Finite Group Databasebeta

← #49132740

Subgroups — #49132740

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
4#31827100C211yesmaximal, characteristic, complemented
5#31826715C211yesmaximal, characteristic, complemented
6#31826691C211yesmaximal, characteristic, complemented
7#31826677C211yesmaximal, characteristic, complemented
8#31826657C211yesmaximal, characteristic, complemented
9#31826614C211yesmaximal, characteristic, complemented
10#31826609C211yesmaximal, characteristic, complemented
11#31826597C211yesmaximal, characteristic, complemented
12#31826597C211yesmaximal, characteristic, complemented
13#31826578C211yesmaximal, characteristic, complemented
14#31826538C211yesmaximal, characteristic, complemented
15#496×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic, complemented
17C2×D4×D283—21no—
18C2×D4×D283C2211yescharacteristic
19C2×D4×D283C2211yescharacteristic, complemented
20C2×D4×D283C2211yescharacteristic, complemented
21#22583637—24no—
22C2×#17985379C2211yescharacteristic, complemented
23C2×D2264C2211yescharacteristic
24C2×#17985377C2211yescharacteristic
25C2×#17985377C2211yescharacteristic, complemented
26C2×#17985377C2211yescharacteristic, complemented
27D8×D283—24no—
28C2×#17985375C2211yescharacteristic, complemented
29C2×C8×D283C2211yescharacteristic, complemented
30#22583592—21no—
31#22583586C2211yescharacteristic, complemented
32#22583583—21no—
33#22583581—21no—
34#22583581C2211yescharacteristic
35#22583581C2211yescharacteristic, complemented
36#22583581C2211yescharacteristic, complemented
37#22583580—21no—
38#22583578—21no—
39#22583578—21no—
40C4×D1132C2211yescharacteristic
41C4×D1132C2211yescharacteristic, complemented
42C4×D1132C2211yescharacteristic, complemented
43C22⋊C4×D283—21no—
44#22583542C2211yescharacteristic, complemented
45C4⋊(C283⋊C8)C2211yescharacteristic, complemented
46#22583534C2211yescharacteristic, complemented
47#22583532C2211yescharacteristic, complemented
48#22583532C2211yescharacteristic, complemented
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, complemented
59#124×C283C2211yescharacteristic, complemented
60#97×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62C23×D283—21no—
63C23×D283—21no—
64#17985393—22no—
65#17985393—41no—
66C2×D1132—21no—
67C2×D1132D411yescharacteristic
68C2×D1132C2311yescharacteristic
69C2×D1132C2311yescharacteristic
70C2×C22⋊D283—21no—
71C2×C22⋊D283—21no—
72C2×C22⋊D283—21no—
73C2×C22⋊D283—21no—
74C2×C22⋊D283—21no—
75C2×C22⋊D283—21no—
76D4×D283—22no—
77D4×D283—22no—
78D4×D283—22no—
79D4×D283—41no—
80D4×D283—41no—
81D4×D283—41no—
82D4×D283—44no—
83C2×C4×D283—21no—
84C2×C4×D283—21no—
85C2×C4×D283—21no—
86C2×C4×D283C2311yescharacteristic
87C2×C4×D283C2311yescharacteristic
88#17985379—22no—
89D2264D414yescomplemented
90#17985377—22no—
91#17985377—22no—
92#17985377—22no—
93#17985375—22no—
94C2×C283⋊C8C2311yescharacteristic
95C2×C283⋊C8C2311yescharacteristic
96C8×D283—22no—
97C4⋊Dic283D411yescharacteristic
98C4⋊Dic283C2311yescharacteristic
99C4⋊Dic283C2311yescharacteristic
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)⋊C4—21no—
107(C2×D283)⋊C4D411yescharacteristic
108(C2×D283)⋊C4D411yescharacteristic
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—21no—
126C22×D283—42no—
127C22×D283—42no—
128C22×D283—42no—
129C22×D283—42no—
130C22×D283—42no—
131C22×D283—42no—
132C22×D283—81no—
133C22×D283—81no—
134C22×D283—81no—
135C22×D283C2×D411yescharacteristic
