Finite Group Databasebeta

← #49132742

Subgroups — #49132742

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×#22583635C211yesmaximal, characteristic, complemented
2C22×#17985377C211yesmaximal, characteristic, complemented
3D4×D1132C211yesmaximal, characteristic, complemented
4#31826855C211yesmaximal, characteristic, complemented
5#31826716C211yesmaximal, characteristic, complemented
6#31826696C211yesmaximal, characteristic, complemented
7#31826677C211yesmaximal, characteristic, complemented
8#31826660C211yesmaximal, characteristic, complemented
9#31826615C211yesmaximal, characteristic, complemented
10#31826610C211yesmaximal, characteristic, complemented
11#31826599C211yesmaximal, characteristic, complemented
12#31826598C211yesmaximal, characteristic, complemented
13#31826575C211yesmaximal, characteristic, complemented
14#31826537C211yesmaximal, characteristic, complemented
15#546×C283C211yesmaximal, characteristic, complemented
16C22×D1132C2211yescharacteristic
17C22×D1132C2211yescharacteristic, complemented
18C2×D4×D283—21no—
19#22583635—24no—
20C2×#17985379C2211yescharacteristic, complemented
21C2×#17985379C2211yescharacteristic, complemented
22C2×#17985379C2211yescharacteristic, complemented
23C2×#17985377—24no—
24C2×#17985377C2211yescharacteristic
25C2×#17985377C2211yescharacteristic
26C2×#17985377C2211yescharacteristic, complemented
27C2×#17985377C2211yescharacteristic, complemented
28C2×#17985377C2211yescharacteristic, complemented
29C2×#17985377C2211yescharacteristic, complemented
30C2×#17985376C2211yescharacteristic, complemented
31C22×C283⋊C8C2211yescharacteristic, complemented
32#22583607C2211yescharacteristic, complemented
33#22583585—21no—
34#22583584C2211yescharacteristic
35#22583584C2211yescharacteristic, complemented
36#22583580—21no—
37#22583580—21no—
38C4×D1132C2211yescharacteristic, complemented
39#22583541C2211yescharacteristic, complemented
40C4⋊(C283⋊C8)C2211yescharacteristic, complemented
41#22583534C2211yescharacteristic, complemented
42#22583531C2211yescharacteristic, complemented
43#22583531C2211yescharacteristic, complemented
44#22583530C2211yescharacteristic, complemented
45#22583530C2211yescharacteristic, complemented
46#22583530C2211yescharacteristic, complemented
47C22⋊(C283⋊C8)C2211yescharacteristic, complemented
48C22⋊(C283⋊C8)C2211yescharacteristic, complemented
49#22583521C2211yescharacteristic, complemented
50C2×#41×C283C2211yescharacteristic, complemented
51C22×D4×C283C2211yescharacteristic, complemented
52#124×C283C2211yescharacteristic, complemented
53#123×C283—21no—
54C22⋊Q8×C283C2211yescharacteristic, complemented
55#121×C283—21no—
56#121×C283C2211yescharacteristic, complemented
57#120×C283—21no—
58C4×D4×C283C2211yescharacteristic
59C4×D4×C283C2211yescharacteristic, complemented
60C2×C22⋊C4×C283—21no—
61C23×D283—21no—
62C23×D283—21no—
63C2×D1132—22no—
64C2×D1132—22no—
65C2×D1132—21no—
66C2×D1132—21no—
67C2×D1132—21no—
68C2×D1132—21no—
69C2×D1132—41no—
70C2×D1132—41no—
71C2×D1132C2311yescharacteristic
