Finite Group Databasebeta

← #37583995

Subgroups — #37583995

Order 24448.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259346C211yesmaximal, characteristic, complemented
2C2×D8×D191C211yesmaximal, characteristic, complemented
3#27133308C211yesmaximal, characteristic, complemented
4#27133298C211yesmaximal, characteristic, complemented
5#27133050C211yesmaximal, characteristic, complemented
6#27132915C211yesmaximal, characteristic, complemented
7#27132891C211yesmaximal, characteristic, complemented
8#27132879C211yesmaximal, characteristic, complemented
9#27132815C211yesmaximal, characteristic, complemented
10#27132805C211yesmaximal, characteristic, complemented
11#27132798C211yesmaximal, characteristic, complemented
12#27132795C211yesmaximal, characteristic, complemented
13#27132773C211yesmaximal, characteristic, complemented
14#27132733C211yesmaximal, characteristic, complemented
15#459×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic, complemented
17C2×D4×D191—21no—
18C2×D4×D191C2211yescharacteristic
19C2×D4×D191C2211yescharacteristic, complemented
20C2×D4×D191C2211yescharacteristic, complemented
21#20259346—24no—
22C2×#15690276C2211yescharacteristic, complemented
23C2×#15690275C2211yescharacteristic, complemented
24C2×#15690275C2211yescharacteristic, complemented
25C2×D1528C2211yescharacteristic
26C2×#15690272C2211yescharacteristic
27C2×#15690272C2211yescharacteristic, complemented
28C2×#15690272C2211yescharacteristic, complemented
29D8×D191—24no—
30C2×#15690270C2211yescharacteristic, complemented
31C2×C8×D191C2211yescharacteristic, complemented
32#20259310C2211yescharacteristic, complemented
33#20259306—21no—
34#20259302C2211yescharacteristic, complemented
35#20259297C2211yescharacteristic, complemented
36#20259296—21no—
37#20259293—21no—
38#20259293C2211yescharacteristic
39#20259293C2211yescharacteristic, complemented
40#20259292—21no—
41#20259290—21no—
42#20259290—21no—
43C4×D764C2211yescharacteristic
44C4×D764C2211yescharacteristic, complemented
45C2×(C2×D191)⋊C4—21no—
46#20259253C2211yescharacteristic, complemented
47#20259247C2211yescharacteristic, complemented
48#20259244C2211yescharacteristic, complemented
49#20259243C2211yescharacteristic, complemented
50#20259242C2211yescharacteristic, complemented
51(C2×D191)⋊C8C2211yescharacteristic, complemented
52(C2×D191)⋊C8C2211yescharacteristic, complemented
53#20259233C2211yescharacteristic, complemented
54C2×D8×C191C2211yescharacteristic, complemented
55C2×D8×C191C2211yescharacteristic, complemented
56#127×C191C2211yescharacteristic, complemented
57C4×D4×C191C2211yescharacteristic, complemented
58C4⋊C8×C191C2211yescharacteristic, complemented
59#102×C191C2211yescharacteristic, complemented
60#102×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62C23×D191—21no—
63C23×D191—21no—
64#15690286—22no—
65#15690286—41no—
66C2×Dic382C2311yescharacteristic
67C2×D764—21no—
68C2×D764D411yescharacteristic
69C2×D764C2311yescharacteristic
70C2×D764C2311yescharacteristic
71C2×C22⋊D191—21no—
72C2×C22⋊D191—21no—
73C2×C22⋊D191—21no—
74C2×C22⋊D191—21no—
75C2×C22⋊D191—21no—
76C2×C22⋊D191—21no—
