Finite Group Databasebeta

← #37581807

Subgroups — #37581807

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×#20259370C211yesmaximal, characteristic, complemented
2C2×#20259350C211yesmaximal, characteristic, complemented
3C22×#15690274C211yesmaximal, characteristic, complemented
4#27133232C211yesmaximal, characteristic, complemented
5#27132811C214yesmaximal, complemented
6#27132803C214yesmaximal, complemented
7#27132657C211yesmaximal, characteristic, complemented
8#27132631C211yesmaximal, characteristic, complemented
9#419×C191C211yesmaximal, characteristic, complemented
10#20259370—24no—
11#20259370—24no—
12C2×#15690288—21no—
13C2×#15690288C2212yescomplemented
14C2×#15690287—22no—
15C2×#15690287C2211yescharacteristic
16C2×#15690287C2211yescharacteristic
17C2×Q8×D191—21no—
18C2×Q8×D191C2212yescomplemented
19C22×D764C2211yescharacteristic
20#20259350—24no—
21C2×#15690276C2212yes—
22C2×#15690274—24no—
23C2×#15690274C2214yes—
24C2×D1528C2212yes—
25C22×C191⋊C8C2211yescharacteristic
26#20259319—21no—
27#20259310—21no—
28#20259297C2214yes—
29#20259292—22no—
30#20259245C2212yescomplemented
31#20259242C2212yescomplemented
32(C2×D191)⋊C8C2214yescomplemented
33C22×Q8×C191C2211yescharacteristic
34C2×M16×C191C2211yescharacteristic
35#117×C191C2211yescharacteristic
36#103×C191C2214yescomplemented
37C23×D191—21no—
38#15690288—28no—
39#15690288—24no—
40#15690288—24no—
41#15690287—216no—
42#15690287—22no—
43#15690287—22no—
44#15690287—22no—
45#15690287D414yescomplemented
46C2×Dic382—22no—
47C2×Dic382D411yescharacteristic
48C2×Dic382C2311yescharacteristic
49Q8×D191—28no—
50Q8×D191—24no—
51Q8×D191—24no—
52C2×D764—22no—
53C2×D764—24no—
54C2×D764—22no—
55C2×D764—22no—
56C2×D764—21no—
57C2×D764D411yescharacteristic
58C2×D764C2312yes—
59C2×D764C2311yescharacteristic
60C2×C22⋊D191—21no—
61C2×C22⋊D191D412yes—
62C2×C4×D191—22no—
63C2×C4×D191—22no—
64C2×C4×D191—21no—
65C2×C4×D191C2312yes—
66#15690276—24no—
67#15690274—28no—
68D1528—24no—
69C2×C191⋊C8—22no—
70C2×C191⋊C8C2312yes—
71(C2×D191)⋊C4—24no—
72C2×Q8×C191—21no—
73C2×Q8×C191—22no—
74C2×Q8×C191D414yescomplemented
75C2×Q8×C191C2312yes—
76C22×C4×C191—21no—
77C22×C4×C191C2311yescharacteristic
78M16×C191—22no—
79C2×C8×C191C2312yes—
80C4⋊C4×C191C2312yes—
81C22⋊C4×C191—21no—
82C42×C191—21no—
83C22×D191—21no—
84C22×D191—22no—
85C22×D191—42no—
86C22×D191—82no—
87C22×D191C2×D412yes—
88Dic382—28no—
89Dic382—22no—
90Dic382—21no—
91Dic382C2×D412yes—
92D764—28no—
93D764—22no—
94D764—21no—
95D764—22no—
96D764—22no—
97D764—42no—
98D764—84no—
99D764C2×D412yes—
100C22⋊D191—24no—
101C22⋊D191—24no—
102C2×Dic191—21no—
103C2×Dic191C2×D412yes—
104C4×D191—28no—
105C4×D191—28no—
106C4×D191—24no—
107C4×D191—22no—
108C4×D191—22no—
109C191⋊C8—22no—
110C23×C191C2×D411yescharacteristic
111Q8×C191—24no—
112Q8×C191—22no—
113Q8×C191C2×D414yes—
114C2×C4×C191—24no—
115C2×C4×C191—22no—
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#12012yes—
124C22×C191C22×D411yescharacteristic
125C2×D191—22no—
126C2×D191—22no—
127C2×D191—24no—
128C2×D191—42no—
129C2×D191—42no—
130C2×D191—44no—
131C2×D191—84no—
132C4×C191—24no—
133C4×C191—42no—
134C4×C191#12012yes—
135C4×C191C22×D411yescharacteristic
136C4×C191C22×D411yescharacteristic
137Dic191—22no—
138Dic191—22no—
139C2×C191—22no—
140C2×C191C2×#12011yescharacteristic
141C2×C191#57712yes—
142D191—42no—
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#419D19111yescharacteristic, complemented
