Finite Group Databasebeta

← #49132858

Subgroups — #49132858

Order 36224.

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×#22583638C211yesmaximal, characteristic, complemented
2C2×#22583637C211yesmaximal, characteristic, complemented
3#31827117C211yesmaximal, characteristic, complemented
4#31827099C211yesmaximal, characteristic, complemented
5#31826836C211yesmaximal, characteristic, complemented
6#31826712C211yesmaximal, characteristic, complemented
7#31826669C211yesmaximal, characteristic, complemented
8#31826659C211yesmaximal, characteristic, complemented
9#31826617C211yesmaximal, characteristic, complemented
10#31826615C211yesmaximal, characteristic, complemented
11#31826613C211yesmaximal, characteristic, complemented
12#31826608C211yesmaximal, characteristic, complemented
13#31826547C211yesmaximal, characteristic, complemented
14#31826541C211yesmaximal, characteristic, complemented
15#468×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic, complemented
17C2×#17985392C2211yescharacteristic, complemented
18C22×D1132C2211yescharacteristic
19C2×D4×D283—21no—
20C2×D4×D283C2211yescharacteristic
21#22583638—24no—
22#22583637—24no—
23C2×#17985381C2211yescharacteristic
24C2×#17985381C2211yescharacteristic, complemented
25C2×#17985380C2211yescharacteristic, complemented
26C2×#17985379C2211yescharacteristic
27C2×#17985379C2211yescharacteristic, complemented
28C2×D2264C2211yescharacteristic
29C2×D2264C2211yescharacteristic, complemented
30C2×#17985377C2211yescharacteristic, complemented
31C2×#17985375C2211yescharacteristic, complemented
32#22583608C2211yescharacteristic, complemented
33#22583606C2211yescharacteristic
34#22583595—21no—
35#22583592—21no—
36#22583590C2211yescharacteristic, complemented
37#22583585C2211yescharacteristic
38#22583583—21no—
39#22583583—21no—
40#22583580—21no—
41#22583578—21no—
42#22583572C2211yescharacteristic
43#22583572C2211yescharacteristic, complemented
44C4×C22⋊D283C2211yescharacteristic
45C4⋊C4×D283—21no—
46Dic283⋊C8C2211yescharacteristic, complemented
47#22583533C2211yescharacteristic, complemented
48#22583530C2211yescharacteristic
49#22583530C2211yescharacteristic, complemented
50#22583530C2211yescharacteristic, complemented
51C22⋊(C283⋊C8)C2211yescharacteristic, complemented
52(C2×D283)⋊C8C2211yescharacteristic, complemented
53C8×Dic283C2211yescharacteristic, complemented
54C2×SD16×C283C2211yescharacteristic, complemented
55C2×M16×C283C2211yescharacteristic, complemented
56C22⋊Q8×C283C2211yescharacteristic, complemented
57#121×C283C2211yescharacteristic
58#107×C283C2211yescharacteristic, complemented
59#103×C283C2211yescharacteristic, complemented
60#102×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62C23×D283—21no—
63#17985393—22no—
64#17985393—41no—
65#17985392—22no—
66#17985392—41no—
67C2×Dic566C2311yescharacteristic
68C2×D1132—22no—
69C2×D1132—21no—
70C2×D1132—21no—
71C2×D1132—21no—
72C2×D1132—41no—
73C2×D1132C2311yescharacteristic
74C2×D1132C2311yescharacteristic
75C2×D1132C2311yescharacteristic
