Finite Group Databasebeta

← #37582462

Subgroups — #37582462

Order 24448.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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×#20259297C211yesmaximal, characteristic
2#117×D191C211yesmaximal, characteristic, complemented
3#27132990C212yesmaximal, complemented
4#27132989C212yesmaximal, complemented
5#27132983C212yesmaximal, complemented
6#27132978C212yesmaximal, complemented
7#27132703C211yesmaximal, characteristic, complemented
8#27132669C211yesmaximal, characteristic, complemented
9#27132667C211yesmaximal, characteristic, complemented
10#27132626C211yesmaximal, characteristic, complemented
11#428×C191C211yesmaximal, characteristic, complemented
12C22×D764—21no—
13C22×D764C2211yescharacteristic
14C22×C4×D191C2211yescharacteristic
15C2×#15690271C2211yescharacteristic
16#20259297—21no—
17#20259297—21no—
18#20259297C2212yes—
19#20259297C2212yes—
20#20259281C2211yescharacteristic
21#20259280C2211yescharacteristic, complemented
22#20259280C2211yescharacteristic, complemented
23#20259279—21no—
24#20259278—21no—
25C4⋊C4×D191C2211yescharacteristic, complemented
26C4⋊C4×D191C2211yescharacteristic, complemented
27C2×(C2×D191)⋊C4—21no—
28C22⋊C4×D191—21no—
29C42×D191—21no—
30#20259255C2212yescomplemented
31#20259253C2212yescomplemented
32#20259244C2212yescomplemented
33#20259244C2212yescomplemented
34#20259242C2212yescomplemented
35#20259242C2212yescomplemented
36(C2×D191)⋊C8C2212yescomplemented
37(C2×D191)⋊C8C2212yescomplemented
38C2×M16×C191C2211yescharacteristic
39#117×C191C2211yescharacteristic
40C2×C4⋊C4×C191C2211yescharacteristic
41#107×C191C2212yescomplemented
42#106×C191C2212yescomplemented
43C23×D191—21no—
44C23×D191D411yescharacteristic
45C2×D764—22no—
46C2×D764—22no—
47C2×D764—21no—
48C2×D764—22no—
49C2×D764—22no—
50C2×D764—44no—
51C2×D764C2312yes—
52C22×Dic191D411yescharacteristic
53C2×C4×D191—21no—
54C2×C4×D191—21no—
55C2×C4×D191—22no—
56C2×C4×D191—22no—
57C2×C4×D191D411yescharacteristic
58C2×C4×D191D411yescharacteristic
59C2×C4×D191C2311yescharacteristic
60C2×C4×D191C2311yescharacteristic
61#15690271—22no—
62C2×C191⋊C8C2312yes—
63C4⋊Dic191C2311yescharacteristic
64C4⋊Dic191C2311yescharacteristic
65Dic191⋊C4—21no—
66Dic191⋊C4—21no—
67C22⋊Dic191—21no—
68(C2×D191)⋊C4—21no—
69(C2×D191)⋊C4—21no—
70(C2×D191)⋊C4—22no—
71(C2×D191)⋊C4—22no—
72C4×Dic191—21no—
73C4×Dic191—21no—
74C4×Dic191—21no—
75C22×C4×C191—21no—
76C22×C4×C191C2311yescharacteristic
77M16×C191—22no—
78C2×C8×C191C2312yes—
79C4⋊C4×C191—21no—
80C4⋊C4×C191C2311yescharacteristic
81C4⋊C4×C191C2312yes—
82C4⋊C4×C191C2311yescharacteristic
83C22⋊C4×C191—21no—
84C42×C191—21no—
85C22×D191—22no—
86C22×D191—22no—
87C22×D191—22no—
88C22×D191—41no—
89C22×D191—42no—
90C22×D191—42no—
91C22×D191—42no—
92C22×D191C2×D411yescharacteristic
93C22×D191C2×D411yescharacteristic
94D764—22no—
95D764—22no—
96D764—44no—
97D764—44no—
98D764—42no—
99C2×Dic191—22no—
100C2×Dic191—21no—
101C2×Dic191—21no—
102C2×Dic191C2×D411yescharacteristic
103C2×Dic191C2×D411yescharacteristic
104C4×D191—22no—
105C4×D191—22no—
106C4×D191—21no—
107C4×D191—21no—
108C4×D191—21no—
109C4×D191—21no—
110C4×D191—42no—
111C4×D191—42no—
112C191⋊C8—22no—
113C23×C191C2×D411yescharacteristic
114C2×C4×C191—21no—
115C2×C4×C191—21no—
116C2×C4×C191—22no—
117C2×C4×C191—42no—
118C2×C4×C191C2×D411yescharacteristic
119C2×C4×C191#4112yes—
120C2×C4×C191#4112yes—
121C2×C4×C191C2411yescharacteristic
122C8×C191—22no—
123C22×C191—21no—
124C22×C191—21no—
125C22×C191#12312yes—
126C22×C191C22×D411yescharacteristic
127C2×D191—21no—
128C2×D191—21no—
129C2×D191—21no—
130C2×D191—21no—
131C2×D191—21no—
132C2×D191—44no—
133C2×D191—42no—
134C2×D191—42no—
135C2×D191—42no—
136C2×D191—84no—
137C2×D191AGL(1,ℤ/8)12yes—
138C4×C191—42no—
139C4×C191—41no—
140C4×C191—41no—
141C4×C191#12312yes—
142C4×C191C2×#4111yescharacteristic
143C4×C191C2×#4111yescharacteristic
144Dic191—21no—
145Dic191—21no—
146Dic191—41no—
147Dic191—41no—
148C2×C191—22no—
149C2×C191C2×#12311yescharacteristic
150C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C191#57511yescharacteristic
