Finite Group Databasebeta

← #49132757

Subgroups — #49132757

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
2C22×#17985379C211yesmaximal, characteristic, complemented
3D4×D1132C211yesmaximal, characteristic, complemented
4#31826855C211yesmaximal, characteristic, complemented
5#31826713C211yesmaximal, characteristic, complemented
6#31826699C211yesmaximal, characteristic, complemented
7#31826677C211yesmaximal, characteristic, complemented
8#31826665C211yesmaximal, characteristic, complemented
9#31826615C211yesmaximal, characteristic, complemented
10#31826610C211yesmaximal, characteristic, complemented
11#31826604C211yesmaximal, characteristic, complemented
12#31826598C211yesmaximal, characteristic, complemented
13#31826563C211yesmaximal, characteristic, complemented
14#31826537C211yesmaximal, characteristic, complemented
15#548×C283C211yesmaximal, characteristic, complemented
16C22×D1132C2211yescharacteristic
17C22×D1132C2211yescharacteristic
18C2×D4×D283—21no—
19#22583635—24no—
20C2×#17985379—24no—
21C2×#17985379C2211yescharacteristic
22C2×#17985379C2211yescharacteristic
23C2×#17985379C2211yescharacteristic, complemented
24C2×#17985379C2211yescharacteristic, complemented
25C2×#17985379C2211yescharacteristic, complemented
26C2×#17985379C2211yescharacteristic, complemented
27C2×#17985377C2211yescharacteristic
28C2×#17985377C2211yescharacteristic, complemented
29C2×#17985377C2211yescharacteristic, complemented
30C2×#17985376C2211yescharacteristic, complemented
31C22×C283⋊C8C2211yescharacteristic, complemented
32#22583607C2211yescharacteristic
33#22583585—21no—
34#22583584C2211yescharacteristic
35#22583584C2211yescharacteristic
36#22583580—21no—
37#22583580—21no—
38C4×D1132C2211yescharacteristic
39#22583543C2211yescharacteristic, complemented
40C4⋊(C283⋊C8)C2211yescharacteristic, complemented
41#22583534C2211yescharacteristic, complemented
42#22583534C2211yescharacteristic, complemented
43#22583531C2211yescharacteristic, complemented
44#22583530C2211yescharacteristic
45#22583530C2211yescharacteristic, complemented
46#22583530C2211yescharacteristic, complemented
47C22⋊(C283⋊C8)C2211yescharacteristic, complemented
48C22⋊(C283⋊C8)C2211yescharacteristic, complemented
49#22583521C2211yescharacteristic, complemented
50C2×#41×C283C2211yescharacteristic, complemented
51C22×Q8×C283C2211yescharacteristic, complemented
52#124×C283—21no—
53#124×C283C2211yescharacteristic, complemented
54C22⋊Q8×C283—21no—
55C22⋊Q8×C283C2211yescharacteristic, complemented
56#121×C283—21no—
57#121×C283C2211yescharacteristic, complemented
58C4×Q8×C283C2211yescharacteristic, complemented
59C4×D4×C283—21no—
60C4×D4×C283C2211yescharacteristic
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—22no—
81#17985379—22no—
82#17985379—41no—
83#17985379D414yescomplemented
84#17985377—22no—
85#17985377—22no—
86#17985377—41no—
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—
97#41×C283—22no—
98#41×C283—41no—
99C2×Q8×C283—21no—
100C2×Q8×C283—22no—
101C2×Q8×C283—41no—
102C2×Q8×C283C2311yescharacteristic
103C2×Q8×C283C2311yescharacteristic
104C2×D4×C283—21no—
105C2×D4×C283—21no—
106C2×D4×C283D411yescharacteristic
107C2×D4×C283C2311yescharacteristic
108C22×C4×C283—21no—
109C22×C4×C283—21no—
110C22×C4×C283C2311yescharacteristic
111C22×C4×C283C2311yescharacteristic
112C4⋊C4×C283—21no—
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×C283D411yescharacteristic
121C22⋊C4×C283D411yescharacteristic
122C42×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—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×C283C2×D411yescharacteristic