76C2×C22⋊D283—21no—
77C2×C22⋊D283—21no—
78C2×C22⋊D283—21no—
79C2×C22⋊D283—21no—
80C2×C22⋊D283D411yescharacteristic
81D4×D283—22no—
82D4×D283—41no—
83D4×D283—44no—
84C2×C4×D283—21no—
85C2×C4×D283—21no—
86C2×C4×D283—21no—
87C2×C4×D283—21no—
88C2×C4×D283C2311yescharacteristic
89#17985381—22no—
90#17985381—22no—
91#17985380—41no—
92#17985379—22no—
93#17985379—41no—
94D2264—22no—
95D2264D414yescomplemented
96#17985377—22no—
97#17985375—22no—
98C2×C283⋊C8C2311yescharacteristic
99C2×C283⋊C8C2311yescharacteristic
100C4⋊Dic283—21no—
101Dic283⋊C4—21no—
102Dic283⋊C4—21no—
103Dic283⋊C4D411yescharacteristic
104C22⋊Dic283D411yescharacteristic
105(C2×D283)⋊C4—21no—
106(C2×D283)⋊C4—21no—
107(C2×D283)⋊C4—21no—
108(C2×D283)⋊C4—21no—
109(C2×D283)⋊C4D411yescharacteristic
110C4×Dic283C2311yescharacteristic
111C2×Q8×C283C2311yescharacteristic
112C2×D4×C283—21no—
113C2×D4×C283C2311yescharacteristic
114C22×C4×C283C2311yescharacteristic
115SD16×C283—22no—
116M16×C283—22no—
117C2×C8×C283C2311yescharacteristic
118C2×C8×C283C2311yescharacteristic
119C4⋊C4×C283—21no—
120C4⋊C4×C283C2311yescharacteristic
121C4⋊C4×C283C2311yescharacteristic
122C22⋊C4×C283—21no—
123C22⋊C4×C283—21no—
124C22×D283—21no—
125C22×D283—21no—
126C22×D283—21no—
127C22×D283—42no—
128C22×D283—42no—
129C22×D283—42no—
130C22×D283—42no—
131C22×D283—81no—
132C22×D283—81no—
133C22×D283C2×D411yescharacteristic
134Dic566—21no—
135Dic566—21no—
136D1132—21no—
137D1132—21no—
138D1132—21no—
139D1132—21no—
140D1132—42no—
141D1132—42no—
142D1132—42no—
143D1132C2×D412yes—
144D1132C2×D412yescomplemented
145C22⋊D283—22no—
146C22⋊D283—42no—
147C22⋊D283—42no—
148C22⋊D283—42no—
149C22⋊D283—81no—
150C2×Dic283—21no—
151C2×Dic283C2×D411yescharacteristic
152C2×Dic283C2×D411yescharacteristic
153C2×Dic283C2×D411yescharacteristic
154C4×D283—21no—
155C4×D283—21no—
156C4×D283—42no—
157C4×D283—42no—
158C4×D283—42no—
159C4×D283—42no—
160C283⋊C8—21no—
161C283⋊C8—21no—
162C23×C283—21no—
163C23×C283C2×D411yescharacteristic
164Q8×C283—21no—
165Q8×C283—21no—
166D4×C283—21no—
167D4×C283—21no—
168D4×C283—42no—
169C2×C4×C283—21no—
170C2×C4×C283—21no—
171C2×C4×C283—21no—
172C2×C4×C283—21no—
173C2×C4×C283—21no—
174C2×C4×C283C2×D411yescharacteristic
175C2×C4×C283C2411yescharacteristic
176C8×C283—21no—
177C8×C283C2×D412yescomplemented
178C22×C283—21no—
179C22×C283—21no—
180C22×C283—21no—
181C22×C283—41no—
182C22×C283—41no—
183C22×C283—41no—
184C22×C283C22×D411yescharacteristic
185C2×D283—21no—
186C2×D283—22no—
187C2×D283—22no—
188C2×D283—21no—
189C2×D283—21no—
190C2×D283—41no—
191C2×D283—41no—
192C2×D283—41no—
193C2×D283—41no—
194C2×D283—41no—
195C2×D283—41no—
196C2×D283—41no—
197C2×D283—84no—
198C2×D283—84no—
199C4×C283—21no—
200C4×C283—41no—
201C4×C283—41no—
202C4×C283—41no—
203C4×C283C22×D411yescharacteristic
204C4×C283H(5,𝔽2)11yescharacteristic
205Dic283—21no—
206Dic283—21no—
207Dic283—41no—
208Dic283AGL(1,ℤ/8)12yescomplemented
209C2×C283—41no—
210C2×C283—81no—
211C2×C283D4211yescharacteristic