152D191—22no—
153D191—41no—
154D191—82no—
155C191#309411yescharacteristic, complemented, Sylow
156#3094—1911nomaximal, Sylow
157C2×#121—1911no—
158C2×#117—1911no—
159#482—1912no—
160#482—1912no—
161#481—1912no—
162#480—1912no—
163#428—1911no—
164#428D19111yescharacteristic, complemented
165#418—1911no—
166#418—1911no—
167#407—1911no—
168C22×D4—3821no—
169C22×D4—1911no—
170C23×C4—1911no—
171C2×M16—1911no—
172C2×M16C2×D19111yescharacteristic
173#121—3821no—
174#121—3821no—
175#121—1912no—
176#121—1912no—
177#117—3821no—
178#117—3821no—
179#117—1911no—
180#117—1911no—
181#117—1911no—
182#117C2×D19111yescharacteristic
183C2×C4⋊C4—1911no—
184C2×C4⋊C4—1911no—
185C2×C4⋊C4C2×D19111yescharacteristic
186C2×C22⋊C4—3821no—
187C2×C22⋊C4—3821no—
188C2×C42—3821no—
189#107—1912no—
190#107C2×D19112yescomplemented
191#106—1912no—
192#106C2×D19112yescomplemented
193#102—1912no—
194#102—1912no—
195#102—1912no—
196#102—1912no—
197C22⋊C8—1912no—
198C22⋊C8—1912no—
199C24—3821no—
200C24—1911no—
201C2×D4—3822no—
202C2×D4—3822no—
203C2×D4—3821no—
204C2×D4—3822no—
205C2×D4—3822no—
206C2×D4—1912no—
207C2×D4—7644no—
208C22×C4—21no—
209C22×C4—3821no—
210C22×C4—3821no—
211C22×C4—3822no—
212C22×C4—3822no—
213C22×C4—1911no—
214C22×C4—1911no—
215C22×C4—1911no—
216C22×C4—1911no—
217C22×C4—1911no—
218C22×C4C22×D19111yescharacteristic
219M16—22no—
220M16—3822no—
221C2×C8—1912no—
222C2×C8C22×D19112yes—
223C4⋊C4—21no—
224C4⋊C4—3821no—
225C4⋊C4—3821no—
226C4⋊C4—1911no—
227C4⋊C4—1911no—
228C4⋊C4C22×D19111yescharacteristic
229C4⋊C4C22×D19111yescharacteristic
230C4⋊C4C22×D19112yes—
231C22⋊C4—21no—
232C22⋊C4—3821no—
233C22⋊C4—3821no—
234C22⋊C4—3822no—
235C22⋊C4—3822no—
236C22⋊C4—3821no—
237C42—21no—
238C42—3821no—
239C42—3821no—
240C42—3821no—
241C23—3822no—
242C23—3822no—
243C23—3822no—
244C23—1911no—
245C23—1911no—
246C23—7641no—
247C23—7642no—
248C23—7642no—
249C23—7642no—
250C23D4×D19111yescharacteristic
251D4—3822no—
252D4—3822no—
253D4—7644no—
254D4—7644no—
255D4—7642no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—3822no—
261C2×C4—3821no—
262C2×C4—3821no—
263C2×C4—3821no—
264C2×C4—3821no—
265C2×C4—3822no—
266C2×C4—3822no—
267C2×C4—3821no—
268C2×C4—3821no—
269C2×C4—1911no—
270C2×C4—1911no—
271C2×C4—7642no—
272C2×C4—7642no—
273C2×C4D4×D19111yescharacteristic
274C2×C4#1569028712yes—
275C2×C4#1569028812yes—
276C2×C4C23×D19111yescharacteristic
277C8—22no—
278C8—3822no—
279C22—21no—
280C22—21no—
281C22—3821no—
282C22—3821no—
283C22—3821no—
284C22—3821no—
285C22—3821no—
286C22—1912no—
287C22—15284no—
288C22—7644no—
289C22—7642no—
290C22—7642no—
291C22—7642no—
292C22#2025930712yes—
293C22C2×D4×D19111yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—3821no—
298C4—3821no—
299C4—7641no—
300C4—7641no—
301C4#2025930712yes—
302C4C2×#1569028711yescharacteristic
303C4C2×#1569028811yescharacteristic
304C2—22no—
305C2—3822no—
306C2—15282no—
307C2—7641no—
308C2C2×#2025930711yescharacteristic
309C2AGL(1,ℤ/8)×D19111yescharacteristic
310C2#2713346211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#428×C191C2
socleC22×C191C22×D4
prime radicalC2×#20259297C2
hypercenter#428D191
hypocenterC191#3094
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #117×D191C2semidirect
3. #27132990C2semidirect
4. #27132989C2semidirect
5. #27132983C2semidirect
6. #27132978C2semidirect
7. #27132703C2semidirect
8. #27132669C2semidirect
9. #27132667C2semidirect
10. #27132626C2semidirect
11. #428×C191C2semidirect
21. #20259280C22semidirect
22. #20259280C22semidirect
25. C4⋊C4×D191C22semidirect
26. C4⋊C4×D191C22semidirect
30. #20259255C22semidirect
31. #20259253C22semidirect
32. #20259244C22semidirect
33. #20259244C22semidirect
34. #20259242C22semidirect
35. #20259242C22semidirect
36. (C2×D191)⋊C8C22semidirect
37. (C2×D191)⋊C8C22semidirect
41. #107×C191C22semidirect
42. #106×C191C22semidirect
155. C191#3094semidirect
164. #428D191semidirect
190. #107C2×D191semidirect
192. #106C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202592972
#117×D1912
#271329902
#271329892
#271329832
#271329782
#271327032
#271326692
#271326672
#271326262
#428×C1912
#3094191