Finite Group Databasebeta

← #49132747

Subgroups — #49132747

Order 36224.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583637C211yesmaximal, characteristic, complemented
2C2×D8×D283C211yesmaximal, characteristic, complemented
3#31827108C211yesmaximal, characteristic, complemented
4D4×D1132C211yesmaximal, characteristic, complemented
5#31826855C211yesmaximal, characteristic, complemented
6#31826717C211yesmaximal, characteristic, complemented
7#31826697C211yesmaximal, characteristic, complemented
8#31826661C211yesmaximal, characteristic, complemented
9#31826614C211yesmaximal, characteristic, complemented
10#31826613C211yesmaximal, characteristic, complemented
11#31826600C211yesmaximal, characteristic, complemented
12#31826597C211yesmaximal, characteristic, complemented
13#31826575C211yesmaximal, characteristic, complemented
14#31826535C211yesmaximal, characteristic, complemented
15#463×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic, complemented
17C22×D1132—21no—
18C2×D4×D283C2211yescharacteristic
19C2×D4×D283C2211yescharacteristic, complemented
20C2×D4×D283C2211yescharacteristic, complemented
21#22583637—24no—
22C2×#17985379C2211yescharacteristic, complemented
23C2×#17985379C2211yescharacteristic, complemented
24C2×D2264C2211yescharacteristic, complemented
25C2×D2264C2211yescharacteristic, complemented
26C2×#17985377C2211yescharacteristic
27C2×#17985377C2211yescharacteristic, complemented
28C2×#17985377C2211yescharacteristic, complemented
29D8×D283—24no—
30C2×#17985375C2211yescharacteristic, complemented
31C2×C8×D283C2211yescharacteristic, complemented
32#22583607C2211yescharacteristic, complemented
33#22583592—21no—
34#22583586C2211yescharacteristic, complemented
35#22583585C2211yescharacteristic
36#22583585C2211yescharacteristic, complemented
37#22583584—21no—
38#22583583—21no—
39#22583581C2211yescharacteristic, complemented
40#22583580—21no—
41#22583580—21no—
42#22583580—21no—
43C4×D1132C2211yescharacteristic
44C4×D1132C2211yescharacteristic, complemented
45C22⋊C4×D283—21no—
46#22583541C2211yescharacteristic, complemented
47#22583532C2211yescharacteristic, complemented
48#22583532C2211yescharacteristic, complemented
49#22583531C2211yescharacteristic, complemented
50#22583530C2211yescharacteristic, complemented
51(C2×D283)⋊C8C2211yescharacteristic, complemented
52(C2×D283)⋊C8C2211yescharacteristic, complemented
53#22583521C2211yescharacteristic, complemented
54C2×SD16×C283C2211yescharacteristic, complemented
55C2×D8×C283C2211yescharacteristic
56#124×C283C2211yescharacteristic, complemented
57C4×D4×C283C2211yescharacteristic, complemented
58C4⋊C8×C283C2211yescharacteristic, complemented
59#103×C283C2211yescharacteristic, complemented
60#102×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62C23×D283—21no—
63C23×D283—21no—
64#17985393—22no—
65#17985393—41no—
66C2×D1132—21no—
67C2×D1132—21no—
68C2×D1132—21no—
69C2×D1132—21no—
70C2×D1132—21no—
71C2×D1132—44no—