212C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C283#57711yescharacteristic
214D283—41no—
215D283—42no—
216D283—42no—
217D283—81no—
218C283#327211yescharacteristic, complemented, Sylow
219#3272—2831nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2831no—
221C2×AGL(1,ℤ/8)—2831no—
222#547—2831no—
223D42—2831no—
224#492—2831no—
225#468—2831no—
226#468D28311yescharacteristic, complemented
227#461—2831no—
228#459—2831no—
229#450—2831no—
230#450—2831no—
231#449—2831no—
232#449—2831no—
233C4×D8—2831no—
234#436—2831no—
235C2×#41—2831no—
236C2×#41—2831no—
237C22×D4—2831no—
238C22×D4—2831no—
239C22×D4—5661no—
240AGL(1,ℤ/8)—5664no—
241AGL(1,ℤ/8)—5664no—
242C2×SD16—2831no—
243C2×SD16—2831no—
244C2×SD16—2831no—
245C2×SD16—2831no—
246C2×SD16—2831no—
247C2×SD16C2×D28311yescharacteristic, complemented
248C2×D8—2831no—
249C2×D8—2831no—
250C2×D8—2831no—
251C2×M16—2831no—
252C2×M16C2×D28311yescharacteristic, complemented
253C4⋊Q8—2831no—
254#127—2831no—
255#123—5661no—
256#123—5661no—
257C22⋊Q8—2831no—
258C22⋊Q8C2×D28311yescharacteristic, complemented
259#121—2831no—
260#121—5661no—
261#121—5661no—
262#121C2×D28311yescharacteristic
263#120—5661no—
264#120—5661no—
265C4×D4—2831no—
266C4×D4—2831no—
267C4×D4—2831no—
268C2×C4⋊C4—5661no—
269#107C2×D28311yescharacteristic, complemented
270C4⋊C8—2831no—
271#103—2831no—
272#103C2×D28311yescharacteristic, complemented
273#102—2831no—
274#102—2831no—
275#102—2831no—
276#102C2×D28311yescharacteristic, complemented
277C22⋊C8—2831no—
278C22⋊C8—2831no—
279C4×C8—2831no—
280C24—5661no—
281C24—5661no—
282#41—5662no—
283#41—5662no—
284#41—11321no—
285#41—11321no—
286C2×Q8—2831no—
287C2×Q8C22×D28311yescharacteristic
288C2×D4—21no—
289C2×D4—2831no—
290C2×D4—2831no—
291C2×D4—2831no—
292C2×D4—2831no—
293C2×D4—5662no—
294C2×D4—5661no—
295C2×D4—5661no—
296C2×D4—5661no—
297C2×D4—5661no—
298C2×D4—5661no—
299C2×D4—5662no—
300C2×D4—5661no—
301C2×D4—5661no—
302C2×D4—11321no—
303C2×D4—11321no—
304C2×D4—11324no—
305C2×D4C22×D28311yescharacteristic
306C22×C4—2831no—
307C22×C4—5661no—
308C22×C4—5661no—
309C22×C4—5661no—
310C22×C4—5661no—
311C22×C4C22×D28311yescharacteristic
312SD16—22no—
313SD16—5662no—
314SD16—5662no—
315SD16—5662no—
316SD16—11321no—
317SD16—11321no—
318D8—2834no—
319D8—5662no—
320D8—5662no—
321M16—22no—
322M16—5662no—
323C2×C8—2831no—
324C2×C8—2831no—
325C2×C8C22×D28311yescharacteristic
326C2×C8C22×D28311yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—2831no—
329C4⋊C4—5661no—
330C4⋊C4—5661no—
331C4⋊C4—5661no—
332C4⋊C4C22×D28311yescharacteristic
333C4⋊C4C22×D28311yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—2831no—
337C22⋊C4—2831no—
338C22⋊C4—5661no—
339C22⋊C4—5661no—
340C22⋊C4—5661no—
341C22⋊C4—5661no—
342C42—2831no—
343C23—21no—
344C23—2831no—
345C23—5661no—
346C23—5661no—
347C23—5661no—
348C23—11322no—
349C23—11322no—
350C23—11322no—
351C23—11322no—
352C23—22641no—
353C23—22641no—
354C23D4×D28311yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—5661no—
358Q8—5661no—
359D4—21no—
360D4—21no—
361D4—42no—
