Finite Group Databasebeta

← #37583333

Subgroups — #37583333

Order 24448.

466 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 113 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#27133173C211yesmaximal, characteristic, complemented
2C2×#20259297C211yesmaximal, characteristic, complemented
3#27132993C211yesmaximal, characteristic, complemented
4#27132990C212yesmaximal, complemented
5#27132915C212yesmaximal, complemented
6#27132910C212yesmaximal, complemented
7#27132816C211yesmaximal, characteristic, complemented
8#27132809C212yesmaximal, complemented
9#27132796C211yesmaximal, characteristic, complemented
10#27132625C211yesmaximal, characteristic, complemented
11#480×C191C211yesmaximal, characteristic, complemented
12C2×#15690286C2211yescharacteristic, complemented
13C22×D764—21no—
14C22×D764C2211yescharacteristic
15C22×C4×D191C2211yescharacteristic
16C2×#15690276C2212yescomplemented
17C2×#15690272C2212yescomplemented
18C2×#15690270C2212yescomplemented
19#20259304—21no—
20#20259302C2211yescharacteristic, complemented
21#20259301C2211yescharacteristic
22#20259297—21no—
23#20259297—21no—
24#20259297C2212yes—
25#20259297C2212yes—
26#20259293C2211yescharacteristic, complemented
27#20259290—21no—
28D4×Dic191—21no—
29(C22⋊D191)⋊C4—21no—
30#20259280C2211yescharacteristic, complemented
31C2×(C2×D191)⋊C4—21no—
32#20259253C2212yescomplemented
33#20259245C2212yescomplemented
34#20259244C2212yescomplemented
35#20259242C2212yescomplemented
36C22⋊(C191⋊C8)C2211yescharacteristic
37(C2×D191)⋊C8C2212yescomplemented
38#121×C191C2211yescharacteristic
39C2×C4⋊C4×C191C2211yescharacteristic
40#107×C191C2212yescomplemented
41#102×C191C2212yescomplemented
42C22⋊C8×C191C2211yescharacteristic
43C23×D191—21no—
44C23×D191D411yescharacteristic
45#15690286—42no—
46C2×Dic382C2311yescharacteristic
47C2×D764—22no—
48C2×D764—22no—
49C2×D764—21no—
50C2×D764—42no—
51C2×D764—44no—
52C2×D764C2312yes—
53C2×C22⋊D191—21no—
54C2×C22⋊D191—21no—
55C22×Dic191—21no—
56C22×Dic191D411yescharacteristic
57C2×C4×D191—42no—
58C2×C4×D191D411yescharacteristic
59C2×C4×D191D411yescharacteristic
60C2×C4×D191C2311yescharacteristic
61C2×C4×D191C2311yescharacteristic
62#15690276—42no—
63#15690272—42no—
64#15690270—42no—
65C2×C191⋊C8C2312yes—
66C4⋊Dic191—21no—
67C4⋊Dic191C2311yescharacteristic
68Dic191⋊C4—21no—
69C22⋊Dic191—21no—
70C22⋊Dic191—21no—
71(C2×D191)⋊C4—21no—
72(C2×D191)⋊C4—21no—
73(C2×D191)⋊C4—22no—
74(C2×D191)⋊C4—22no—
75C4×Dic191—21no—
76C2×D4×C191—21no—
77C2×D4×C191C2311yescharacteristic
78C22×C4×C191—21no—
79C22×C4×C191C2311yescharacteristic
80C2×C8×C191C2312yes—
81C4⋊C4×C191—21no—
82C4⋊C4×C191C2312yes—
83C4⋊C4×C191C2311yescharacteristic
84C22⋊C4×C191—21no—
85C22×D191—22no—
86C22×D191—22no—
87C22×D191—41no—
88C22×D191—41no—
89C22×D191—42no—
90C22×D191—42no—
91C22×D191—42no—
92C22×D191C2×D411yescharacteristic
93C22×D191C2×D411yescharacteristic
94Dic382—22no—
95D764—22no—
96D764—22no—
97D764—42no—
98D764—44no—
99D764—44no—
100C22⋊D191—42no—
101C22⋊D191—42no—
102C2×Dic191—21no—
103C2×Dic191—21no—
104C2×Dic191—41no—
105C2×Dic191—42no—
106C2×Dic191C2×D411yescharacteristic
107C2×Dic191C2×D411yescharacteristic
108C4×D191—22no—
109C4×D191—22no—
110C4×D191—41no—
111C4×D191—41no—
112C191⋊C8—22no—
113C23×C191—21no—
114C23×C191C2×D411yescharacteristic
115D4×C191—22no—
116D4×C191—42no—
117C2×C4×C191—22no—
118C2×C4×C191—21no—
119C2×C4×C191—22no—
120C2×C4×C191—42no—
121C2×C4×C191C2×D411yescharacteristic
122C2×C4×C191C2411yescharacteristic
123C8×C191—22no—
124C22×C191—21no—
125C22×C191—21no—
126C22×C191—41no—
127C22×C191—41no—
128C22×C191—41no—
129C22×C191AGL(1,ℤ/8)12yes—
130C22×C191C22×D411yescharacteristic
131C2×D191—21no—
132C2×D191—21no—
133C2×D191—21no—
134C2×D191—21no—
135C2×D191—21no—
136C2×D191—21no—
137C2×D191—44no—
138C2×D191—42no—
139C2×D191—42no—
140C2×D191—42no—
141C2×D191—84no—
142C4×C191—21no—
143C4×C191—42no—
144C4×C191—41no—
145C4×C191H(5,𝔽2)12yes—
146Dic191—21no—
147Dic191—21no—
148Dic191—41no—
149Dic191—41no—
150C2×C191—22no—
151C2×C191—81no—
152C2×C191#53411yescharacteristic
153C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