72C2×D1132D411yescharacteristic
73C2×D1132C2311yescharacteristic
74C2×D1132C2311yescharacteristic
75C2×D1132C2311yescharacteristic
76C2×C22⋊D283—21no—
77C2×C22⋊D283—21no—
78C2×C22⋊D283—21no—
79C2×C22⋊D283—21no—
80D4×D283—22no—
81D4×D283—22no—
82D4×D283—22no—
83D4×D283—41no—
84D4×D283—41no—
85D4×D283—41no—
86C2×C4×D283—21no—
87C2×C4×D283—21no—
88C2×C4×D283C2311yescharacteristic
89C2×C4×D283C2311yescharacteristic
90#17985379—22no—
91#17985379—41no—
92D2264—22no—
93D2264—22no—
94#17985377—22no—
95#17985377—41no—
96#17985377D414yescomplemented
97#17985375—22no—
98C2×C283⋊C8C2311yescharacteristic
99C2×C283⋊C8C2311yescharacteristic
100C8×D283—22no—
101C4⋊Dic283D411yescharacteristic
102C4⋊Dic283C2311yescharacteristic
103Dic283⋊C4—21no—
104C22⋊Dic283—21no—
105(C2×D283)⋊C4—21no—
106(C2×D283)⋊C4—21no—
107(C2×D283)⋊C4—21no—
108(C2×D283)⋊C4—21no—
109(C2×D283)⋊C4D411yescharacteristic
110(C2×D283)⋊C4D411yescharacteristic
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—21no—
127C22×D283—42no—
128C22×D283—42no—
129C22×D283—42no—
130C22×D283—42no—
131C22×D283—42no—
132C22×D283—42no—
133C22×D283—81no—
134C22×D283—81no—
135C22×D283—81no—
136C22×D283C2×D411yescharacteristic
137C22×D283C2×D411yescharacteristic
138D1132—22no—
139D1132—21no—
140D1132—21no—
141D1132—21no—
142D1132—21no—
143D1132—21no—
144D1132—42no—
145D1132—42no—
146D1132—42no—
147D1132—42no—
148D1132—81no—
149D1132C2×D412yescomplemented
150C22⋊D283—42no—
151C22⋊D283—42no—
152C22⋊D283—42no—
153C22⋊D283—81no—
154C2×Dic283—21no—
155C2×Dic283C2×D411yescharacteristic
156C2×Dic283C2×D411yescharacteristic
157C4×D283—21no—
158C4×D283—21no—
159C4×D283—21no—
160C4×D283—21no—
161C4×D283—42no—
162C4×D283—81no—
163C283⋊C8—21no—
164C283⋊C8C2×D412yescomplemented
165C23×C283—21no—
166C23×C283—21no—
167Q8×C283—21no—
168Q8×C283—21no—
169D4×C283—21no—
170D4×C283—21no—
171D4×C283—21no—
172D4×C283C2×D412yes—
173C2×C4×C283—21no—
174C2×C4×C283—21no—
175C2×C4×C283—42no—
176C2×C4×C283C2×D411yescharacteristic
177C2×C4×C283C2×D411yescharacteristic
178C2×C4×C283C2411yescharacteristic
179C8×C283—21no—
180C8×C283—21no—
181C22×C283—22no—
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—41no—
197C2×D283—41no—
198C2×D283—41no—
199C2×D283—41no—
200C2×D283—41no—
201C2×D283—41no—
202C2×D283—84no—
203C2×D283—84no—
204C2×D283—84no—
205C2×D283AGL(1,ℤ/8)12yescomplemented
206C4×C283—21no—
207C4×C283—21no—
208C4×C283—41no—
209C4×C283—41no—
210C4×C283C22×D411yescharacteristic
211C4×C283H(5,𝔽2)11yescharacteristic
212Dic283—21no—
213Dic283—21no—
214Dic283—41no—
215C2×C283—42no—
216C2×C283—81no—
217C2×C283D4211yescharacteristic
218C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C283#57611yescharacteristic
220D283—22no—
221D283—41no—
222D283—42no—
223D283—81no—
224D283—81no—
225C283#326411yescharacteristic, complemented, Sylow
226#3264—2831nomaximal, Sylow
227C2×AGL(1,ℤ/8)—2831no—
228C22×D8—2831no—
229#546—2831no—
230D42—2831no—
231#496—2831no—
