Finite Group Databasebeta

← #37581374

Subgroups — #37581374

Order 24448.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D191C211yesmaximal, characteristic
2#27133139C211yesmaximal, characteristic, complemented
3#27133135C211yesmaximal, characteristic
4#27132766C211yesmaximal, characteristic, complemented
5#27132763C211yesmaximal, characteristic, complemented
6#27132761C211yesmaximal, characteristic, complemented
7#27132760C211yesmaximal, characteristic, complemented
8C4×#15690276C211yesmaximal, characteristic, complemented
9SD16×Dic191C211yesmaximal, characteristic, complemented
10#27132750C211yesmaximal, characteristic, complemented
11#27132698C211yesmaximal, characteristic, complemented
12#27132670C211yesmaximal, characteristic, complemented
13#27132666C211yesmaximal, characteristic
14#27132617C211yesmaximal, characteristic, complemented
15C4×SD16×C191C211yesmaximal, characteristic, complemented
16C2×Q8×D191C2211yescharacteristic
17C2×D4×D191C2211yescharacteristic
18C2×#15690276C2211yescharacteristic
19C2×#15690275C2211yescharacteristic
20C2×#15690274C2211yescharacteristic
21SD16×D191C418yescomplemented
22C2×#15690270C2211yescharacteristic
23C2×C8×D191C2211yescharacteristic
24Q8×Dic191C2211yescharacteristic, complemented
25C4×Dic382—21no—
26C4×Dic382C2211yescharacteristic, complemented
27#20259287—21no—
28#20259287C2211yescharacteristic, complemented
29C4×D764C2211yescharacteristic
30D4×Dic191C2211yescharacteristic
31#20259284C2211yescharacteristic
32C4×C22⋊D191—21no—
33(C22⋊D191)⋊C4—21no—
34#20259280—21no—
35#20259280C2211yescharacteristic
36#20259280C2211yescharacteristic
37#20259279—21no—
38#20259279C2211yescharacteristic
39C2×(C2×D191)⋊C4—21no—
40C22⋊C4×D191—21no—
41#20259255C2211yescharacteristic, complemented
42#20259255C2211yescharacteristic, complemented
43#20259254C2211yescharacteristic, complemented
44#20259247C2211yescharacteristic, complemented
45#20259246C2211yescharacteristic, complemented
46#20259245C2211yescharacteristic, complemented
47#20259244C2211yescharacteristic
48#20259243C2211yescharacteristic
49#20259242C2211yescharacteristic
50#20259233C2211yescharacteristic, complemented
51#20259232C2211yescharacteristic, complemented
52C8×Dic191C2211yescharacteristic, complemented
53C2×SD16×C191C2211yescharacteristic
54C4×Q8×C191C2211yescharacteristic, complemented
55C4×D4×C191C2211yescharacteristic
56#106×C191C2211yescharacteristic, complemented
57#103×C191C2211yescharacteristic, complemented
58#102×C191C2211yescharacteristic
59C4×C8×C191C2211yescharacteristic, complemented
60C23×D191—21no—
61C2×Dic382—21no—
62C2×Dic382C2311yescharacteristic
63Q8×D191—22no—
64Q8×D191C2×C414yescomplemented
65C2×D764C2311yescharacteristic
66C2×C22⋊D191—21no—
67D4×D191—22no—
68D4×D191C2×C414yes—
69C22×Dic191—21no—
70C2×C4×D191—21no—
71C2×C4×D191—21no—
72C2×C4×D191—21no—
73C2×C4×D191C2311yescharacteristic
74#15690276C2×C414yescomplemented
75#15690275C2×C414yescomplemented
76#15690274C2×C414yescomplemented
77#15690270—22no—
78C2×C191⋊C8C2311yescharacteristic
79C2×C191⋊C8C2311yescharacteristic
80C8×D191C2×C414yescomplemented
81C4⋊Dic191—21no—
82C4⋊Dic191C2311yescharacteristic
83C4⋊Dic191C2311yescharacteristic
84Dic191⋊C4—21no—
85Dic191⋊C4—21no—
86Dic191⋊C4D411yescharacteristic
87Dic191⋊C4D411yescharacteristic
88C22⋊Dic191—21no—
89(C2×D191)⋊C4—21no—
90(C2×D191)⋊C4—21no—
91(C2×D191)⋊C4—21no—
92(C2×D191)⋊C4—21no—
93(C2×D191)⋊C4D411yescharacteristic
94(C2×D191)⋊C4D411yescharacteristic
95C4×Dic191—21no—
96C4×Dic191—21no—
97C4×Dic191—21no—
98C4×Dic191—21no—
99C4×Dic191—21no—
