Finite Group Databasebeta

← #49132755

Subgroups — #49132755

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×#22583634C211yesmaximal, characteristic, complemented
2C2×SD16×D283C211yesmaximal, characteristic, complemented
3#31827123C211yesmaximal, characteristic, complemented
4#31827100C211yesmaximal, characteristic, complemented
5#31826708C211yesmaximal, characteristic, complemented
6#31826698C211yesmaximal, characteristic, complemented
7#31826677C211yesmaximal, characteristic, complemented
8#31826662C211yesmaximal, characteristic, complemented
9#31826614C211yesmaximal, characteristic, complemented
10#31826607C211yesmaximal, characteristic, complemented
11#31826602C211yesmaximal, characteristic, complemented
12#31826597C211yesmaximal, characteristic, complemented
13#31826567C211yesmaximal, characteristic, complemented
14#31826538C211yesmaximal, characteristic, complemented
15#496×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic, complemented
17C2×Q8×D283C2211yescharacteristic, complemented
18C2×D4×D283—21no—
19C2×D4×D283C2211yescharacteristic
20C2×D4×D283C2211yescharacteristic
21#22583634—24no—
22C2×#17985381C2211yescharacteristic
23C2×#17985380C2211yescharacteristic, complemented
24C2×#17985380C2211yescharacteristic, complemented
25C2×#17985379C2211yescharacteristic
26SD16×D283—24no—
27C2×#17985377C2211yescharacteristic
28C2×#17985375C2211yescharacteristic, complemented
29C2×C8×D283C2211yescharacteristic, complemented
30#22583596—21no—
31#22583587—21no—
32#22583586C2211yescharacteristic, complemented
33#22583583—21no—
34#22583581—21no—
35#22583581C2211yescharacteristic
36#22583581C2211yescharacteristic
37#22583581C2211yescharacteristic, complemented
38#22583578—21no—
39#22583578—21no—
40C4×Dic566C2211yescharacteristic, complemented
41C4×D1132C2211yescharacteristic
42C4×D1132C2211yescharacteristic
43(C22⋊D283)⋊C4—21no—
44#22583544C2211yescharacteristic, complemented
45C4⋊(C283⋊C8)C2211yescharacteristic, complemented
46#22583535C2211yescharacteristic, complemented
47#22583534C2211yescharacteristic, complemented
48#22583532C2211yescharacteristic
49#22583531C2211yescharacteristic, complemented
50#22583531C2211yescharacteristic, complemented
51#22583531C2211yescharacteristic, complemented
52(C2×D283)⋊C8C2211yescharacteristic, complemented
53(C2×D283)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C283C2211yescharacteristic, complemented
55C2×SD16×C283C2211yescharacteristic, complemented
56C2×D8×C283C2211yescharacteristic, complemented
57C2×D8×C283C2211yescharacteristic, complemented
58#127×C283C2211yescharacteristic
59#124×C283C2211yescharacteristic, complemented
60#97×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62C23×D283—21no—
63#17985391—22no—
64#17985391—41no—
65C2×Dic566—21no—
66C2×Dic566C2311yescharacteristic
67Q8×D283—22no—
68Q8×D283—41no—
69C2×D1132D411yescharacteristic
70C2×D1132C2311yescharacteristic
71C2×C22⋊D283—21no—
72C2×C22⋊D283—21no—
73C2×C22⋊D283—21no—
74C2×C22⋊D283—21no—