232#469—2831no—
233#465—2831no—
234#463D28311yescharacteristic, complemented
235#459—2831no—
236#450—2831no—
237#449—2831no—
238#447—2831no—
239#447—2831no—
240#442—2831no—
241#435—2831no—
242C2×#41—2831no—
243C22×D4—2831no—
244C22×D4—2831no—
245C22×D4—2831no—
246C22×D4—5661no—
247AGL(1,ℤ/8)—5664no—
248C2×SD16—2831no—
249C2×SD16—2831no—
250C2×SD16C2×D28311yescharacteristic, complemented
251C2×D8—2831no—
252C2×D8—2831no—
253C2×D8—2831no—
254C2×D8—2831no—
255C2×D8—2831no—
256C2×D8—5664no—
257C2×D8C2×D28311yescharacteristic
258C2×M16—2831no—
259C22×C8—2831no—
260#127—2831no—
261#124C2×D28311yescharacteristic, complemented
262#123—5661no—
263C22⋊Q8—2831no—
264#121—2831no—
265#121—2831no—
266#121—2831no—
267#121—5661no—
268#121—5661no—
269#120—5661no—
270#120—5661no—
271#120—5661no—
272C4×D4—2831no—
273C4×D4—2831no—
274C4×D4C2×D28311yescharacteristic, complemented
275C2×C22⋊C4—5661no—
276#106—2831no—
277C4⋊C8C2×D28311yescharacteristic, complemented
278#103C2×D28311yescharacteristic, complemented
279#102—2831no—
280#102—2831no—
281#102—2831no—
282#102—2831no—
283#102C2×D28311yescharacteristic, complemented
284C22⋊C8—2831no—
285C22⋊C8—2831no—
286#97—2831no—
287C24—5661no—
288C24—5661no—
289C24—5661no—
290#41—5662no—
291#41—11321no—
292C2×Q8C22×D28311yescharacteristic
293C2×D4—2831no—
294C2×D4—2831no—
295C2×D4—2831no—
296C2×D4—2831no—
297C2×D4—5661no—
298C2×D4—5661no—
299C2×D4—5661no—
300C2×D4—5661no—
301C2×D4—5661no—
302C2×D4—5661no—
303C2×D4—5662no—
304C2×D4—5662no—
305C2×D4—5662no—
306C2×D4—5661no—
307C2×D4—5661no—
308C2×D4—5661no—
309C2×D4—11324no—
310C2×D4—11321no—
311C2×D4—11321no—
312C2×D4—11321no—
313C2×D4C22×D28311yescharacteristic
314C2×D4C22×D28311yescharacteristic
315C22×C4—21no—
316C22×C4—2831no—
317C22×C4—2831no—
318C22×C4—5661no—
319C22×C4—5661no—
320SD16—22no—
321SD16—5662no—
322SD16—11321no—
323D8—22no—
324D8—2834no—
325D8—5662no—
326D8—5662no—
327D8—5662no—
328D8—11321no—
329M16—5662no—
330C2×C8—2831no—
331C2×C8—2831no—
332C2×C8—5662no—
333C2×C8C22×D28311yescharacteristic
334C2×C8C22×D28311yescharacteristic
335C4⋊C4—2831no—
336C4⋊C4—2831no—
337C4⋊C4—5661no—
338C4⋊C4C22×D28311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—2831no—
343C22⋊C4—2831no—
344C22⋊C4—5661no—
345C22⋊C4—5661no—
346C22⋊C4—5661no—
347C22⋊C4—5661no—
348C22⋊C4—5661no—
349C42C22×D28311yescharacteristic
350C23—21no—
351C23—21no—
352C23—2831no—
353C23—2831no—
354C23—5661no—
355C23—5661no—
356C23—5661no—
357C23—11322no—
358C23—11322no—
359C23—11322no—
360C23—11322no—
361C23—11322no—
362C23—11322no—
363C23—22641no—
364C23—22641no—
365C23—22641no—
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—2832no—
372D4—5662no—
373D4—5661no—
374D4—5661no—
375D4—5661no—
376D4—5661no—
377D4—5661no—
378D4—11322no—
379D4—11322no—
380D4—11322no—
381D4—11322no—
382D4—11322no—
383D4—11322no—
384D4—11322no—
385D4—22641no—
386D4—22641no—
387D4D4×D28312yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—2831no—
