Finite Group Databasebeta

← #37581809

Subgroups — #37581809

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×#20259347C211yesmaximal, characteristic, complemented
3C22×#15690276C211yesmaximal, characteristic, complemented
4#27133232C211yesmaximal, characteristic, complemented
5#27132809C214yesmaximal, complemented
6#27132803C214yesmaximal, complemented
7#27132671C211yesmaximal, characteristic, complemented
8#27132629C211yesmaximal, characteristic, complemented
9#416×C191C211yesmaximal, characteristic, complemented
10#20259369—24no—
11#20259369—24no—
12C2×#15690287—21no—
13C2×#15690287C2211yescharacteristic
14C2×#15690286—22no—
15C2×#15690286C2212yescomplemented
16C22×Dic382—21no—
17C22×Dic382C2211yescharacteristic
18C2×Q8×D191C2212yescomplemented
19C22×D764C2211yescharacteristic
20#20259347—24no—
21C2×#15690276—24no—
22C2×#15690276C2214yes—
23C2×#15690274C2212yes—
24C2×#15690272C2212yes—
25C2×#15690271C2211yescharacteristic
26#20259319—21no—
27#20259310—21no—
28#20259297C2214yes—
29#20259292—22no—
30#20259245C2214yescomplemented
31(C2×D191)⋊C8C2214yescomplemented
32C2×#41×C191C2211yescharacteristic
33C22×C8×C191C2211yescharacteristic
34#117×C191C2211yescharacteristic
35#103×C191C2212yescomplemented
36#102×C191C2212yescomplemented
37C23×D191—21no—
38#15690287—28no—
39#15690287—22no—
40#15690287—22no—
41#15690286—216no—
42#15690286—24no—
43#15690286—24no—
44C2×Dic382—28no—
45C2×Dic382—22no—
46C2×Dic382—21no—
47C2×Dic382—22no—
48C2×Dic382D411yescharacteristic
49C2×Dic382D414yescomplemented
50C2×Dic382C2312yes—
51Q8×D191—24no—
52Q8×D191—24no—
53C2×D764—24no—
54C2×D764—22no—
55C2×D764—22no—
56C2×D764—21no—
57C2×D764D411yescharacteristic
58C2×D764C2312yes—
59C2×C22⋊D191—22no—
60C2×C22⋊D191D412yes—
61C22×Dic191—22no—
62C22×Dic191—21no—
63C2×C4×D191—22no—
64C2×C4×D191C2312yes—
65#15690276—28no—
66#15690274—24no—
67#15690272—24no—
68#15690271—22no—
69C2×C191⋊C8C2312yes—
70(C2×D191)⋊C4—24no—
71#41×C191—22no—
72#41×C191D414yescomplemented
73C2×Q8×C191C2311yescharacteristic
74C2×D4×C191—21no—
75C2×D4×C191C2311yescharacteristic
76C22×C4×C191—21no—
77C22×C4×C191C2311yescharacteristic
78C2×C8×C191—22no—
79C2×C8×C191C2312yes—
80C4⋊C4×C191C2312yes—
81C22⋊C4×C191—21no—
82C42×C191—21no—
83C22×D191—22no—
84C22×D191—42no—
85C22×D191—82no—
86C22×D191C2×D412yes—
87Dic382—28no—
88Dic382—22no—
89Dic382—24no—
90Dic382—22no—
91Dic382C2×D414yes—
92D764—22no—
93D764—22no—
94D764—22no—
95D764—42no—
96D764—84no—
97C22⋊D191—28no—
98C22⋊D191—24no—
99C2×Dic191—28no—
100C2×Dic191—24no—
101C2×Dic191—22no—
102C2×Dic191C2×D412yes—
103C4×D191—28no—
104C4×D191—22no—
105C4×D191—22no—
106C191⋊C8—22no—
107C23×C191—21no—
108C23×C191C2×D411yescharacteristic
109Q8×C191—21no—
110Q8×C191C2×D412yes—
111D4×C191—24no—
112D4×C191—21no—
113D4×C191C2×D412yes—
114C2×C4×C191—21no—
115C2×C4×C191—24no—
116C2×C4×C191—22no—
117C2×C4×C191C2×D411yescharacteristic
118C2×C4×C191C2×D412yes—
119C2×C4×C191C2×D412yes—
120C2×C4×C191C2411yescharacteristic
121C8×C191—22no—
122C22×C191—22no—
123C22×C191—22no—
124C22×C191—42no—
125C22×C191#12012yes—
126C22×C191C22×D411yescharacteristic
127C2×D191—22no—
128C2×D191—24no—
129C2×D191—42no—
130C2×D191—44no—
131C2×D191—84no—
132C4×C191—22no—
133C4×C191—42no—
134C4×C191#12012yes—
135C4×C191C22×D411yescharacteristic
136C4×C191C22×D411yescharacteristic
137Dic191—24no—
138Dic191—22no—