100C4×Dic191C2311yescharacteristic
101C2×Q8×C191C2311yescharacteristic
102C2×D4×C191C2311yescharacteristic
103C22×C4×C191—21no—
104SD16×C191C2×C414yescomplemented
105C2×C8×C191C2311yescharacteristic
106C2×C8×C191C2311yescharacteristic
107C4⋊C4×C191—21no—
108C4⋊C4×C191C2311yescharacteristic
109C4⋊C4×C191C2311yescharacteristic
110C22⋊C4×C191—21no—
111C42×C191—21no—
112C42×C191C2311yescharacteristic
113C22×D191—21no—
114C22×D191—24no—
115C22×D191—81no—
116C22×D191—81no—
117C22×D191C2×D411yescharacteristic
118Dic382—21no—
119Dic382—24no—
120Dic382C22×C412yes—
121D764—21no—
122D764C22×C412yes—
123C22⋊D191—24no—
124C2×Dic191—21no—
125C2×Dic191—21no—
126C2×Dic191—21no—
127C2×Dic191—81no—
128C2×Dic191C2×D411yescharacteristic
129C2×Dic191C2×D411yescharacteristic
130C2×Dic191C2×D411yescharacteristic
131C4×D191—24no—
132C4×D191—81no—
133C4×D191—81no—
134C4×D191C22×C412yes—
135C4×D191#4112yes—
136C191⋊C8—21no—
137C191⋊C8C22×C412yes—
138C23×C191—21no—
139Q8×C191—21no—
140Q8×C191C22×C412yes—
141D4×C191—21no—
142D4×C191C22×C412yes—
143C2×C4×C191—21no—
144C2×C4×C191—21no—
145C2×C4×C191—21no—
146C2×C4×C191—81no—
147C2×C4×C191C2×D411yescharacteristic
148C2×C4×C191C2×D411yescharacteristic
149C2×C4×C191C2411yescharacteristic
150C8×C191—21no—
151C8×C191C22×C412yes—
152C22×C191—22no—
153C22×C191—41no—
154C22×C191—41no—
155C22×C191C22×D411yescharacteristic
156C2×D191—21no—
157C2×D191—21no—
158C2×D191—22no—
159C2×D191—41no—
160C2×D191—41no—
161C2×D191—84no—
162C2×D191C4×D412yes—
163C4×C191—22no—
164C4×C191—21no—
165C4×C191—21no—
166C4×C191—41no—
167C4×C191—41no—
168C4×C191C23×C411yescharacteristic
169C4×C191C2×#4111yescharacteristic
170Dic191—22no—
171Dic191—21no—
172Dic191—21no—
173Dic191—41no—
174Dic191—41no—
175Dic191C4×D412yes—
176C2×C191—42no—
177C2×C191C2×C4×D411yescharacteristic
178C2×C191#57711yescharacteristic
179C2×C191#57711yescharacteristic
180D191—22no—
181D191—42no—
182C191#294911yescharacteristic, complemented, Sylow
183#2949—1911nomaximal, Sylow
184C22×SD16—1911no—
185#519—1911no—
186#518—1911no—
187#440—1911no—
188#440—1911no—
189#440—1911no—
190#440—1911no—
191C4×SD16—1911no—
192C4×SD16—1911no—
193C4×SD16—1911no—
194C4×SD16D19111yescharacteristic, complemented
195#427—1911no—
196#419—1911no—
197#418—1911no—
198#405—1911no—
199C22×Q8—1911no—
200C22×D4—1911no—
201C2×SD16—1911no—
202C2×SD16—1911no—
203C2×SD16—1918no—
204C2×SD16—1911no—
205C2×SD16C2×D19111yescharacteristic
206C2×M16—1911no—
207C22×C8—1911no—
208C4×Q8—3821no—
209C4×Q8—3821no—
210C4×Q8—1911no—
211C4×Q8—1911no—
212C4×Q8—1911no—
213C4×Q8C2×D19111yescharacteristic, complemented
214C4×D4—3821no—
215C4×D4—3821no—
216C4×D4—1911no—
217C4×D4—1911no—
218C4×D4—1911no—
219C4×D4C2×D19111yescharacteristic
220#117—3821no—
221#117—3821no—
222#117—1911no—
223#117—1911no—
224#117—1911no—
225C2×C22⋊C4—3821no—
226C2×C22⋊C4—3821no—
227#107—1911no—
228#107—1911no—
229#106—1911no—
230#106C2×D19111yescharacteristic, complemented
231#103—1911no—
232#103—1911no—
233#103—1911no—
234#103C2×D19111yescharacteristic, complemented
235#102—1911no—
236#102—1911no—
237#102—1911no—
238#102C2×D19111yescharacteristic
239#97—1911no—
240#97—1911no—
241C4×C8—1911no—
242C4×C8C2×D19111yescharacteristic, complemented
243C24—3821no—
244C2×Q8—3821no—
245C2×Q8—3822no—
246C2×Q8—1914no—
247C2×Q8—1911no—
248C2×Q8C22×D19111yescharacteristic
249C2×D4—3822no—
250C2×D4—3821no—
251C2×D4—1914no—
252C2×D4—1911no—
253C2×D4C22×D19111yescharacteristic
254C22×C4—21no—
255C22×C4—3821no—
256C22×C4—3821no—
257C22×C4—3821no—
258C22×C4—3821no—
259C22×C4—1911no—
260SD16—1914no—
