Finite Group Databasebeta

← #37584121

Subgroups — #37584121

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×#20259346C212yesmaximal, complemented
2#27133317C211yesmaximal, characteristic, complemented
3D4×D764C211yesmaximal, characteristic, complemented
4#27133036C211yesmaximal, characteristic, complemented
5#27132910C212yesmaximal, complemented
6#27132868C211yesmaximal, characteristic, complemented
7#27132858C211yesmaximal, characteristic, complemented
8#27132815C212yesmaximal, complemented
9#27132809C212yesmaximal, complemented
10#27132742C211yesmaximal, characteristic, complemented
11C4×D8×C191C211yesmaximal, characteristic, complemented
12C2×#15690286C2212yescomplemented
13C22×D764—21no—
14C2×D4×D191C2212yes—
15#20259346—28no—
16C2×#15690276C2212yescomplemented
17C2×#15690276C2212yescomplemented
18C2×#15690275C2212yes—
19C2×#15690272C2212yescomplemented
20C2×#15690270C2212yescomplemented
21#20259322C2211yescharacteristic, complemented
22#20259319C2211yescharacteristic
23#20259306—22no—
24#20259302C2212yescomplemented
25#20259297C2212yes—
26#20259296—21no—
27#20259296—21no—
28#20259292—22no—
29C4×D764C2211yescharacteristic
30C2×C4⋊Dic191—21no—
31C4⋊(C191⋊C8)C2211yescharacteristic, complemented
32#20259245C2212yescomplemented
33#20259242C2212yescomplemented
34(C2×D191)⋊C8C2212yescomplemented
35C2×D8×C191C2211yescharacteristic
36C4×D4×C191C2211yescharacteristic
37C4×D4×C191C2211yescharacteristic, complemented
38#107×C191C2211yescharacteristic, complemented
39#102×C191C2211yescharacteristic
40#102×C191C2211yescharacteristic, complemented
41C4×C8×C191C2211yescharacteristic, complemented
42C23×D191—22no—
43#15690286—24no—
44#15690286—42no—
45C2×Dic382C2312yes—
46C2×D764—22no—
47C2×D764—21no—
48C2×D764—21no—
49C2×D764—44no—
50C2×D764D411yescharacteristic
51C2×D764C2312yes—
52C2×C22⋊D191—22no—
53C2×C22⋊D191—22no—
54D4×D191—24no—
55D4×D191—42no—
56C22×Dic191—22no—
57C2×C4×D191C2312yes—
58#15690276—24no—
59#15690276—42no—
60#15690275—24no—
61#15690272—24no—
62#15690270—24no—
63C2×C191⋊C8C2312yes—
64C4⋊Dic191—22no—
65C4⋊Dic191—21no—
66C4⋊Dic191—21no—
67C4⋊Dic191D411yescharacteristic
68(C2×D191)⋊C4—22no—
69(C2×D191)⋊C4—22no—
70(C2×D191)⋊C4D412yes—
71C2×D4×C191C2311yescharacteristic
72C2×D4×C191C2311yescharacteristic
73C22×C4×C191—21no—
74C22×C4×C191—21no—
75D8×C191D414yescomplemented
76C2×C8×C191C2311yescharacteristic
77C2×C8×C191C2311yescharacteristic
78C4⋊C4×C191C2311yescharacteristic
79C4⋊C4×C191C2311yescharacteristic
80C22⋊C4×C191—21no—
81C22⋊C4×C191—21no—
82C42×C191C2311yescharacteristic
83C22×D191—22no—
84C22×D191—44no—
85C22×D191—44no—
86C22×D191—82no—
87C22×D191C2×D412yes—
88Dic382—22no—
89Dic382—22no—
90D764—22no—
91D764—22no—
92D764—22no—
93D764—44no—
94D764—42no—
95D764—81no—
96C22⋊D191—44no—
97C22⋊D191—44no—
98C2×Dic191—22no—
99C2×Dic191—44no—
100C2×Dic191C2×D412yes—
101C4×D191—22no—
102C4×D191—22no—
103C191⋊C8—22no—
104C23×C191—21no—
105C23×C191—21no—
106D4×C191—21no—
107D4×C191—21no—
108D4×C191C2×D412yes—
109D4×C191C2×D412yescomplemented
110C2×C4×C191—21no—
111C2×C4×C191—21no—
112C2×C4×C191—42no—
113C2×C4×C191—42no—
114C2×C4×C191C2×D411yescharacteristic
115C2×C4×C191C2×D411yescharacteristic
116C2×C4×C191C2411yescharacteristic
117C8×C191—21no—
118C8×C191C2×D412yescomplemented
119C22×C191—22no—
120C22×C191—22no—
121C22×C191—41no—
122C22×C191—41no—
123C22×C191—41no—
124C22×C191—41no—
125C22×C191C22×D411yescharacteristic
126C2×D191—22no—
127C2×D191—22no—
128C2×D191—22no—
129C2×D191—42no—
130C2×D191—42no—
131C2×D191—42no—
132C2×D191—88no—
133C4×C191—21no—
134C4×C191—41no—
135C4×C191—41no—
136C4×C191AGL(1,ℤ/8)12yescomplemented
137C4×C191C22×D411yescharacteristic
138C4×C191H(5,𝔽2)11yescharacteristic
139Dic191—22no—
140Dic191—42no—
141C2×C191—42no—
142C2×C191—42no—
143C2×C191D4211yescharacteristic
144C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C191#57711yescharacteristic
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×D8D19111yescharacteristic, complemented