154#416—1911no—
155#408—1911no—
1562−1+4—3824no—
1572−1+4—3824no—
158C2×#41—3822no—
159C2×#41—3821no—
160C2×#41—1912no—
161C2×#41—1911no—
162C2×#41—1911no—
163C22×Q8—3821no—
164C22×Q8—1912no—
165C22×Q8C2×D19111yescharacteristic
166C22×D4—1911no—
167AGL(1,ℤ/8)—3824no—
168C2×SD16—3824no—
169C2×SD16—1914no—
170C2×SD16—1912no—
171C2×D8—1912no—
172C2×M16C2×D19111yescharacteristic
173C22×C8—1911no—
174#127—3821no—
175#124—3821no—
176#121—1914no—
177#120—3822no—
178#117C2×D19111yescharacteristic
179#103—1912no—
180#103C2×D19114yescomplemented
181#102—1912no—
182C22⋊C8—1914no—
183C24—3821no—
184#41—38216no—
185#41—3828no—
186#41—3824no—
187#41—3824no—
188#41—3822no—
189#41—3822no—
190#41—3822no—
191#41—1914no—
192C2×Q8—21no—
193C2×Q8—22no—
194C2×Q8—3822no—
195C2×Q8—3828no—
196C2×Q8—3824no—
197C2×Q8—3824no—
198C2×Q8—1911no—
199C2×Q8—1911no—
200C2×Q8D76414yescomplemented
201C2×Q8C22×D19112yes—
202C2×D4—3822no—
203C2×D4—3824no—
204C2×D4—3821no—
205C2×D4—3822no—
206C2×D4—3821no—
207C2×D4—3822no—
208C2×D4—1912no—
209C2×D4—1911no—
210C2×D4—1912no—
211C2×D4—1911no—
212C22×C4—21no—
213C22×C4—3822no—
214C22×C4—3822no—
215C22×C4—3821no—
216C22×C4—1912no—
217C22×C4C22×D19111yescharacteristic
218SD16—3824no—
219SD16—3828no—
220D8—3824no—
221M16—22no—
222C2×C8—3822no—
223C2×C8—1912no—
224C2×C8C22×D19112yes—
225C4⋊C4C22×D19112yes—
226C22⋊C4—21no—
227C22⋊C4—3824no—
228C42—21no—
229C23—3821no—
230C23—3822no—
231C23—1912no—
232C23—15282no—
233C23—7642no—
234C23D4×D19111yescharacteristic
235Q8—22no—
236Q8—24no—
237Q8—3828no—
238Q8—3821no—
239Q8—3822no—
240Q8—1912no—
241Q8C2×D76414yes—
242D4—3828no—
243D4—3824no—
244D4—3824no—
245D4—3822no—
246D4—3821no—
247D4—3822no—
248D4—3822no—
249D4—1912no—
250D4—15284no—
251D4—7642no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—3828no—
256C2×C4—3828no—
257C2×C4—3821no—
258C2×C4—3822no—
259C2×C4—3822no—
260C2×C4—3824no—
261C2×C4—1912no—
262C2×C4D4×D19112yes—
263C2×C4D4×D19111yescharacteristic
264C2×C4C2×D76412yes—
265C2×C4C23×D19111yescharacteristic
266C8—22no—
267C8—3822no—
268C22—22no—
269C22—3824no—
270C22—3822no—
271C22—3822no—
272C22—15284no—
273C22—7644no—
274C22—7642no—
275C22—7642no—
276C22#2025929212yes—
277C22C2×D4×D19111yescharacteristic
278C4—24no—
279C4—42no—
280C4—3822no—
281C4—3822no—
282C4#2025929212yes—
283C4C2×D4×D19111yescharacteristic
284C4C22×D76411yescharacteristic
285C2—22no—
286C2—15282no—
287C2—7642no—
288C2—7642no—
289C2C2×#2025929211yescharacteristic
290C2#2713347212yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#419×C191C2
socleC22×C191C22×D4
hypercenter#419D191
hypocenterC191#3010
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C2×#20259370C2semidirect
2. C2×#20259350C2semidirect
3. C22×#15690274C2semidirect
4. #27133232C2semidirect
5. #27132811C2semidirect
6. #27132803C2semidirect
7. #27132657C2semidirect
8. #27132631C2semidirect
9. #419×C191C2semidirect
13. C2×#15690288C22semidirect
18. C2×Q8×D191C22semidirect
30. #20259245C22semidirect
31. #20259242C22semidirect
32. (C2×D191)⋊C8C22semidirect
36. #103×C191C22semidirect
45. #15690287D4semidirect
74. C2×Q8×C191D4semidirect
145. C191#3010semidirect
153. #419D191semidirect
180. #103C2×D191semidirect
200. C2×Q8D764semidirect

Maximal subgroups

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

subgroupindex
C2×#202593702
C2×#202593502
C22×#156902742
#271332322
#271328112
#271328032
#271326572
#271326312
#419×C1912
#3010191