Finite Group Databasebeta

← #37583287

Subgroups — #37583287

Order 24448.

472 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 125 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#27133174C211yesmaximal, characteristic, complemented
2C2×#20259296C211yesmaximal, characteristic, complemented
3#27132999C211yesmaximal, characteristic, complemented
4#27132988C212yesmaximal, complemented
5#27132982C211yesmaximal, characteristic, complemented
6#27132899C212yesmaximal, complemented
7#27132892C212yesmaximal, complemented
8#27132816C211yesmaximal, characteristic, complemented
9#27132810C212yesmaximal, complemented
10#27132799C211yesmaximal, characteristic, complemented
11C2×C22⋊C8×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic, complemented
13C22×D764C2211yescharacteristic
14C22×C22⋊D191—21no—
15C2×#15690276C2212yescomplemented
16C2×D1528C2212yescomplemented
17#20259305—21no—
18#20259301C2211yescharacteristic
19#20259301C2211yescharacteristic, complemented
20#20259296—21no—
21#20259296—21no—
22#20259296C2211yescharacteristic
23#20259296C2212yes—
24#20259296C2212yes—
25#20259296C2211yescharacteristic, complemented
26#20259291—21no—
27C4×C22⋊D191—21no—
28C4×C22⋊D191—21no—
29#20259281C2211yescharacteristic, complemented
30C2×C4⋊Dic191C2211yescharacteristic
31C2×(C2×D191)⋊C4—21no—
32#20259256C2212yescomplemented
33#20259254C2212yescomplemented
34#20259247C2212yescomplemented
35#20259244C2212yescomplemented
36#20259244C2212yescomplemented
37#20259244C2212yescomplemented
38C23×C4×C191C2211yescharacteristic
39C22×C8×C191C2212yescomplemented
40C22⋊C8×C191C2211yescharacteristic
41C22⋊C8×C191C2211yescharacteristic
42C22⋊C8×C191C2212yescomplemented
43C23×D191—21no—
44#15690287—42no—
45C2×Dic382C2311yescharacteristic
46C2×D764—21no—
47C2×D764—42no—
48C2×D764C2311yescharacteristic
49C2×D764C2312yes—
50C2×C22⋊D191—22no—
51C2×C22⋊D191—22no—
52C2×C22⋊D191—21no—
53C2×C22⋊D191—21no—
54C2×C22⋊D191—21no—
55C2×C22⋊D191—44no—
56C22×Dic191—21no—
57C2×C4×D191—21no—
58#15690276—42no—
59D1528—42no—
60C4⋊Dic191—21no—
61C4⋊Dic191C2311yescharacteristic
62C4⋊Dic191C2311yescharacteristic
63C4⋊Dic191C2312yes—
64Dic191⋊C4—21no—
65Dic191⋊C4—21no—
66C22⋊Dic191—21no—
67C22⋊Dic191—21no—
68C22⋊Dic191—21no—
69(C2×D191)⋊C4—22no—
70(C2×D191)⋊C4—22no—
71(C2×D191)⋊C4—21no—
72(C2×D191)⋊C4—21no—
73C4×Dic191—21no—
74C24×C191D411yescharacteristic
75C22×C4×C191—42no—
76C22×C4×C191D411yescharacteristic
77C22×C4×C191D411yescharacteristic
78C22×C4×C191D411yescharacteristic
79C22×C4×C191C2311yescharacteristic
80C22×C4×C191C2311yescharacteristic
81C22×C4×C191C2311yescharacteristic
82C2×C8×C191—42no—
83C2×C8×C191C2312yes—
84C2×C8×C191C2312yes—
85C22×D191—21no—
86C22×D191—22no—
87C22×D191—42no—
88C22×D191—42no—
89C22×D191—42no—
90Dic382—22no—
91D764—22no—
