Finite Group Databasebeta

← #49132762

Subgroups — #49132762

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
4D4×D1132C211yesmaximal, characteristic, complemented
5#31826855C211yesmaximal, characteristic, complemented
6#31826714C211yesmaximal, characteristic, complemented
7#31826700C211yesmaximal, characteristic, complemented
8#31826666C211yesmaximal, characteristic, complemented
9#31826614C211yesmaximal, characteristic, complemented
10#31826611C211yesmaximal, characteristic, complemented
11#31826605C211yesmaximal, characteristic, complemented
12#31826597C211yesmaximal, characteristic, complemented
13#31826565C211yesmaximal, characteristic, complemented
14#31826535C211yesmaximal, characteristic, complemented
15#463×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic, complemented
17C2×Q8×D283C2211yescharacteristic, complemented
18C22×D1132—21no—
19C2×D4×D283C2211yescharacteristic
20C2×D4×D283C2211yescharacteristic
21#22583634—24no—
22C2×#17985381C2211yescharacteristic, complemented
23C2×#17985381C2211yescharacteristic, complemented
24C2×#17985380C2211yescharacteristic
25C2×#17985379C2211yescharacteristic, complemented
26C2×#17985379C2211yescharacteristic, complemented
27SD16×D283—24no—
28C2×#17985377C2211yescharacteristic, complemented
29C2×#17985377C2211yescharacteristic, complemented
30C2×#17985375C2211yescharacteristic, complemented
31C2×C8×D283C2211yescharacteristic, complemented
32#22583607C2211yescharacteristic
33#22583596—21no—
34#22583587—21no—
35#22583586C2211yescharacteristic, complemented
36#22583585C2211yescharacteristic
37#22583585C2211yescharacteristic
38#22583584—21no—
39#22583583—21no—
40#22583581C2211yescharacteristic, complemented
41#22583580—21no—
42#22583580—21no—
43C4×Dic566C2211yescharacteristic, complemented
44C4×D1132C2211yescharacteristic
45(C22⋊D283)⋊C4—21no—
46#22583543C2211yescharacteristic, complemented
47#22583533C2211yescharacteristic, complemented
48#22583532C2211yescharacteristic, complemented
49#22583532C2211yescharacteristic, complemented
50#22583531C2211yescharacteristic
51(C2×D283)⋊C8C2211yescharacteristic, complemented
52(C2×D283)⋊C8C2211yescharacteristic, complemented
53#22583521C2211yescharacteristic, complemented
54C2×SD16×C283C2211yescharacteristic
55C2×D8×C283C2211yescharacteristic
56#124×C283C2211yescharacteristic, complemented
57C4×D4×C283C2211yescharacteristic
58C4⋊C8×C283C2211yescharacteristic, complemented
59#103×C283C2211yescharacteristic, complemented
60#102×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×D1132—21no—
70C2×D1132—21no—
71C2×D1132—21no—
72C2×D1132—21no—
73C2×D1132—44no—
74C2×D1132D411yescharacteristic
75C2×D1132C2311yescharacteristic
76C2×D1132C2311yescharacteristic
77C2×C22⋊D283—21no—
78C2×C22⋊D283—21no—
79C2×C22⋊D283—21no—
80C2×C22⋊D283—21no—
81D4×D283—22no—
82D4×D283—22no—
83D4×D283—41no—
84D4×D283—41no—
85C22×Dic283—21no—
86C2×C4×D283—21no—
87C2×C4×D283C2311yescharacteristic
88C2×C4×D283C2311yescharacteristic
89#17985381—22no—
90#17985381—22no—
91#17985380D414yescomplemented
92#17985379—22no—
93#17985379—41no—
94#17985377—22no—
95#17985377—41no—
96#17985375—22no—
97C2×C283⋊C8C2311yescharacteristic
98C2×C283⋊C8C2311yescharacteristic
99C8×D283—22no—
100C4⋊Dic283D411yescharacteristic
101C4⋊Dic283C2311yescharacteristic
102Dic283⋊C4—21no—
103Dic283⋊C4—21no—
104C22⋊Dic283—21no—
105(C2×D283)⋊C4—21no—
