Finite Group Databasebeta

← #37581789

Subgroups — #37581789

Order 24448.

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

normal subgroups only

This group has 784 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×#20259367C211yesmaximal, characteristic, complemented
2C2×#20259350C211yesmaximal, characteristic, complemented
3C22×#15690272C211yesmaximal, characteristic, complemented
4#27133232C211yesmaximal, characteristic, complemented
5#27132809C214yesmaximal, complemented
6#27132798C214yesmaximal, complemented
7#27132657C211yesmaximal, characteristic, complemented
8#27132631C211yesmaximal, characteristic, complemented
9#418×C191C211yesmaximal, characteristic, complemented
10#20259367—24no—
11#20259367—24no—
12C2×#15690287C2211yescharacteristic
13C2×#15690287C2211yescharacteristic
14C2×#15690286—21no—
15C2×#15690286C2212yescomplemented
16C22×D764C2211yescharacteristic, complemented
17C22×C22⋊D191—22no—
18C2×D4×D191—21no—
19C2×D4×D191C2212yescomplemented
20#20259350—24no—
21C2×#15690276C2212yescomplemented
22C2×D1528C2212yescomplemented
23C2×#15690272—24no—
24C2×#15690272C2214yescomplemented
25C22×C191⋊C8C2211yescharacteristic, complemented
26#20259319—21no—
27#20259310—21no—
28#20259297C2214yescomplemented
29#20259292—22no—
30#20259245C2212yescomplemented
31#20259242C2212yescomplemented
32(C2×D191)⋊C8C2214yescomplemented
33C22×D4×C191C2211yescharacteristic
34C2×M16×C191C2211yescharacteristic, complemented
35#117×C191C2211yescharacteristic, complemented
36#102×C191C2214yescomplemented
37C23×D191—22no—
38C23×D191—21no—
39#15690287—22no—
40#15690287—22no—
41#15690287—22no—
42#15690287D414yescomplemented
43#15690286—28no—
44#15690286—24no—
45#15690286—24no—
46C2×Dic382D411yescharacteristic
47C2×Dic382C2311yescharacteristic
48C2×D764—24no—
49C2×D764—22no—
50C2×D764—22no—
51C2×D764—21no—
52C2×D764D411yescharacteristic
53C2×D764C2312yes—
54C2×D764C2311yescharacteristic
55C2×C22⋊D191—216no—
56C2×C22⋊D191—22no—
57C2×C22⋊D191—22no—
58C2×C22⋊D191—21no—
59C2×C22⋊D191D412yes—
60D4×D191—28no—
61D4×D191—24no—
62D4×D191—24no—
63C22×Dic191—22no—
64C2×C4×D191—21no—
65C2×C4×D191C2312yes—
66#15690276—24no—
67D1528—24no—
68#15690272—28no—
69C2×C191⋊C8—22no—
70C2×C191⋊C8C2312yes—
71(C2×D191)⋊C4—24no—
72C24×C191—21no—
73C2×D4×C191—21no—
74C2×D4×C191—22no—
75C2×D4×C191D414yescomplemented
76C2×D4×C191C2312yes—
77C22×C4×C191C2311yescharacteristic
78M16×C191—22no—
79C2×C8×C191C2312yes—
80C4⋊C4×C191C2312yes—
81C22⋊C4×C191—21no—
82C42×C191—21no—
83C22×D191—21no—
84C22×D191—28no—
85C22×D191—22no—
86C22×D191—42no—
87C22×D191—84no—
88C22×D191—82no—
89C22×D191C2×D412yes—
90Dic382—22no—
91Dic382—21no—
92Dic382C2×D412yes—
93D764—22no—
94D764—21no—
95D764—22no—
96D764—22no—
97D764—42no—
98D764—84no—
99D764C2×D412yes—
100C22⋊D191—28no—
101C22⋊D191—28no—
102C22⋊D191—24no—
103C22⋊D191—24no—
104C2×Dic191—28no—
105C2×Dic191—21no—
106C2×Dic191C2×D412yes—
107C4×D191—24no—
108C4×D191—22no—
109C4×D191—22no—
110C191⋊C8—22no—
111C23×C191—22no—
112C23×C191—24no—
113C23×C191—82no—
114C23×C191C2×D411yescharacteristic
115D4×C191—24no—
116D4×C191—22no—
117D4×C191C2×D414yes—
118C2×C4×C191—22no—
119C2×C4×C191C2×D411yescharacteristic
120C2×C4×C191C2×D412yes—
121C2×C4×C191C2×D412yes—
122C2×C4×C191C2411yescharacteristic
123C8×C191—22no—
124C22×C191—22no—
125C22×C191—24no—
126C22×C191—44no—
127C22×C191—84no—
128C22×C191#12012yes—
129C22×C191C22×D411yescharacteristic
130C2×D191—22no—
131C2×D191—22no—
132C2×D191—24no—
133C2×D191—42no—
134C2×D191—42no—
135C2×D191—44no—
136C2×D191—88no—
137C2×D191—84no—
138C4×C191—42no—
139C4×C191#12012yes—
140C4×C191C22×D411yescharacteristic
141C4×C191C22×D411yescharacteristic
142Dic191—22no—
143Dic191—22no—
144C2×C191—22no—
145C2×C191—44no—
146C2×C191C2×#12011yescharacteristic
147C2×C191#57612yes—
148D191—42no—
149D191—42no—
150D191—82no—
151C191#300811yescharacteristic, complemented, Sylow
152#3008—1911nomaximal, Sylow
153C2×H(5,𝔽2)—1911no—
