Finite Group Databasebeta

← #37581808

Subgroups — #37581808

Order 24448.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#20259369C211yesmaximal, characteristic, complemented
2C2×#20259350C211yesmaximal, characteristic, complemented
3C22×#15690276C211yesmaximal, characteristic, complemented
4#27133227C211yesmaximal, characteristic, complemented
5#27132810C214yesmaximal, complemented
6#27132804C214yesmaximal, complemented
7#27132673C211yesmaximal, characteristic, complemented
8#27132660C211yesmaximal, characteristic, complemented
9#408×C191C211yesmaximal, characteristic, complemented
10#20259369—24no—
11#20259369—24no—
12C2×#15690287C2211yescharacteristic
13C2×#15690287C2212yescomplemented
14C2×#15690286—22no—
15C2×#15690286—21no—
16C22×Dic382C2211yescharacteristic
17C22×Dic382C2212yescomplemented
18C2×Q8×D191—21no—
19C22×D764C2211yescharacteristic
20#20259350—24no—
21C2×#15690276—24no—
22C2×#15690276C2212yes—
23C2×#15690276C2214yes—
24C2×D1528C2212yes—
25#20259318—21no—
26#20259311—21no—
27#20259296C2214yes—
28#20259290—22no—
29#20259281C2211yescharacteristic
30#20259247C2214yescomplemented
31#20259247C2212yescomplemented
32#20259244C2212yescomplemented
33C2×#41×C191C2211yescharacteristic
34C2×M16×C191C2211yescharacteristic
35C22×C8×C191C2211yescharacteristic
36C22⋊C8×C191C2214yescomplemented
37C23×D191—21no—
38#15690287—24no—
39#15690287—24no—
40#15690287—22no—
41#15690287D414yescomplemented
42#15690286—216no—
43#15690286—28no—
44C2×Dic382—24no—
45C2×Dic382—24no—
46C2×Dic382—21no—
47C2×Dic382—22no—
48C2×Dic382—22no—
49C2×Dic382D414yescomplemented
50C2×Dic382C2312yes—
51C2×Dic382C2311yescharacteristic
52Q8×D191—28no—
53C2×D764—22no—
54C2×D764—22no—
55C2×D764—21no—
56C2×D764C2312yes—
57C2×D764C2311yescharacteristic
58C2×C22⋊D191—21no—
59C2×C22⋊D191—22no—
60C2×C22⋊D191—24no—
61C22×Dic191—21no—
62C22×Dic191—22no—
63C2×C4×D191—21no—
64C2×C4×D191—22no—
65#15690276—28no—
66#15690276—24no—
67D1528—24no—
68C4⋊Dic191C2312yes—
69C22⋊Dic191—21no—
70(C2×D191)⋊C4—24no—
71C4×Dic191—21no—
72#41×C191—22no—
73#41×C191—22no—
74C2×Q8×C191D411yescharacteristic
75C2×D4×C191D412yes—
76C2×D4×C191D411yescharacteristic
77C22×C4×C191C2312yes—
78C22×C4×C191C2311yescharacteristic
79M16×C191—22no—
80C2×C8×C191—22no—
81C2×C8×C191C2312yes—
82C2×C8×C191C2312yes—
83C22×D191—21no—
84C22×D191—22no—
85C22×D191—42no—
86C22×D191—82no—
87Dic382—28no—
88Dic382—24no—
89Dic382—21no—
90Dic382—22no—
91Dic382C2×D412yes—
92Dic382C2×D414yes—
93D764—21no—
94D764—22no—
95D764—22no—
96D764—42no—
97D764C2×D412yes—
98C22⋊D191—28no—
99C22⋊D191—24no—
100C22⋊D191—84no—
101C2×Dic191—28no—
102C2×Dic191—24no—
103C2×Dic191—21no—
104C2×Dic191—22no—
105C2×Dic191—22no—
106C4×D191—28no—
107C4×D191—24no—
108C23×C191C2×D412yes—
109C23×C191C2×D411yescharacteristic
110Q8×C191—22no—
111D4×C191—24no—
112D4×C191—22no—
113C2×C4×C191—22no—
114C2×C4×C191—22no—
115C2×C4×C191C2×D412yes—
116C2×C4×C191C2×D412yes—
117C2×C4×C191C2×D412yes—
118C2×C4×C191C2×D411yescharacteristic
119C2×C4×C191C2411yescharacteristic
120C8×C191—22no—
121C8×C191—22no—
122C22×C191—22no—
123C22×C191—22no—
124C22×C191—24no—
125C22×C191#12012yes—
126C22×C191C22×D411yescharacteristic
127C2×D191—22no—
128C2×D191—42no—
129C2×D191—42no—
130C2×D191—44no—
131C2×D191—84no—
132C4×C191—22no—
133C4×C191#12012yes—
134C4×C191C22×D411yescharacteristic
135C4×C191C22×D411yescharacteristic
136Dic191—24no—
137Dic191—22no—
138Dic191—42no—
139C2×C191—22no—
140C2×C191—42no—
141C2×C191C2×#12011yescharacteristic
142C2×C191#57712yes—
143D191—42no—
144D191—82no—
145C191#301011yescharacteristic, complemented, Sylow