72C2×D1132C2311yescharacteristic
73C2×D1132C2311yescharacteristic
74C2×C22⋊D283—21no—
75C2×C22⋊D283—21no—
76D4×D283—44no—
77C2×C4×D283—21no—
78#17985379—22no—
79#17985379—22no—
80#17985379—41no—
81#17985377—22no—
82#17985377—22no—
83#17985377—22no—
84#17985377—22no—
85#17985377—41no—
86#17985377D414yescomplemented
87#17985376—22no—
88C2×C283⋊C8—22no—
89C2×C283⋊C8C2311yescharacteristic
90C2×C283⋊C8C2311yescharacteristic
91C2×C283⋊C8C2311yescharacteristic
92C2×C283⋊C8C2311yescharacteristic
93C4⋊Dic283C2311yescharacteristic
94(C2×D283)⋊C4—21no—
95(C2×D283)⋊C4—21no—
96(C2×D283)⋊C4—21no—
97C24×C283—21no—
98#41×C283—22no—
99#41×C283—41no—
100C2×Q8×C283C2311yescharacteristic
101C2×D4×C283—22no—
102C2×D4×C283—21no—
103C2×D4×C283—21no—
104C2×D4×C283—21no—
105C2×D4×C283—41no—
106C2×D4×C283D411yescharacteristic
107C2×D4×C283C2311yescharacteristic
108C2×D4×C283C2311yescharacteristic
109C22×C4×C283—21no—
110C22×C4×C283—21no—
111C22×C4×C283C2311yescharacteristic
112C22×C4×C283C2311yescharacteristic
113C4⋊C4×C283—21no—
114C4⋊C4×C283D411yescharacteristic
115C4⋊C4×C283C2311yescharacteristic
116C22⋊C4×C283—21no—
117C22⋊C4×C283—21no—
118C22⋊C4×C283—21no—
119C22⋊C4×C283—21no—
120C22⋊C4×C283—21no—
121C22⋊C4×C283D411yescharacteristic
122C22⋊C4×C283D411yescharacteristic
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—81no—
132C22×D283—81no—
133D1132—21no—
134D1132—21no—
135D1132—21no—
136D1132—21no—
137D1132—21no—
138D1132—42no—
139D1132—42no—
140D1132—42no—
141D1132—81no—
142D1132C2×D412yes—
143C22⋊D283—42no—
144C22⋊D283—42no—
145C2×Dic283—21no—
146C4×D283—42no—
147C283⋊C8—21no—
148C283⋊C8—21no—
149C283⋊C8—21no—
150C283⋊C8C2×D412yescomplemented
151C23×C283—21no—
152C23×C283—21no—
153C23×C283—42no—
154C23×C283—42no—
155C23×C283—81no—
156C23×C283C2×D411yescharacteristic
157C23×C283C2×D411yescharacteristic
158Q8×C283—21no—
159Q8×C283—21no—
160D4×C283—22no—
161D4×C283—21no—
162D4×C283—21no—
163D4×C283—21no—
164D4×C283—42no—
165D4×C283—42no—
166D4×C283—81no—
167D4×C283C2×D412yescomplemented
168C2×C4×C283—21no—
169C2×C4×C283—21no—
170C2×C4×C283—21no—
171C2×C4×C283—21no—
172C2×C4×C283—21no—
173C2×C4×C283—21no—
174C2×C4×C283—42no—
175C2×C4×C283—81no—
176C2×C4×C283C2×D411yescharacteristic
177C2×C4×C283C2×D411yescharacteristic
178C2×C4×C283C2×D411yescharacteristic
179C2×C4×C283C2×D411yescharacteristic
180C2×C4×C283C2411yescharacteristic
181C22×C283—21no—
182C22×C283—21no—
183C22×C283—21no—
184C22×C283—22no—
185C22×C283—21no—
186C22×C283—21no—
187C22×C283—41no—
188C22×C283—41no—
189C22×C283—41no—
190C22×C283—41no—
191C22×C283—41no—
192C22×C283—84no—
193C22×C283AGL(1,ℤ/8)12yescomplemented
194C22×C283C22×D411yescharacteristic
195C2×D283—22no—
196C2×D283—41no—
197C2×D283—41no—
198C2×D283—41no—
199C2×D283—41no—
200C2×D283—41no—
201C2×D283—41no—
202C2×D283—41no—
203C2×D283—41no—
204C2×D283—84no—
205C2×D283—84no—
206C4×C283—21no—
207C4×C283—21no—
208C4×C283—21no—
209C4×C283—21no—
210C4×C283—41no—
211C4×C283—41no—
212C4×C283C22×D411yescharacteristic
213C4×C283H(5,𝔽2)11yescharacteristic
