Finite Group Databasebeta

← #37584122

Subgroups — #37584122

Order 24448.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#20259350C212yesmaximal, complemented
2#27133318C211yesmaximal, characteristic, complemented
3D4×D764C211yesmaximal, characteristic, complemented
4#27133036C211yesmaximal, characteristic, complemented
5#27132909C212yesmaximal, complemented
6#27132869C211yesmaximal, characteristic, complemented
7#27132859C211yesmaximal, characteristic, complemented
8#27132816C212yesmaximal, complemented
9#27132810C212yesmaximal, complemented
10C4×D1528C211yesmaximal, characteristic, complemented
11#436×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2212yescomplemented
13C22×D764C2212yes—
14C2×D4×D191—21no—
15#20259350—28no—
16C2×#15690276C2212yes—
17C2×#15690276C2212yescomplemented
18C2×#15690276C2212yescomplemented
19C2×D1528C2211yescharacteristic
20C2×D1528C2212yescomplemented
21#20259322C2211yescharacteristic, complemented
22#20259319C2211yescharacteristic
23#20259304—22no—
24#20259301C2212yescomplemented
25#20259297—21no—
26#20259296C2212yes—
27#20259293—21no—
28#20259292—22no—
29C4×D764C2211yescharacteristic
30C4×D764C2211yescharacteristic, complemented
31C4⋊C4×D191—21no—
32#20259256C2211yescharacteristic, complemented
33#20259247C2212yescomplemented
34#20259244C2211yescharacteristic
35#20259244C2211yescharacteristic, complemented
36#20259244C2212yescomplemented
37C2×M16×C191C2212yescomplemented
38C4×D4×C191C2211yescharacteristic
39C4⋊C8×C191C2211yescharacteristic, complemented
40C22⋊C8×C191C2212yescomplemented
41C4×C8×C191C2211yescharacteristic, complemented
42C23×D191—22no—
43#15690287—24no—
44#15690287—42no—
45C2×Dic382C2312yes—
46C2×D764—24no—
47C2×D764—21no—
48C2×D764—22no—
49C2×D764—21no—
50C2×D764—42no—
51C2×D764C2312yes—
52C2×D764C2311yescharacteristic
53C2×D764C2311yescharacteristic
54C2×C22⋊D191—22no—
55C2×C22⋊D191—22no—
56D4×D191—44no—
57C2×C4×D191—21no—
58C2×C4×D191—22no—
59C2×C4×D191—21no—
60#15690276—24no—
61#15690276—24no—
62#15690276—42no—
63D1528—24no—
64D1528D414yescomplemented
65C4⋊Dic191—21no—
66C4⋊Dic191—21no—
67C4⋊Dic191C2311yescharacteristic
68C4⋊Dic191C2311yescharacteristic
69Dic191⋊C4—22no—
70C22⋊Dic191—22no—
71(C2×D191)⋊C4—21no—
72(C2×D191)⋊C4—21no—
73(C2×D191)⋊C4—22no—
74C2×D4×C191D411yescharacteristic
75C22×C4×C191C2312yes—
76M16×C191—24no—
77C2×C8×C191C2312yes—
78C2×C8×C191C2311yescharacteristic
79C2×C8×C191C2311yescharacteristic
80C4⋊C4×C191D411yescharacteristic
81C22⋊C4×C191D412yes—
82C42×C191C2311yescharacteristic
83C22×D191—21no—
84C22×D191—21no—
85C22×D191—22no—
86C22×D191—44no—
87C22×D191—44no—
88C22×D191—82no—
89Dic382—22no—
90Dic382—22no—
91D764—22no—
92D764—21no—
93D764—21no—
94D764—22no—
95D764—44no—
96D764—42no—
97D764—81no—
98D764C2×D412yes—
99D764C2×D412yescomplemented
100C22⋊D191—44no—
101C22⋊D191—44no—
102C2×Dic191—21no—
103C2×Dic191—21no—
104C2×Dic191—22no—
105C4×D191—44no—
106C4×D191—42no—
107C4×D191—42no—
108C23×C191C2×D412yes—
109D4×C191—22no—
110C2×C4×C191—22no—
111C2×C4×C191—22no—
112C2×C4×C191C2×D411yescharacteristic
113C2×C4×C191C2×D412yes—
114C2×C4×C191C2×D411yescharacteristic
115C2×C4×C191C2411yescharacteristic
116C8×C191—22no—
117C8×C191—21no—
118C8×C191C2×D412yescomplemented
119C22×C191—22no—
120C22×C191—22no—
121C22×C191—22no—
122C22×C191C22×D411yescharacteristic
123C2×D191—22no—
124C2×D191—22no—
125C2×D191—41no—
126C2×D191—41no—
127C2×D191—41no—
128C2×D191—41no—
129C2×D191—42no—
130C2×D191—42no—
131C2×D191—42no—
132C2×D191—88no—
133C4×C191—22no—
134C4×C191—21no—
135C4×C191AGL(1,ℤ/8)12yescomplemented
136C4×C191C22×D411yescharacteristic
137C4×C191H(5,𝔽2)11yescharacteristic
138Dic191—42no—
139Dic191—41no—
140Dic191—41no—
141C2×C191—42no—
142C2×C191D4211yescharacteristic
143C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C191#57711yescharacteristic
145D191—42no—
146D191—42no—
147D191—82no—
148C191#327211yescharacteristic, complemented, Sylow
149#3272—1911nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1912no—
151#547—1911no—
152D42—1911no—
153#492—1911no—
154#468—1912no—
155#461—1911no—
156#459—1911no—
157#450—1912no—
158#449—1912no—
