Finite Group Databasebeta

← #37584123

Subgroups — #37584123

Order 24448.

546 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 125 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×#20259349C212yesmaximal, complemented
2#27133316C211yesmaximal, characteristic, complemented
3D4×D764C211yesmaximal, characteristic, complemented
4#27133037C211yesmaximal, characteristic, complemented
5#27132912C212yesmaximal, complemented
6#27132859C211yesmaximal, characteristic, complemented
7#27132812C212yesmaximal, complemented
8#27132811C212yesmaximal, complemented
9C4×#15690272C211yesmaximal, characteristic, complemented
10#27132740C211yesmaximal, characteristic, complemented
11#461×C191C211yesmaximal, characteristic, complemented
12C2×#15690288C2212yescomplemented
13C22×D764—21no—
14C2×D4×D191C2212yes—
15#20259349—28no—
16C2×#15690274C2212yescomplemented
17C2×#15690274C2212yescomplemented
18C2×D1528C2212yescomplemented
19C2×#15690272C2211yescharacteristic
20C2×#15690270C2212yescomplemented
21#20259319C2211yescharacteristic
22#20259307—22no—
23#20259298C2212yescomplemented
24#20259297—21no—
25#20259297C2212yes—
26#20259296—21no—
27#20259292—22no—
28C4×D764C2211yescharacteristic
29C4×D764C2211yescharacteristic, complemented
30C4⋊C4×D191—21no—
31#20259255C2211yescharacteristic, complemented
32#20259244C2212yescomplemented
33#20259243C2211yescharacteristic
34#20259242C2211yescharacteristic, complemented
35(C2×D191)⋊C8C2212yescomplemented
36C4×C191⋊C8C2211yescharacteristic, complemented
37C2×SD16×C191C2212yes—
38C4⋊Q8×C191C2211yescharacteristic, complemented
39C4×D4×C191C2211yescharacteristic
40C4⋊C8×C191C2211yescharacteristic, complemented
41#103×C191C2212yescomplemented
42C23×D191—22no—
43#15690288—24no—
44#15690288—42no—
45C2×D764—22no—
46C2×D764—22no—
47C2×D764—21no—
48C2×D764—21no—
49C2×D764—44no—
50C2×D764D411yescharacteristic
51C2×D764C2312yes—
52C2×D764C2311yescharacteristic
53C2×C22⋊D191—22no—
54D4×D191—24no—
55D4×D191—42no—
56C2×C4×D191—22no—
57C2×C4×D191—21no—
58C2×C4×D191C2312yes—
59#15690274—24no—
60#15690274—42no—
61D1528—24no—
62#15690272D414yescomplemented
63#15690270—24no—
64C2×C191⋊C8C2311yescharacteristic
65C2×C191⋊C8C2311yescharacteristic
66C4⋊Dic191D411yescharacteristic
67C4⋊Dic191C2311yescharacteristic
68Dic191⋊C4—22no—
69(C2×D191)⋊C4—22no—
70(C2×D191)⋊C4—22no—
71(C2×D191)⋊C4—21no—
72(C2×D191)⋊C4D412yes—
73C2×Q8×C191C2312yes—
74C2×D4×C191C2311yescharacteristic
75C22×C4×C191—21no—
76SD16×C191—24no—
77C2×C8×C191C2312yes—
78C4⋊C4×C191—21no—
79C4⋊C4×C191—21no—
80C4⋊C4×C191C2311yescharacteristic
81C22⋊C4×C191—21no—
82C42×C191C2311yescharacteristic
83C22×D191—21no—
84C22×D191—22no—
85C22×D191—44no—
86C22×D191—44no—
87C22×D191—82no—
88C22×D191C2×D412yes—
89D764—22no—
90D764—22no—
91D764—21no—
92D764—22no—
93D764—44no—
94D764—44no—
95D764—42no—
96D764—81no—
97D764C2×D412yescomplemented
98C22⋊D191—44no—
99C2×Dic191—21no—
100C2×Dic191C2×D412yes—
101C4×D191—22no—
102C4×D191—22no—
103C4×D191—44no—
104C4×D191—42no—
105C191⋊C8—21no—
106C191⋊C8C2×D412yescomplemented
107C23×C191—21no—
108Q8×C191—22no—
109Q8×C191—22no—
110D4×C191—21no—
111D4×C191C2×D412yes—
112C2×C4×C191—21no—
113C2×C4×C191—22no—
114C2×C4×C191—42no—
115C2×C4×C191C2×D411yescharacteristic
116C2×C4×C191C2×D411yescharacteristic
117C2×C4×C191C2411yescharacteristic
118C8×C191—22no—
119C22×C191—22no—
120C22×C191—41no—
121C22×C191—41no—
122C22×C191C22×D411yescharacteristic
123C2×D191—22no—
124C2×D191—22no—
125C2×D191—22no—
126C2×D191—22no—
127C2×D191—41no—
128C2×D191—41no—
129C2×D191—42no—
130C2×D191—42no—
131C2×D191—42no—
132C2×D191—88no—
133C4×C191—21no—
134C4×C191—42no—
135C4×C191—41no—
136C4×C191AGL(1,ℤ/8)12yescomplemented
137C4×C191C22×D411yescharacteristic
138C4×C191H(5,𝔽2)11yescharacteristic
139Dic191—22no—
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#461D19111yescharacteristic, complemented