214Dic283—41no—
215C2×C283—22no—
216C2×C283—41no—
217C2×C283—42no—
218C2×C283—81no—
219C2×C283D4211yescharacteristic
220C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C283#57611yescharacteristic
222D283—42no—
223D283—81no—
224D283—81no—
225C283#326411yescharacteristic, complemented, Sylow
226#3264—2831nomaximal, Sylow
227C2×AGL(1,ℤ/8)—2831no—
228C22×D8—2831no—
229#546D28311yescharacteristic, complemented
230D42—2831no—
231#496—2831no—
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×#41C2×D28311yescharacteristic, complemented
243C22×D4—2831no—
244C22×D4—2831no—
245C22×D4—5661no—
246C22×D4C2×D28311yescharacteristic, complemented
247AGL(1,ℤ/8)—5664no—
248C2×SD16—2831no—
249C2×SD16—2831no—
250C2×SD16—2831no—
251C2×D8—2831no—
252C2×D8—2831no—
253C2×D8—2831no—
254C2×D8—2831no—
255C2×D8—2831no—
256C2×D8—2831no—
257C2×D8—5664no—
258C2×M16—2831no—
259C22×C8—2831no—
260#127—2831no—
261#124C2×D28311yescharacteristic, complemented
262#123—21no—
263C22⋊Q8C2×D28311yescharacteristic, complemented
264#121—21no—
265#121—2831no—
266#121—2831no—
267#121—5661no—
268#121C2×D28311yescharacteristic, complemented
269#120—21no—
270#120—5661no—
271#120—5661no—
272C4×D4—2831no—
273C4×D4C2×D28311yescharacteristic
274C4×D4C2×D28311yescharacteristic, complemented
275C2×C22⋊C4—21no—
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#97—2831no—
287C24—21no—
288C24—5661no—
289C24—5661no—
290#41—22no—
291#41—41no—
292C2×Q8C22×D28311yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—2831no—
299C2×D4—2831no—
300C2×D4—2831no—
301C2×D4—5662no—
302C2×D4—5662no—
303C2×D4—5661no—
304C2×D4—5661no—
305C2×D4—5661no—
306C2×D4—5661no—
307C2×D4—5661no—
308C2×D4—5661no—
309C2×D4—11321no—
310C2×D4—11321no—
311C2×D4—11324no—
312C2×D4C22⋊D28311yescharacteristic
313C2×D4C22×D28311yescharacteristic
314C2×D4C22×D28311yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—5661no—
318C22×C4C22×D28311yescharacteristic
319C22×C4C22×D28311yescharacteristic
320SD16—5662no—
321SD16—5662no—
322SD16—11321no—
323D8—2834no—
324D8—5662no—
325D8—5662no—
326D8—5662no—
327D8—5662no—
328D8—11321no—
329M16—5662no—
330C2×C8—2831no—
331C2×C8—2831no—
332C2×C8—2831no—
333C2×C8—2831no—
334C2×C8—5662no—
335C4⋊C4—21no—
336C4⋊C4—2831no—
337C4⋊C4C22⋊D28311yescharacteristic
338C4⋊C4C22×D28311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—5661no—
345C22⋊C4—5661no—
346C22⋊C4—5661no—
347C22⋊C4C22⋊D28311yescharacteristic
348C22⋊C4C22⋊D28311yescharacteristic
349C42C22×D28311yescharacteristic
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—5661no—
356C23—5661no—
357C23—5661no—
358C23—11322no—
359C23—11322no—
360C23—11322no—
361C23—11322no—
362C23—22641no—
363C23—22641no—
364C23C2×C22⋊D28311yescharacteristic
365C23C2×C22⋊D28311yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—42no—
374D4—81no—
375D4—2832no—
376D4—5661no—
377D4—5661no—
378D4—5661no—
379D4—5661no—
380D4—5661no—
381D4—11322no—
382D4—11322no—
383D4—11322no—
384D4—11322no—
385D4—11322no—
386D4—22641no—