159C4×D8—1911no—
160#436D19111yescharacteristic, complemented
161C2×#41—1912no—
162C22×D4—3821no—
163C22×D4—1912no—
164AGL(1,ℤ/8)—3828no—
165C2×SD16—1912no—
166C2×SD16—1912no—
167C2×SD16—1912no—
168C2×D8—1912no—
169C2×D8—1911no—
170C2×M16C2×D19112yescomplemented
171C4⋊Q8—1911no—
172#127—1911no—
173#123—3822no—
174C22⋊Q8—1912no—
175#121—3821no—
176#121—3821no—
177#121—1912no—
178#120—3822no—
179C4×D4—1911no—
180C4×D4—1911no—
181C4×D4C2×D19111yescharacteristic
182C2×C4⋊C4—3821no—
183#107—1911no—
184C4⋊C8C2×D19111yescharacteristic, complemented
185#103—1912no—
186#102—1911no—
187#102—1911no—
188#102—1912no—
189C22⋊C8C2×D19112yescomplemented
190C4×C8C2×D19111yescharacteristic, complemented
191C24—3822no—
192#41—3824no—
193#41—7642no—
194C2×Q8—1912no—
195C2×D4—3824no—
196C2×D4—3822no—
197C2×D4—3822no—
198C2×D4—3821no—
199C2×D4—3822no—
200C2×D4—3821no—
201C2×D4—1912no—
202C2×D4—1911no—
203C2×D4—1911no—
204C2×D4—7642no—
205C2×D4—7644no—
206C2×D4D76411yescharacteristic
207C22×C4—3821no—
208C22×C4—3822no—
209C22×C4—3821no—
210C22×C4C22×D19112yes—
211SD16—3824no—
212SD16—3824no—
213SD16—7642no—
214D8—3824no—
215D8—1914no—
216M16—24no—
217C2×C8C22×D19111yescharacteristic
218C2×C8C22×D19112yes—
219C2×C8C22×D19111yescharacteristic
220C4⋊C4—3822no—
221C4⋊C4—3821no—
222C4⋊C4—3821no—
223C4⋊C4—1911no—
224C4⋊C4—1911no—
225C4⋊C4D76411yescharacteristic
226C22⋊C4—3822no—
227C22⋊C4—3821no—
228C22⋊C4—3821no—
229C22⋊C4—3822no—
230C22⋊C4D76412yes—
231C42C22×D19111yescharacteristic
232C23—3821no—
233C23—3821no—
234C23—3822no—
235C23—15282no—
236C23—7644no—
237C23—7644no—
238C23C2×D76412yes—
239Q8—3822no—
240Q8—3822no—
241D4—22no—
242D4—3822no—
243D4—3821no—
244D4—3821no—
245D4—3822no—
246D4—1912no—
247D4—1912no—
248D4—15281no—
249D4—7644no—
250D4—7644no—
251D4—7644no—
252D4—7642no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—3821no—
256C2×C4—3821no—
257C2×C4—3822no—
258C2×C4—7644no—
259C2×C4—7642no—
260C2×C4—7642no—
261C2×C4C2×D76411yescharacteristic
262C2×C4C2×D76412yes—
263C2×C4C2×D76411yescharacteristic
264C2×C4C23×D19111yescharacteristic
265C8—22no—
266C8—21no—
267C8D4×D19112yescomplemented
268C22—22no—
269C22—22no—
270C22—22no—
271C22—3822no—
272C22—3822no—
273C22—15288no—
274C22—7642no—
275C22—7642no—
276C22—7642no—
277C22—7641no—
278C22—7641no—
279C22—7641no—
280C22—7641no—
281C22C22×D76411yescharacteristic
282C4—21no—
283C4—22no—
284C4—7641no—
285C4—7641no—
286C4—7642no—
287C4#2025935012yescomplemented
288C4C2×D4×D19111yescharacteristic
289C4#2025936811yescharacteristic
290C2—42no—
291C2—15282no—
292C2—7642no—
293C2—7642no—
294C2D4×D76411yescharacteristic
295C2C2×#2025935011yescharacteristic
296C2#2713347511yescharacteristic
subgroupG/H
centerC22C22×D764
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#436×C191C2
socleC22×C191C22×D4
hypercenter#436D191
hypocenterC191#3272
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#20259350C2semidirect
2. #27133318C2semidirect
3. D4×D764C2semidirect
4. #27133036C2semidirect
5. #27132909C2semidirect
6. #27132869C2semidirect
7. #27132859C2semidirect
8. #27132816C2semidirect
9. #27132810C2semidirect
10. C4×D1528C2semidirect
11. #436×C191C2semidirect
12. C2×#15690287C22semidirect
17. C2×#15690276C22semidirect
18. C2×#15690276C22semidirect
20. C2×D1528C22semidirect
21. #20259322C22semidirect
24. #20259301C22semidirect
30. C4×D764C22semidirect
32. #20259256C22semidirect
33. #20259247C22semidirect
35. #20259244C22semidirect
36. #20259244C22semidirect
37. C2×M16×C191C22semidirect
39. C4⋊C8×C191C22semidirect
40. C22⋊C8×C191C22semidirect
41. C4×C8×C191C22semidirect
64. D1528D4semidirect
99. D764C2×D4semidirect
118. C8×C191C2×D4semidirect
135. C4×C191AGL(1,ℤ/8)semidirect
148. C191#3272semidirect
160. #436D191semidirect
170. C2×M16C2×D191semidirect
184. C4⋊C8C2×D191semidirect
189. C22⋊C8C2×D191semidirect
190. C4×C8C2×D191semidirect
267. C8D4×D191semidirect
287. C4#20259350semidirect

Maximal subgroups

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

subgroupindex
C2×#202593502
#271333182
D4×D7642
#271330362
#271329092
#271328692
#271328592
#271328162
#271328102
C4×D15282
#436×C1912
#3272191