75C2×C22⋊D283—21no—
76C2×C22⋊D283—21no—
77D4×D283—22no—
78D4×D283—22no—
79D4×D283—41no—
80D4×D283—41no—
81D4×D283—44no—
82C22×Dic283—21no—
83C2×C4×D283—21no—
84C2×C4×D283—21no—
85C2×C4×D283C2311yescharacteristic
86C2×C4×D283C2311yescharacteristic
87#17985381D414yescomplemented
88#17985380—22no—
89#17985380—22no—
90#17985379—22no—
91#17985377—22no—
92#17985375—22no—
93C2×C283⋊C8C2311yescharacteristic
94C2×C283⋊C8C2311yescharacteristic
95C8×D283—22no—
96C4⋊Dic283D411yescharacteristic
97C4⋊Dic283C2311yescharacteristic
98C4⋊Dic283C2311yescharacteristic
99Dic283⋊C4—21no—
100Dic283⋊C4—21no—
101C22⋊Dic283—21no—
102C22⋊Dic283—21no—
103C22⋊Dic283—21no—
104(C2×D283)⋊C4—21no—
105(C2×D283)⋊C4—21no—
106(C2×D283)⋊C4D411yescharacteristic
107(C2×D283)⋊C4D411yescharacteristic
108C4×Dic283—21no—
109C2×Q8×C283C2311yescharacteristic
110C2×D4×C283—21no—
111C2×D4×C283—21no—
112C2×D4×C283C2311yescharacteristic
113C2×D4×C283C2311yescharacteristic
114C2×D4×C283C2311yescharacteristic
115SD16×C283—22no—
116SD16×C283—41no—
117D8×C283—22no—
118D8×C283—41no—
119C2×C8×C283C2311yescharacteristic
120C2×C8×C283C2311yescharacteristic
121C22⋊C4×C283—21no—
122C22⋊C4×C283—21no—
123C42×C283C2311yescharacteristic
124C22×D283—21no—
125C22×D283—42no—
126C22×D283—42no—
127C22×D283—42no—
128C22×D283—42no—
129C22×D283—81no—
130C22×D283—81no—
131C22×D283C2×D411yescharacteristic
132C22×D283C2×D411yescharacteristic
133Dic566—21no—
134Dic566—42no—
135Dic566C2×D412yescomplemented
136D1132—22no—
137D1132—21no—
138D1132C2×D412yes—
139C22⋊D283—42no—
140C22⋊D283—42no—
141C22⋊D283—42no—
142C22⋊D283—42no—
143C22⋊D283—42no—
144C22⋊D283—42no—
145C2×Dic283—21no—
146C2×Dic283—21no—
147C2×Dic283—21no—
148C2×Dic283—42no—
149C2×Dic283C2×D411yescharacteristic
150C2×Dic283C2×D411yescharacteristic
151C4×D283—21no—
152C4×D283—21no—
153C4×D283—21no—
154C4×D283—21no—
155C4×D283—42no—
156C4×D283—42no—
157C283⋊C8—21no—
158C283⋊C8—21no—
159C23×C283—21no—
160C23×C283—21no—
161C23×C283—21no—
162Q8×C283—21no—
163Q8×C283—21no—
164D4×C283—21no—
165D4×C283—21no—
166D4×C283—21no—
167D4×C283—21no—
168D4×C283—21no—
169D4×C283—21no—
170D4×C283—42no—
171D4×C283—81no—
172C2×C4×C283—21no—
173C2×C4×C283C2×D411yescharacteristic
174C2×C4×C283C2×D411yescharacteristic
175C2×C4×C283C2411yescharacteristic
176C8×C283—21no—
177C8×C283C2×D412yescomplemented
178C22×C283—41no—
179C22×C283—41no—
180C22×C283—41no—
181C22×C283—41no—
182C22×C283—41no—
183C22×C283—41no—
184C22×C283—41no—
185C22×C283—41no—
186C22×C283—41no—
187C22×C283C22×D411yescharacteristic
188C2×D283—21no—
189C2×D283—21no—
190C2×D283—21no—
191C2×D283—22no—
192C2×D283—21no—
193C2×D283—21no—
194C2×D283—41no—
195C2×D283—41no—
196C2×D283—84no—
197C2×D283—84no—
198C2×D283AGL(1,ℤ/8)12yescomplemented
199C4×C283—21no—
200C4×C283—21no—
201C4×C283—41no—
202C4×C283C22×D411yescharacteristic
203C4×C283H(5,𝔽2)11yescharacteristic
204Dic283—21no—
205Dic283—21no—
206Dic283—22no—
207Dic283—41no—
208Dic283—41no—
209C2×C283—81no—
210C2×C283—81no—
211C2×C283—81no—