387D4D4×D28312yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—81no—
396C2×C4—5661no—
397C2×C4—11322no—
398C2×C4C2×C22⋊D28311yescharacteristic
399C2×C4C2×C22⋊D28311yescharacteristic
400C2×C4C2×C22⋊D28311yescharacteristic
401C2×C4C2×C22⋊D28311yescharacteristic
402C2×C4C23×D28311yescharacteristic
403C8—2832no—
404C8—5661no—
405C8—5661no—
406C8—5661no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—5662no—
420C22—11321no—
421C22—11321no—
422C22—11321no—
423C22—11321no—
424C22—11321no—
425C22—11321no—
426C22—11321no—
427C22—11321no—
428C22—22644no—
429C22—22644no—
430C22#2258363512yescomplemented
431C22C22×C22⋊D28311yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—41no—
438C4—11321no—
439C4C2×D4×D28311yescharacteristic
440C4#2258365511yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—11322no—
446C2—22641no—
447C2—22641no—
448C2D4×C22⋊D28311yescharacteristic
449C2C2×#2258363511yescharacteristic
450C2#3182726811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#546×C283C2
socleC22×C283C22×D4
hypercenter#546D283
hypocenterC283#3264
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#22583635C2semidirect
2. C22×#17985377C2semidirect
3. D4×D1132C2semidirect
4. #31826855C2semidirect
5. #31826716C2semidirect
6. #31826696C2semidirect
7. #31826677C2semidirect
8. #31826660C2semidirect
9. #31826615C2semidirect
10. #31826610C2semidirect
11. #31826599C2semidirect
12. #31826598C2semidirect
13. #31826575C2semidirect
14. #31826537C2semidirect
15. #546×C283C2semidirect
17. C22×D1132C22semidirect
20. C2×#17985379C22semidirect
21. C2×#17985379C22semidirect
22. C2×#17985379C22semidirect
26. C2×#17985377C22semidirect
27. C2×#17985377C22semidirect
28. C2×#17985377C22semidirect
29. C2×#17985377C22semidirect
30. C2×#17985376C22semidirect
31. C22×C283⋊C8C22semidirect
32. #22583607C22semidirect
35. #22583584C22semidirect
38. C4×D1132C22semidirect
39. #22583541C22semidirect
40. C4⋊(C283⋊C8)C22semidirect
41. #22583534C22semidirect
42. #22583531C22semidirect
43. #22583531C22semidirect
44. #22583530C22semidirect
45. #22583530C22semidirect
46. #22583530C22semidirect
47. C22⋊(C283⋊C8)C22semidirect
48. C22⋊(C283⋊C8)C22semidirect
49. #22583521C22semidirect
50. C2×#41×C283C22semidirect
51. C22×D4×C283C22semidirect
52. #124×C283C22semidirect
54. C22⋊Q8×C283C22semidirect
56. #121×C283C22semidirect
59. C4×D4×C283C22semidirect
86. #17985377D4semidirect
150. C283⋊C8C2×D4semidirect
167. D4×C283C2×D4semidirect
193. C22×C283AGL(1,ℤ/8)semidirect
225. C283#3264semidirect
229. #546D283semidirect
242. C2×#41C2×D283semidirect
246. C22×D4C2×D283semidirect
261. #124C2×D283semidirect
263. C22⋊Q8C2×D283semidirect
268. #121C2×D283semidirect
274. C4×D4C2×D283semidirect
387. D4D4×D283semidirect
430. C22#22583635semidirect

Maximal subgroups

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

subgroupindex
C2×#225836352
C22×#179853772
D4×D11322
#318268552
#318267162
#318266962
#318266772
#318266602
#318266152
#318266102
#318265992
#318265982
#318265752
#318265372
#546×C2832
#3264283