362D4—2832no—
363D4—2832no—
364D4—5662no—
365D4—5661no—
366D4—5661no—
367D4—5661no—
368D4—5661no—
369D4—11322no—
370D4—11322no—
371D4—11322no—
372D4—11322no—
373D4—11322no—
374D4—11322no—
375D4—22641no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—2831no—
382C2×C4—2831no—
383C2×C4—2831no—
384C2×C4—5661no—
385C2×C4—5661no—
386C2×C4—5661no—
387C2×C4—11322no—
388C2×C4—11322no—
389C2×C4—11322no—
390C2×C4—11322no—
391C2×C4D4×D28311yescharacteristic
392C2×C4C23×D28311yescharacteristic
393C8—21no—
394C8—5661no—
395C8—5661no—
396C8D4×D28312yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—5661no—
404C22—5661no—
405C22—5661no—
406C22—5662no—
407C22—5662no—
408C22—11321no—
409C22—11321no—
410C22—11321no—
411C22—11321no—
412C22—11321no—
413C22—11321no—
414C22—11321no—
415C22—22644no—
416C22—22644no—
417C22C2×D4×D28311yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—2832no—
423C4—5661no—
424C4—5661no—
425C4—11321no—
426C4C2×D4×D28311yescharacteristic
427C4#2258365611yescharacteristic
428C2—41no—
429C2—81no—
430C2—11321no—
431C2—11322no—
432C2—11322no—
433C2—22641no—
434C2#3182709911yescharacteristic
435C2AGL(1,ℤ/8)×D28311yescharacteristic
436C2#3182727411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#468×C283C2
socleC22×C283C22×D4
hypercenter#468D283
hypocenterC283#3272
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#22583638C2semidirect
2. C2×#22583637C2semidirect
3. #31827117C2semidirect
4. #31827099C2semidirect
5. #31826836C2semidirect
6. #31826712C2semidirect
7. #31826669C2semidirect
8. #31826659C2semidirect
9. #31826617C2semidirect
10. #31826615C2semidirect
11. #31826613C2semidirect
12. #31826608C2semidirect
13. #31826547C2semidirect
14. #31826541C2semidirect
15. #468×C283C2semidirect
16. C2×#17985393C22semidirect
17. C2×#17985392C22semidirect
24. C2×#17985381C22semidirect
25. C2×#17985380C22semidirect
27. C2×#17985379C22semidirect
29. C2×D2264C22semidirect
30. C2×#17985377C22semidirect
31. C2×#17985375C22semidirect
32. #22583608C22semidirect
36. #22583590C22semidirect
43. #22583572C22semidirect
46. Dic283⋊C8C22semidirect
47. #22583533C22semidirect
49. #22583530C22semidirect
50. #22583530C22semidirect
51. C22⋊(C283⋊C8)C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. C8×Dic283C22semidirect
54. C2×SD16×C283C22semidirect
55. C2×M16×C283C22semidirect
56. C22⋊Q8×C283C22semidirect
58. #107×C283C22semidirect
59. #103×C283C22semidirect
60. #102×C283C22semidirect
95. D2264D4semidirect
144. D1132C2×D4semidirect
177. C8×C283C2×D4semidirect
208. Dic283AGL(1,ℤ/8)semidirect
218. C283#3272semidirect
226. #468D283semidirect
247. C2×SD16C2×D283semidirect
252. C2×M16C2×D283semidirect
258. C22⋊Q8C2×D283semidirect
269. #107C2×D283semidirect
272. #103C2×D283semidirect
276. #102C2×D283semidirect
396. C8D4×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836382
C2×#225836372
#318271172
#318270992
#318268362
#318267122
#318266692
#318266592
#318266172
#318266152
#318266132
#318266082
#318265472
#318265412
#468×C2832
#3272283