106(C2×D283)⋊C4—21no—
107(C2×D283)⋊C4—21no—
108(C2×D283)⋊C4D411yescharacteristic
109(C2×D283)⋊C4D411yescharacteristic
110C4×Dic283—21no—
111C2×Q8×C283C2311yescharacteristic
112C2×D4×C283C2311yescharacteristic
113C2×D4×C283C2311yescharacteristic
114C22×C4×C283—21no—
115SD16×C283—22no—
116D8×C283—22no—
117C2×C8×C283C2311yescharacteristic
118C2×C8×C283C2311yescharacteristic
119C4⋊C4×C283C2311yescharacteristic
120C22⋊C4×C283—21no—
121C22⋊C4×C283—21no—
122C22⋊C4×C283—21no—
123C42×C283C2311yescharacteristic
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
133C22×D283C2×D411yescharacteristic
134Dic566—21no—
135Dic566—42no—
136Dic566C2×D412yescomplemented
137D1132—22no—
138D1132—21no—
139D1132—21no—
140D1132—21no—
141D1132—21no—
142D1132—42no—
143D1132—42no—
144D1132—42no—
145D1132—81no—
146C22⋊D283—42no—
147C22⋊D283—42no—
148C22⋊D283—42no—
149C22⋊D283—42no—
150C2×Dic283—21no—
151C2×Dic283—21no—
152C2×Dic283—42no—
153C2×Dic283C2×D411yescharacteristic
154C2×Dic283C2×D411yescharacteristic
155C4×D283—21no—
156C4×D283—21no—
157C4×D283—21no—
158C4×D283—21no—
159C4×D283—42no—
160C283⋊C8—21no—
161C283⋊C8C2×D412yescomplemented
162C23×C283—21no—
163C23×C283—21no—
164Q8×C283—21no—
165Q8×C283—21no—
166D4×C283—21no—
167D4×C283—21no—
168D4×C283—21no—
169D4×C283C2×D412yes—
170C2×C4×C283—21no—
171C2×C4×C283—21no—
172C2×C4×C283—42no—
173C2×C4×C283C2×D411yescharacteristic
174C2×C4×C283C2×D411yescharacteristic
175C2×C4×C283C2411yescharacteristic
176C8×C283—21no—
177C8×C283—21no—
178C22×C283—22no—
179C22×C283—41no—
180C22×C283—41no—
181C22×C283—41no—
182C22×C283—41no—
183C22×C283—41no—
184C22×C283C22×D411yescharacteristic
185C2×D283—21no—
186C2×D283—21no—
187C2×D283—21no—
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—84no—
197C2×D283—84no—
198C2×D283AGL(1,ℤ/8)12yescomplemented
199C4×C283—21no—
200C4×C283—21no—
201C4×C283—41no—
202C4×C283—41no—
203C4×C283C22×D411yescharacteristic
204C4×C283H(5,𝔽2)11yescharacteristic
205Dic283—21no—
206Dic283—21no—
207Dic283—22no—
208Dic283—41no—
209C2×C283—42no—
210C2×C283—81no—
211C2×C283D4211yescharacteristic
212C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C283#57711yescharacteristic
214D283—22no—
215D283—41no—
216D283—81no—
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#463D28311yescharacteristic, complemented
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—2831no—
246C2×SD16—5664no—
247C2×SD16C2×D28311yescharacteristic
248C2×D8—2831no—
249C2×D8—2831no—
250C2×D8C2×D28311yescharacteristic
251C2×M16—2831no—
252C22×C8—2831no—
253#127—2831no—
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—5661no—
268C4×D4C2×D28311yescharacteristic
269#107—2831no—
270C4⋊C8C2×D28311yescharacteristic, complemented
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—
279#97—2831no—
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—2831no—
290C2×D4—2831no—
291C2×D4—2831no—
292C2×D4—5661no—
293C2×D4—5661no—
294C2×D4—5661no—
295C2×D4—5661no—
296C2×D4—5662no—
297C2×D4—5662no—
298C2×D4—5661no—
299C2×D4—5661no—
300C2×D4—5661no—
301C2×D4—5661no—
302C2×D4—11324no—
303C2×D4—11321no—
304C2×D4—11321no—
305C2×D4C22×D28311yescharacteristic
306C2×D4C22×D28311yescharacteristic
307C22×C4—21no—
308C22×C4—2831no—
309C22×C4—2831no—
310C22×C4—5661no—
311C22×C4—5661no—
312SD16—22no—
313SD16—2834no—
314SD16—5662no—
315SD16—5662no—
316SD16—5662no—
