Finite Group Databasebeta

← #49132759

Subgroups — #49132759

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×#22583635C211yesmaximal, characteristic, complemented
2C2×SD16×D283C211yesmaximal, characteristic, complemented
3#31827122C211yesmaximal, characteristic, complemented
4#31827099C211yesmaximal, characteristic, complemented
5#31826853C211yesmaximal, characteristic, complemented
6#31826709C211yesmaximal, characteristic, complemented
7#31826700C211yesmaximal, characteristic, complemented
8#31826679C211yesmaximal, characteristic, complemented
9#31826663C211yesmaximal, characteristic, complemented
10#31826618C211yesmaximal, characteristic, complemented
11#31826605C211yesmaximal, characteristic, complemented
12#31826597C211yesmaximal, characteristic, complemented
13#31826568C211yesmaximal, characteristic, complemented
14#31826539C211yesmaximal, characteristic, complemented
15#449×C283C211yesmaximal, characteristic, complemented
16C2×Q8×D283C2211yescharacteristic, complemented
17C22×D1132C2211yescharacteristic
18C2×D4×D283—21no—
19C2×D4×D283C2211yescharacteristic
20#22583635—24no—
21C2×#17985381C2211yescharacteristic
22C2×#17985381C2211yescharacteristic, complemented
23C2×#17985380C2211yescharacteristic, complemented
24C2×#17985379C2211yescharacteristic
25C2×#17985379C2211yescharacteristic, complemented
26SD16×D283—24no—
27C2×#17985377C2211yescharacteristic
28C2×#17985377C2211yescharacteristic, complemented
29C2×#17985376C2211yescharacteristic, complemented
30C2×C8×D283C2211yescharacteristic, complemented
31#22583606C2211yescharacteristic
32#22583597—21no—
33#22583597C2211yescharacteristic, complemented
34#22583589—21no—
35#22583589C2211yescharacteristic, complemented
36#22583585C2211yescharacteristic
37#22583583—21no—
38#22583581—21no—
39#22583581C2211yescharacteristic, complemented
40#22583580—21no—
41#22583578—21no—
42Q8×Dic283C2211yescharacteristic, complemented
43#22583572C2211yescharacteristic
44C4×C22⋊D283—21no—
45C4×C22⋊D283C2211yescharacteristic
46#22583543C2211yescharacteristic, complemented
47Dic283⋊C8C2211yescharacteristic, complemented
48#22583533C2211yescharacteristic, complemented
49#22583532C2211yescharacteristic
50#22583532C2211yescharacteristic, complemented
51#22583531C2211yescharacteristic, complemented
52(C2×D283)⋊C8C2211yescharacteristic, complemented
53#22583520C2211yescharacteristic, complemented
54C2×#41×C283C2211yescharacteristic, complemented
55C2×SD16×C283C2211yescharacteristic, complemented
56C2×D8×C283C2211yescharacteristic, complemented
57#121×C283C2211yescharacteristic
58#103×C283C2211yescharacteristic, complemented
59#102×C283C2211yescharacteristic, complemented
60C22⋊C8×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62C23×D283—21no—
63C2×Dic566—21no—
64C2×Dic566C2311yescharacteristic
65Q8×D283—22no—
66Q8×D283—41no—
67C2×D1132—22no—
68C2×D1132—21no—
69C2×D1132—21no—
70C2×D1132—41no—
71C2×D1132C2311yescharacteristic
72C2×D1132C2311yescharacteristic
73C2×C22⋊D283—21no—
74C2×C22⋊D283—21no—
75C2×C22⋊D283—21no—
76C2×C22⋊D283—21no—
77C2×C22⋊D283D411yescharacteristic
78D4×D283—22no—
79D4×D283—41no—
80D4×D283—44no—
81C2×C4×D283—21no—
82C2×C4×D283—21no—
83C2×C4×D283C2311yescharacteristic
84#17985381—22no—
85#17985381—41no—
86#17985380—22no—
87#17985379—22no—
88#17985379D414yescomplemented
89#17985377—22no—
90#17985377—22no—
91#17985376—22no—
92C2×C283⋊C8C2311yescharacteristic
93C2×C283⋊C8C2311yescharacteristic
94C8×D283—22no—
95C4⋊Dic283—21no—
96C4⋊Dic283C2311yescharacteristic
97Dic283⋊C4—21no—
98Dic283⋊C4D411yescharacteristic
99C22⋊Dic283—21no—
100C22⋊Dic283—21no—
101C22⋊Dic283—21no—
102C22⋊Dic283—21no—
103C22⋊Dic283D411yescharacteristic
104(C2×D283)⋊C4—21no—
105(C2×D283)⋊C4—21no—
106(C2×D283)⋊C4D411yescharacteristic
