Finite Group Databasebeta

← #37583339

Subgroups — #37583339

Order 24448.

466 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#27133176C211yesmaximal, characteristic, complemented
2C2×#20259297C211yesmaximal, characteristic, complemented
3#27132989C212yesmaximal, complemented
4#27132976C211yesmaximal, characteristic, complemented
5#27132917C212yesmaximal, complemented
6#27132912C212yesmaximal, complemented
7#27132813C211yesmaximal, characteristic, complemented
8#27132811C212yesmaximal, complemented
9#27132799C211yesmaximal, characteristic, complemented
10#27132625C211yesmaximal, characteristic, complemented
11#483×C191C211yesmaximal, characteristic, complemented
12C2×#15690288C2211yescharacteristic, complemented
13C22×D764—21no—
14C22×D764C2211yescharacteristic
15C22×C4×D191C2211yescharacteristic
16C2×#15690274C2212yescomplemented
17C2×D1528C2212yescomplemented
18C2×#15690270C2212yescomplemented
19#20259307—21no—
20#20259298C2211yescharacteristic, complemented
21#20259297—21no—
22#20259297—21no—
23#20259297C2212yes—
24#20259297C2212yes—
25#20259297C2211yescharacteristic, complemented
26#20259296C2211yescharacteristic
27#20259292—21no—
28#20259284—21no—
29(C22⋊D191)⋊C4—21no—
30#20259280C2211yescharacteristic, complemented
31C2×(C2×D191)⋊C4—21no—
32#20259255C2212yescomplemented
33#20259244C2212yescomplemented
34#20259242C2212yescomplemented
35#20259242C2212yescomplemented
36C22⋊(C191⋊C8)C2211yescharacteristic
37(C2×D191)⋊C8C2212yescomplemented
38C22⋊Q8×C191C2211yescharacteristic
39C2×C4⋊C4×C191C2211yescharacteristic
40#106×C191C2212yescomplemented
41#103×C191C2212yescomplemented
42C22⋊C8×C191C2211yescharacteristic
43C23×D191—21no—
44C23×D191D411yescharacteristic
45#15690288—42no—
46C2×D764—22no—
47C2×D764—22no—
48C2×D764—21no—
49C2×D764—21no—
50C2×D764—21no—
51C2×D764—42no—
52C2×D764—44no—
53C2×D764C2311yescharacteristic
54C2×D764C2312yes—
55C2×C22⋊D191—21no—
56C22×Dic191D411yescharacteristic
57C2×C4×D191—21no—
58C2×C4×D191—42no—
59C2×C4×D191D411yescharacteristic
60C2×C4×D191D411yescharacteristic
61C2×C4×D191C2311yescharacteristic
62C2×C4×D191C2311yescharacteristic
63#15690274—42no—
64D1528—42no—
65#15690270—42no—
66C2×C191⋊C8C2312yes—
67C4⋊Dic191C2311yescharacteristic
68Dic191⋊C4—21no—
69(C2×D191)⋊C4—21no—
70(C2×D191)⋊C4—21no—
71(C2×D191)⋊C4—22no—
72(C2×D191)⋊C4—22no—
73(C2×D191)⋊C4—21no—
74(C2×D191)⋊C4—21no—
75C4×Dic191—21no—
76C2×Q8×C191C2311yescharacteristic
77C22×C4×C191—21no—
78C22×C4×C191C2311yescharacteristic
79C2×C8×C191C2312yes—
80C4⋊C4×C191—21no—
81C4⋊C4×C191—21no—
82C4⋊C4×C191C2312yes—
83C4⋊C4×C191C2311yescharacteristic
84C22⋊C4×C191—21no—
85C22×D191—22no—
86C22×D191—21no—
87C22×D191—22no—
88C22×D191—41no—
89C22×D191—41no—
90C22×D191—42no—
91C22×D191—42no—
92C22×D191—42no—
93C22×D191C2×D411yescharacteristic
94C22×D191C2×D411yescharacteristic
95D764—22no—
96D764—22no—
97D764—22no—
98D764—42no—
99D764—42no—
100D764—44no—
101D764—44no—
102D764—42no—
103C22⋊D191—42no—
104C2×Dic191—21no—
105C2×Dic191—41no—
106C2×Dic191C2×D411yescharacteristic
107C2×Dic191C2×D411yescharacteristic
108C4×D191—22no—
109C4×D191—22no—
110C4×D191—41no—
111C4×D191—41no—
112C4×D191—42no—
113C191⋊C8—22no—
114C23×C191C2×D411yescharacteristic
115Q8×C191—22no—
116C2×C4×C191—22no—
117C2×C4×C191—21no—
118C2×C4×C191—21no—
119C2×C4×C191—22no—
120C2×C4×C191—42no—
121C2×C4×C191C2×D411yescharacteristic
122C2×C4×C191C2411yescharacteristic
123C8×C191—22no—
124C22×C191—21no—
125C22×C191—21no—
126C22×C191AGL(1,ℤ/8)12yes—
127C22×C191C22×D411yescharacteristic
128C2×D191—21no—
129C2×D191—21no—
130C2×D191—21no—
131C2×D191—21no—
132C2×D191—21no—
133C2×D191—21no—
134C2×D191—44no—
135C2×D191—41no—
136C2×D191—42no—
137C2×D191—41no—
138C2×D191—42no—
139C2×D191—41no—
140C2×D191—42no—
141C2×D191—84no—
142C4×C191—21no—
143C4×C191—41no—
144C4×C191—42no—
145C4×C191—41no—
146C4×C191H(5,𝔽2)12yes—
147Dic191—21no—
148Dic191—21no—
149Dic191—41no—
150C2×C191—22no—
151C2×C191#53411yescharacteristic
152C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C191#57511yescharacteristic
154D191—41no—