136C22×D283C2×D411yescharacteristic
137D1132—22no—
138D1132—21no—
139D1132—21no—
140D1132—42no—
141D1132C2×D412yes—
142D1132C2×D412yescomplemented
143C22⋊D283—42no—
144C22⋊D283—42no—
145C22⋊D283—42no—
146C22⋊D283—42no—
147C22⋊D283—42no—
148C22⋊D283—81no—
149C2×Dic283—21no—
150C2×Dic283—21no—
151C2×Dic283C2×D411yescharacteristic
152C2×Dic283C2×D411yescharacteristic
153C4×D283—21no—
154C4×D283—21no—
155C4×D283—21no—
156C4×D283—21no—
157C4×D283—42no—
158C4×D283—42no—
159C4×D283—81no—
160C283⋊C8—21no—
161C283⋊C8—21no—
162C23×C283—21no—
163C23×C283—21no—
164C23×C283—21no—
165Q8×C283—21no—
166Q8×C283—21no—
167D4×C283—21no—
168D4×C283—21no—
169D4×C283—21no—
170D4×C283—21no—
171D4×C283—21no—
172D4×C283—21no—
173D4×C283—42no—
174D4×C283—81no—
175C2×C4×C283—21no—
176C2×C4×C283C2×D411yescharacteristic
177C2×C4×C283C2×D411yescharacteristic
178C2×C4×C283C2411yescharacteristic
179C8×C283—21no—
180C8×C283C2×D412yescomplemented
181C22×C283—41no—
182C22×C283—41no—
183C22×C283—41no—
184C22×C283—41no—
185C22×C283—41no—
186C22×C283—41no—
187C22×C283—41no—
188C22×C283—41no—
189C22×C283—41no—
190C22×C283C22×D411yescharacteristic
191C2×D283—21no—
192C2×D283—21no—
193C2×D283—21no—
194C2×D283—22no—
195C2×D283—22no—
196C2×D283—21no—
197C2×D283—21no—
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×C283C22×D411yescharacteristic
210C4×C283H(5,𝔽2)11yescharacteristic
211Dic283—21no—
212Dic283—21no—
213Dic283—41no—
214Dic283—41no—
215C2×C283—81no—
216C2×C283—81no—
217C2×C283—81no—
218C2×C283D4211yescharacteristic
219C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C283#57611yescharacteristic
221D283—22no—
222D283—41no—
223D283—42no—
224D283—42no—
225C283#326411yescharacteristic, complemented, Sylow
226#3264—2831nomaximal, Sylow
227C2×AGL(1,ℤ/8)—2831no—
228C22×D8—2831no—
229#546—2831no—
230D42—2831no—
231#496D28311yescharacteristic, complemented
232#469—2831no—
233#465—2831no—
234#463—2831no—
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×SD16C2×D28311yescharacteristic, complemented
250C2×SD16C2×D28311yescharacteristic, complemented
251C2×D8—2831no—
252C2×D8—2831no—
253C2×D8—2831no—
254C2×D8—2831no—
255C2×D8—5664no—
256C2×D8C2×D28311yescharacteristic, complemented
257C2×D8C2×D28311yescharacteristic, complemented
258C2×M16—2831no—
259C22×C8—2831no—
260#127C2×D28311yescharacteristic, complemented
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×D4—2831no—
275C2×C22⋊C4—5661no—
276#106—2831no—
277C4⋊C8—2831no—
278#103—2831no—
279#102—2831no—
280#102—2831no—
281#102—2831no—
282#102—2831no—
283#102—2831no—
284C22⋊C8—2831no—
285C22⋊C8—2831no—
286#97C2×D28311yescharacteristic, complemented
287C24—5661no—
288C24—5661no—
289C24—5661no—
290#41—5662no—
291#41—11321no—
292C2×Q8C22×D28311yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—2831no—
296C2×D4—2831no—
297C2×D4—2831no—
298C2×D4—5661no—
299C2×D4—5661no—
300C2×D4—5661no—
301C2×D4—5661no—
302C2×D4—5662no—
303C2×D4—5662no—
304C2×D4—5662no—
305C2×D4—5661no—
306C2×D4—5661no—
307C2×D4—5661no—
308C2×D4—11321no—
309C2×D4—11321no—
310C2×D4—11321no—
311C2×D4—11324no—
312C2×D4C22×D28311yescharacteristic
313C2×D4C22×D28311yescharacteristic
314C2×D4C22×D28311yescharacteristic
315C22×C4—2831no—
316C22×C4—2831no—
317C22×C4—5661no—
318C22×C4—5661no—
319C22×C4—5661no—
320SD16—22no—
321SD16—41no—
322SD16—5662no—
323D8—22no—
324D8—41no—
325D8—2834no—
326D8—5662no—
327D8—5662no—
328D8—5662no—