77D4×D191—22no—
78D4×D191—22no—
79D4×D191—22no—
80D4×D191—41no—
81D4×D191—41no—
82D4×D191—41no—
83D4×D191—44no—
84C22×Dic191—21no—
85C2×C4×D191—21no—
86C2×C4×D191C2311yescharacteristic
87C2×C4×D191C2311yescharacteristic
88#15690276—22no—
89#15690275—22no—
90#15690275—41no—
91D1528—22no—
92#15690272—22no—
93#15690272—41no—
94#15690272D414yescomplemented
95#15690270—22no—
96C2×C191⋊C8C2311yescharacteristic
97C2×C191⋊C8C2311yescharacteristic
98C8×D191—22no—
99C4⋊Dic191—21no—
100C4⋊Dic191D411yescharacteristic
101C4⋊Dic191C2311yescharacteristic
102C22⋊Dic191—21no—
103C22⋊Dic191—21no—
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4—21no—
106(C2×D191)⋊C4—21no—
107(C2×D191)⋊C4—21no—
108(C2×D191)⋊C4—21no—
109(C2×D191)⋊C4D411yescharacteristic
110(C2×D191)⋊C4D411yescharacteristic
111C2×D4×C191—21no—
112C2×D4×C191—21no—
113C2×D4×C191C2311yescharacteristic
114C2×D4×C191C2311yescharacteristic
115C2×D4×C191C2311yescharacteristic
116C22×C4×C191—21no—
117D8×C191—22no—
118D8×C191—22no—
119C2×C8×C191C2311yescharacteristic
120C2×C8×C191C2311yescharacteristic
121C4⋊C4×C191C2311yescharacteristic
122C22⋊C4×C191—21no—
123C42×C191C2311yescharacteristic
124C22×D191—21no—
125C22×D191—21no—
126C22×D191—42no—
127C22×D191—42no—
128C22×D191—42no—
129C22×D191—42no—
130C22×D191—42no—
131C22×D191—42no—
132C22×D191—81no—
133C22×D191—81no—
134C22×D191—81no—
135C22×D191C2×D411yescharacteristic
136C22×D191C2×D411yescharacteristic
137Dic382—21no—
138Dic382—21no—
139D764—22no—
140D764—21no—
141D764—21no—
142D764—21no—
143D764—81no—
144D764C2×D412yes—
145C22⋊D191—42no—
146C22⋊D191—42no—
147C22⋊D191—42no—
148C22⋊D191—42no—
149C22⋊D191—42no—
150C22⋊D191—42no—
151C2×Dic191—21no—
152C2×Dic191—21no—
153C2×Dic191—42no—
154C2×Dic191C2×D411yescharacteristic
155C2×Dic191C2×D411yescharacteristic
156C4×D191—21no—
157C4×D191—21no—
158C4×D191—21no—
159C4×D191—21no—
160C4×D191—42no—
161C191⋊C8—21no—
162C191⋊C8C2×D412yescomplemented
163C23×C191—21no—
164C23×C191—21no—
165C23×C191—21no—
166D4×C191—21no—
167D4×C191—21no—
168D4×C191—21no—
169D4×C191—21no—
170D4×C191—21no—
171D4×C191—42no—
172D4×C191—81no—
173D4×C191C2×D412yescomplemented
174C2×C4×C191—21no—
175C2×C4×C191—81no—
176C2×C4×C191C2×D411yescharacteristic
177C2×C4×C191C2×D411yescharacteristic
178C2×C4×C191C2411yescharacteristic
179C8×C191—21no—
180C8×C191—21no—
181C22×C191—22no—
182C22×C191—41no—
183C22×C191—41no—
184C22×C191—41no—
185C22×C191—41no—
186C22×C191—41no—
187C22×C191—41no—
188C22×C191—41no—
189C22×C191—41no—
190C22×C191C22×D411yescharacteristic
191C2×D191—21no—
192C2×D191—21no—
193C2×D191—21no—
194C2×D191—22no—
195C2×D191—21no—
196C2×D191—21no—
197C2×D191—41no—
198C2×D191—41no—
199C2×D191—41no—
200C2×D191—41no—
201C2×D191—41no—
202C2×D191—84no—
203C2×D191—84no—
204C2×D191—84no—
205C2×D191AGL(1,ℤ/8)12yescomplemented
206C4×C191—21no—
207C4×C191—21no—
208C4×C191—41no—
209C4×C191C22×D411yescharacteristic
210C4×C191H(5,𝔽2)11yescharacteristic
211Dic191—21no—
212Dic191—21no—
213Dic191—41no—
214Dic191—41no—
215C2×C191—42no—
216C2×C191—81no—
217C2×C191—81no—
218C2×C191D4211yescharacteristic
