Finite Group Databasebeta

← #37583999

Subgroups — #37583999

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×#20259349C211yesmaximal, characteristic, complemented
2C2×D8×D191C211yesmaximal, characteristic, complemented
3#27133306C211yesmaximal, characteristic, complemented
4#27133298C211yesmaximal, characteristic, complemented
5#27132913C211yesmaximal, characteristic, complemented
6#27132889C211yesmaximal, characteristic, complemented
7#27132875C211yesmaximal, characteristic, complemented
8#27132855C211yesmaximal, characteristic, complemented
9#27132812C211yesmaximal, characteristic, complemented
10#27132807C211yesmaximal, characteristic, complemented
11#27132795C211yesmaximal, characteristic, complemented
12#27132795C211yesmaximal, characteristic, complemented
13#27132776C211yesmaximal, characteristic, complemented
14#27132736C211yesmaximal, characteristic, complemented
15#496×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic, complemented
17C2×D4×D191—21no—
18C2×D4×D191C2211yescharacteristic
19C2×D4×D191C2211yescharacteristic, complemented
20C2×D4×D191C2211yescharacteristic, complemented
21#20259349—24no—
22C2×#15690274C2211yescharacteristic, complemented
23C2×D1528C2211yescharacteristic
24C2×#15690272C2211yescharacteristic
25C2×#15690272C2211yescharacteristic, complemented
26C2×#15690272C2211yescharacteristic, complemented
27D8×D191—24no—
28C2×#15690270C2211yescharacteristic, complemented
29C2×C8×D191C2211yescharacteristic, complemented
30#20259304—21no—
31#20259298C2211yescharacteristic, complemented
32#20259295—21no—
33#20259293—21no—
34#20259293C2211yescharacteristic
35#20259293C2211yescharacteristic, complemented
36#20259293C2211yescharacteristic, complemented
37#20259292—21no—
38#20259290—21no—
39#20259290—21no—
40C4×D764C2211yescharacteristic
41C4×D764C2211yescharacteristic, complemented
42C4×D764C2211yescharacteristic, complemented
43C22⋊C4×D191—21no—
44#20259254C2211yescharacteristic, complemented
45C4⋊(C191⋊C8)C2211yescharacteristic, complemented
46#20259246C2211yescharacteristic, complemented
47#20259244C2211yescharacteristic, complemented
48#20259244C2211yescharacteristic, complemented
49#20259243C2211yescharacteristic, complemented
50#20259243C2211yescharacteristic, complemented
51#20259243C2211yescharacteristic, complemented
52(C2×D191)⋊C8C2211yescharacteristic, complemented
53(C2×D191)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C191C2211yescharacteristic, complemented
55C2×SD16×C191C2211yescharacteristic, complemented
56C2×D8×C191C2211yescharacteristic, complemented
57C2×D8×C191C2211yescharacteristic, complemented
58#127×C191C2211yescharacteristic, complemented
59#124×C191C2211yescharacteristic, complemented
60#97×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62C23×D191—21no—
63C23×D191—21no—
64#15690288—22no—
65#15690288—41no—
66C2×D764—21no—
67C2×D764D411yescharacteristic
68C2×D764C2311yescharacteristic
69C2×D764C2311yescharacteristic
70C2×C22⋊D191—21no—
71C2×C22⋊D191—21no—
72C2×C22⋊D191—21no—
73C2×C22⋊D191—21no—
74C2×C22⋊D191—21no—
75C2×C22⋊D191—21no—
76D4×D191—22no—
77D4×D191—22no—
78D4×D191—22no—
79D4×D191—41no—
80D4×D191—41no—
81D4×D191—41no—
82D4×D191—44no—
83C2×C4×D191—21no—
84C2×C4×D191—21no—
85C2×C4×D191—21no—
