Finite Group Databasebeta

← #37583323

Subgroups — #37583323

Order 24448.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22⋊Q8×D191C211yesmaximal, characteristic, complemented
2C2×#20259293C211yesmaximal, characteristic
3#27132999C211yesmaximal, characteristic, complemented
4#27132996C211yesmaximal, characteristic, complemented
5#27132975C212yesmaximal, complemented
6#27132908C212yesmaximal, complemented
7#27132906C212yesmaximal, complemented
8#27132814C211yesmaximal, characteristic, complemented
9#27132800C212yesmaximal, complemented
10#27132625C211yesmaximal, characteristic, complemented
11#450×C191C211yesmaximal, characteristic, complemented
12C2×Q8×D191C2211yescharacteristic, complemented
13C22×C22⋊D191—21no—
14C22×C4×D191C2211yescharacteristic
15C2×#15690275C2212yescomplemented
16C2×#15690270C2212yescomplemented
17#20259302C2211yescharacteristic, complemented
18#20259301C2211yescharacteristic
19#20259300—21no—
20#20259299—21no—
21#20259298C2211yescharacteristic, complemented
22#20259293—21no—
23#20259293—21no—
24#20259293C2212yes—
25#20259293C2212yes—
26C2×C4⋊Dic191C2211yescharacteristic
27C4⋊C4×D191—21no—
28C4⋊C4×D191C2211yescharacteristic, complemented
29C2×C22⋊Dic191—21no—
30C22⋊C4×D191—21no—
31#20259256C2212yescomplemented
32#20259255C2212yescomplemented
33#20259247C2212yescomplemented
34#20259246C2212yescomplemented
35#20259243C2212yescomplemented
36C22⋊(C191⋊C8)C2211yescharacteristic
37(C2×D191)⋊C8C2212yescomplemented
38C22×D4×C191C2211yescharacteristic
39C2×SD16×C191C2212yescomplemented
40C22⋊Q8×C191C2211yescharacteristic
41#102×C191C2212yescomplemented
42C22⋊C8×C191C2211yescharacteristic
43C23×D191D411yescharacteristic
44C2×Dic382—21no—
45C2×Dic382C2311yescharacteristic
46Q8×D191—42no—
47C2×C22⋊D191—22no—
48C2×C22⋊D191—22no—
49C2×C22⋊D191—44no—
50C22×Dic191—21no—
51C22×Dic191D411yescharacteristic
52C2×C4×D191—21no—
53C2×C4×D191—21no—
54C2×C4×D191—42no—
55C2×C4×D191D411yescharacteristic
56C2×C4×D191D411yescharacteristic
57C2×C4×D191C2311yescharacteristic
58C2×C4×D191C2311yescharacteristic
59#15690275—42no—
60#15690270—42no—
61C2×C191⋊C8C2312yes—
62C4⋊Dic191—21no—
63C4⋊Dic191—21no—
64C4⋊Dic191C2311yescharacteristic
65C4⋊Dic191C2312yes—
66Dic191⋊C4—21no—
67Dic191⋊C4—21no—
68Dic191⋊C4—21no—
69C22⋊Dic191—22no—
70C22⋊Dic191—22no—
71C22⋊Dic191—21no—
72(C2×D191)⋊C4—21no—
73(C2×D191)⋊C4—21no—
74C24×C191—21no—
75C2×Q8×C191C2311yescharacteristic
76C2×D4×C191—21no—
77C2×D4×C191—42no—
78C2×D4×C191C2312yes—
79C22×C4×C191C2311yescharacteristic
80SD16×C191—42no—
81C2×C8×C191C2312yes—
82C4⋊C4×C191—21no—
83C4⋊C4×C191C2311yescharacteristic
84C22⋊C4×C191—21no—
85C22×D191—22no—
86C22×D191—22no—
87C22×D191—41no—
88C22×D191C2×D411yescharacteristic
89C22×D191C2×D411yescharacteristic
90Dic382—22no—
91Dic382—42no—
92C22⋊D191—44no—
93C22⋊D191—44no—
94C2×Dic191—21no—
95C2×Dic191—21no—
96C2×Dic191—22no—
97C2×Dic191—41no—
98C2×Dic191—42no—
99C2×Dic191C2×D411yescharacteristic
100C2×Dic191C2×D411yescharacteristic
101C4×D191—22no—
102C4×D191—22no—
103C4×D191—41no—
104C4×D191—41no—
105C4×D191—42no—
106C4×D191—42no—
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×C191#13712yes—
130C22×C191C22×D411yescharacteristic
131C2×D191—21no—
132C2×D191—21no—
133C2×D191—21no—
134C2×D191—21no—
135C2×D191—21no—
136C2×D191—44no—
137C2×D191AGL(1,ℤ/8)12yes—
138C4×C191—21no—
139C4×C191—41no—
140C4×C191—41no—
141C4×C191H(5,𝔽2)12yes—
142Dic191—21no—
143Dic191—21no—
144Dic191—41no—
145Dic191—41no—
146Dic191—42no—
147C2×C191—22no—
148C2×C191—82no—
149C2×C191#53411yescharacteristic
150C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C191C2×#13711yescharacteristic
152D191—22no—
153D191—41no—
154C191#319311yescharacteristic, complemented, Sylow
155#3193—1911nomaximal, Sylow
156C2×C22⋊Q8—1911no—
157C2×#121—1911no—
158#484—1911no—
159#484—1911no—
160#480—1912no—
161#468—1912no—
