Finite Group Databasebeta

← #37584136

Subgroups — #37584136

Order 24448.

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

normal subgroups only

This group has 556 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×#20259348C212yesmaximal, complemented
2#27133334C211yesmaximal, characteristic, complemented
3#27133039C211yesmaximal, characteristic, complemented
4#27132914C212yesmaximal, complemented
5#27132860C211yesmaximal, characteristic, complemented
6#27132858C211yesmaximal, characteristic, complemented
7#27132806C212yesmaximal, complemented
8#27132797C212yesmaximal, complemented
9C4×#15690275C211yesmaximal, characteristic, complemented
10#27132741C211yesmaximal, characteristic, complemented
11D42×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2212yescomplemented
13#20259348—28no—
14C2×#15690275C2211yescharacteristic
15C2×#15690275C2212yescomplemented
16C2×#15690272C2212yes—
17C2×#15690272C2212yescomplemented
18C2×#15690272C2212yescomplemented
19C2×#15690271C2212yescomplemented
20#20259319C2211yescharacteristic, complemented
21#20259318—21no—
22#20259296C2212yescomplemented
23#20259295—22no—
24C4×Dic382C2211yescharacteristic, complemented
25#20259284—21no—
26#20259253C2211yescharacteristic, complemented
27C4⋊(C191⋊C8)C2211yescharacteristic, complemented
28#20259245C2211yescharacteristic, complemented
29#20259243C2211yescharacteristic
30#20259243C2212yescomplemented
31#20259242C2212yescomplemented
32C22⋊(C191⋊C8)C2212yescomplemented
33C4×C191⋊C8C2211yescharacteristic, complemented
34C22×D4×C191—21no—
35C22×D4×C191C2212yes—
36#127×C191C2211yescharacteristic
37#121×C191—21no—
38#121×C191C2212yes—
39#120×C191—22no—
40C4×D4×C191C2211yescharacteristic
41C4×D4×C191C2211yescharacteristic
42#15690287—24no—
43#15690287—42no—
44C2×Dic382C2311yescharacteristic
45C2×D764—21no—
46C2×D764—21no—
47C2×D764C2312yes—
48C2×C22⋊D191—22no—
49C2×C22⋊D191—22no—
50C2×C4×D191—22no—
51#15690275—24no—
52#15690275D414yescomplemented
53#15690272—24no—
54#15690272—24no—
55#15690272—42no—
56#15690271—24no—
57C2×C191⋊C8C2312yes—
58C2×C191⋊C8C2311yescharacteristic
59C2×C191⋊C8C2311yescharacteristic
60C4⋊Dic191C2311yescharacteristic
61Dic191⋊C4—21no—
62(C2×D191)⋊C4—22no—
63C4×Dic191—21no—
64C24×C191—22no—
65C2×D4×C191—24no—
66C2×D4×C191—22no—
67C2×D4×C191—22no—
68C2×D4×C191—21no—
69C2×D4×C191—21no—
70C2×D4×C191—42no—
71C2×D4×C191—44no—
72C2×D4×C191D411yescharacteristic
73C2×D4×C191C2312yes—
74C2×D4×C191C2311yescharacteristic
75C22×C4×C191—21no—
76C22×C4×C191C2312yes—
77C4⋊C4×C191D411yescharacteristic
78C4⋊C4×C191C2311yescharacteristic
79C22⋊C4×C191—21no—
80C22⋊C4×C191—22no—
81C22⋊C4×C191D412yes—
82C42×C191C2311yescharacteristic
83C22×D191—22no—
84Dic382—21no—
85Dic382C2×D412yescomplemented
86D764—22no—
87D764—22no—
88D764—42no—
89D764—81no—
90C22⋊D191—44no—
91C22⋊D191—82no—
92C2×Dic191—21no—
93C2×Dic191—21no—
94C4×D191—44no—
95C191⋊C8—22no—
96C191⋊C8—21no—
97C191⋊C8C2×D412yescomplemented
98C23×C191—21no—
99C23×C191—22no—
100C23×C191—44no—
101C23×C191—44no—
102C23×C191—82no—
103C23×C191C2×D412yes—
104D4×C191—22no—
105D4×C191—22no—
106D4×C191—21no—
107D4×C191—22no—
108D4×C191—44no—
109D4×C191—44no—
110D4×C191—42no—
111D4×C191—81no—
112D4×C191C2×D412yes—
113C2×C4×C191—22no—
114C2×C4×C191—22no—
115C2×C4×C191—21no—
116C2×C4×C191—42no—
117C2×C4×C191C2×D411yescharacteristic
118C2×C4×C191C2×D412yes—
119C2×C4×C191C2×D411yescharacteristic
120C2×C4×C191C2411yescharacteristic
121C22×C191—22no—
122C22×C191—22no—
123C22×C191—22no—
124C22×C191—22no—
125C22×C191—41no—
126C22×C191—41no—
127C22×C191—42no—
128C22×C191—42no—
129C22×C191—42no—
130C22×C191—88no—
131C22×C191C22×D411yescharacteristic
132C2×D191—42no—
133C2×D191—42no—
134C2×D191—42no—
135C4×C191—22no—
136C4×C191—21no—
137C4×C191—41no—
138C4×C191AGL(1,ℤ/8)12yescomplemented
139C4×C191C22×D411yescharacteristic
140C4×C191H(5,𝔽2)11yescharacteristic
141Dic191—22no—
142Dic191—41no—
143C2×C191—42no—
144C2×C191—42no—
145C2×C191—82no—
146C2×C191D4211yescharacteristic
147C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
