Finite Group Databasebeta

← #37582996

Subgroups — #37582996

Order 24448.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#20259355C211yesmaximal, characteristic, complemented
2C2×#20259350C211yesmaximal, characteristic, complemented
3C2×#20259344C211yesmaximal, characteristic, complemented
4C22×#15690276C211yesmaximal, characteristic, complemented
5#27133239C211yesmaximal, characteristic, complemented
6#27133232C211yesmaximal, characteristic, complemented
7#27133057C212yesmaximal, complemented
8#27133052C212yesmaximal, complemented
9#27133047C212yesmaximal, complemented
10#27133036C212yesmaximal, complemented
11#405×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C2×#15690287C2211yescharacteristic
14C22×Dic382C2211yescharacteristic
15C22×D764C2211yescharacteristic
16#20259355—24no—
17#20259350—24no—
18#20259344—24no—
19C2×Dic764C2211yescharacteristic, complemented
20C2×Dic764C2212yescomplemented
21C2×#15690276—24no—
22C2×#15690276C2212yes—
23C2×#15690276C2212yes—
24C2×#15690276C2212yes—
25C2×#15690276C2211yescharacteristic, complemented
26C2×#15690276C2211yescharacteristic, complemented
27C2×#15690276C2212yescomplemented
28C2×D1528C2212yes—
29C2×D1528C2211yescharacteristic, complemented
30#20259322C2212yescomplemented
31#20259319C2212yes—
32#20259310C2212yes—
33#20259310C2212yescomplemented
34#20259306—22no—
35#20259302—22no—
36#20259297—22no—
37#20259292—22no—
38C2×M16×C191C2211yescharacteristic
39C22×C8×C191C2211yescharacteristic
40#117×C191C2211yescharacteristic
41#97×C191C2212yescomplemented
42C4×C8×C191C2212yescomplemented
43C23×D191—21no—
44#15690287—22no—
45#15690287—22no—
46#15690287—22no—
47#15690287—22no—
48C2×Dic382—21no—
49C2×Dic382—22no—
50C2×Dic382—22no—
51C2×Dic382C2312yes—
52C2×Dic382C2311yescharacteristic
53C2×Dic382C2311yescharacteristic
54C2×D764—22no—
55C2×D764—22no—
56C2×D764—21no—
57C2×D764—24no—
58C2×D764C2311yescharacteristic
59C2×D764C2312yes—
60C2×D764C2311yescharacteristic
61C2×C22⋊D191—21no—
62C2×C22⋊D191—21no—
63C22×Dic191—21no—
64C2×C4×D191—21no—
65C2×C4×D191—21no—
66Dic764—22no—
67Dic764—24no—
68#15690276—24no—
69#15690276—24no—
70#15690276—24no—
71#15690276—22no—
72#15690276—22no—
73#15690276—24no—
74D1528—24no—
75D1528—22no—
76C4⋊Dic191—24no—
77(C2×D191)⋊C4—24no—
78(C2×D191)⋊C4—24no—
79C22×C4×C191C2311yescharacteristic
80M16×C191D414yescomplemented
81C2×C8×C191D414yescomplemented
82C2×C8×C191C2312yes—
83C2×C8×C191C2311yescharacteristic
84C2×C8×C191C2311yescharacteristic
85C4⋊C4×C191D412yes—
86C22⋊C4×C191D412yes—
87C42×C191C2312yes—
88C22×D191—21no—
89C22×D191—22no—
90C22×D191—21no—
91C22×D191—42no—
92C22×D191—82no—
93Dic382—22no—
94Dic382—21no—
95Dic382—22no—
96Dic382—21no—
97Dic382—21no—
98Dic382—21no—
99Dic382—42no—
100D764—21no—
101D764—22no—
102D764—21no—
103D764—21no—
104D764—22no—
105D764—21no—
106D764—42no—
107D764—84no—
108C22⋊D191—42no—
109C22⋊D191—42no—
110C2×Dic191—21no—
111C2×Dic191—22no—
112C2×Dic191—21no—
113C2×Dic191—42no—
114C4×D191—42no—
115C4×D191—42no—
116C23×C191C2×D411yescharacteristic
117C2×C4×C191C2×D414yes—
118C2×C4×C191C2×D411yescharacteristic
119C2×C4×C191C2×D412yes—
120C2×C4×C191C2×D412yes—
121C2×C4×C191C2411yescharacteristic
122C8×C191C2×D412yes—
123C8×C191C2×D412yes—
124C8×C191C2×D414yes—
125C22×C191—22no—
126C22×C191#12712yes—
127C22×C191C22×D411yescharacteristic
128C2×D191—41no—
129C2×D191—41no—
130C2×D191—42no—
131C2×D191—42no—
132C2×D191—44no—
133C2×D191—42no—
134C2×D191—84no—
135C4×C191—24no—
136C4×C191#12712yes—
137C4×C191C22×D411yescharacteristic
138C4×C191C22×D411yescharacteristic
139Dic191—41no—
140Dic191—41no—
141Dic191—42no—
142C2×C191—22no—
143C2×C191C2×#12711yescharacteristic
144C2×C191#57712yes—
145D191—81no—
146D191—81no—
147D191—82no—
148C191#315411yescharacteristic, complemented, Sylow
149#3154—1911nomaximal, Sylow
150C2×#137—1911no—
151C2×AGL(1,ℤ/8)—1911no—
152C2×#135—1911no—
153C22×SD16—1911no—
154#536—1911no—