219C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C191#57611yescharacteristic
221D191—22no—
222D191—41no—
223D191—42no—
224D191—81no—
225C191#326411yescharacteristic, complemented, Sylow
226#3264—1911nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1911no—
228C22×D8—1911no—
229#546—1911no—
230D42—1911no—
231#496—1911no—
232#469—1911no—
233#465—1911no—
234#463—1911no—
235#459D19111yescharacteristic, complemented
236#450—1911no—
237#449—1911no—
238#447—1911no—
239#447—1911no—
240#442—1911no—
241#435—1911no—
242C2×#41—1911no—
243C22×D4—3821no—
244C22×D4—1911no—
245C22×D4—1911no—
246C22×D4—1911no—
247AGL(1,ℤ/8)—3824no—
248C2×SD16—1911no—
249C2×SD16—1911no—
250C2×SD16—1911no—
251C2×D8—3824no—
252C2×D8—1911no—
253C2×D8—1911no—
254C2×D8—1911no—
255C2×D8—1911no—
256C2×D8C2×D19111yescharacteristic, complemented
257C2×D8C2×D19111yescharacteristic, complemented
258C2×M16—1911no—
259C22×C8—1911no—
260#127C2×D19111yescharacteristic, complemented
261#124—1911no—
262#123—3821no—
263C22⋊Q8—1911no—
264#121—3821no—
265#121—3821no—
266#121—1911no—
267#121—1911no—
268#121—1911no—
269#120—3821no—
270#120—3821no—
271#120—3821no—
272C4×D4—1911no—
273C4×D4—1911no—
274C4×D4C2×D19111yescharacteristic, complemented
275C2×C22⋊C4—3821no—
276#106—1911no—
277C4⋊C8C2×D19111yescharacteristic, complemented
278#103—1911no—
279#102—1911no—
280#102—1911no—
281#102—1911no—
282#102C2×D19111yescharacteristic, complemented
283#102C2×D19111yescharacteristic, complemented
284C22⋊C8—1911no—
285C22⋊C8—1911no—
286#97—1911no—
287C24—3821no—
288C24—3821no—
289C24—3821no—
290#41—3822no—
291#41—7641no—
292C2×Q8—1911no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—3821no—
296C2×D4—3822no—
297C2×D4—3822no—
298C2×D4—3822no—
299C2×D4—3821no—
300C2×D4—3821no—
301C2×D4—3821no—
302C2×D4—3821no—
303C2×D4—3821no—
304C2×D4—3821no—
305C2×D4—1911no—
306C2×D4—1911no—
307C2×D4—1911no—
308C2×D4—7641no—
309C2×D4—7641no—
310C2×D4—7641no—
311C2×D4—7644no—
312C2×D4C22×D19111yescharacteristic
313C2×D4C22×D19111yescharacteristic
314C2×D4C22×D19111yescharacteristic
315C22×C4—21no—
316C22×C4—3821no—
317C22×C4—3821no—
318C22×C4—1911no—
319C22×C4—1911no—
320SD16—3822no—
321SD16—3822no—
322SD16—7641no—
323D8—22no—
324D8—22no—
325D8—3822no—
326D8—3822no—
327D8—1914no—
328D8—7641no—
329M16—3822no—
330C2×C8—3822no—
331C2×C8—1911no—
332C2×C8—1911no—
333C2×C8C22×D19111yescharacteristic
334C2×C8C22×D19111yescharacteristic
335C4⋊C4—3821no—
336C4⋊C4—1911no—
337C4⋊C4—1911no—
338C4⋊C4C22×D19111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—3821no—
341C22⋊C4—3821no—
342C22⋊C4—3821no—
343C22⋊C4—3821no—
344C22⋊C4—3821no—
345C22⋊C4—3821no—
346C22⋊C4—3821no—
347C22⋊C4—1911no—
348C22⋊C4—1911no—
349C42C22×D19111yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—3821no—
354C23—3821no—
355C23—1911no—
356C23—1911no—
357C23—15281no—
358C23—15281no—
359C23—15281no—
360C23—7642no—
361C23—7642no—
362C23—7642no—
363C23—7642no—
364C23—7642no—
365C23—7642no—
366Q8—3821no—
367Q8—3821no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—42no—
374D4—81no—
375D4—3822no—
376D4—3821no—
377D4—3821no—
378D4—3821no—
379D4—1912no—
380D4—15281no—
381D4—7642no—
