Finite Group Databasebeta

← #37584017

Subgroups — #37584017

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×#20259349C211yesmaximal, characteristic, complemented
2C2×SD16×D191C211yesmaximal, characteristic, complemented
3#27133325C211yesmaximal, characteristic, complemented
4D4×D764C211yesmaximal, characteristic, complemented
5#27133052C211yesmaximal, characteristic, complemented
6#27132910C211yesmaximal, characteristic, complemented
7#27132900C211yesmaximal, characteristic, complemented
8#27132878C211yesmaximal, characteristic, complemented
9#27132863C211yesmaximal, characteristic, complemented
10#27132815C211yesmaximal, characteristic, complemented
11#27132811C211yesmaximal, characteristic, complemented
12#27132803C211yesmaximal, characteristic, complemented
13#27132798C211yesmaximal, characteristic, complemented
14#27132736C211yesmaximal, characteristic, complemented
15#440×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic, complemented
17C2×Q8×D191C2211yescharacteristic, complemented
18C22×D764—21no—
19C2×D4×D191C2211yescharacteristic
20C2×D4×D191C2211yescharacteristic
21#20259349—24no—
22C2×#15690276C2211yescharacteristic, complemented
23C2×#15690276C2211yescharacteristic, complemented
24C2×#15690275C2211yescharacteristic
25C2×#15690274C2211yescharacteristic, complemented
26C2×#15690274C2211yescharacteristic, complemented
27SD16×D191—24no—
28C2×D1528C2211yescharacteristic, complemented
29C2×D1528C2211yescharacteristic, complemented
30C2×#15690272C2211yescharacteristic
31C2×#15690270C2211yescharacteristic, complemented
32C2×C8×D191C2211yescharacteristic, complemented
33#20259319C2211yescharacteristic
34#20259310—21no—
35#20259310C2211yescharacteristic, complemented
36#20259302—21no—
37#20259302C2211yescharacteristic, complemented
38#20259297—21no—
39#20259297C2211yescharacteristic
40#20259297C2211yescharacteristic
41#20259297C2211yescharacteristic, complemented
42#20259296—21no—
43#20259292—21no—
44#20259292—21no—
45C4×D764—21no—
46C4×D764C2211yescharacteristic
47C4⋊(C191⋊C8)C2211yescharacteristic, complemented
48#20259245C2211yescharacteristic, complemented
49#20259242C2211yescharacteristic, complemented
50#20259242C2211yescharacteristic, complemented
51#20259242C2211yescharacteristic, complemented
52(C2×D191)⋊C8C2211yescharacteristic, complemented
53(C2×D191)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C191C2211yescharacteristic
55C4×Q8×C191C2211yescharacteristic, complemented
56C4×D4×C191C2211yescharacteristic
57#107×C191C2211yescharacteristic, complemented
58#103×C191C2211yescharacteristic, complemented
59#102×C191C2211yescharacteristic
60#97×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62C23×D191—21no—
63#15690288—22no—
64#15690288—41no—
65C2×Dic382—21no—
66C2×Dic382C2311yescharacteristic
67Q8×D191—22no—
68Q8×D191—41no—
69C2×D764—21no—
70C2×D764—21no—
71C2×D764—21no—
72C2×D764—21no—
73C2×D764—21no—
74C2×D764—21no—
75C2×D764—44no—
76C2×D764D411yescharacteristic
77C2×D764C2311yescharacteristic
78C2×D764C2311yescharacteristic
79C2×D764C2311yescharacteristic
80C2×C22⋊D191—21no—
81C2×C22⋊D191—21no—
82D4×D191—22no—
83D4×D191—22no—
84D4×D191—41no—
85D4×D191—41no—
86C2×C4×D191—21no—
87C2×C4×D191—21no—
88C2×C4×D191C2311yescharacteristic
89C2×C4×D191C2311yescharacteristic
90#15690276—22no—
91#15690276—41no—
92#15690275—22no—
93#15690274—22no—
94#15690274—22no—
95D1528—22no—
96D1528—41no—
97#15690272—22no—
98#15690270—22no—
99C2×C191⋊C8C2311yescharacteristic
100C2×C191⋊C8C2311yescharacteristic
101C8×D191—22no—
102C4⋊Dic191—21no—
103C4⋊Dic191D411yescharacteristic
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)⋊C4—21no—
110(C2×D191)⋊C4D411yescharacteristic
111(C2×D191)⋊C4D411yescharacteristic
112C2×Q8×C191C2311yescharacteristic
113C2×D4×C191C2311yescharacteristic
114C22×C4×C191—21no—