148C2×C191#57611yescharacteristic
149D191—82no—
150C191#327411yescharacteristic, complemented, Sylow
151#3274—1911nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1912no—
153#550—1911no—
154D42D19111yescharacteristic, complemented
155#493—1911no—
156#469—1912no—
157#460—1911no—
158#459—1911no—
159#449—1912no—
160#447—1912no—
161C4×SD16—1911no—
162#436—1911no—
163C2×#41—1912no—
164C22×D4—21no—
165C22×D4C2×D19112yes—
166AGL(1,ℤ/8)—3828no—
167C2×SD16—1912no—
168C2×SD16—1911no—
169C2×D8—1912no—
170C2×D8—1912no—
171C2×D8—1912no—
172C2×M16—1912no—
173#127—3821no—
174#127—1911no—
175#127C2×D19111yescharacteristic
176#121—21no—
177#121—3822no—
178#121—1912no—
179#121C2×D19112yes—
180#120—22no—
181C4×Q8—1911no—
182C4×D4—3821no—
183C4×D4C2×D19111yescharacteristic
184C4×D4C2×D19111yescharacteristic
185#106—1911no—
186C4⋊C8—1911no—
187#103—1911no—
188#102—1911no—
189#102—1912no—
190#102—1912no—
191C22⋊C8—1912no—
192C4×C8—1911no—
193C24—22no—
194#41—3824no—
195#41—7642no—
196C2×Q8—1911no—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—42no—
203C2×D4—44no—
204C2×D4—3822no—
205C2×D4—3822no—
206C2×D4—3821no—
207C2×D4—3821no—
208C2×D4—1912no—
209C2×D4C22⋊D19111yescharacteristic
210C2×D4C22×D19112yes—
211C2×D4C22×D19111yescharacteristic
212C22×C4—21no—
213C22×C4—3822no—
214C22×C4C22×D19112yes—
215SD16—3824no—
216SD16—1914no—
217D8—3824no—
218D8—3824no—
219D8—7642no—
220M16—3824no—
221C2×C8—1911no—
222C2×C8—1912no—
223C2×C8—1911no—
224C4⋊C4—3821no—
225C4⋊C4—1911no—
226C4⋊C4C22⋊D19111yescharacteristic
227C4⋊C4C22×D19111yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—3822no—
231C22⋊C4C22⋊D19112yes—
232C42—3821no—
233C42C22×D19111yescharacteristic
234C23—21no—
235C23—22no—
236C23—44no—
237C23—44no—
238C23—82no—
239C23—3822no—
240C23C2×C22⋊D19112yes—
241Q8—3821no—
242Q8—1912no—
243D4—22no—
244D4—22no—
245D4—21no—
246D4—22no—
247D4—44no—
248D4—44no—
249D4—42no—
250D4—81no—
251D4—3822no—
252D4—3822no—
253D4—15281no—
254D4—15282no—
255D4—7644no—
256D4—7642no—
257D4D4×D19112yes—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—3821no—
263C2×C4—3821no—
264C2×C4—7644no—
265C2×C4C2×C22⋊D19111yescharacteristic
266C2×C4C2×C22⋊D19112yes—
267C2×C4C2×C22⋊D19111yescharacteristic
268C2×C4C23×D19111yescharacteristic
269C8—3822no—
270C8—3821no—
271C8—1912no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—88no—
282C22—7642no—
283C22—7642no—
284C22—7642no—
285C22C22×C22⋊D19111yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—3822no—
290C4—7641no—
291C4#2025934812yescomplemented
292C4C2×D4×D19111yescharacteristic
293C4#2025936711yescharacteristic
294C2—42no—
295C2—42no—
296C2—82no—
297C2—15282no—
298C2D4×C22⋊D19111yescharacteristic
299C2C2×#2025934811yescharacteristic
300C2#2713346611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingD42×C191C2
socleC22×C191C22×D4
hypercenterD42D191
hypocenterC191#3274
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#20259348C2semidirect
2. #27133334C2semidirect
3. #27133039C2semidirect
4. #27132914C2semidirect
5. #27132860C2semidirect
6. #27132858C2semidirect
7. #27132806C2semidirect
8. #27132797C2semidirect
9. C4×#15690275C2semidirect
10. #27132741C2semidirect
11. D42×C191C2semidirect
12. C2×#15690287C22semidirect
15. C2×#15690275C22semidirect
17. C2×#15690272C22semidirect
18. C2×#15690272C22semidirect
19. C2×#15690271C22semidirect
20. #20259319C22semidirect
22. #20259296C22semidirect
24. C4×Dic382C22semidirect
26. #20259253C22semidirect
27. C4⋊(C191⋊C8)C22semidirect
28. #20259245C22semidirect
30. #20259243C22semidirect
31. #20259242C22semidirect
32. C22⋊(C191⋊C8)C22semidirect
33. C4×C191⋊C8C22semidirect
52. #15690275D4semidirect
85. Dic382C2×D4semidirect
97. C191⋊C8C2×D4semidirect
138. C4×C191AGL(1,ℤ/8)semidirect
150. C191#3274semidirect
154. D42D191semidirect
291. C4#20259348semidirect

Maximal subgroups

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

subgroupindex
C2×#202593482
#271333342
#271330392
#271329142
#271328602
#271328582
#271328062
#271327972
C4×#156902752
#271327412
D42×C1912
#3274191