212C2×C283D4211yescharacteristic
213C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C283#57711yescharacteristic
215D283—22no—
216D283—41no—
217D283—42no—
218C283#326511yescharacteristic, complemented, Sylow
219#3265—2831nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2831no—
221C22×SD16—2831no—
222#548—2831no—
223D42—2831no—
224#496D28311yescharacteristic, complemented
225#468—2831no—
226#466—2831no—
227#463—2831no—
228#460—2831no—
229#450—2831no—
230#449—2831no—
231#448—2831no—
232#447—2831no—
233#440—2831no—
234#435—2831no—
235C2×#41—2831no—
236C22×Q8—2831no—
237C22×D4—2831no—
238C22×D4—2831no—
239C22×D4—5661no—
240AGL(1,ℤ/8)—5664no—
241C2×SD16—2831no—
242C2×SD16—2831no—
243C2×SD16—2831no—
244C2×SD16—2831no—
245C2×SD16—5664no—
246C2×SD16C2×D28311yescharacteristic, complemented
247C2×SD16C2×D28311yescharacteristic, complemented
248C2×D8—2831no—
249C2×D8C2×D28311yescharacteristic, complemented
250C2×D8C2×D28311yescharacteristic, complemented
251C2×M16—2831no—
252C22×C8—2831no—
253#127C2×D28311yescharacteristic
254#124—5661no—
255#124C2×D28311yescharacteristic, complemented
256C22⋊Q8—2831no—
257C22⋊Q8—5661no—
258#121—2831no—
259#121—2831no—
260#121—2831no—
261#121—5661no—
262#121—5661no—
263#120—5661no—
264#120—5661no—
265C4×Q8—2831no—
266C4×D4—2831no—
267C4×D4—2831no—
268C4×D4—5661no—
269#107—2831no—
270C4⋊C8—2831no—
271#103—2831no—
272#103—2831no—
273#102—2831no—
274#102—2831no—
275#102—2831no—
276#102—2831no—
277C22⋊C8—2831no—
278C22⋊C8—2831no—
279#97C2×D28311yescharacteristic, complemented
280C24—5661no—
281C24—5661no—
282#41—5662no—
283#41—11321no—
284C2×Q8—2831no—
285C2×Q8—5661no—
286C2×Q8—5662no—
287C2×Q8—11321no—
288C2×Q8C22×D28311yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—2831no—
292C2×D4—2831no—
293C2×D4—5661no—
294C2×D4—5661no—
295C2×D4—5662no—
296C2×D4—5662no—
297C2×D4—5661no—
298C2×D4—5661no—
299C2×D4—5661no—
300C2×D4—5661no—
301C2×D4—11321no—
302C2×D4—11321no—
303C2×D4—11324no—
304C2×D4C22×D28311yescharacteristic
305C2×D4C22×D28311yescharacteristic
306C2×D4C22×D28311yescharacteristic
307C22×C4—2831no—
308C22×C4—2831no—
309C22×C4—5661no—
310C22×C4—5661no—
311C22×C4—5661no—
312SD16—22no—
313SD16—41no—
314SD16—2834no—
315SD16—5662no—
316SD16—5662no—
317SD16—5662no—
318D8—22no—
319D8—41no—
320D8—5662no—
321M16—5662no—
322C2×C8—2831no—
323C2×C8—2831no—
324C2×C8—5662no—
325C2×C8C22×D28311yescharacteristic
326C2×C8C22×D28311yescharacteristic
327C4⋊C4—2831no—
328C4⋊C4—2831no—
329C4⋊C4—2831no—
330C4⋊C4—5661no—
331C4⋊C4—5661no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—2831no—
335C22⋊C4—2831no—
336C22⋊C4—5661no—
337C22⋊C4—5661no—
338C22⋊C4—5661no—
339C22⋊C4—5661no—
340C22⋊C4—5661no—
341C42—5661no—
342C42C22×D28311yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—2831no—
347C23—2831no—
348C23—5661no—
349C23—11322no—
350C23—11322no—
351C23—11322no—
352C23—11322no—
353C23—22641no—
354C23—22641no—
355Q8—21no—
356Q8—21no—
357Q8—2832no—
358Q8—5661no—