155D191—41no—
156D191—81no—
157D191—82no—
158C191#319411yescharacteristic, complemented, Sylow
159#3194—1911nomaximal, Sylow
160#525—1911no—
161C2×#121—1911no—
162#483D19111yescharacteristic, complemented
163#482—1912no—
164#480—1911no—
165#469—1912no—
166#468—1912no—
167#450—1911no—
168#449—1912no—
169#447—1911no—
170#407—1911no—
171C2×#41—1911no—
172C22×D4—3821no—
173C22×D4—1911no—
174C23×C4—1911no—
175C2×SD16—1912no—
176C2×D8—1912no—
177C2×M16—1912no—
178#123—3821no—
179C22⋊Q8—1911no—
180C22⋊Q8C2×D19111yescharacteristic
181#121—3821no—
182#121—3821no—
183#121—1911no—
184#121—1912no—
185#121—1912no—
186#121—1911no—
187#120—3821no—
188C4×D4—3821no—
189C4×D4—3821no—
190#117—1911no—
191C2×C4⋊C4C2×D19111yescharacteristic
192C2×C22⋊C4—3821no—
193#107—1912no—
194#106C2×D19112yescomplemented
195#103C2×D19112yescomplemented
196#102—1912no—
197#102—1912no—
198#102—1912no—
199C22⋊C8—1911no—
200C22⋊C8—1912no—
201C22⋊C8C2×D19111yescharacteristic
202C24—3821no—
203C24—1911no—
204#41—7642no—
205C2×Q8C22×D19111yescharacteristic
206C2×D4—3821no—
207C2×D4—3821no—
208C2×D4—3822no—
209C2×D4—3822no—
210C2×D4—3821no—
211C2×D4—3821no—
212C2×D4—1911no—
213C2×D4—1912no—
214C2×D4—7642no—
215C2×D4—7644no—
216C22×C4—21no—
217C22×C4—3821no—
218C22×C4—1911no—
219C22×C4—1911no—
220C22×C4—1911no—
221C22×C4—1911no—
222C22×C4—1911no—
223C22×C4—7642no—
224C22×C4C22×D19111yescharacteristic
225SD16—7642no—
226D8—7642no—
227M16—7642no—
228C2×C8—1912no—
229C2×C8C22×D19112yes—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—3821no—
233C4⋊C4—1911no—
234C4⋊C4C22×D19111yescharacteristic
235C4⋊C4C22×D19112yes—
236C22⋊C4—21no—
237C22⋊C4—3821no—
238C22⋊C4—3821no—
239C22⋊C4—3822no—
240C22⋊C4—3822no—
241C22⋊C4—3821no—
242C22⋊C4—3821no—
243C42—3821no—
244C23—3822no—
245C23—3821no—
246C23—3822no—
247C23—1911no—
248C23—1911no—
249C23—7641no—
250C23—7641no—
251C23—7642no—
252C23—7642no—
253C23—7642no—
254C23D4×D19111yescharacteristic
255Q8—22no—
256D4—3822no—
257D4—3822no—
258D4—3822no—
259D4—7642no—
260D4—7644no—
261D4—7644no—
262D4—7642no—
263D4—7642no—
264D4—7642no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—3822no—
271C2×C4—3822no—
272C2×C4—3821no—
273C2×C4—1911no—
274C2×C4—1911no—
275C2×C4—7641no—
276C2×C4—7641no—
277C2×C4—7642no—
278C2×C4—7641no—
279C2×C4D4×D19111yescharacteristic
280C2×C4C23×D19111yescharacteristic
281C8—22no—
282C8—3822no—
283C22—21no—
284C22—21no—
285C22—3821no—
286C22—3821no—
287C22—3821no—
288C22—3821no—
289C22—3821no—
290C22—3821no—
291C22—15284no—
292C22—7644no—
293C22—7642no—
294C22—7641no—
295C22—7641no—
296C22—7642no—
297C22—7641no—
298C22—7642no—
299C22#2025934912yes—
300C22C2×D4×D19111yescharacteristic
301C4—21no—
302C4—42no—
303C4—41no—
304C4—41no—
305C4—3821no—
306C4—3821no—
307C4—7641no—
308C4#2025936812yes—
309C2—22no—
310C2—15282no—
311C2—15281no—
312C2—7641no—
313C2—7641no—
314C2#2713322411yescharacteristic
315C2C2×#2025934911yescharacteristic
316C2#2713346211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#483×C191C2
socleC22×C191C22×D4
hypercenter#483D191
hypocenterC191#3194
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #27133176C2semidirect
2. C2×#20259297C2semidirect
3. #27132989C2semidirect
4. #27132976C2semidirect
5. #27132917C2semidirect
6. #27132912C2semidirect
7. #27132813C2semidirect
8. #27132811C2semidirect
9. #27132799C2semidirect
10. #27132625C2semidirect
11. #483×C191C2semidirect
12. C2×#15690288C22semidirect
16. C2×#15690274C22semidirect
17. C2×D1528C22semidirect
18. C2×#15690270C22semidirect
20. #20259298C22semidirect
25. #20259297C22semidirect
30. #20259280C22semidirect
32. #20259255C22semidirect
33. #20259244C22semidirect
34. #20259242C22semidirect
35. #20259242C22semidirect
37. (C2×D191)⋊C8C22semidirect
40. #106×C191C22semidirect
41. #103×C191C22semidirect
158. C191#3194semidirect
162. #483D191semidirect
194. #106C2×D191semidirect
195. #103C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271331762
C2×#202592972
#271329892
#271329762
#271329172
#271329122
#271328132
#271328112
#271327992
#271326252
#483×C1912
#3194191