317SD16—11321no—
318D8—22no—
319D8—5662no—
320D8—11321no—
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—21no—
334C22⋊C4—21no—
335C22⋊C4—2831no—
336C22⋊C4—2831no—
337C22⋊C4—5661no—
338C22⋊C4—5661no—
339C22⋊C4—5661no—
340C22⋊C4—5661no—
341C42—5661no—
342C42C22×D28311yescharacteristic
343C23—21no—
344C23—21no—
345C23—2831no—
346C23—2831no—
347C23—5661no—
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—5662no—
364D4—5661no—
365D4—5661no—
366D4—5661no—
367D4—5661no—
368D4—11322no—
369D4—11322no—
370D4—11322no—
371D4—11322no—
372D4—11322no—
373D4—11322no—
374D4—11322no—
375D4—22641no—
376D4D4×D28312yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—2831no—
381C2×C4—2831no—
382C2×C4—5661no—
383C2×C4—5661no—
384C2×C4—5661no—
385C2×C4—5661no—
386C2×C4—5661no—
387C2×C4—5661no—
388C2×C4—11322no—
389C2×C4—11322no—
390C2×C4D4×D28311yescharacteristic
391C2×C4D4×D28311yescharacteristic
392C2×C4C23×D28311yescharacteristic
393C8—21no—
394C8—21no—
395C8—2832no—
396C8—5661no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—2832no—
404C22—5661no—
405C22—5661no—
406C22—5661no—
407C22—5661no—
408C22—5661no—
409C22—11321no—
410C22—11321no—
411C22—11321no—
412C22—11321no—
413C22—11321no—
414C22—11321no—
415C22—22644no—
416C22—22644no—
417C22C2×D4×D28311yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—41no—
422C4—5661no—
423C4—5661no—
424C4—5662no—
425C4—11321no—
426C4C2×D4×D28311yescharacteristic
427C4#2258365611yescharacteristic
428C2—42no—
429C2—81no—
430C2—5662no—
431C2—11321no—
432C2—22641no—
433C2—22641no—
434C2#3182709911yescharacteristic
435C2AGL(1,ℤ/8)×D28311yescharacteristic
436C2#3182727211yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#463×C283C2
socleC22×C283C22×D4
hypercenter#463D283
hypocenterC283#3265
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#22583634C2semidirect
2. C2×SD16×D283C2semidirect
3. #31827123C2semidirect
4. D4×D1132C2semidirect
5. #31826855C2semidirect
6. #31826714C2semidirect
7. #31826700C2semidirect
8. #31826666C2semidirect
9. #31826614C2semidirect
10. #31826611C2semidirect
11. #31826605C2semidirect
12. #31826597C2semidirect
13. #31826565C2semidirect
14. #31826535C2semidirect
15. #463×C283C2semidirect
16. C2×#17985391C22semidirect
17. C2×Q8×D283C22semidirect
22. C2×#17985381C22semidirect
23. C2×#17985381C22semidirect
25. C2×#17985379C22semidirect
26. C2×#17985379C22semidirect
28. C2×#17985377C22semidirect
29. C2×#17985377C22semidirect
30. C2×#17985375C22semidirect
31. C2×C8×D283C22semidirect
35. #22583586C22semidirect
40. #22583581C22semidirect
43. C4×Dic566C22semidirect
46. #22583543C22semidirect
47. #22583533C22semidirect
48. #22583532C22semidirect
49. #22583532C22semidirect
51. (C2×D283)⋊C8C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. #22583521C22semidirect
56. #124×C283C22semidirect
58. C4⋊C8×C283C22semidirect
59. #103×C283C22semidirect
60. #102×C283C22semidirect
91. #17985380D4semidirect
136. Dic566C2×D4semidirect
161. C283⋊C8C2×D4semidirect
198. C2×D283AGL(1,ℤ/8)semidirect
218. C283#3265semidirect
227. #463D283semidirect
255. #124C2×D283semidirect
270. C4⋊C8C2×D283semidirect
272. #103C2×D283semidirect
276. #102C2×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
D4×D11322
#318268552
#318267142
#318267002
#318266662
#318266142
#318266112
#318266052
#318265972
#318265652
#318265352
#463×C2832
#3265283