153C23×C283C2×D411yescharacteristic
154Q8×C283—21no—
155Q8×C283—21no—
156Q8×C283—21no—
157Q8×C283—42no—
158Q8×C283C2×D412yescomplemented
159D4×C283—22no—
160D4×C283—21no—
161D4×C283—21no—
162D4×C283—42no—
163D4×C283—42no—
164C2×C4×C283—21no—
165C2×C4×C283—21no—
166C2×C4×C283—21no—
167C2×C4×C283—21no—
168C2×C4×C283—21no—
169C2×C4×C283—21no—
170C2×C4×C283—21no—
171C2×C4×C283—42no—
172C2×C4×C283—42no—
173C2×C4×C283C2×D411yescharacteristic
174C2×C4×C283C2×D411yescharacteristic
175C2×C4×C283C2×D411yescharacteristic
176C2×C4×C283C2×D411yescharacteristic
177C2×C4×C283C2411yescharacteristic
178C22×C283—21no—
179C22×C283—21no—
180C22×C283—21no—
181C22×C283—21no—
182C22×C283—21no—
183C22×C283—41no—
184C22×C283—41no—
185C22×C283—41no—
186C22×C283AGL(1,ℤ/8)12yescomplemented
187C22×C283C22×D411yescharacteristic
188C2×D283—22no—
189C2×D283—41no—
190C2×D283—41no—
191C2×D283—41no—
192C2×D283—41no—
193C2×D283—41no—
194C2×D283—41no—
195C2×D283—41no—
196C2×D283—41no—
197C2×D283—84no—
198C2×D283—84no—
199C4×C283—21no—
200C4×C283—21no—
201C4×C283—22no—
202C4×C283—21no—
203C4×C283—21no—
204C4×C283—41no—
205C4×C283—41no—
206C4×C283C22×D411yescharacteristic
207C4×C283H(5,𝔽2)11yescharacteristic
208Dic283—41no—
209C2×C283—22no—
210C2×C283—41no—
211C2×C283—81no—
212C2×C283D4211yescharacteristic
213C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C283#57711yescharacteristic
215D283—42no—
216D283—81no—
217D283—81no—
218C283#326511yescharacteristic, complemented, Sylow
219#3265—2831nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2831no—
221C22×SD16—2831no—
222#548D28311yescharacteristic, complemented
223D42—2831no—
224#496—2831no—
225#468—2831no—
226#466—2831no—
227#463—2831no—
228#460—2831no—
229#450—2831no—
230#449—2831no—
231#448—2831no—
232#447—2831no—
233#440—2831no—
234#435—2831no—
235C2×#41C2×D28311yescharacteristic, complemented
236C22×Q8C2×D28311yescharacteristic, complemented
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—2831no—
247C2×SD16—5664no—
248C2×D8—2831no—
249C2×D8—2831no—
250C2×D8—2831no—
251C2×M16—2831no—
252C22×C8—2831no—
253#127—2831no—
254#124—21no—
255#124C2×D28311yescharacteristic, complemented
256C22⋊Q8—21no—
257C22⋊Q8C2×D28311yescharacteristic, complemented
258#121—21no—
259#121—2831no—
260#121—2831no—
261#121—5661no—
262#121C2×D28311yescharacteristic, complemented
263#120—5661no—
264#120—5661no—
265C4×Q8C2×D28311yescharacteristic, complemented
266C4×D4—21no—
267C4×D4—2831no—
268C4×D4C2×D28311yescharacteristic
269#107—2831no—
270C4⋊C8—2831no—
271#103—2831no—
272#103—2831no—
273#102—2831no—
274#102—2831no—
275#102—2831no—
276#102—2831no—
277C22⋊C8—2831no—
278C22⋊C8—2831no—
279#97—2831no—
280C24—5661no—
281C24—5661no—
282#41—22no—
283#41—41no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D28311yescharacteristic
288C2×Q8C22×D28311yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—2831no—
292C2×D4—2831no—
293C2×D4—2831no—
294C2×D4—5662no—
295C2×D4—5662no—
296C2×D4—5661no—
297C2×D4—5661no—
298C2×D4—5661no—
299C2×D4—5661no—
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—21no—
309C22×C4—5661no—
310C22×C4C22×D28311yescharacteristic
311C22×C4C22×D28311yescharacteristic
312SD16—2834no—
313SD16—5662no—
314SD16—5662no—
315SD16—5662no—
316SD16—5662no—
317SD16—11321no—
318D8—5662no—
319D8—5662no—
320D8—11321no—
321M16—5662no—
322C2×C8—2831no—
323C2×C8—2831no—
324C2×C8—2831no—
325C2×C8—2831no—
326C2×C8—5662no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—2831no—