115SD16×C191D414yescomplemented
116C2×C8×C191C2311yescharacteristic
117C2×C8×C191C2311yescharacteristic
118C4⋊C4×C191—21no—
119C4⋊C4×C191C2311yescharacteristic
120C4⋊C4×C191C2311yescharacteristic
121C22⋊C4×C191—21no—
122C42×C191—21no—
123C42×C191C2311yescharacteristic
124C22×D191—21no—
125C22×D191—21no—
126C22×D191—21no—
127C22×D191—42no—
128C22×D191—42no—
129C22×D191—42no—
130C22×D191—42no—
131C22×D191—81no—
132C22×D191—81no—
133C22×D191C2×D411yescharacteristic
134C22×D191C2×D411yescharacteristic
135Dic382—21no—
136Dic382—21no—
137Dic382—42no—
138D764—22no—
139D764—21no—
140D764—21no—
141D764—21no—
142D764—21no—
143D764—21no—
144D764—21no—
145D764—42no—
146D764—42no—
147D764—42no—
148D764—42no—
149D764—42no—
150D764—81no—
151C22⋊D191—42no—
152C22⋊D191—42no—
153C2×Dic191—21no—
154C2×Dic191C2×D411yescharacteristic
155C2×Dic191C2×D411yescharacteristic
156C4×D191—21no—
157C4×D191—21no—
158C4×D191—21no—
159C4×D191—21no—
160C4×D191—42no—
161C4×D191—42no—
162C191⋊C8—21no—
163C191⋊C8—21no—
164C23×C191—21no—
165Q8×C191—21no—
166Q8×C191C2×D412yescomplemented
167D4×C191—21no—
168D4×C191C2×D412yes—
169C2×C4×C191—21no—
170C2×C4×C191—21no—
171C2×C4×C191—21no—
172C2×C4×C191—42no—
173C2×C4×C191C2×D411yescharacteristic
174C2×C4×C191C2×D411yescharacteristic
175C2×C4×C191C2411yescharacteristic
176C8×C191—21no—
177C8×C191C2×D412yescomplemented
178C22×C191—22no—
179C22×C191—41no—
180C22×C191—41no—
181C22×C191C22×D411yescharacteristic
182C2×D191—21no—
183C2×D191—21no—
184C2×D191—21no—
185C2×D191—21no—
186C2×D191—21no—
187C2×D191—41no—
188C2×D191—41no—
189C2×D191—41no—
190C2×D191—41no—
191C2×D191—41no—
192C2×D191—41no—
193C2×D191—41no—
194C2×D191—41no—
195C2×D191—41no—
196C2×D191—84no—
197C2×D191—84no—
198C2×D191AGL(1,ℤ/8)12yescomplemented
199C4×C191—22no—
200C4×C191—21no—
201C4×C191—21no—
202C4×C191—41no—
203C4×C191—41no—
204C4×C191C22×D411yescharacteristic
205C4×C191H(5,𝔽2)11yescharacteristic
206Dic191—21no—
207Dic191—21no—
208Dic191—41no—
209C2×C191—42no—
210C2×C191D4211yescharacteristic
211C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
212C2×C191#57711yescharacteristic
213D191—22no—
214D191—41no—
215D191—81no—
216D191—81no—
217D191—81no—
218C191#326511yescharacteristic, complemented, Sylow
219#3265—1911nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1911no—
221C22×SD16—1911no—
222#548—1911no—
223D42—1911no—
224#496—1911no—
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#440D19111yescharacteristic, complemented
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×SD16—1911no—
247C2×SD16C2×D19111yescharacteristic
248C2×D8—1911no—
249C2×D8—1911no—
250C2×D8—1911no—
251C2×M16—1911no—
252C22×C8—1911no—
253#127—1911no—
254#124—3821no—
255#124—1911no—
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×Q8C2×D19111yescharacteristic, complemented
266C4×D4—3821no—
267C4×D4—1911no—
268C4×D4C2×D19111yescharacteristic
269#107C2×D19111yescharacteristic, complemented
270C4⋊C8—1911no—
271#103—1911no—
272#103C2×D19111yescharacteristic, complemented
273#102—1911no—
274#102—1911no—
275#102—1911no—
276#102C2×D19111yescharacteristic
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—3821no—
290C2×D4—3821no—
291C2×D4—3821no—
292C2×D4—3821no—
293C2×D4—3822no—
294C2×D4—3822no—
295C2×D4—3821no—
296C2×D4—3821no—
297C2×D4—3821no—
298C2×D4—3821no—
299C2×D4—1911no—
300C2×D4—1911no—
301C2×D4—1911no—
302C2×D4—1911no—
303C2×D4—7644no—
304C2×D4—7641no—
305C2×D4—7641no—
306C2×D4C22×D19111yescharacteristic
307C22×C4—21no—
308C22×C4—3821no—
309C22×C4—3821no—
310C22×C4—1911no—
311C22×C4—1911no—
312SD16—3822no—
313SD16—3822no—
314SD16—3822no—
315SD16—3822no—
316SD16—7641no—
317SD16D76414yescomplemented
318D8—3822no—
319D8—3822no—
320D8—7641no—
321M16—3822no—
322C2×C8—3822no—
323C2×C8—1911no—
324C2×C8—1911no—
325C2×C8C22×D19111yescharacteristic