154C2×C191#57511yescharacteristic
155D191—41no—
156D191—41no—
157D191—82no—
158C191#319411yescharacteristic, complemented, Sylow
159#3194—1911nomaximal, Sylow
160#525—1911no—
161C2×#121—1911no—
162#483—1911no—
163#482—1912no—
164#480D19111yescharacteristic, complemented
165#469—1912no—
166#468—1912no—
167#450—1911no—
168#449—1912no—
169#447—1911no—
170#407—1911no—
171C2×#41—1911no—
172C22×D4—3821no—
173C22×D4—1911no—
174C23×C4—1911no—
175C2×SD16—1912no—
176C2×D8—1912no—
177C2×M16—1912no—
178#123—3821no—
179C22⋊Q8—1911no—
180C22⋊Q8—1911no—
181#121—3821no—
182#121—3821no—
183#121—1911no—
184#121—1912no—
185#121—1912no—
186#121C2×D19111yescharacteristic
187#120—3821no—
188C4×D4—3821no—
189C4×D4—3821no—
190#117—1911no—
191C2×C4⋊C4C2×D19111yescharacteristic
192C2×C22⋊C4—3821no—
193#107C2×D19112yescomplemented
194#106—1912no—
195#103—1912no—
196#102—1912no—
197#102—1912no—
198#102C2×D19112yescomplemented
199C22⋊C8—1911no—
200C22⋊C8—1912no—
201C22⋊C8C2×D19111yescharacteristic
202C24—3821no—
203C24—1911no—
204#41—7642no—
205C2×Q8—1911no—
206C2×D4—21no—
207C2×D4—3822no—
208C2×D4—3822no—
209C2×D4—3821no—
210C2×D4—3821no—
211C2×D4—3821no—
212C2×D4—1912no—
213C2×D4—7642no—
214C2×D4—7644no—
215C2×D4C22×D19111yescharacteristic
216C22×C4—21no—
217C22×C4—3821no—
218C22×C4—1911no—
219C22×C4—1911no—
220C22×C4—1911no—
221C22×C4—1911no—
222C22×C4—1911no—
223C22×C4—7642no—
224C22×C4C22×D19111yescharacteristic
225SD16—7642no—
226D8—7642no—
227M16—7642no—
228C2×C8—1912no—
229C2×C8C22×D19112yes—
230C4⋊C4—21no—
231C4⋊C4—3821no—
232C4⋊C4—3821no—
233C4⋊C4—1911no—
234C4⋊C4C22×D19111yescharacteristic
235C4⋊C4C22×D19112yes—
236C22⋊C4—21no—
237C22⋊C4—3821no—
238C22⋊C4—3821no—
239C22⋊C4—3822no—
240C22⋊C4—3822no—
241C22⋊C4—3821no—
242C22⋊C4—3821no—
243C42—3821no—
244C23—21no—
245C23—3822no—
246C23—3822no—
247C23—1911no—
248C23—1911no—
249C23—7641no—
250C23—7641no—
251C23—7642no—
252C23—7642no—
253C23—7642no—
254C23D4×D19111yescharacteristic
255Q8—3822no—
256D4—22no—
257D4—42no—
258D4—3822no—
259D4—3822no—
260D4—7642no—
261D4—7644no—
262D4—7644no—
263D4—7642no—
264D4—7642no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—3822no—
270C2×C4—3822no—
271C2×C4—3821no—
272C2×C4—3821no—
273C2×C4—1911no—
274C2×C4—1911no—
275C2×C4—7641no—
276C2×C4—7641no—
277C2×C4—7641no—
278C2×C4—7642no—
279C2×C4D4×D19111yescharacteristic
280C2×C4C23×D19111yescharacteristic
281C8—22no—
282C8—3822no—
283C22—21no—
284C22—21no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—3821no—
289C22—3821no—
290C22—3821no—
291C22—3821no—
292C22—3821no—
293C22—3821no—
294C22—15284no—
295C22—7644no—
296C22—7642no—
297C22—7642no—
298C22—7642no—
299C22#2025934612yes—
300C22C2×D4×D19111yescharacteristic
301C4—21no—
302C4—42no—
303C4—41no—
304C4—3821no—
305C4—3821no—
306C4—7641no—
307C4—7641no—
308C4#2025936812yes—
309C2—22no—
310C2—81no—
311C2—15282no—
312C2—7641no—
313C2—7641no—
314C2#2713322411yescharacteristic
315C2C2×#2025934611yescharacteristic
316C2#2713346211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#480×C191C2
socleC22×C191C22×D4
hypercenter#480D191
hypocenterC191#3194
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #27133173C2semidirect
2. C2×#20259297C2semidirect
3. #27132993C2semidirect
4. #27132990C2semidirect
5. #27132915C2semidirect
6. #27132910C2semidirect
7. #27132816C2semidirect
8. #27132809C2semidirect
9. #27132796C2semidirect
10. #27132625C2semidirect
11. #480×C191C2semidirect
12. C2×#15690286C22semidirect
16. C2×#15690276C22semidirect
17. C2×#15690272C22semidirect
18. C2×#15690270C22semidirect
20. #20259302C22semidirect
26. #20259293C22semidirect
30. #20259280C22semidirect
32. #20259253C22semidirect
33. #20259245C22semidirect
34. #20259244C22semidirect
35. #20259242C22semidirect
37. (C2×D191)⋊C8C22semidirect
40. #107×C191C22semidirect
41. #102×C191C22semidirect
158. C191#3194semidirect
164. #480D191semidirect
193. #107C2×D191semidirect
198. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271331732
C2×#202592972
#271329932
#271329902
#271329152
#271329102
#271328162
#271328092
#271327962
#271326252
#480×C1912
#3194191