107C4×Dic283—21no—
108C4×Dic283C2311yescharacteristic
109#41×C283—22no—
110#41×C283—41no—
111C2×Q8×C283C2311yescharacteristic
112C2×D4×C283—21no—
113C2×D4×C283—21no—
114C2×D4×C283C2311yescharacteristic
115C2×D4×C283C2311yescharacteristic
116C22×C4×C283—21no—
117C22×C4×C283C2311yescharacteristic
118SD16×C283—22no—
119D8×C283—41no—
120C2×C8×C283C2311yescharacteristic
121C2×C8×C283C2311yescharacteristic
122C4⋊C4×C283C2311yescharacteristic
123C22⋊C4×C283—21no—
124C22×D283—21no—
125C22×D283—21no—
126C22×D283—42no—
127C22×D283—42no—
128C22×D283—42no—
129C22×D283—42no—
130C22×D283—81no—
131C22×D283—81no—
132C22×D283C2×D411yescharacteristic
133Dic566—21no—
134Dic566—21no—
135Dic566—42no—
136D1132—21no—
137D1132—21no—
138D1132—21no—
139D1132—42no—
140D1132—42no—
141D1132C2×D412yes—
142C22⋊D283—22no—
143C22⋊D283—42no—
144C22⋊D283—42no—
145C22⋊D283—42no—
146C22⋊D283—81no—
147C2×Dic283—21no—
148C2×Dic283—21no—
149C2×Dic283C2×D411yescharacteristic
150C2×Dic283C2×D411yescharacteristic
151C2×Dic283C2×D411yescharacteristic
152C4×D283—21no—
153C4×D283—21no—
154C4×D283—42no—
155C4×D283—42no—
156C283⋊C8—21no—
157C283⋊C8C2×D412yescomplemented
158C23×C283—21no—
159C23×C283—21no—
160C23×C283C2×D411yescharacteristic
161Q8×C283—21no—
162Q8×C283C2×D412yescomplemented
163D4×C283—21no—
164D4×C283—21no—
165D4×C283—21no—
166D4×C283—21no—
167D4×C283—42no—
168D4×C283—42no—
169C2×C4×C283—21no—
170C2×C4×C283—21no—
171C2×C4×C283—21no—
172C2×C4×C283—21no—
173C2×C4×C283—42no—
174C2×C4×C283C2×D411yescharacteristic
175C2×C4×C283C2411yescharacteristic
176C8×C283—21no—
177C8×C283—21no—
178C22×C283—21no—
179C22×C283—21no—
180C22×C283—21no—
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—22no—
191C2×D283—41no—
192C2×D283—41no—
193C2×D283—41no—
194C2×D283—41no—
195C2×D283—41no—
196C2×D283—84no—
197C2×D283—84no—
198C2×D283AGL(1,ℤ/8)12yescomplemented
199C4×C283—21no—
200C4×C283—22no—
201C4×C283—41no—
202C4×C283C22×D411yescharacteristic
203C4×C283H(5,𝔽2)11yescharacteristic
204Dic283—21no—
205Dic283—21no—
206Dic283—21no—
207Dic283—41no—
208Dic283—41no—
209C2×C283—41no—
210C2×C283—81no—
211C2×C283—81no—
212C2×C283D4211yescharacteristic
213C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C283#57711yescharacteristic
215D283—22no—
216D283—42no—
217D283—81no—
218C283#326511yescharacteristic, complemented, Sylow
219#3265—2831nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2831no—
221C22×SD16—2831no—
222#548—2831no—
223D42—2831no—
224#496—2831no—
225#468—2831no—
226#466—2831no—
227#463—2831no—
228#460—2831no—
229#450—2831no—
230#449D28311yescharacteristic, complemented
231#448—2831no—
232#447—2831no—
233#440—2831no—
234#435—2831no—
235C2×#41C2×D28311yescharacteristic, complemented
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—2831no—
246C2×SD16—5664no—
247C2×SD16C2×D28311yescharacteristic, complemented
248C2×D8—2831no—
249C2×D8—2831no—
250C2×D8C2×D28311yescharacteristic, complemented
251C2×M16—2831no—
252C22×C8—2831no—
253#127—2831no—
254#124—2831no—
255#124—5661no—
256C22⋊Q8—2831no—
257C22⋊Q8—5661no—
258#121—2831no—
259#121—2831no—
260#121—5661no—
261#121—5661no—
262#121C2×D28311yescharacteristic
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#103C2×D28311yescharacteristic, complemented
273#102—2831no—
274#102—2831no—
275#102—2831no—
276#102C2×D28311yescharacteristic, complemented
277C22⋊C8—2831no—
278C22⋊C8C2×D28311yescharacteristic, complemented
279#97—2831no—
280C24—5661no—
281C24—5661no—
282#41—22no—
283#41—41no—
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—2831no—
294C2×D4—5662no—
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
306C2×D4C22×D28311yescharacteristic
307C22×C4—21no—
308C22×C4—2831no—
309C22×C4—5661no—
310C22×C4—5661no—