326C2×C8C22×D19111yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—3821no—
329C4⋊C4—1911no—
330C4⋊C4C22×D19111yescharacteristic
331C4⋊C4C22×D19111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—3821no—
334C22⋊C4—3821no—
335C22⋊C4—3821no—
336C22⋊C4—3821no—
337C22⋊C4—3821no—
338C22⋊C4—3821no—
339C22⋊C4—1911no—
340C22⋊C4—1911no—
341C42—21no—
342C42C22×D19111yescharacteristic
343C23—21no—
344C23—3821no—
345C23—3821no—
346C23—3821no—
347C23—1911no—
348C23—1911no—
349C23—15281no—
350C23—15281no—
351C23—7642no—
352C23—7642no—
353C23—7642no—
354C23—7642no—
355Q8—21no—
356Q8—3821no—
357Q8—3821no—
358Q8—7642no—
359Q8C2×D76412yescomplemented
360D4—21no—
361D4—3822no—
362D4—3821no—
363D4—3821no—
364D4—3821no—
365D4—3821no—
366D4—3821no—
367D4—3821no—
368D4—15281no—
369D4—7642no—
370D4—7642no—
371D4—7642no—
372D4—7642no—
373D4—7642no—
374D4—7642no—
375D4—7642no—
376D4C2×D76412yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—3821no—
382C2×C4—3821no—
383C2×C4—3821no—
384C2×C4—3821no—
385C2×C4—3821no—
386C2×C4—1911no—
387C2×C4—1911no—
388C2×C4—7642no—
389C2×C4—7642no—
390C2×C4D4×D19111yescharacteristic
391C2×C4D4×D19111yescharacteristic
392C2×C4C23×D19111yescharacteristic
393C8—21no—
394C8—3821no—
395C8—3821no—
396C8C2×D76412yescomplemented
397C22—22no—
398C22—41no—
399C22—41no—
400C22—3821no—
401C22—3821no—
402C22—3821no—
403C22—3821no—
404C22—3821no—
405C22—1912no—
406C22—15284no—
407C22—15284no—
408C22—7641no—
409C22—7641no—
410C22—7641no—
411C22—7641no—
412C22—7641no—
413C22—7641no—
414C22—7641no—
415C22—7641no—
416C22—7641no—
417C22C2×D4×D19111yescharacteristic
418C4—21no—
419C4—21no—
420C4—22no—
421C4—41no—
422C4—41no—
423C4—3821no—
424C4—3821no—
425C4—7641no—
426C4C22×D76411yescharacteristic
427C4#2025936811yescharacteristic
428C2—42no—
429C2—3822no—
430C2—15281no—
431C2—15281no—
432C2—15281no—
433C2—7641no—
434C2D4×D76411yescharacteristic
435C2AGL(1,ℤ/8)×D19111yescharacteristic
436C2#2713347211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#440×C191C2
socleC22×C191C22×D4
hypercenter#440D191
hypocenterC191#3265
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. C2×#20259349C2semidirect
2. C2×SD16×D191C2semidirect
3. #27133325C2semidirect
4. D4×D764C2semidirect
5. #27133052C2semidirect
6. #27132910C2semidirect
7. #27132900C2semidirect
8. #27132878C2semidirect
9. #27132863C2semidirect
10. #27132815C2semidirect
11. #27132811C2semidirect
12. #27132803C2semidirect
13. #27132798C2semidirect
14. #27132736C2semidirect
15. #440×C191C2semidirect
16. C2×#15690288C22semidirect
17. C2×Q8×D191C22semidirect
22. C2×#15690276C22semidirect
23. C2×#15690276C22semidirect
25. C2×#15690274C22semidirect
26. C2×#15690274C22semidirect
28. C2×D1528C22semidirect
29. C2×D1528C22semidirect
31. C2×#15690270C22semidirect
32. C2×C8×D191C22semidirect
35. #20259310C22semidirect
37. #20259302C22semidirect
41. #20259297C22semidirect
47. C4⋊(C191⋊C8)C22semidirect
48. #20259245C22semidirect
49. #20259242C22semidirect
50. #20259242C22semidirect
51. #20259242C22semidirect
52. (C2×D191)⋊C8C22semidirect
53. (C2×D191)⋊C8C22semidirect
55. C4×Q8×C191C22semidirect
57. #107×C191C22semidirect
58. #103×C191C22semidirect
60. #97×C191C22semidirect
115. SD16×C191D4semidirect
166. Q8×C191C2×D4semidirect
177. C8×C191C2×D4semidirect
198. C2×D191AGL(1,ℤ/8)semidirect
218. C191#3265semidirect
233. #440D191semidirect
265. C4×Q8C2×D191semidirect
269. #107C2×D191semidirect
272. #103C2×D191semidirect
279. #97C2×D191semidirect
317. SD16D764semidirect
359. Q8C2×D764semidirect
396. C8C2×D764semidirect

Maximal subgroups

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

subgroupindex
C2×#202593492
C2×SD16×D1912
#271333252
D4×D7642
#271330522
#271329102
#271329002
#271328782
#271328632
#271328152
#271328112
#271328032
#271327982
#271327362
#440×C1912
#3265191