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—2831no—
338C4⋊C4—5661no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—2831no—
342C22⋊C4—2831no—
343C22⋊C4—5661no—
344C22⋊C4—5661no—
345C22⋊C4—5661no—
346C22⋊C4—5661no—
347C22⋊C4—5661no—
348C22⋊C4—5661no—
349C42C22×D28311yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—2831no—
354C23—2831no—
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—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—81no—
376D4—2832no—
377D4—2832no—
378D4—5662no—
379D4—5661no—
380D4—5661no—
381D4—11322no—
382D4—11322no—
383D4—11322no—
384D4—11322no—
385D4—11322no—
386D4—11322no—
387D4—22641no—
388C2×C4—21no—
389C2×C4—2831no—
390C2×C4—2831no—
391C2×C4—5661no—
392C2×C4—5661no—
393C2×C4—5661no—
394C2×C4—5661no—
395C2×C4—5661no—
396C2×C4—5661no—
397C2×C4—11322no—
398C2×C4—11322no—
399C2×C4—22641no—
400C2×C4D4×D28311yescharacteristic
401C2×C4D4×D28311yescharacteristic
402C2×C4C23×D28311yescharacteristic
403C8—21no—
404C8—5661no—
405C8—5661no—
406C8D4×D28312yescomplemented
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—2832no—
417C22—5661no—
418C22—5661no—
419C22—5661no—
420C22—5661no—
421C22—5661no—
422C22—5662no—
423C22—5662no—
424C22—11321no—
425C22—11321no—
426C22—11321no—
427C22—11321no—
428C22—22644no—
429C22—22644no—
430C22—22644no—
431C22C2×D4×D28311yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—5661no—
436C4—5661no—
437C4—11321no—
438C4—11321no—
439C4C2×D4×D28311yescharacteristic
440C4#2258365511yescharacteristic
441C2—81no—
442C2—81no—
443C2—81no—
444C2—5662no—
445C2—11321no—
446C2—11322no—
447C2—11322no—
448C2#3182710011yescharacteristic
449C2AGL(1,ℤ/8)×D28311yescharacteristic
450C2#3182726911yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#496×C283C2
socleC22×C283C22×D4
hypercenter#496D283
hypocenterC283#3264
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×#22583637C2semidirect
2. C2×D8×D283C2semidirect
3. #31827108C2semidirect
4. #31827100C2semidirect
5. #31826715C2semidirect
6. #31826691C2semidirect
7. #31826677C2semidirect
8. #31826657C2semidirect
9. #31826614C2semidirect
10. #31826609C2semidirect
11. #31826597C2semidirect
12. #31826597C2semidirect
13. #31826578C2semidirect
14. #31826538C2semidirect
15. #496×C283C2semidirect
16. C2×#17985393C22semidirect
19. C2×D4×D283C22semidirect
20. C2×D4×D283C22semidirect
22. C2×#17985379C22semidirect
25. C2×#17985377C22semidirect
26. C2×#17985377C22semidirect
28. C2×#17985375C22semidirect
29. C2×C8×D283C22semidirect
31. #22583586C22semidirect
35. #22583581C22semidirect
36. #22583581C22semidirect
41. C4×D1132C22semidirect
42. C4×D1132C22semidirect
44. #22583542C22semidirect
45. C4⋊(C283⋊C8)C22semidirect
46. #22583534C22semidirect
47. #22583532C22semidirect
48. #22583532C22semidirect
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
58. #127×C283C22semidirect
59. #124×C283C22semidirect
60. #97×C283C22semidirect
89. D2264D4semidirect
142. D1132C2×D4semidirect
180. C8×C283C2×D4semidirect
205. C2×D283AGL(1,ℤ/8)semidirect
225. C283#3264semidirect
231. #496D283semidirect
249. C2×SD16C2×D283semidirect
250. C2×SD16C2×D283semidirect
256. C2×D8C2×D283semidirect
257. C2×D8C2×D283semidirect
260. #127C2×D283semidirect
261. #124C2×D283semidirect
286. #97C2×D283semidirect
406. C8D4×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
#318271002
#318267152
#318266912
#318266772
#318266572
#318266142
#318266092
#318265972
#318265782
#318265382
#496×C2832
#3264283