86C2×C4×D191C2311yescharacteristic
87C2×C4×D191C2311yescharacteristic
88#15690274—22no—
89D1528D414yescomplemented
90#15690272—22no—
91#15690272—22no—
92#15690272—22no—
93#15690270—22no—
94C2×C191⋊C8C2311yescharacteristic
95C2×C191⋊C8C2311yescharacteristic
96C8×D191—22no—
97C4⋊Dic191D411yescharacteristic
98C4⋊Dic191C2311yescharacteristic
99C4⋊Dic191C2311yescharacteristic
100Dic191⋊C4—21no—
101C22⋊Dic191—21no—
102C22⋊Dic191—21no—
103C22⋊Dic191—21no—
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4—21no—
106(C2×D191)⋊C4—21no—
107(C2×D191)⋊C4D411yescharacteristic
108(C2×D191)⋊C4D411yescharacteristic
109C2×Q8×C191C2311yescharacteristic
110C2×D4×C191—21no—
111C2×D4×C191—21no—
112C2×D4×C191C2311yescharacteristic
113C2×D4×C191C2311yescharacteristic
114C2×D4×C191C2311yescharacteristic
115SD16×C191—22no—
116SD16×C191—41no—
117D8×C191—22no—
118D8×C191—41no—
119C2×C8×C191C2311yescharacteristic
120C2×C8×C191C2311yescharacteristic
121C22⋊C4×C191—21no—
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
137D764—22no—
138D764—21no—
139D764—21no—
140D764—42no—
141D764C2×D412yes—
142D764C2×D412yescomplemented
143C22⋊D191—42no—
144C22⋊D191—42no—
145C22⋊D191—42no—
146C22⋊D191—42no—
147C22⋊D191—42no—
148C22⋊D191—81no—
149C2×Dic191—21no—
150C2×Dic191—21no—
151C2×Dic191C2×D411yescharacteristic
152C2×Dic191C2×D411yescharacteristic
153C4×D191—21no—
154C4×D191—21no—
155C4×D191—21no—
156C4×D191—21no—
157C4×D191—42no—
158C4×D191—42no—
159C4×D191—81no—
160C191⋊C8—21no—
161C191⋊C8—21no—
162C23×C191—21no—
163C23×C191—21no—
164C23×C191—21no—
165Q8×C191—21no—
166Q8×C191—21no—
167D4×C191—21no—
168D4×C191—21no—
169D4×C191—21no—
170D4×C191—21no—
171D4×C191—21no—
172D4×C191—21no—
173D4×C191—42no—
174D4×C191—81no—
175C2×C4×C191—21no—
176C2×C4×C191C2×D411yescharacteristic
177C2×C4×C191C2×D411yescharacteristic
178C2×C4×C191C2411yescharacteristic
179C8×C191—21no—
180C8×C191C2×D412yescomplemented
181C22×C191—41no—
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—22no—
196C2×D191—21no—
197C2×D191—21no—
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—81no—
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—42no—
225C191#326411yescharacteristic, complemented, Sylow
226#3264—1911nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1911no—
228C22×D8—1911no—
229#546—1911no—
230D42—1911no—
231#496D19111yescharacteristic, complemented
232#469—1911no—
233#465—1911no—
234#463—1911no—
235#459—1911no—
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×SD16C2×D19111yescharacteristic, complemented
250C2×SD16C2×D19111yescharacteristic, complemented
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#124C2×D19111yescharacteristic, complemented
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×D4—1911no—
275C2×C22⋊C4—3821no—
276#106—1911no—
277C4⋊C8—1911no—
278#103—1911no—
279#102—1911no—
280#102—1911no—
281#102—1911no—
282#102—1911no—
283#102—1911no—
284C22⋊C8—1911no—
285C22⋊C8—1911no—
286#97C2×D19111yescharacteristic, complemented
287C24—3821no—
288C24—3821no—
289C24—3821no—
290#41—3822no—
291#41—7641no—
292C2×Q8C22×D19111yescharacteristic
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—3821no—
316C22×C4—3821no—
317C22×C4—3821no—
318C22×C4—1911no—
319C22×C4—1911no—
320SD16—22no—
321SD16—41no—
