Finite Group Databasebeta

← #37584014

Subgroups — #37584014

Order 24448.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259346C211yesmaximal, characteristic, complemented
2C2×SD16×D191C211yesmaximal, characteristic, complemented
3#27133321C211yesmaximal, characteristic, complemented
4#27133298C211yesmaximal, characteristic, complemented
5#27132906C211yesmaximal, characteristic, complemented
6#27132896C211yesmaximal, characteristic, complemented
7#27132875C211yesmaximal, characteristic, complemented
8#27132860C211yesmaximal, characteristic, complemented
9#27132812C211yesmaximal, characteristic, complemented
10#27132805C211yesmaximal, characteristic, complemented
11#27132800C211yesmaximal, characteristic, complemented
12#27132795C211yesmaximal, characteristic, complemented
13#27132765C211yesmaximal, characteristic, complemented
14#27132736C211yesmaximal, characteristic, complemented
15#496×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic, complemented
17C2×Q8×D191C2211yescharacteristic, complemented
18C2×D4×D191—21no—
19C2×D4×D191C2211yescharacteristic
20C2×D4×D191C2211yescharacteristic
21#20259346—24no—
22C2×#15690276C2211yescharacteristic
23C2×#15690275C2211yescharacteristic, complemented
24C2×#15690275C2211yescharacteristic, complemented
25C2×#15690274C2211yescharacteristic
26SD16×D191—24no—
27C2×#15690272C2211yescharacteristic
28C2×#15690270C2211yescharacteristic, complemented
29C2×C8×D191C2211yescharacteristic, complemented
30#20259308—21no—
31#20259299—21no—
32#20259298C2211yescharacteristic, complemented
33#20259295—21no—
34#20259293—21no—
35#20259293C2211yescharacteristic
36#20259293C2211yescharacteristic
37#20259293C2211yescharacteristic, complemented
38#20259290—21no—
39#20259290—21no—
40C4×Dic382C2211yescharacteristic, complemented
41C4×D764C2211yescharacteristic
42C4×D764C2211yescharacteristic
43(C22⋊D191)⋊C4—21no—
44#20259256C2211yescharacteristic, complemented
45C4⋊(C191⋊C8)C2211yescharacteristic, complemented
46#20259247C2211yescharacteristic, complemented
47#20259246C2211yescharacteristic, complemented
48#20259244C2211yescharacteristic
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
59#124×C191C2211yescharacteristic, complemented
60#97×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62C23×D191—21no—
63#15690286—22no—
64#15690286—41no—
65C2×Dic382—21no—
66C2×Dic382C2311yescharacteristic
67Q8×D191—22no—
68Q8×D191—41no—
69C2×D764D411yescharacteristic
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—41no—
80D4×D191—41no—
81D4×D191—44no—
82C22×Dic191—21no—
83C2×C4×D191—21no—
84C2×C4×D191—21no—
85C2×C4×D191C2311yescharacteristic
86C2×C4×D191C2311yescharacteristic
87#15690276D414yescomplemented
88#15690275—22no—
89#15690275—22no—
90#15690274—22no—
91#15690272—22no—
92#15690270—22no—
93C2×C191⋊C8C2311yescharacteristic
94C2×C191⋊C8C2311yescharacteristic
95C8×D191—22no—
96C4⋊Dic191D411yescharacteristic
97C4⋊Dic191C2311yescharacteristic
98C4⋊Dic191C2311yescharacteristic
99Dic191⋊C4—21no—
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)⋊C4D411yescharacteristic
107(C2×D191)⋊C4D411yescharacteristic
108C4×Dic191—21no—
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—42no—
126C22×D191—42no—
127C22×D191—42no—
128C22×D191—42no—
129C22×D191—81no—
130C22×D191—81no—
131C22×D191C2×D411yescharacteristic
132C22×D191C2×D411yescharacteristic
133Dic382—21no—
134Dic382—42no—
135Dic382C2×D412yescomplemented
136D764—22no—
137D764—21no—
138D764C2×D412yes—
139C22⋊D191—42no—
140C22⋊D191—42no—
141C22⋊D191—42no—
142C22⋊D191—42no—
143C22⋊D191—42no—