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#416D19111yescharacteristic, complemented
155#408—1911no—
1562−1+4—3824no—
1572−1+4—3824no—
158C2×#41—3821no—
159C2×#41—3822no—
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—1912no—
170C2×SD16—1914no—
171C2×D8—1912no—
172C2×M16—1911no—
173C22×C8C2×D19111yescharacteristic
174#127—3821no—
175#124—3821no—
176#121—1914no—
177#120—3822no—
178#117C2×D19111yescharacteristic
179#103—1914no—
180#103C2×D19112yescomplemented
181#102C2×D19112yescomplemented
182C22⋊C8—1914no—
183C24—3821no—
184#41—22no—
185#41—3828no—
186#41—38216no—
187#41—3824no—
188#41—3824no—
189#41—3822no—
190#41—3822no—
191#41D76414yescomplemented
192C2×Q8—3828no—
193C2×Q8—3822no—
194C2×Q8—3824no—
195C2×Q8—3824no—
196C2×Q8—3821no—
197C2×Q8—3822no—
198C2×Q8—1914no—
199C2×Q8—1912no—
200C2×Q8—1911no—
201C2×Q8C22×D19111yescharacteristic
202C2×D4—21no—
203C2×D4—3824no—
204C2×D4—3822no—
205C2×D4—3822no—
206C2×D4—3821no—
207C2×D4—3822no—
208C2×D4—1912no—
209C2×D4—1911no—
210C2×D4—1912no—
211C2×D4C22×D19111yescharacteristic
212C22×C4—21no—
213C22×C4—3822no—
214C22×C4—3821no—
215C22×C4—3822no—
216C22×C4—1912no—
217C22×C4C22×D19111yescharacteristic
218SD16—3828no—
219SD16—3824no—
220D8—3824no—
221M16—3822no—
222C2×C8—22no—
223C2×C8—1912no—
224C2×C8C22×D19112yes—
225C4⋊C4C22×D19112yes—
226C22⋊C4—21no—
227C22⋊C4—3824no—
228C42—21no—
229C23—21no—
230C23—3822no—
231C23—1912no—
232C23—15282no—
233C23—7642no—
234C23D4×D19111yescharacteristic
235Q8—21no—
236Q8—3828no—
237Q8—3822no—
238Q8—3824no—
239Q8—3822no—
240Q8—1914no—
241Q8C2×D76412yes—
242D4—24no—
243D4—21no—
244D4—3828no—
245D4—3824no—
246D4—3822no—
247D4—3822no—
248D4—3822no—
249D4—15284no—
250D4—7642no—
251D4C2×D76412yes—
252C2×C4—21no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—3828no—
256C2×C4—3828no—
257C2×C4—3822no—
258C2×C4—3824no—
259C2×C4—3822no—
260C2×C4—3822no—
261C2×C4—1912no—
262C2×C4D4×D19112yes—
263C2×C4D4×D19111yescharacteristic
264C2×C4C2×D76412yes—
265C2×C4C23×D19111yescharacteristic
266C8—22no—
267C8—3822no—
268C22—22no—
269C22—22no—
270C22—42no—
271C22—3824no—
272C22—3822no—
273C22—15284no—
274C22—7644no—
275C22—7642no—
276C22#2025929212yes—
277C22C2×D4×D19111yescharacteristic
278C4—22no—
279C4—42no—
280C4—3824no—
281C4—3822no—
282C4#2025929212yes—
283C4C2×D4×D19111yescharacteristic
284C4C22×D76411yescharacteristic
285C2—22no—
286C2—42no—
287C2—15282no—
288C2—7642no—
289C2C2×#2025929211yescharacteristic
290C2#2713347012yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#416×C191C2
socleC22×C191C22×D4
hypercenter#416D191
hypocenterC191#3010
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#20259369C2semidirect
2. C2×#20259347C2semidirect
3. C22×#15690276C2semidirect
4. #27133232C2semidirect
5. #27132809C2semidirect
6. #27132803C2semidirect
7. #27132671C2semidirect
8. #27132629C2semidirect
9. #416×C191C2semidirect
15. C2×#15690286C22semidirect
18. C2×Q8×D191C22semidirect
30. #20259245C22semidirect
31. (C2×D191)⋊C8C22semidirect
35. #103×C191C22semidirect
36. #102×C191C22semidirect
49. C2×Dic382D4semidirect
72. #41×C191D4semidirect
145. C191#3010semidirect
154. #416D191semidirect
180. #103C2×D191semidirect
181. #102C2×D191semidirect
191. #41D764semidirect

Maximal subgroups

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

subgroupindex
C2×#202593692
C2×#202593472
C22×#156902762
#271332322
#271328092
#271328032
#271326712
#271326292
#416×C1912
#3010191