322SD16—3822no—
323D8—22no—
324D8—41no—
325D8—3822no—
326D8—3822no—
327D8—3822no—
328D8—1914no—
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⋊C4—1911no—
339C22⋊C4—21no—
340C22⋊C4—21no—
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—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—81no—
376D4—3822no—
377D4—3821no—
378D4—3821no—
379D4—1912no—
380D4—1912no—
381D4—15281no—
382D4—7642no—
383D4—7642no—
384D4—7642no—
385D4—7642no—
386D4—7642no—
387D4—7642no—
388C2×C4—21no—
389C2×C4—3821no—
390C2×C4—3821no—
391C2×C4—3821no—
392C2×C4—3821no—
393C2×C4—3821no—
394C2×C4—3821no—
395C2×C4—1911no—
396C2×C4—1911no—
397C2×C4—15281no—
398C2×C4—7642no—
399C2×C4—7642no—
400C2×C4D4×D19111yescharacteristic
401C2×C4D4×D19111yescharacteristic
402C2×C4C23×D19111yescharacteristic
403C8—21no—
404C8—3821no—
405C8—3821no—
406C8D4×D19112yescomplemented
407C22—41no—
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—3822no—
423C22—1912no—
424C22—15284no—
425C22—15284no—
426C22—15284no—
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#2025936711yescharacteristic
441C2—81no—
442C2—81no—
443C2—81no—
444C2—3822no—
445C2—7641no—
446C2—7642no—
447C2—7642no—
448C2#2713329811yescharacteristic
449C2AGL(1,ℤ/8)×D19111yescharacteristic
450C2#2713346711yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#496×C191C2
socleC22×C191C22×D4
hypercenter#496D191
hypocenterC191#3264
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#20259349C2semidirect
2. C2×D8×D191C2semidirect
3. #27133306C2semidirect
4. #27133298C2semidirect
5. #27132913C2semidirect
6. #27132889C2semidirect
7. #27132875C2semidirect
8. #27132855C2semidirect
9. #27132812C2semidirect
10. #27132807C2semidirect
11. #27132795C2semidirect
12. #27132795C2semidirect
13. #27132776C2semidirect
14. #27132736C2semidirect
15. #496×C191C2semidirect
16. C2×#15690288C22semidirect
19. C2×D4×D191C22semidirect
20. C2×D4×D191C22semidirect
22. C2×#15690274C22semidirect
25. C2×#15690272C22semidirect
26. C2×#15690272C22semidirect
28. C2×#15690270C22semidirect
29. C2×C8×D191C22semidirect
31. #20259298C22semidirect
35. #20259293C22semidirect
36. #20259293C22semidirect
41. C4×D764C22semidirect
42. C4×D764C22semidirect
44. #20259254C22semidirect
45. C4⋊(C191⋊C8)C22semidirect
46. #20259246C22semidirect
47. #20259244C22semidirect
48. #20259244C22semidirect
49. #20259243C22semidirect
50. #20259243C22semidirect
51. #20259243C22semidirect
52. (C2×D191)⋊C8C22semidirect
53. (C2×D191)⋊C8C22semidirect
54. C2×SD16×C191C22semidirect
55. C2×SD16×C191C22semidirect
56. C2×D8×C191C22semidirect
57. C2×D8×C191C22semidirect
58. #127×C191C22semidirect
59. #124×C191C22semidirect
60. #97×C191C22semidirect
89. D1528D4semidirect
142. D764C2×D4semidirect
180. C8×C191C2×D4semidirect
205. C2×D191AGL(1,ℤ/8)semidirect
225. C191#3264semidirect
231. #496D191semidirect
249. C2×SD16C2×D191semidirect
250. C2×SD16C2×D191semidirect
256. C2×D8C2×D191semidirect
257. C2×D8C2×D191semidirect
260. #127C2×D191semidirect
261. #124C2×D191semidirect
286. #97C2×D191semidirect
406. C8D4×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593492
C2×D8×D1912
#271333062
#271332982
#271329132
#271328892
#271328752
#271328552
#271328122
#271328072
#271327952
#271327762
#271327362
#496×C1912
#3264191