144C22⋊D191—42no—
145C2×Dic191—21no—
146C2×Dic191—21no—
147C2×Dic191—21no—
148C2×Dic191—42no—
149C2×Dic191C2×D411yescharacteristic
150C2×Dic191C2×D411yescharacteristic
151C4×D191—21no—
152C4×D191—21no—
153C4×D191—21no—
154C4×D191—21no—
155C4×D191—42no—
156C4×D191—42no—
157C191⋊C8—21no—
158C191⋊C8—21no—
159C23×C191—21no—
160C23×C191—21no—
161C23×C191—21no—
162Q8×C191—21no—
163Q8×C191—21no—
164D4×C191—21no—
165D4×C191—21no—
166D4×C191—21no—
167D4×C191—21no—
168D4×C191—21no—
169D4×C191—21no—
170D4×C191—42no—
171D4×C191—81no—
172C2×C4×C191—21no—
173C2×C4×C191C2×D411yescharacteristic
174C2×C4×C191C2×D411yescharacteristic
175C2×C4×C191C2411yescharacteristic
176C8×C191—21no—
177C8×C191C2×D412yescomplemented
178C22×C191—41no—
179C22×C191—41no—
180C22×C191—41no—
181C22×C191—41no—
182C22×C191—41no—
183C22×C191—41no—
184C22×C191—41no—
185C22×C191—41no—
186C22×C191—41no—
187C22×C191C22×D411yescharacteristic
188C2×D191—21no—
189C2×D191—21no—
190C2×D191—21no—
191C2×D191—22no—
192C2×D191—21no—
193C2×D191—21no—
194C2×D191—41no—
195C2×D191—41no—
196C2×D191—84no—
197C2×D191—84no—
198C2×D191AGL(1,ℤ/8)12yescomplemented
199C4×C191—21no—
200C4×C191—21no—
201C4×C191—41no—
202C4×C191C22×D411yescharacteristic
203C4×C191H(5,𝔽2)11yescharacteristic
204Dic191—21no—
205Dic191—21no—
206Dic191—22no—
207Dic191—41no—
208Dic191—41no—
209C2×C191—81no—
210C2×C191—81no—
211C2×C191—81no—
212C2×C191D4211yescharacteristic
213C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C191#57711yescharacteristic
215D191—22no—
216D191—41no—
217D191—42no—
218C191#326511yescharacteristic, complemented, Sylow
219#3265—1911nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1911no—
221C22×SD16—1911no—
222#548—1911no—
223D42—1911no—
224#496D19111yescharacteristic, complemented
225#468—1911no—
226#466—1911no—
227#463—1911no—
228#460—1911no—
229#450—1911no—
230#449—1911no—
231#448—1911no—
232#447—1911no—
233#440—1911no—
234#435—1911no—
235C2×#41—1911no—
236C22×Q8—1911no—
237C22×D4—3821no—
238C22×D4—1911no—
239C22×D4—1911no—
240AGL(1,ℤ/8)—3824no—
241C2×SD16—3824no—
242C2×SD16—1911no—
243C2×SD16—1911no—
244C2×SD16—1911no—
245C2×SD16—1911no—
246C2×SD16C2×D19111yescharacteristic, complemented
247C2×SD16C2×D19111yescharacteristic, complemented
248C2×D8—1911no—
249C2×D8C2×D19111yescharacteristic, complemented
250C2×D8C2×D19111yescharacteristic, complemented
251C2×M16—1911no—
252C22×C8—1911no—
253#127C2×D19111yescharacteristic
254#124—3821no—
255#124C2×D19111yescharacteristic, complemented
256C22⋊Q8—3821no—
257C22⋊Q8—1911no—
258#121—3821no—
259#121—3821no—
260#121—1911no—
261#121—1911no—
262#121—1911no—
263#120—3821no—
264#120—3821no—
265C4×Q8—1911no—
266C4×D4—3821no—
267C4×D4—1911no—
268C4×D4—1911no—
269#107—1911no—
270C4⋊C8—1911no—
271#103—1911no—
272#103—1911no—
273#102—1911no—
274#102—1911no—
275#102—1911no—
276#102—1911no—
277C22⋊C8—1911no—
278C22⋊C8—1911no—
279#97C2×D19111yescharacteristic, complemented
280C24—3821no—
281C24—3821no—
282#41—3822no—
283#41—7641no—
284C2×Q8—3821no—
285C2×Q8—3822no—
286C2×Q8—1911no—
287C2×Q8—7641no—
288C2×Q8C22×D19111yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—3822no—
292C2×D4—3822no—
293C2×D4—3821no—
294C2×D4—3821no—
295C2×D4—3821no—
296C2×D4—3821no—
297C2×D4—3821no—
298C2×D4—3821no—
299C2×D4—1911no—
300C2×D4—1911no—
301C2×D4—7641no—
302C2×D4—7641no—
303C2×D4—7644no—
304C2×D4C22×D19111yescharacteristic
305C2×D4C22×D19111yescharacteristic
306C2×D4C22×D19111yescharacteristic
307C22×C4—3821no—
308C22×C4—3821no—
309C22×C4—3821no—
310C22×C4—1911no—
311C22×C4—1911no—
312SD16—22no—
313SD16—41no—
