Finite Group Databasebeta

← #37583459

Subgroups — #37583459

Order 24448.

488 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 123 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#27133215C211yesmaximal, characteristic, complemented
2C2×#20259318C211yesmaximal, characteristic
3#27133034C212yesmaximal, complemented
4#27133008C212yesmaximal, complemented
5#27132862C212yesmaximal, complemented
6#27132861C212yesmaximal, complemented
7#27132816C211yesmaximal, characteristic, complemented
8#27132814C211yesmaximal, characteristic, complemented
9#27132813C211yesmaximal, characteristic, complemented
10#27132713C211yesmaximal, characteristic, complemented
11#450×C191C211yesmaximal, characteristic, complemented
12C22×D764C2211yescharacteristic
13C22×C22⋊D191—21no—
14C22×C22⋊D191—21no—
15C2×#15690276C2212yescomplemented
16C2×#15690275C2212yescomplemented
17C2×#15690274C2212yescomplemented
18#20259321C2211yescharacteristic, complemented
19#20259320C2211yescharacteristic, complemented
20#20259318—21no—
21#20259318—21no—
22#20259318C2212yes—
23#20259318C2212yes—
24#20259312—21no—
25#20259301C2211yescharacteristic
26#20259299—21no—
27Q8×Dic191C2211yescharacteristic, complemented
28#20259287—21no—
29#20259287C2211yescharacteristic, complemented
30#20259278—21no—
31C2×C4×Dic191C2211yescharacteristic
32Dic191⋊C8C2212yescomplemented
33#20259244C2212yescomplemented
34#20259243C2212yescomplemented
35#20259242C2212yescomplemented
36C22⋊(C191⋊C8)C2211yescharacteristic
37C8×Dic191C2212yescomplemented
38C22×D4×C191C2211yescharacteristic
39C2×SD16×C191C2212yescomplemented
40C22⋊Q8×C191C2211yescharacteristic
41#102×C191C2212yescomplemented
42C22⋊C8×C191C2211yescharacteristic
43C23×D191—21no—
44C2×Dic382—21no—
45C2×Dic382C2311yescharacteristic
46C2×D764—21no—
47C2×D764—42no—
48C2×D764C2312yes—
49C2×C22⋊D191—22no—
50C2×C22⋊D191—22no—
51C2×C22⋊D191—22no—
52C2×C22⋊D191—22no—
53C2×C22⋊D191—44no—
54C2×C22⋊D191—44no—
55C22×Dic191D411yescharacteristic
56C22×Dic191D411yescharacteristic
57#15690276—42no—
58#15690275—42no—
59#15690274—42no—
60C2×C191⋊C8C2312yes—
61C4⋊Dic191—21no—
62C4⋊Dic191C2311yescharacteristic
63Dic191⋊C4—21no—
64Dic191⋊C4—21no—
65Dic191⋊C4—21no—
66Dic191⋊C4—21no—
67Dic191⋊C4—21no—
68C22⋊Dic191—21no—
69C4×Dic191—21no—
70C4×Dic191—21no—
71C4×Dic191D411yescharacteristic
72C4×Dic191D411yescharacteristic
73C4×Dic191C2311yescharacteristic
74C4×Dic191C2311yescharacteristic
75C24×C191—21no—
76C2×Q8×C191C2311yescharacteristic
77C2×D4×C191—21no—
78C2×D4×C191—42no—
79C2×D4×C191C2312yes—
80C22×C4×C191C2311yescharacteristic
81SD16×C191—42no—
82C2×C8×C191C2312yes—
83C4⋊C4×C191—21no—
84C4⋊C4×C191C2311yescharacteristic
85C22⋊C4×C191—21no—
86C22×D191—22no—
87C22×D191—42no—
88C22×D191—42no—
89C22×D191—42no—
90Dic382—22no—
91Dic382—42no—
92D764—22no—
93D764—22no—
94D764—42no—
95C22⋊D191—44no—
96C22⋊D191—44no—
97C22⋊D191—44no—
98C22⋊D191—44no—
99C2×Dic191—22no—
100C2×Dic191—21no—
101C2×Dic191—21no—
102C2×Dic191SD1614yescomplemented
103C2×Dic191C2×D411yescharacteristic
104C2×Dic191C2×D411yescharacteristic
105C2×Dic191C2×D411yescharacteristic
106C2×Dic191C2×D411yescharacteristic
107C191⋊C8—22no—
108C23×C191—22no—
109C23×C191—42no—
110C23×C191—42no—
111C23×C191—42no—
112C23×C191C2×D411yescharacteristic
113Q8×C191—22no—
114D4×C191—22no—
115D4×C191—22no—
116D4×C191—42no—
117C2×C4×C191—22no—
118C2×C4×C191—21no—
119C2×C4×C191—21no—
120C2×C4×C191C2×D411yescharacteristic
121C2×C4×C191C2411yescharacteristic
122C8×C191—22no—
123C22×C191—21no—
124C22×C191—21no—
125C22×C191—42no—
126C22×C191—42no—
127C22×C191—42no—
128C22×C191—84no—
129C22×C191C2×SD1612yes—
130C22×C191C22×D411yescharacteristic
131C2×D191—42no—
132C2×D191—42no—
133C2×D191—42no—
134C2×D191—84no—
135C4×C191—21no—
136C4×C191—41no—
137C4×C191—41no—
138C4×C191H(5,𝔽2)12yes—
139Dic191—21no—
140Dic191—41no—
141Dic191—41no—
142Dic191C2×SD1612yes—
143Dic191C2×SD1612yes—
144Dic191AGL(1,ℤ/8)12yes—
145C2×C191—22no—
146C2×C191—82no—
147C2×C191#53511yescharacteristic
148C2×C191C22×SD1611yescharacteristic