359Q8—11322no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—81no—
368D4—2832no—
369D4—5662no—
370D4—5661no—
371D4—11322no—
372D4—11322no—
373D4—11322no—
374D4—11322no—
375D4—11322no—
376D4—11322no—
377C2×C4—21no—
378C2×C4—2831no—
379C2×C4—2831no—
380C2×C4—5661no—
381C2×C4—5661no—
382C2×C4—5661no—
383C2×C4—5661no—
384C2×C4—5661no—
385C2×C4—5661no—
386C2×C4—5661no—
387C2×C4—11322no—
388C2×C4—11322no—
389C2×C4—11322no—
390C2×C4D4×D28311yescharacteristic
391C2×C4D4×D28311yescharacteristic
392C2×C4C23×D28311yescharacteristic
393C8—21no—
394C8—5661no—
395C8—5661no—
396C8D4×D28312yescomplemented
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—2832no—
407C22—5661no—
408C22—5661no—
409C22—5661no—
410C22—5661no—
411C22—5661no—
412C22—5662no—
413C22—11321no—
414C22—11321no—
415C22—22644no—
416C22—22644no—
417C22C2×D4×D28311yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—5661no—
422C4—5661no—
423C4—5662no—
424C4—11321no—
425C4—11321no—
426C4C2×D4×D28311yescharacteristic
427C4#2258365511yescharacteristic
428C2—81no—
429C2—81no—
430C2—81no—
431C2—5662no—
432C2—11321no—
433C2—11322no—
434C2#3182710011yescharacteristic
435C2AGL(1,ℤ/8)×D28311yescharacteristic
436C2#3182727511yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#496×C283C2
socleC22×C283C22×D4
hypercenter#496D283
hypocenterC283#3265
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#22583634C2semidirect
2. C2×SD16×D283C2semidirect
3. #31827123C2semidirect
4. #31827100C2semidirect
5. #31826708C2semidirect
6. #31826698C2semidirect
7. #31826677C2semidirect
8. #31826662C2semidirect
9. #31826614C2semidirect
10. #31826607C2semidirect
11. #31826602C2semidirect
12. #31826597C2semidirect
13. #31826567C2semidirect
14. #31826538C2semidirect
15. #496×C283C2semidirect
16. C2×#17985391C22semidirect
17. C2×Q8×D283C22semidirect
23. C2×#17985380C22semidirect
24. C2×#17985380C22semidirect
28. C2×#17985375C22semidirect
29. C2×C8×D283C22semidirect
32. #22583586C22semidirect
37. #22583581C22semidirect
40. C4×Dic566C22semidirect
44. #22583544C22semidirect
45. C4⋊(C283⋊C8)C22semidirect
46. #22583535C22semidirect
47. #22583534C22semidirect
49. #22583531C22semidirect
50. #22583531C22semidirect
51. #22583531C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. (C2×D283)⋊C8C22semidirect
54. C2×SD16×C283C22semidirect
55. C2×SD16×C283C22semidirect
56. C2×D8×C283C22semidirect
57. C2×D8×C283C22semidirect
59. #124×C283C22semidirect
60. #97×C283C22semidirect
87. #17985381D4semidirect
135. Dic566C2×D4semidirect
177. C8×C283C2×D4semidirect
198. C2×D283AGL(1,ℤ/8)semidirect
218. C283#3265semidirect
224. #496D283semidirect
246. C2×SD16C2×D283semidirect
247. C2×SD16C2×D283semidirect
249. C2×D8C2×D283semidirect
250. C2×D8C2×D283semidirect
255. #124C2×D283semidirect
279. #97C2×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×#225836342
C2×SD16×D2832
#318271232
#318271002
#318267082
#318266982
#318266772
#318266622
#318266142
#318266072
#318266022
#318265972
#318265672
#318265382
#496×C2832
#3265283