392C2×C4—2831no—
393C2×C4—5661no—
394C2×C4—5661no—
395C2×C4—5661no—
396C2×C4—5661no—
397C2×C4—5661no—
398C2×C4—11322no—
399C2×C4—22641no—
400C2×C4D4×D28311yescharacteristic
401C2×C4D4×D28311yescharacteristic
402C2×C4C23×D28311yescharacteristic
403C8—21no—
404C8—21no—
405C8—2832no—
406C8—5661no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—2832no—
414C22—5661no—
415C22—5661no—
416C22—5661no—
417C22—5661no—
418C22—5661no—
419C22—5662no—
420C22—11321no—
421C22—11321no—
422C22—11321no—
423C22—11321no—
424C22—11321no—
425C22—11321no—
426C22—11321no—
427C22—11321no—
428C22—22644no—
429C22—22644no—
430C22—22644no—
431C22C2×D4×D28311yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—5661no—
437C4—5661no—
438C4—11321no—
439C4C2×D4×D28311yescharacteristic
440C4#2258365611yescharacteristic
441C2—42no—
442C2—81no—
443C2—5662no—
444C2—11321no—
445C2—11322no—
446C2—22641no—
447C2—22641no—
448C2#3182709911yescharacteristic
449C2AGL(1,ℤ/8)×D28311yescharacteristic
450C2#3182726711yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#463×C283C2
socleC22×C283C22×D4
hypercenter#463D283
hypocenterC283#3264
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×#22583637C2semidirect
2. C2×D8×D283C2semidirect
3. #31827108C2semidirect
4. D4×D1132C2semidirect
5. #31826855C2semidirect
6. #31826717C2semidirect
7. #31826697C2semidirect
8. #31826661C2semidirect
9. #31826614C2semidirect
10. #31826613C2semidirect
11. #31826600C2semidirect
12. #31826597C2semidirect
13. #31826575C2semidirect
14. #31826535C2semidirect
15. #463×C283C2semidirect
16. C2×#17985393C22semidirect
19. C2×D4×D283C22semidirect
20. C2×D4×D283C22semidirect
22. C2×#17985379C22semidirect
23. C2×#17985379C22semidirect
24. C2×D2264C22semidirect
25. C2×D2264C22semidirect
27. C2×#17985377C22semidirect
28. C2×#17985377C22semidirect
30. C2×#17985375C22semidirect
31. C2×C8×D283C22semidirect
32. #22583607C22semidirect
34. #22583586C22semidirect
36. #22583585C22semidirect
39. #22583581C22semidirect
44. C4×D1132C22semidirect
46. #22583541C22semidirect
47. #22583532C22semidirect
48. #22583532C22semidirect
49. #22583531C22semidirect
50. #22583530C22semidirect
51. (C2×D283)⋊C8C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. #22583521C22semidirect
54. C2×SD16×C283C22semidirect
56. #124×C283C22semidirect
57. C4×D4×C283C22semidirect
58. C4⋊C8×C283C22semidirect
59. #103×C283C22semidirect
60. #102×C283C22semidirect
96. #17985377D4semidirect
149. D1132C2×D4semidirect
164. C283⋊C8C2×D4semidirect
205. C2×D283AGL(1,ℤ/8)semidirect
225. C283#3264semidirect
234. #463D283semidirect
250. C2×SD16C2×D283semidirect
261. #124C2×D283semidirect
274. C4×D4C2×D283semidirect
277. C4⋊C8C2×D283semidirect
278. #103C2×D283semidirect
283. #102C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836372
C2×D8×D2832
#318271082
D4×D11322
#318268552
#318267172
#318266972
#318266612
#318266142
#318266132
#318266002
#318265972
#318265752
#318265352
#463×C2832
#3264283