92D764—22no—
93D764—22no—
94D764—42no—
95C22⋊D191—44no—
96C22⋊D191—42no—
97C22⋊D191—42no—
98C22⋊D191—42no—
99C22⋊D191—44no—
100C2×Dic191—21no—
101C2×Dic191—21no—
102C2×Dic191—22no—
103C2×Dic191—21no—
104C2×Dic191—42no—
105C4×D191—42no—
106C23×C191—22no—
107C23×C191—22no—
108C23×C191—41no—
109C23×C191C2×D411yescharacteristic
110C23×C191C2×D411yescharacteristic
111C23×C191C2×D411yescharacteristic
112C2×C4×C191—22no—
113C2×C4×C191—22no—
114C2×C4×C191—22no—
115C2×C4×C191—41no—
116C2×C4×C191—41no—
117C2×C4×C191—41no—
118C2×C4×C191C2×D411yescharacteristic
119C2×C4×C191C2×D411yescharacteristic
120C2×C4×C191C2×D411yescharacteristic
121C2×C4×C191C2411yescharacteristic
122C8×C191—22no—
123C8×C191—22no—
124C22×C191—21no—
125C22×C191—21no—
126C22×C191—21no—
127C22×C191—21no—
128C22×C191—21no—
129C22×C191—21no—
130C22×C191—21no—
131C22×C191—44no—
132C22×C191#13512yes—
133C22×C191AGL(1,ℤ/8)12yes—
134C22×C191C22×D411yescharacteristic
135C2×D191—41no—
136C2×D191—42no—
137C2×D191—41no—
138C2×D191—42no—
139C2×D191—42no—
140C2×D191—41no—
141C2×D191—84no—
142C4×C191—21no—
143C4×C191—21no—
144C4×C191—21no—
145C4×C191H(5,𝔽2)12yes—
146Dic191—41no—
147Dic191—41no—
148Dic191—42no—
149Dic191—41no—
150C2×C191—22no—
151C2×C191—22no—
152C2×C191—41no—
153C2×C191#53411yescharacteristic
154C2×C191C2×#13511yescharacteristic
155C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
156D191—81no—
157D191—82no—
158C191#318811yescharacteristic, complemented, Sylow
159#3188—1911nomaximal, Sylow
160#525—1911no—
161C2×#121—1911no—
162#484—1911no—
163#482—1912no—
164#481—1911no—
165#466—1912no—
166#465—1912no—
167#450—1911no—
168#449—1912no—
169#447—1911no—
170C2×C22⋊C8D19111yescharacteristic, complemented
171C2×#41—1911no—
172C22×D4—3821no—
173C22×D4—1911no—
174C23×C4C2×D19111yescharacteristic
175C2×SD16—1912no—
176C2×D8—1912no—
177C22×C8C2×D19112yescomplemented
178#123—3821no—
179C22⋊Q8—1911no—
180C22⋊Q8—1911no—
181#121—3821no—
182#121—3821no—
183#121—1912no—
184#121—1912no—
185#121—1911no—
186#121—1911no—
187#120—3821no—
188C4×D4—3821no—
189C4×D4—3821no—
190#117—1911no—
191C2×C4⋊C4—1911no—
192C2×C22⋊C4—3821no—
193#107—1912no—
194#106—1912no—
195#103—1912no—
196#102—1912no—
197#102—1912no—
198#102—1912no—
199C22⋊C8C2×D19111yescharacteristic
200C22⋊C8C2×D19111yescharacteristic
201C22⋊C8C2×D19112yescomplemented
202C24—3821no—
203C24D76411yescharacteristic
204#41—7642no—
205C2×Q8—1911no—
206C2×D4—3821no—
207C2×D4—3821no—
208C2×D4—3821no—
209C2×D4—3822no—
210C2×D4—3822no—
211C2×D4—3821no—
212C2×D4—1911no—
213C2×D4—1912no—
214C2×D4—7642no—
215C2×D4—7644no—
216C22×C4—42no—
217C22×C4—3821no—
218C22×C4—3821no—
219C22×C4D76411yescharacteristic
220C22×C4D76411yescharacteristic
221C22×C4D76411yescharacteristic
222C22×C4C22×D19111yescharacteristic