154C2×AGL(1,ℤ/8)—1911no—
155C22×D8—1911no—
156#535—1911no—
157#449—1914no—
158#447—1914no—
159#418D19111yescharacteristic, complemented
160#416—1911no—
161#408—1911no—
162H(5,𝔽2)—3824no—
163H(5,𝔽2)—3824no—
164C2×#41—3821no—
165C2×#41—1912no—
166C2×#41—1911no—
167C2×#41—1911no—
168C22×D4—3821no—
169C22×D4—3822no—
170C22×D4—1912no—
171C22×D4—1911no—
172C22×D4C2×D19111yescharacteristic
173AGL(1,ℤ/8)—3824no—
174C2×SD16—1912no—
175C2×D8—3824no—
176C2×D8—1914no—
177C2×D8—1912no—
178C2×M16C2×D19111yescharacteristic, complemented
179C22×C8—1911no—
180#127—3821no—
181#124—3821no—
182#121—1914no—
183#120—3822no—
184#117C2×D19111yescharacteristic, complemented
185#103—1912no—
186#102—1912no—
187#102C2×D19114yescomplemented
188C22⋊C8—1914no—
189C24—21no—
190C24—3822no—
191C24—3821no—
192#41—3828no—
193#41—3824no—
194#41—3824no—
195#41—3822no—
196#41—3822no—
197#41—3822no—
198#41—1914no—
199C2×Q8—1911no—
200C2×Q8—1911no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—3824no—
204C2×D4—3828no—
205C2×D4—3824no—
206C2×D4—38216no—
207C2×D4—3822no—
208C2×D4—3821no—
209C2×D4—3824no—
210C2×D4—3822no—
211C2×D4—3822no—
212C2×D4—3821no—
213C2×D4—3822no—
214C2×D4—1912no—
215C2×D4—1911no—
216C2×D4—1912no—
217C2×D4—1911no—
218C2×D4D76414yescomplemented
219C2×D4C22×D19112yes—
220C22×C4—3822no—
221C22×C4—3821no—
222C22×C4—1912no—
223C22×C4C22×D19111yescharacteristic
224SD16—3824no—
225D8—3824no—
226D8—3828no—
227M16—22no—
228C2×C8—3822no—
229C2×C8—1912no—
230C2×C8C22×D19112yes—
231C4⋊C4C22×D19112yes—
232C22⋊C4—21no—
233C22⋊C4—3824no—
234C42—21no—
235C23—22no—
236C23—24no—
237C23—82no—
238C23—3828no—
239C23—3821no—
240C23—3822no—
241C23—1912no—
242C23—15282no—
243C23—15284no—
244C23—7642no—
245C23D4×D19111yescharacteristic
246Q8—3821no—
247Q8—3822no—
248Q8—1912no—
249D4—22no—
250D4—24no—
251D4—3828no—
252D4—3828no—
253D4—3824no—
254D4—3824no—
255D4—3822no—
256D4—3821no—
257D4—3822no—
258D4—3822no—
259D4—1912no—
260D4—15284no—
261D4—7642no—
262D4C2×D76414yes—
263C2×C4—22no—
264C2×C4—3828no—
265C2×C4—3821no—
266C2×C4—3822no—
267C2×C4—3822no—
268C2×C4—3824no—
269C2×C4—1912no—
270C2×C4D4×D19112yes—
271C2×C4D4×D19111yescharacteristic
272C2×C4C2×D76412yes—
273C2×C4C23×D19111yescharacteristic
274C8—22no—
275C8—3822no—
276C22—22no—
277C22—24no—
278C22—44no—
279C22—84no—
280C22—3824no—
281C22—3822no—
282C22—3822no—
283C22—15284no—
284C22—15288no—
285C22—7644no—
286C22—7642no—
287C22—7642no—
288C22#2025929212yes—
289C22C2×D4×D19111yescharacteristic
290C4—42no—
291C4—3822no—
292C4—3822no—
293C4#2025929212yes—
294C4C2×D4×D19111yescharacteristic
295C4C22×D76411yescharacteristic
296C2—22no—
297C2—44no—
298C2—15282no—
299C2—7642no—
300C2—7642no—
301C2C2×#2025929211yescharacteristic
302C2#2713346712yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#418×C191C2
socleC22×C191C22×D4
hypercenter#418D191
hypocenterC191#3008
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#20259367C2semidirect
2. C2×#20259350C2semidirect
3. C22×#15690272C2semidirect
4. #27133232C2semidirect
5. #27132809C2semidirect
6. #27132798C2semidirect
7. #27132657C2semidirect
8. #27132631C2semidirect
9. #418×C191C2semidirect
15. C2×#15690286C22semidirect
16. C22×D764C22semidirect
19. C2×D4×D191C22semidirect
21. C2×#15690276C22semidirect
22. C2×D1528C22semidirect
24. C2×#15690272C22semidirect
25. C22×C191⋊C8C22semidirect
28. #20259297C22semidirect
30. #20259245C22semidirect
31. #20259242C22semidirect
32. (C2×D191)⋊C8C22semidirect
34. C2×M16×C191C22semidirect
35. #117×C191C22semidirect
36. #102×C191C22semidirect
42. #15690287D4semidirect
75. C2×D4×C191D4semidirect
151. C191#3008semidirect
159. #418D191semidirect
178. C2×M16C2×D191semidirect
184. #117C2×D191semidirect
187. #102C2×D191semidirect
218. C2×D4D764semidirect

Maximal subgroups

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

subgroupindex
C2×#202593672
C2×#202593502
C22×#156902722
#271332322
#271328092
#271327982
#271326572
#271326312
#418×C1912
#3008191