382D4—7642no—
383D4—7642no—
384D4—7642no—
385D4—7642no—
386D4—7642no—
387D4D4×D19112yescomplemented
388C2×C4—21no—
389C2×C4—81no—
390C2×C4—3821no—
391C2×C4—3821no—
392C2×C4—3821no—
393C2×C4—3821no—
394C2×C4—3821no—
395C2×C4—3821no—
396C2×C4—1911no—
397C2×C4—1911no—
398C2×C4—7642no—
399C2×C4—7642no—
400C2×C4D4×D19111yescharacteristic
401C2×C4D4×D19111yescharacteristic
402C2×C4C23×D19111yescharacteristic
403C8—21no—
404C8—21no—
405C8—3821no—
406C8—1912no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—3821no—
417C22—3821no—
418C22—3821no—
419C22—3821no—
420C22—3821no—
421C22—3822no—
422C22—1912no—
423C22—15284no—
424C22—15284no—
425C22—15284no—
426C22—7641no—
427C22—7641no—
428C22—7641no—
429C22—7641no—
430C22—7641no—
431C22C2×D4×D19111yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—3821no—
436C4—3821no—
437C4—7641no—
438C4—7641no—
439C4C2×D4×D19111yescharacteristic
440C4#2025936811yescharacteristic
441C2—42no—
442C2—81no—
443C2—81no—
444C2—3822no—
445C2—15281no—
446C2—7641no—
447C2—7642no—
448C2#2713329711yescharacteristic
449C2AGL(1,ℤ/8)×D19111yescharacteristic
450C2#2713346411yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#459×C191C2
socleC22×C191C22×D4
hypercenter#459D191
hypocenterC191#3264
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#20259346C2semidirect
2. C2×D8×D191C2semidirect
3. #27133308C2semidirect
4. #27133298C2semidirect
5. #27133050C2semidirect
6. #27132915C2semidirect
7. #27132891C2semidirect
8. #27132879C2semidirect
9. #27132815C2semidirect
10. #27132805C2semidirect
11. #27132798C2semidirect
12. #27132795C2semidirect
13. #27132773C2semidirect
14. #27132733C2semidirect
15. #459×C191C2semidirect
16. C2×#15690286C22semidirect
19. C2×D4×D191C22semidirect
20. C2×D4×D191C22semidirect
22. C2×#15690276C22semidirect
23. C2×#15690275C22semidirect
24. C2×#15690275C22semidirect
27. C2×#15690272C22semidirect
28. C2×#15690272C22semidirect
30. C2×#15690270C22semidirect
31. C2×C8×D191C22semidirect
32. #20259310C22semidirect
34. #20259302C22semidirect
35. #20259297C22semidirect
39. #20259293C22semidirect
44. C4×D764C22semidirect
46. #20259253C22semidirect
47. #20259247C22semidirect
48. #20259244C22semidirect
49. #20259243C22semidirect
50. #20259242C22semidirect
51. (C2×D191)⋊C8C22semidirect
52. (C2×D191)⋊C8C22semidirect
53. #20259233C22semidirect
54. C2×D8×C191C22semidirect
55. C2×D8×C191C22semidirect
56. #127×C191C22semidirect
57. C4×D4×C191C22semidirect
58. C4⋊C8×C191C22semidirect
59. #102×C191C22semidirect
60. #102×C191C22semidirect
94. #15690272D4semidirect
162. C191⋊C8C2×D4semidirect
173. D4×C191C2×D4semidirect
205. C2×D191AGL(1,ℤ/8)semidirect
225. C191#3264semidirect
235. #459D191semidirect
256. C2×D8C2×D191semidirect
257. C2×D8C2×D191semidirect
260. #127C2×D191semidirect
274. C4×D4C2×D191semidirect
277. C4⋊C8C2×D191semidirect
282. #102C2×D191semidirect
283. #102C2×D191semidirect
387. D4D4×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593462
C2×D8×D1912
#271333082
#271332982
#271330502
#271329152
#271328912
#271328792
#271328152
#271328052
#271327982
#271327952
#271327732
#271327332
#459×C1912
#3264191