223C22×C4C22×D19111yescharacteristic
224C22×C4C22×D19111yescharacteristic
225SD16—7642no—
226D8—7642no—
227C2×C8—42no—
228C2×C8C22×D19112yes—
229C2×C8C22×D19112yes—
230C4⋊C4—3821no—
231C4⋊C4—3821no—
232C4⋊C4—3821no—
233C4⋊C4—1911no—
234C4⋊C4—1911no—
235C4⋊C4—1912no—
236C22⋊C4—3821no—
237C22⋊C4—3821no—
238C22⋊C4—3821no—
239C22⋊C4—3822no—
240C22⋊C4—3822no—
241C22⋊C4—3821no—
242C22⋊C4—3821no—
243C42—3821no—
244C23—22no—
245C23—22no—
246C23—41no—
247C23—3821no—
248C23—3822no—
249C23—7642no—
250C23—7642no—
251C23—7642no—
252C23C2×D76411yescharacteristic
253C23C2×D76411yescharacteristic
254C23C2×D76411yescharacteristic
255Q8—3822no—
256D4—3822no—
257D4—3822no—
258D4—3822no—
259D4—7642no—
260D4—7644no—
261D4—7642no—
262D4—7642no—
263D4—7644no—
264D4—7642no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—3821no—
272C2×C4—3821no—
273C2×C4—3821no—
274C2×C4—3822no—
275C2×C4—7642no—
276C2×C4—7642no—
277C2×C4C2×D76411yescharacteristic
278C2×C4C2×D76411yescharacteristic
279C2×C4C2×D76411yescharacteristic
280C2×C4C23×D19111yescharacteristic
281C8—22no—
282C8—22no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—44no—
291C22—15284no—
292C22—7641no—
293C22—7642no—
294C22—7642no—
295C22—7641no—
296C22—7641no—
297C22—7642no—
298C22#2025934412yes—
299C22#2025935012yes—
300C22C22×D76411yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—7641no—
305C4—7642no—
306C4—7641no—
307C4—7641no—
308C4#2025936712yes—
309C2—22no—
310C2—22no—
311C2—41no—
312C2—15282no—
313C2—15281no—
314C2#2713322511yescharacteristic
315C2C2×#2025934411yescharacteristic
316C2C2×#2025935011yescharacteristic
subgroupG/H
centerC22C22×D764
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×C22⋊C8×C191C2
socleC22×C191C22×D4
hypercenterC2×C22⋊C8D191
hypocenterC191#3188
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. #27133174C2semidirect
2. C2×#20259296C2semidirect
3. #27132999C2semidirect
4. #27132988C2semidirect
5. #27132982C2semidirect
6. #27132899C2semidirect
7. #27132892C2semidirect
8. #27132816C2semidirect
9. #27132810C2semidirect
10. #27132799C2semidirect
11. C2×C22⋊C8×C191C2semidirect
12. C2×#15690287C22semidirect
15. C2×#15690276C22semidirect
16. C2×D1528C22semidirect
19. #20259301C22semidirect
25. #20259296C22semidirect
29. #20259281C22semidirect
32. #20259256C22semidirect
33. #20259254C22semidirect
34. #20259247C22semidirect
35. #20259244C22semidirect
36. #20259244C22semidirect
37. #20259244C22semidirect
39. C22×C8×C191C22semidirect
42. C22⋊C8×C191C22semidirect
158. C191#3188semidirect
170. C2×C22⋊C8D191semidirect
177. C22×C8C2×D191semidirect
201. C22⋊C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271331742
C2×#202592962
#271329992
#271329882
#271329822
#271328992
#271328922
#271328162
#271328102
#271327992
C2×C22⋊C8×C1912
#3188191