330C4⋊C4C22⋊D28311yescharacteristic
331C4⋊C4C22×D28311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—5661no—
337C22⋊C4—5661no—
338C22⋊C4—5661no—
339C22⋊C4C22⋊D28311yescharacteristic
340C22⋊C4C22⋊D28311yescharacteristic
341C42—21no—
342C42C22×D28311yescharacteristic
343C23—21no—
344C23—5661no—
345C23—5661no—
346C23—5661no—
347C23—11322no—
348C23—11322no—
349C23—11322no—
350C23—11322no—
351C23—22641no—
352C23—22641no—
353C23C2×C22⋊D28311yescharacteristic
354C23C2×C22⋊D28311yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8D4×D28312yescomplemented
360D4—22no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—2832no—
366D4—5661no—
367D4—5661no—
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—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—5661no—
387C2×C4—11322no—
388C2×C4C2×C22⋊D28311yescharacteristic
389C2×C4C2×C22⋊D28311yescharacteristic
390C2×C4C2×C22⋊D28311yescharacteristic
391C2×C4C2×C22⋊D28311yescharacteristic
392C2×C4C23×D28311yescharacteristic
393C8—2832no—
394C8—5661no—
395C8—5661no—
396C8—5661no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—5662no—
406C22—11321no—
407C22—11321no—
408C22—11321no—
409C22—11321no—
410C22—11321no—
411C22—11321no—
412C22—11321no—
413C22—11321no—
414C22—22644no—
415C22—22644no—
416C22#2258363512yescomplemented
417C22C22×C22⋊D28311yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—22no—
423C4—41no—
424C4—41no—
425C4—11321no—
426C4C2×D4×D28311yescharacteristic
427C4#2258365511yescharacteristic
428C2—22no—
429C2—41no—
430C2—81no—
431C2—11322no—
432C2—22641no—
433C2—22641no—
434C2D4×C22⋊D28311yescharacteristic
435C2C2×#2258363511yescharacteristic
436C2#3182727311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#548×C283C2
socleC22×C283C22×D4
hypercenter#548D283
hypocenterC283#3265
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#22583635C2semidirect
2. C22×#17985379C2semidirect
3. D4×D1132C2semidirect
4. #31826855C2semidirect
5. #31826713C2semidirect
6. #31826699C2semidirect
7. #31826677C2semidirect
8. #31826665C2semidirect
9. #31826615C2semidirect
10. #31826610C2semidirect
11. #31826604C2semidirect
12. #31826598C2semidirect
13. #31826563C2semidirect
14. #31826537C2semidirect
15. #548×C283C2semidirect
23. C2×#17985379C22semidirect
24. C2×#17985379C22semidirect
25. C2×#17985379C22semidirect
26. C2×#17985379C22semidirect
28. C2×#17985377C22semidirect
29. C2×#17985377C22semidirect
30. C2×#17985376C22semidirect
31. C22×C283⋊C8C22semidirect
39. #22583543C22semidirect
40. C4⋊(C283⋊C8)C22semidirect
41. #22583534C22semidirect
42. #22583534C22semidirect
43. #22583531C22semidirect
45. #22583530C22semidirect
46. #22583530C22semidirect
47. C22⋊(C283⋊C8)C22semidirect
48. C22⋊(C283⋊C8)C22semidirect
49. #22583521C22semidirect
50. C2×#41×C283C22semidirect
51. C22×Q8×C283C22semidirect
53. #124×C283C22semidirect
55. C22⋊Q8×C283C22semidirect
57. #121×C283C22semidirect
58. C4×Q8×C283C22semidirect
83. #17985379D4semidirect
150. C283⋊C8C2×D4semidirect
158. Q8×C283C2×D4semidirect
186. C22×C283AGL(1,ℤ/8)semidirect
218. C283#3265semidirect
222. #548D283semidirect
235. C2×#41C2×D283semidirect
236. C22×Q8C2×D283semidirect
255. #124C2×D283semidirect
257. C22⋊Q8C2×D283semidirect
262. #121C2×D283semidirect
265. C4×Q8C2×D283semidirect
359. Q8D4×D283semidirect
416. 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×#179853792
D4×D11322
#318268552
#318267132
#318266992
#318266772
#318266652
#318266152
#318266102
#318266042
#318265982
#318265632
#318265372
#548×C2832
#3265283