155#535—1911no—
156#497—1912no—
157#496—1912no—
158#495—1912no—
159#492—1912no—
160#405D19111yescharacteristic, complemented
161C2×#41—1911no—
162C2×#41—1911no—
163C22×Q8—1911no—
164C22×D4—1911no—
165#137—3824no—
166AGL(1,ℤ/8)—3824no—
167#135—3824no—
168C2×Q16—1911no—
169C2×Q16—1912no—
170C2×SD16—3824no—
171C2×SD16—1912no—
172C2×SD16—1912no—
173C2×SD16—1912no—
174C2×SD16—1911no—
175C2×SD16—1911no—
176C2×SD16—1912no—
177C2×D8—1912no—
178C2×D8—1911no—
179C2×M16C2×D19111yescharacteristic
180C22×C8C2×D19111yescharacteristic
181C4⋊Q8—1912no—
182#127—1912no—
183#124—1912no—
184#124—1912no—
185#123—3822no—
186C22⋊Q8—3822no—
187#121—3822no—
188#120—3822no—
189#117C2×D19111yescharacteristic
190#97C2×D19112yescomplemented
191C4×C8C2×D19112yescomplemented
192C24—3821no—
193#41—3822no—
194#41—3822no—
195#41—3822no—
196#41—3822no—
197C2×Q8—3821no—
198C2×Q8—3822no—
199C2×Q8—3822no—
200C2×Q8—1911no—
201C2×Q8—1912no—
202C2×Q8—1911no—
203C2×D4—3821no—
204C2×D4—3821no—
205C2×D4—3822no—
206C2×D4—3822no—
207C2×D4—3824no—
208C2×D4—3821no—
209C2×D4—1911no—
210C2×D4—1912no—
211C2×D4—1911no—
212C22×C4—3821no—
213C22×C4—3821no—
214C22×C4—3821no—
215C22×C4C22×D19111yescharacteristic
216Q16—3822no—
217Q16—3824no—
218SD16—3824no—
219SD16—3824no—
220SD16—3822no—
221SD16—3822no—
222SD16—3824no—
223SD16—3824no—
224D8—3822no—
225D8—3824no—
226M16D76414yescomplemented
227C2×C8D76414yescomplemented
228C2×C8C22×D19111yescharacteristic
229C2×C8C22×D19112yes—
230C2×C8C22×D19111yescharacteristic
231C4⋊C4—3824no—
232C4⋊C4D76412yes—
233C22⋊C4—3824no—
234C22⋊C4—3824no—
235C22⋊C4D76412yes—
236C42C22×D19112yes—
237C23—3821no—
238C23—3822no—
239C23—3821no—
240C23—15282no—
241C23—7642no—
242C23C2×D76411yescharacteristic
243Q8—3822no—
244Q8—3821no—
245Q8—3822no—
246Q8—3821no—
247Q8—3821no—
248Q8—3821no—
249Q8—7642no—
250D4—3821no—
251D4—3822no—
252D4—3821no—
253D4—3821no—
254D4—3822no—
255D4—3821no—
256D4—15284no—
257D4—7642no—
258D4—7642no—
259D4—7642no—
260C2×C4—3821no—
261C2×C4—3822no—
262C2×C4—3821no—
263C2×C4—7642no—
264C2×C4—7642no—
265C2×C4—7642no—
266C2×C4C2×D76414yes—
267C2×C4C2×D76412yes—
268C2×C4C2×D76412yes—
269C2×C4C2×D76411yescharacteristic
270C2×C4C23×D19111yescharacteristic
271C8C2×D76412yes—
272C8C2×D76412yes—
273C8C2×D76414yes—
274C22—22no—
275C22—15284no—
276C22—7641no—
277C22—7641no—
278C22—7642no—
279C22—7642no—
280C22—7644no—
281C22—7642no—
282C22#2025931912yes—
283C22C22×D76411yescharacteristic
284C4—24no—
285C4—7641no—
286C4—7641no—
287C4—7642no—
288C4#2025931912yes—
289C4C22×D76411yescharacteristic
290C4C22×D76411yescharacteristic
291C2—22no—
292C2—15282no—
293C2—15281no—
294C2—15281no—
295C2C2×#2025931911yescharacteristic
296C2#2713347512yes—
subgroupG/H
centerC22C22×D764
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#405×C191C2
socleC22×C191C22×D4
hypercenter#405D191
hypocenterC191#3154
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#20259355C2semidirect
2. C2×#20259350C2semidirect
3. C2×#20259344C2semidirect
4. C22×#15690276C2semidirect
5. #27133239C2semidirect
6. #27133232C2semidirect
7. #27133057C2semidirect
8. #27133052C2semidirect
9. #27133047C2semidirect
10. #27133036C2semidirect
11. #405×C191C2semidirect
19. C2×Dic764C22semidirect
20. C2×Dic764C22semidirect
25. C2×#15690276C22semidirect
26. C2×#15690276C22semidirect
27. C2×#15690276C22semidirect
29. C2×D1528C22semidirect
30. #20259322C22semidirect
33. #20259310C22semidirect
41. #97×C191C22semidirect
42. C4×C8×C191C22semidirect
80. M16×C191D4semidirect
81. C2×C8×C191D4semidirect
148. C191#3154semidirect
160. #405D191semidirect
190. #97C2×D191semidirect
191. C4×C8C2×D191semidirect
226. M16D764semidirect
227. C2×C8D764semidirect

Maximal subgroups

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

subgroupindex
C2×#202593552
C2×#202593502
C2×#202593442
C22×#156902762
#271332392
#271332322
#271330572
#271330522
#271330472
#271330362
#405×C1912
#3154191