314SD16—3822no—
315SD16—3822no—
316SD16—3822no—
317SD16—1914no—
318D8—22no—
319D8—41no—
320D8—3822no—
321M16—3822no—
322C2×C8—3822no—
323C2×C8—1911no—
324C2×C8—1911no—
325C2×C8C22×D19111yescharacteristic
326C2×C8C22×D19111yescharacteristic
327C4⋊C4—3821no—
328C4⋊C4—3821no—
329C4⋊C4—1911no—
330C4⋊C4—1911no—
331C4⋊C4—1911no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—3821no—
335C22⋊C4—3821no—
336C22⋊C4—3821no—
337C22⋊C4—3821no—
338C22⋊C4—3821no—
339C22⋊C4—1911no—
340C22⋊C4—1911no—
341C42—3821no—
342C42C22×D19111yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—3821no—
347C23—1911no—
348C23—1911no—
349C23—15281no—
350C23—15281no—
351C23—7642no—
352C23—7642no—
353C23—7642no—
354C23—7642no—
355Q8—21no—
356Q8—21no—
357Q8—3821no—
358Q8—1912no—
359Q8—7642no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—81no—
368D4—3822no—
369D4—3821no—
370D4—1912no—
371D4—7642no—
372D4—7642no—
373D4—7642no—
374D4—7642no—
375D4—7642no—
376D4—7642no—
377C2×C4—21no—
378C2×C4—3821no—
379C2×C4—3821no—
380C2×C4—3821no—
381C2×C4—3821no—
382C2×C4—3821no—
383C2×C4—3821no—
384C2×C4—3821no—
385C2×C4—1911no—
386C2×C4—1911no—
387C2×C4—7642no—
388C2×C4—7642no—
389C2×C4—7642no—
390C2×C4D4×D19111yescharacteristic
391C2×C4D4×D19111yescharacteristic
392C2×C4C23×D19111yescharacteristic
393C8—21no—
394C8—3821no—
395C8—3821no—
396C8D4×D19112yescomplemented
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—3821no—
407C22—3821no—
408C22—3821no—
409C22—3821no—
410C22—3821no—
411C22—3822no—
412C22—1912no—
413C22—15284no—
414C22—15284no—
415C22—7641no—
416C22—7641no—
417C22C2×D4×D19111yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—3821no—
422C4—3821no—
423C4—3822no—
424C4—7641no—
425C4—7641no—
426C4C2×D4×D19111yescharacteristic
427C4#2025936711yescharacteristic
428C2—81no—
429C2—81no—
430C2—81no—
431C2—3822no—
432C2—7641no—
433C2—7642no—
434C2#2713329811yescharacteristic
435C2AGL(1,ℤ/8)×D19111yescharacteristic
436C2#2713347311yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#496×C191C2
socleC22×C191C22×D4
hypercenter#496D191
hypocenterC191#3265
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×#20259346C2semidirect
2. C2×SD16×D191C2semidirect
3. #27133321C2semidirect
4. #27133298C2semidirect
5. #27132906C2semidirect
6. #27132896C2semidirect
7. #27132875C2semidirect
8. #27132860C2semidirect
9. #27132812C2semidirect
10. #27132805C2semidirect
11. #27132800C2semidirect
12. #27132795C2semidirect
13. #27132765C2semidirect
14. #27132736C2semidirect
15. #496×C191C2semidirect
16. C2×#15690286C22semidirect
17. C2×Q8×D191C22semidirect
23. C2×#15690275C22semidirect
24. C2×#15690275C22semidirect
28. C2×#15690270C22semidirect
29. C2×C8×D191C22semidirect
32. #20259298C22semidirect
37. #20259293C22semidirect
40. C4×Dic382C22semidirect
44. #20259256C22semidirect
45. C4⋊(C191⋊C8)C22semidirect
46. #20259247C22semidirect
47. #20259246C22semidirect
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
59. #124×C191C22semidirect
60. #97×C191C22semidirect
87. #15690276D4semidirect
135. Dic382C2×D4semidirect
177. C8×C191C2×D4semidirect
198. C2×D191AGL(1,ℤ/8)semidirect
218. C191#3265semidirect
224. #496D191semidirect
246. C2×SD16C2×D191semidirect
247. C2×SD16C2×D191semidirect
249. C2×D8C2×D191semidirect
250. C2×D8C2×D191semidirect
255. #124C2×D191semidirect
279. #97C2×D191semidirect
396. C8D4×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593462
C2×SD16×D1912
#271333212
#271332982
#271329062
#271328962
#271328752
#271328602
#271328122
#271328052
#271328002
#271327952
#271327652
#271327362
#496×C1912
#3265191