156#459—1911no—
157#450—1912no—
158#449—1912no—
159C4×D8—1911no—
160#436—1911no—
161C2×#41—1912no—
162C22×D4—3821no—
163C22×D4—1912no—
164AGL(1,ℤ/8)—3828no—
165C2×SD16—1912no—
166C2×SD16—1912no—
167C2×SD16C2×D19112yes—
168C2×D8—1911no—
169C2×D8—1912no—
170C2×M16—1912no—
171C4⋊Q8C2×D19111yescharacteristic, complemented
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#103C2×D19112yescomplemented
186#102—1911no—
187#102—1912no—
188#102—1911no—
189C22⋊C8—1912no—
190C4×C8—1911no—
191C24—3822no—
192#41—3824no—
193#41—7642no—
194C2×Q8C22×D19112yes—
195C2×D4—3822no—
196C2×D4—3822no—
197C2×D4—3824no—
198C2×D4—3822no—
199C2×D4—3821no—
200C2×D4—3821no—
201C2×D4—1911no—
202C2×D4—1912no—
203C2×D4—1911no—
204C2×D4—7644no—
205C2×D4—7642no—
206C2×D4C22×D19111yescharacteristic
207C22×C4—21no—
208C22×C4—3822no—
209C22×C4—3821no—
210C22×C4—1912no—
211SD16—24no—
212SD16—3824no—
213SD16—7642no—
214D8—3824no—
215D8—1914no—
216M16—3824no—
217C2×C8—1911no—
218C2×C8—1911no—
219C2×C8C22×D19112yes—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—3822no—
223C4⋊C4—1911no—
224C4⋊C4—1911no—
225C4⋊C4C22×D19111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—3822no—
228C22⋊C4—3822no—
229C22⋊C4—3821no—
230C22⋊C4—1912no—
231C42C22×D19111yescharacteristic
232C23—21no—
233C23—3821no—
234C23—3822no—
235C23—1912no—
236C23—15282no—
237C23—7644no—
238C23—7644no—
239Q8—22no—
240Q8—22no—
241D4—21no—
242D4—3822no—
243D4—3822no—
244D4—3821no—
245D4—3822no—
246D4—1912no—
247D4—15281no—
248D4—7644no—
249D4—7644no—
250D4—7644no—
251D4—7642no—
252D4D4×D19112yes—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—3822no—
257C2×C4—3822no—
258C2×C4—3821no—
259C2×C4—1912no—
260C2×C4—7644no—
261C2×C4—7642no—
262C2×C4D4×D19111yescharacteristic
263C2×C4D4×D19111yescharacteristic
264C2×C4C23×D19111yescharacteristic
265C8—22no—
266C8—3821no—
267C8—1912no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—3822no—
272C22—3822no—
273C22—3822no—
274C22—3822no—
275C22—15288no—
276C22—7642no—
277C22—7642no—
278C22—7642no—
279C22—7641no—
280C22—7641no—
281C22C2×D4×D19111yescharacteristic
282C4—21no—
283C4—41no—
284C4—42no—
285C4—3822no—
286C4—7641no—
287C4#2025934912yescomplemented
288C4C2×D4×D19111yescharacteristic
289C4#2025936811yescharacteristic
290C2—42no—
291C2—15282no—
292C2—7642no—
293C2—7642no—
294C2#2713329711yescharacteristic
295C2C2×#2025934911yescharacteristic
296C2#2713347211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#461×C191C2
socleC22×C191C22×D4
hypercenter#461D191
hypocenterC191#3272
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#20259349C2semidirect
2. #27133316C2semidirect
3. D4×D764C2semidirect
4. #27133037C2semidirect
5. #27132912C2semidirect
6. #27132859C2semidirect
7. #27132812C2semidirect
8. #27132811C2semidirect
9. C4×#15690272C2semidirect
10. #27132740C2semidirect
11. #461×C191C2semidirect
12. C2×#15690288C22semidirect
16. C2×#15690274C22semidirect
17. C2×#15690274C22semidirect
18. C2×D1528C22semidirect
20. C2×#15690270C22semidirect
23. #20259298C22semidirect
29. C4×D764C22semidirect
31. #20259255C22semidirect
32. #20259244C22semidirect
34. #20259242C22semidirect
35. (C2×D191)⋊C8C22semidirect
36. C4×C191⋊C8C22semidirect
38. C4⋊Q8×C191C22semidirect
40. C4⋊C8×C191C22semidirect
41. #103×C191C22semidirect
62. #15690272D4semidirect
97. D764C2×D4semidirect
106. C191⋊C8C2×D4semidirect
136. C4×C191AGL(1,ℤ/8)semidirect
148. C191#3272semidirect
155. #461D191semidirect
171. C4⋊Q8C2×D191semidirect
184. C4⋊C8C2×D191semidirect
185. #103C2×D191semidirect
287. C4#20259349semidirect

Maximal subgroups

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

subgroupindex
C2×#202593492
#271333162
D4×D7642
#271330372
#271329122
#271328592
#271328122
#271328112
C4×#156902722
#271327402
#461×C1912
#3272191