311C22×C4C22×D28311yescharacteristic
312SD16—22no—
313SD16—2834no—
314SD16—5662no—
315SD16—5662no—
316SD16—5662no—
317SD16—11321no—
318D8—41no—
319D8—5662no—
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—5661no—
330C4⋊C4—5661no—
331C4⋊C4C22×D28311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—2831no—
334C22⋊C4—2831no—
335C22⋊C4—5661no—
336C22⋊C4—5661no—
337C22⋊C4—5661no—
338C22⋊C4—5661no—
339C22⋊C4—5661no—
340C22⋊C4—5661no—
341C42—2831no—
342C42—5661no—
343C23—21no—
344C23—21no—
345C23—2831no—
346C23—5661no—
347C23—5661no—
348C23—11322no—
349C23—11322no—
350C23—11322no—
351C23—11322no—
352C23—22641no—
353C23—22641no—
354C23D4×D28311yescharacteristic
355Q8—21no—
356Q8—5661no—
357Q8—5661no—
358Q8—11322no—
359Q8D4×D28312yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—42no—
365D4—42no—
366D4—2832no—
367D4—5662no—
368D4—5661no—
369D4—5661no—
370D4—5661no—
371D4—11322no—
372D4—11322no—
373D4—11322no—
374D4—11322no—
375D4—11322no—
376D4—22641no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—2831no—
383C2×C4—2831no—
384C2×C4—2831no—
385C2×C4—5661no—
386C2×C4—5661no—
387C2×C4—5661no—
388C2×C4—5661no—
389C2×C4—11322no—
390C2×C4—11322no—
391C2×C4D4×D28311yescharacteristic
392C2×C4C23×D28311yescharacteristic
393C8—21no—
394C8—21no—
395C8—2832no—
396C8—5661no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—2832no—
407C22—5661no—
408C22—5661no—
409C22—5662no—
410C22—11321no—
411C22—11321no—
412C22—11321no—
413C22—11321no—
414C22—11321no—
415C22—22644no—
416C22—22644no—
417C22C2×D4×D28311yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—5661no—
422C4—5661no—
423C4—5661no—
424C4—11321no—
425C4—11321no—
426C4C2×D4×D28311yescharacteristic
427C4#2258365511yescharacteristic
428C2—41no—
429C2—81no—
430C2—81no—
431C2—5662no—
432C2—11322no—
433C2—22641no—
434C2#3182709711yescharacteristic
435C2AGL(1,ℤ/8)×D28311yescharacteristic
436C2#3182727211yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#449×C283C2
socleC22×C283C22×D4
hypercenter#449D283
hypocenterC283#3265
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#22583635C2semidirect
2. C2×SD16×D283C2semidirect
3. #31827122C2semidirect
4. #31827099C2semidirect
5. #31826853C2semidirect
6. #31826709C2semidirect
7. #31826700C2semidirect
8. #31826679C2semidirect
9. #31826663C2semidirect
10. #31826618C2semidirect
11. #31826605C2semidirect
12. #31826597C2semidirect
13. #31826568C2semidirect
14. #31826539C2semidirect
15. #449×C283C2semidirect
16. C2×Q8×D283C22semidirect
22. C2×#17985381C22semidirect
23. C2×#17985380C22semidirect
25. C2×#17985379C22semidirect
28. C2×#17985377C22semidirect
29. C2×#17985376C22semidirect
30. C2×C8×D283C22semidirect
33. #22583597C22semidirect
35. #22583589C22semidirect
39. #22583581C22semidirect
42. Q8×Dic283C22semidirect
46. #22583543C22semidirect
47. Dic283⋊C8C22semidirect
48. #22583533C22semidirect
50. #22583532C22semidirect
51. #22583531C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. #22583520C22semidirect
54. C2×#41×C283C22semidirect
55. C2×SD16×C283C22semidirect
56. C2×D8×C283C22semidirect
58. #103×C283C22semidirect
59. #102×C283C22semidirect
60. C22⋊C8×C283C22semidirect
88. #17985379D4semidirect
157. C283⋊C8C2×D4semidirect
162. Q8×C283C2×D4semidirect
198. C2×D283AGL(1,ℤ/8)semidirect
218. C283#3265semidirect
230. #449D283semidirect
235. C2×#41C2×D283semidirect
247. C2×SD16C2×D283semidirect
250. C2×D8C2×D283semidirect
272. #103C2×D283semidirect
276. #102C2×D283semidirect
278. C22⋊C8C2×D283semidirect
359. Q8D4×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836352
C2×SD16×D2832
#318271222
#318270992
#318268532
#318267092
#318267002
#318266792
#318266632
#318266182
#318266052
#318265972
#318265682
#318265392
#449×C2832
#3265283