149C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
150D191—82no—
151C191#320811yescharacteristic, complemented, Sylow
152#3208—1911nomaximal, Sylow
153#533—1911no—
154C2×#127—1911no—
155#492—1912no—
156#486—1912no—
157#460—1912no—
158#460—1912no—
159#450—1911no—
160#450—1911no—
161#450—1911no—
162#450D19111yescharacteristic, complemented
163#431—1911no—
164C22×D4—3821no—
165C22×D4—3821no—
166C22×D4—1911no—
167C22×D4C2×D19111yescharacteristic
168C2×SD16—1912no—
169C2×SD16—1912no—
170C2×SD16—1912no—
171C2×SD16C2×D19112yescomplemented
172C4⋊Q8—1911no—
173C4⋊Q8—1911no—
174#127—3821no—
175#127—3821no—
176#127—1912no—
177#127—1912no—
178#125—3821no—
179C22⋊Q8—3821no—
180C22⋊Q8—1911no—
181C22⋊Q8C2×D19111yescharacteristic
182C4×Q8—3821no—
183C4×Q8—1911no—
184C4×Q8—1911no—
185#117—3821no—
186C2×C42—1911no—
187C4⋊C8—1912no—
188#102—1912no—
189#102—1912no—
190#102—1912no—
191#102C2×D19112yescomplemented
192C22⋊C8—1911no—
193C22⋊C8C2×D19111yescharacteristic
194C4×C8—1912no—
195C24—21no—
196C24—3821no—
197C2×Q8—3821no—
198C2×Q8—1911no—
199C2×Q8C22×D19111yescharacteristic
200C2×D4—21no—
201C2×D4—42no—
202C2×D4—3822no—
203C2×D4—3822no—
204C2×D4—3822no—
205C2×D4—3822no—
206C2×D4—3821no—
207C2×D4—1912no—
208C2×D4—7642no—
209C2×D4—7644no—
210C2×D4—7644no—
211C2×D4C22×D19112yes—
212C22×C4—1911no—
213C22×C4—1911no—
214C22×C4C22×D19111yescharacteristic
215SD16—42no—
216SD16—7642no—
217SD16—7642no—
218SD16—7642no—
219C2×C8—1912no—
220C2×C8C22×D19112yes—
221C4⋊C4—21no—
222C4⋊C4—3821no—
223C4⋊C4—3821no—
224C4⋊C4—3821no—
225C4⋊C4—3821no—
226C4⋊C4—3821no—
227C4⋊C4—3821no—
228C4⋊C4—1911no—
229C4⋊C4C22×D19111yescharacteristic
230C22⋊C4—21no—
231C22⋊C4—3821no—
232C42—3821no—
233C42—3821no—
234C42—1911no—
235C42—1911no—
236C42—1911no—
237C42—1911no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—3822no—
243C23—7642no—
244C23—7642no—
245C23—7642no—
246C23D4×D19111yescharacteristic
247Q8—22no—
248Q8—3822no—
249Q8—7642no—
250D4—22no—
251D4—22no—
252D4—42no—
253D4—3822no—
254D4—3822no—
255D4—7642no—
256D4—7644no—
257D4—7644no—
258D4—7644no—
259D4—7644no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—3822no—
264C2×C4—3821no—
265C2×C4—3821no—
266C2×C4—1911no—
267C2×C4—1914no—
268C2×C4—1911no—
269C2×C4—1911no—
270C2×C4—1911no—
271C2×C4D4×D19111yescharacteristic
272C2×C4C23×D19111yescharacteristic
273C8—22no—
274C8—3822no—
275C22—21no—
276C22—21no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—84no—
281C22—15284no—
282C22—7642no—
283C22—7642no—
284C22—7642no—
285C22SD16×D19112yes—
286C22C2×D4×D19111yescharacteristic
287C4—21no—
288C4—41no—
289C4—41no—
290C4—3821no—
291C4—1912no—
292C4—1912no—
293C4—1912no—
294C4—7641no—
295C4—7641no—
296C4#2025936712yes—
297C2—22no—
298C2—82no—
299C2—15282no—
300C2#2713322811yescharacteristic
301C2C2×SD16×D19111yescharacteristic
302C2AGL(1,ℤ/8)×D19111yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#450×C191C2
socleC22×C191C22×D4
prime radicalC2×#20259318C2
hypercenter#450D191
hypocenterC191#3208
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. #27133215C2semidirect
3. #27133034C2semidirect
4. #27133008C2semidirect
5. #27132862C2semidirect
6. #27132861C2semidirect
7. #27132816C2semidirect
8. #27132814C2semidirect
9. #27132813C2semidirect
10. #27132713C2semidirect
11. #450×C191C2semidirect
15. C2×#15690276C22semidirect
16. C2×#15690275C22semidirect
17. C2×#15690274C22semidirect
18. #20259321C22semidirect
19. #20259320C22semidirect
27. Q8×Dic191C22semidirect
29. #20259287C22semidirect
32. Dic191⋊C8C22semidirect
33. #20259244C22semidirect
34. #20259243C22semidirect
35. #20259242C22semidirect
37. C8×Dic191C22semidirect
39. C2×SD16×C191C22semidirect
41. #102×C191C22semidirect
102. C2×Dic191SD16semidirect
151. C191#3208semidirect
162. #450D191semidirect
171. C2×SD16C2×D191semidirect
191. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271332152
C2×#202593182
#271330342
#271330082
#271328622
#271328612
#271328162
#271328142
#271328132
#271327132
#450×C1912
#3208191