162#468—1912no—
163#450—1911no—
164#450D19111yescharacteristic, complemented
165#448—1912no—
166#407—1911no—
167C22×Q8—1911no—
168C22×D4—3821no—
169C22×D4C2×D19111yescharacteristic
170C23×C4—1911no—
171C2×SD16—1912no—
172C2×SD16C2×D19112yescomplemented
173C2×M16—1912no—
174C22⋊Q8—3821no—
175C22⋊Q8—3821no—
176C22⋊Q8—1911no—
177C22⋊Q8—1911no—
178C22⋊Q8—1911no—
179C22⋊Q8C2×D19111yescharacteristic
180#121—3821no—
181#121—3821no—
182#121—1912no—
183#121—1912no—
184C2×C4⋊C4—3821no—
185C2×C4⋊C4—1911no—
186C2×C4⋊C4—1911no—
187C2×C22⋊C4—3821no—
188C2×C22⋊C4—3821no—
189#107—1912no—
190#107—1912no—
191#103—1912no—
192#103—1912no—
193#102—1912no—
194#102C2×D19112yescomplemented
195C22⋊C8—1911no—
196C22⋊C8—1912no—
197C22⋊C8C2×D19111yescharacteristic
198C24—21no—
199C24—1911no—
200C2×Q8—3821no—
201C2×Q8—1911no—
202C2×Q8—7642no—
203C2×Q8C22×D19111yescharacteristic
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—3822no—
207C2×D4—3822no—
208C2×D4—7644no—
209C2×D4C22×D19112yes—
210C22×C4—3821no—
211C22×C4—3821no—
212C22×C4—3821no—
213C22×C4—1911no—
214C22×C4—1911no—
215C22×C4—1911no—
216C22×C4—1911no—
217C22×C4—1911no—
218C22×C4—7642no—
219C22×C4C22×D19111yescharacteristic
220SD16—42no—
221SD16—7642no—
222M16—7642no—
223C2×C8—1912no—
224C2×C8C22×D19112yes—
225C4⋊C4—21no—
226C4⋊C4—3821no—
227C4⋊C4—3821no—
228C4⋊C4—3821no—
229C4⋊C4—3821no—
230C4⋊C4—3821no—
231C4⋊C4—1911no—
232C4⋊C4—1912no—
233C4⋊C4C22×D19111yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—3821no—
236C22⋊C4—3821no—
237C22⋊C4—3821no—
238C22⋊C4—3822no—
239C22⋊C4—3822no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—3822no—
245C23—3822no—
246C23—1911no—
247C23—1911no—
248C23—7641no—
249C23D4×D19111yescharacteristic
250Q8—22no—
251Q8—3822no—
252Q8—7642no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—7644no—
257D4—7644no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—3822no—
262C2×C4—3822no—
263C2×C4—3821no—
264C2×C4—3821no—
265C2×C4—3822no—
266C2×C4—1911no—
267C2×C4—1911no—
268C2×C4—7641no—
269C2×C4—7641no—
270C2×C4—7642no—
271C2×C4—7642no—
272C2×C4—7641no—
273C2×C4—7642no—
274C2×C4D4×D19111yescharacteristic
275C2×C4C23×D19111yescharacteristic
276C8—22no—
277C8—3822no—
278C22—21no—
279C22—21no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—84no—
284C22—3821no—
285C22—3821no—
286C22—3821no—
287C22—3821no—
288C22—3821no—
289C22—1912no—
290C22—7644no—
291C22#2025935112yes—
292C22C2×D4×D19111yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—3821no—
297C4—3821no—
298C4—7642no—
299C4—7641no—
300C4—7641no—
301C4#2025936712yes—
302C2—22no—
303C2—82no—
304C2—3822no—
305C2—7641no—
306C2#2713322311yescharacteristic
307C2AGL(1,ℤ/8)×D19111yescharacteristic
308C2C2×#2025935111yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#450×C191C2
socleC22×C191C22×D4
prime radicalC2×#20259293C2
hypercenter#450D191
hypocenterC191#3193
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. C22⋊Q8×D191C2semidirect
3. #27132999C2semidirect
4. #27132996C2semidirect
5. #27132975C2semidirect
6. #27132908C2semidirect
7. #27132906C2semidirect
8. #27132814C2semidirect
9. #27132800C2semidirect
10. #27132625C2semidirect
11. #450×C191C2semidirect
12. C2×Q8×D191C22semidirect
15. C2×#15690275C22semidirect
16. C2×#15690270C22semidirect
17. #20259302C22semidirect
21. #20259298C22semidirect
28. C4⋊C4×D191C22semidirect
31. #20259256C22semidirect
32. #20259255C22semidirect
33. #20259247C22semidirect
34. #20259246C22semidirect
35. #20259243C22semidirect
37. (C2×D191)⋊C8C22semidirect
39. C2×SD16×C191C22semidirect
41. #102×C191C22semidirect
154. C191#3193semidirect
164. #450D191semidirect
172. C2×SD16C2×D191semidirect
194. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1912
C2×#202592932
#271329992
#271329962
#271329752
#271329082
#271329062
#271328142
#271328002
#271326252
#450×C1912
#3193191