146#3010—1911nomaximal, Sylow
147C2×2−1+4—1911no—
148C2×AGL(1,ℤ/8)—1911no—
149C22×SD16—1911no—
150#535—1911no—
151#449—1914no—
152#448—1914no—
153#419—1911no—
154#416—1911no—
155#408D19111yescharacteristic, complemented
1562−1+4—3824no—
1572−1+4—3824no—
158C2×#41—3822no—
159C2×#41—3821no—
160C2×#41—1912no—
161C2×#41—1911no—
162C2×#41C2×D19111yescharacteristic
163C22×Q8—3821no—
164C22×Q8—1912no—
165C22×Q8—1911no—
166C22×D4—1911no—
167AGL(1,ℤ/8)—3824no—
168C2×SD16—3824no—
169C2×SD16—1914no—
170C2×SD16—1912no—
171C2×D8—1912no—
172C2×M16C2×D19111yescharacteristic
173C22×C8C2×D19111yescharacteristic
174#127—3821no—
175#124—3821no—
176#121—1914no—
177#120—3822no—
178#117—1911no—
179#103—1912no—
180#103—1914no—
181#102—1912no—
182C22⋊C8C2×D19114yescomplemented
183C24—3821no—
184#41—22no—
185#41—22no—
186#41—3824no—
187#41—3824no—
188#41—38216no—
189#41—3828no—
190#41—3822no—
191#41—1914no—
192C2×Q8—3824no—
193C2×Q8—3824no—
194C2×Q8—3828no—
195C2×Q8—3822no—
196C2×Q8—3822no—
197C2×Q8—3821no—
198C2×Q8—1914no—
199C2×Q8—1911no—
200C2×Q8—1912no—
201C2×Q8D76411yescharacteristic
202C2×D4—3822no—
203C2×D4—3821no—
204C2×D4—3824no—
205C2×D4—3822no—
206C2×D4—3822no—
207C2×D4—3821no—
208C2×D4—1912no—
209C2×D4—1911no—
210C2×D4D76412yes—
211C2×D4D76411yescharacteristic
212C22×C4—3822no—
213C22×C4—3821no—
214C22×C4—3822no—
215C22×C4—3821no—
216C22×C4C22×D19112yes—
217C22×C4C22×D19111yescharacteristic
218SD16—3828no—
219SD16—3824no—
220D8—3824no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D19112yes—
224C2×C8C22×D19112yes—
225C4⋊C4—1912no—
226C22⋊C4—3821no—
227C22⋊C4—3824no—
228C42—3821no—
229C23—3821no—
230C23—3822no—
231C23—15282no—
232C23—7642no—
233C23C2×D76412yes—
234C23C2×D76411yescharacteristic
235Q8—22no—
236Q8—3828no—
237Q8—3821no—
238Q8—3822no—
239Q8—3824no—
240Q8—1912no—
241Q8—1914no—
242D4—24no—
243D4—22no—
244D4—3828no—
245D4—3824no—
246D4—3821no—
247D4—3822no—
248D4—3822no—
249D4—1912no—
250D4—15284no—
251D4—7642no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—3828no—
255C2×C4—3828no—
256C2×C4—3822no—
257C2×C4—3821no—
258C2×C4—3824no—
259C2×C4—3824no—
260C2×C4—3822no—
261C2×C4D4×D19112yes—
262C2×C4D4×D19112yes—
263C2×C4C2×D76412yes—
264C2×C4C2×D76411yescharacteristic
265C2×C4C23×D19111yescharacteristic
266C8—22no—
267C8—22no—
268C22—22no—
269C22—24no—
270C22—22no—
271C22—3822no—
272C22—15284no—
273C22—7644no—
274C22—7642no—
275C22—7642no—
276C22#2025929212yes—
277C22C22×D76411yescharacteristic
278C4—22no—
279C4—3824no—
280C4—3822no—
281C4—7642no—
282C4#2025929212yes—
283C4C2×D4×D19111yescharacteristic
284C4C2×D4×D19111yescharacteristic
285C2—22no—
286C2—42no—
287C2—15282no—
288C2—7642no—
289C2C2×#2025929211yescharacteristic
290C2#2713347512yes—
subgroupG/H
centerC22C22×D764
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#408×C191C2
socleC22×C191C22×D4
hypercenter#408D191
hypocenterC191#3010
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#20259369C2semidirect
2. C2×#20259350C2semidirect
3. C22×#15690276C2semidirect
4. #27133227C2semidirect
5. #27132810C2semidirect
6. #27132804C2semidirect
7. #27132673C2semidirect
8. #27132660C2semidirect
9. #408×C191C2semidirect
13. C2×#15690287C22semidirect
17. C22×Dic382C22semidirect
30. #20259247C22semidirect
31. #20259247C22semidirect
32. #20259244C22semidirect
36. C22⋊C8×C191C22semidirect
41. #15690287D4semidirect
49. C2×Dic382D4semidirect
145. C191#3010semidirect
155. #408D191semidirect
182. C22⋊C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593692
C2×#202593502
C22×#156902762
#271332272
#271328102
#271328042
#271326732
#271326602
#408×C1912
#3010191