261SD16—1914no—
262SD16—1914no—
263SD16C4×D19114yescomplemented
264M16—3822no—
265C2×C8—1911no—
266C2×C8—1911no—
267C2×C8—1914no—
268C2×C8C22×D19111yescharacteristic
269C2×C8C22×D19111yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—3821no—
272C4⋊C4—3821no—
273C4⋊C4—3821no—
274C4⋊C4—1911no—
275C4⋊C4—1911no—
276C4⋊C4—1911no—
277C4⋊C4—1911no—
278C4⋊C4C22×D19111yescharacteristic
279C4⋊C4C22×D19111yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—3821no—
282C22⋊C4—3821no—
283C22⋊C4—3821no—
284C22⋊C4—3821no—
285C22⋊C4—3821no—
286C22⋊C4—1911no—
287C22⋊C4—1911no—
288C42—21no—
289C42—3821no—
290C42—3821no—
291C42—3821no—
292C42—3821no—
293C42—3821no—
294C42—1911no—
295C42C22×D19111yescharacteristic
296C23—21no—
297C23—3824no—
298C23—3821no—
299C23—1911no—
300C23—15281no—
301C23—15281no—
302Q8—21no—
303Q8—3824no—
304Q8—3821no—
305Q8—1912no—
306Q8C2×C4×D19112yes—
307D4—21no—
308D4—3824no—
309D4—3821no—
310D4—1912no—
311D4C2×C4×D19112yes—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—3824no—
317C2×C4—3821no—
318C2×C4—3821no—
319C2×C4—3821no—
320C2×C4—1911no—
321C2×C4—1911no—
322C2×C4—1911no—
323C2×C4—1912no—
324C2×C4—1912no—
325C2×C4—15281no—
326C2×C4—15281no—
327C2×C4—15281no—
328C2×C4D4×D19111yescharacteristic
329C2×C4D4×D19111yescharacteristic
330C2×C4C23×D19111yescharacteristic
331C8—21no—
332C8—3821no—
333C8—1912no—
334C8C2×C4×D19112yes—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—3821no—
339C22—3821no—
340C22—3822no—
341C22—1912no—
342C22—15284no—
343C22—7641no—
344C22—7641no—
345C22C2×D4×D19111yescharacteristic
346C4—21no—
347C4—21no—
348C4—22no—
349C4—41no—
350C4—41no—
351C4—3821no—
352C4—3821no—
353C4—3822no—
354C4—1912no—
355C4—7641no—
356C4—7641no—
357C4C22×C4×D19111yescharacteristic
358C4#41×D19111yescharacteristic
359C2—42no—
360C2—3822no—
361C2—7642no—
362C2C4×D4×D19111yescharacteristic
363C2#2713346911yescharacteristic
364C2#2713347011yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutatorC4×C191C23×C4
FrattiniC2×C4C23×D191
FittingC4×SD16×C191C2
socleC22×C191C22×D4
prime radicalC2×D4×D191C22
cosocleC2×C4×C191C24
hypercenterC4×SD16D191
hypocenterC191#2949
LCSG ▷ C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #27133139C2semidirect
4. #27132766C2semidirect
5. #27132763C2semidirect
6. #27132761C2semidirect
7. #27132760C2semidirect
8. C4×#15690276C2semidirect
9. SD16×Dic191C2semidirect
10. #27132750C2semidirect
11. #27132698C2semidirect
12. #27132670C2semidirect
14. #27132617C2semidirect
15. C4×SD16×C191C2semidirect
21. SD16×D191C4semidirect
24. Q8×Dic191C22semidirect
26. C4×Dic382C22semidirect
28. #20259287C22semidirect
41. #20259255C22semidirect
42. #20259255C22semidirect
43. #20259254C22semidirect
44. #20259247C22semidirect
45. #20259246C22semidirect
46. #20259245C22semidirect
50. #20259233C22semidirect
51. #20259232C22semidirect
52. C8×Dic191C22semidirect
54. C4×Q8×C191C22semidirect
56. #106×C191C22semidirect
57. #103×C191C22semidirect
59. C4×C8×C191C22semidirect
64. Q8×D191C2×C4semidirect
74. #15690276C2×C4semidirect
75. #15690275C2×C4semidirect
76. #15690274C2×C4semidirect
80. C8×D191C2×C4semidirect
104. SD16×C191C2×C4semidirect
182. C191#2949semidirect
194. C4×SD16D191semidirect
213. C4×Q8C2×D191semidirect
230. #106C2×D191semidirect
234. #103C2×D191semidirect
242. C4×C8C2×D191semidirect
263. SD16C4×D191semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1912
#271331392
#271331352
#271327662
#271327632
#271327612
#271327602
C4×#156902762
SD16×Dic1912
#271327502
#271326982
#271326702
#271326662
#271326172
C4×SD16×C1912
#2949191