191—21no—
125C4×C191—41no—
126C4×C191—41no—
127C4×C191C2×#4112yes—
128Dic191—21no—
129Dic191—21no—
130Dic191—21no—
131Dic191—21no—
132Dic191—42no—
133Dic191—41no—
134C2×C191—22no—
135C2×C191—44no—
136C2×C191#52511yescharacteristic
137C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C191#57511yescharacteristic
139D191—81no—
140C191#311411yescharacteristic, complemented, Sylow
141#3114—1911nomaximal, Sylow
142#529—1911no—
143C2×C4×D4—1911no—
144#483—1911no—
145#480—1911no—
146#478—1912no—
147#463—1912no—
148#450D19111yescharacteristic, complemented
149#447—1911no—
150#442—1912no—
151#440—1912no—
152#432—1911no—
153C22×D4C2×D19111yescharacteristic
154C23×C4—3821no—
155C2×SD16C2×D19112yes—
156C2×D8—1912no—
157#126—3821no—
158#125—1911no—
159#124—1911no—
160#123—3821no—
161C22⋊Q8C2×D19111yescharacteristic
162#121—1911no—
163C4×Q8—1911no—
164C4×D4—3821no—
165C4×D4—3821no—
166C4×D4—3821no—
167C4×D4—1912no—
168C4×D4—1912no—
169C4×D4—1911no—
170#117—3821no—
171C2×C4⋊C4—1911no—
172C2×C22⋊C4—3821no—
173C2×C42—1911no—
174#107—1912no—
175#106—1912no—
176C4⋊C8—1912no—
177#103—1912no—
178#102—1912no—
179#102—1912no—
180#102C2×D19112yes—
181C22⋊C8—1911no—
182C22⋊C8C2×D19111yescharacteristic
183#97—1912no—
184C24—21no—
185C2×Q8C22×D19111yescharacteristic
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—3821no—
189C2×D4—1911no—
190C2×D4C22×D19112yes—
191C22×C4—3821no—
192C22×C4—3821no—
193C22×C4—3822no—
194C22×C4—3822no—
195C22×C4—1911no—
196C22×C4—1911no—
197C22×C4—7644no—
198C22×C4C22×D19111yescharacteristic
199SD16—24no—
200D8—3824no—
201C2×C8—1912no—
202C2×C8C22×D19112yes—
203C4⋊C4—21no—
204C4⋊C4—3821no—
205C4⋊C4—3821no—
206C4⋊C4—3821no—
207C4⋊C4—3821no—
208C4⋊C4—1911no—
209C4⋊C4—1912no—
210C4⋊C4C22×D19111yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—3821no—
213C22⋊C4—3821no—
214C22⋊C4—3821no—
215C22⋊C4—3822no—
216C22⋊C4—3822no—
217C22⋊C4—3821no—
218C42—3821no—
219C42—1911no—
220C42—1911no—
221C42—1911no—
222C42—1911no—
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—3821no—
228C23D4×D19111yescharacteristic
229Q8—22no—
230D4—24no—
231D4—22no—
232D4—3822no—
233D4—7642no—
234D4#1569028614yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—3822no—
239C2×C4—3821no—
240C2×C4—3822no—
241C2×C4—1911no—
242C2×C4—1911no—
243C2×C4—1911no—
244C2×C4—1911no—
245C2×C4—7642no—
246C2×C4—7641no—
247C2×C4—7644no—
248C2×C4—7644no—
249C2×C4—7642no—
250C2×C4D4×D19111yescharacteristic
251C2×C4C23×D19111yescharacteristic
252C8—22no—
253C8—3822no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—7641no—
261C22—7641no—
262C22—7641no—
263C22#2025934912yes—
264C22C2×D4×D19111yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—3821no—
269C4—3821no—
270C4—3821no—
271C4—3821no—
272C4—7642no—
273C4—7641no—
274C4C2×#1569028612yes—
275C2—22no—
276C2—44no—
277C2—15281no—
278C2#2713316911yescharacteristic
279C2C2×#2025934911yescharacteristic
280C2#2713345911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#450×C191C2
socleC22×C191C22×D4
prime radical#27132797C2
hypercenter#450D191
hypocenterC191#3114
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #27133199C2semidirect
2. C2×D4×Dic191C2semidirect
3. #27132991C2semidirect
4. #27132977C2semidirect
5. #27132961C2semidirect
6. #27132880C2semidirect
8. #27132772C2semidirect
9. #27132764C2semidirect
10. #27132717C2semidirect
11. #450×C191C2semidirect
15. #20259313C22semidirect
16. #20259311C22semidirect
19. Q8×Dic191C22semidirect
24. #20259284C22semidirect
30. #20259256C22semidirect
31. #20259253C22semidirect
32. Dic191⋊C8C22semidirect
33. #20259246C22semidirect
34. #20259244C22semidirect
37. #20259232C22semidirect
140. C191#3114semidirect
148. #450D191semidirect

Maximal subgroups

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

subgroupindex
#271331992
C2×D4×Dic1912
#271329912
#271329772
#271329612
#271328802
#271327972
#271327722
#271327642
#271327172
#450×C1912
#3114191