160#436—1911no—
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×D8C2×D19111yescharacteristic
170C2×M16—1912no—
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×D4C2×D19111yescharacteristic
181C4×D4C2×D19111yescharacteristic, complemented
182C2×C4⋊C4—3821no—
183#107C2×D19111yescharacteristic, complemented
184C4⋊C8—1911no—
185#103—1912no—
186#102—1912no—
187#102C2×D19111yescharacteristic
188#102C2×D19111yescharacteristic, complemented
189C22⋊C8—1912no—
190C4×C8C2×D19111yescharacteristic, complemented
191C24—3822no—
192#41—3824no—
193#41—7642no—
194C2×Q8—1912no—
195C2×D4—3822no—
196C2×D4—3824no—
197C2×D4—3822no—
198C2×D4—3822no—
199C2×D4—3821no—
200C2×D4—3821no—
201C2×D4—1911no—
202C2×D4—1912no—
203C2×D4—7644no—
204C2×D4—7642no—
205C2×D4C22×D19111yescharacteristic
206C2×D4C22×D19111yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—3822no—
210C22×C4—1912no—
211SD16—3824no—
212SD16—3824no—
213SD16—7642no—
214D8—3824no—
215D8D76414yescomplemented
216M16—3824no—
217C2×C8—1912no—
218C2×C8C22×D19111yescharacteristic
219C2×C8C22×D19111yescharacteristic
220C4⋊C4—3822no—
221C4⋊C4—3821no—
222C4⋊C4—3821no—
223C4⋊C4—1911no—
224C4⋊C4C22×D19111yescharacteristic
225C4⋊C4C22×D19111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—3822no—
229C22⋊C4—3822no—
230C22⋊C4—1912no—
231C42C22×D19111yescharacteristic
232C23—21no—
233C23—21no—
234C23—3822no—
235C23—1912no—
236C23—15282no—
237C23—7644no—
238C23—7644no—
239Q8—3822no—
240Q8—3822no—
241D4—21no—
242D4—21no—
243D4—3822no—
244D4—3822no—
245D4—3822no—
246D4—15281no—
247D4—7644no—
248D4—7644no—
249D4—7644no—
250D4—7642no—
251D4C2×D76412yes—
252D4C2×D76412yescomplemented
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—3822no—
258C2×C4—3822no—
259C2×C4—3822no—
260C2×C4—1912no—
261C2×C4—7644no—
262C2×C4D4×D19111yescharacteristic
263C2×C4D4×D19111yescharacteristic
264C2×C4C23×D19111yescharacteristic
265C8—21no—
266C8—3822no—
267C8C2×D76412yescomplemented
268C22—22no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—3822no—
275C22—3822no—
276C22—3822no—
277C22—15288no—
278C22—7642no—
279C22—7642no—
280C22—7642no—
281C22C2×D4×D19111yescharacteristic
282C4—21no—
283C4—41no—
284C4—41no—
285C4—3822no—
286C4—7642no—
287C4#2025934612yescomplemented
288C4C22×D76411yescharacteristic
289C4#2025936811yescharacteristic
290C2—42no—
291C2—42no—
292C2—15282no—
293C2—7642no—
294C2D4×D76411yescharacteristic
295C2C2×#2025934611yescharacteristic
296C2#2713347011yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC4×D8×C191C2
socleC22×C191C22×D4
hypercenterC4×D8D191
hypocenterC191#3272
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#20259346C2semidirect
2. #27133317C2semidirect
3. D4×D764C2semidirect
4. #27133036C2semidirect
5. #27132910C2semidirect
6. #27132868C2semidirect
7. #27132858C2semidirect
8. #27132815C2semidirect
9. #27132809C2semidirect
10. #27132742C2semidirect
11. C4×D8×C191C2semidirect
12. C2×#15690286C22semidirect
16. C2×#15690276C22semidirect
17. C2×#15690276C22semidirect
19. C2×#15690272C22semidirect
20. C2×#15690270C22semidirect
21. #20259322C22semidirect
24. #20259302C22semidirect
31. C4⋊(C191⋊C8)C22semidirect
32. #20259245C22semidirect
33. #20259242C22semidirect
34. (C2×D191)⋊C8C22semidirect
37. C4×D4×C191C22semidirect
38. #107×C191C22semidirect
40. #102×C191C22semidirect
41. C4×C8×C191C22semidirect
75. D8×C191D4semidirect
109. D4×C191C2×D4semidirect
118. C8×C191C2×D4semidirect
136. C4×C191AGL(1,ℤ/8)semidirect
148. C191#3272semidirect
159. C4×D8D191semidirect
181. C4×D4C2×D191semidirect
183. #107C2×D191semidirect
188. #102C2×D191semidirect
190. C4×C8C2×D191semidirect
215. D8D764semidirect
252. D4C2×D764semidirect
267. C8C2×D764semidirect
287. C4#20259346semidirect

Maximal subgroups

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

subgroupindex
C2×#202593462
#271333172
D4×D7642
#271330362
#271329102
#271328682
#271328582
#271328152
#271328092
#271327422
C4×D8×C1912
#3272191