Finite Group Databasebeta

← #49132859

Subgroups — #49132859

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×#22583637C211yesmaximal, characteristic, complemented
2C2×#22583636C211yesmaximal, characteristic, complemented
3#31827117C211yesmaximal, characteristic, complemented
4D4×C22⋊D283C211yesmaximal, characteristic, complemented
5#31826836C211yesmaximal, characteristic, complemented
6#31826713C211yesmaximal, characteristic, complemented
7#31826710C211yesmaximal, characteristic, complemented
8#31826668C211yesmaximal, characteristic, complemented
9#31826659C211yesmaximal, characteristic, complemented
10#31826617C211yesmaximal, characteristic, complemented
11#31826612C211yesmaximal, characteristic, complemented
12#31826609C211yesmaximal, characteristic, complemented
13#31826545C211yesmaximal, characteristic, complemented
14#31826540C211yesmaximal, characteristic, complemented
15#450×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic, complemented
17C2×#17985392C2211yescharacteristic, complemented
18C22×C22⋊D283—21no—
19C2×D4×D283C2211yescharacteristic
20#22583637—24no—
21#22583636—24no—
22C2×#17985381C2211yescharacteristic, complemented
23C2×#17985380C2211yescharacteristic
24C2×#17985380C2211yescharacteristic, complemented
25C2×#17985379C2211yescharacteristic, complemented
26C2×D2264C2211yescharacteristic, complemented
27C2×#17985377C2211yescharacteristic
28C2×#17985377C2211yescharacteristic, complemented
29C2×#17985376C2211yescharacteristic, complemented
30C2×#17985375C2211yescharacteristic, complemented
31#22583608C2211yescharacteristic, complemented
32#22583606C2211yescharacteristic
33#22583595—21no—
34#22583592—21no—
35#22583590C2211yescharacteristic, complemented
36#22583584C2211yescharacteristic
37#22583583—21no—
38#22583582—21no—
39#22583581C2211yescharacteristic
40#22583579—21no—
41#22583578—21no—
42D4×Dic283C2211yescharacteristic
43#22583572C2211yescharacteristic, complemented
44C4×C22⋊D283C2211yescharacteristic
45C4⋊C4×D283—21no—
46#22583543C2211yescharacteristic, complemented
47Dic283⋊C8C2211yescharacteristic, complemented
48#22583535C2211yescharacteristic, complemented
49#22583534C2211yescharacteristic, complemented
50#22583532C2211yescharacteristic, complemented
51#22583530C2211yescharacteristic, complemented
52(C2×D283)⋊C8C2211yescharacteristic, complemented
53C8×Dic283C2211yescharacteristic, complemented
54C22×D4×C283C2211yescharacteristic
55C2×SD16×C283C2211yescharacteristic
56C2×SD16×C283C2211yescharacteristic, complemented
57C22⋊Q8×C283C2211yescharacteristic, complemented
58#102×C283C2211yescharacteristic
59#102×C283C2211yescharacteristic, complemented
60C22⋊C8×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62#17985393—22no—
63#17985393—41no—
64#17985392—22no—
65#17985392—41no—
66C2×Dic566C2311yescharacteristic
67C2×D1132—21no—
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—
76C2×C22⋊D283—44no—
77C2×C22⋊D283D411yescharacteristic
78D4×D283—22no—
79D4×D283—41no—
80C22×Dic283—21no—
81C2×C4×D283—21no—
82C2×C4×D283—21no—
83C2×C4×D283—21no—
84C2×C4×D283C2311yescharacteristic
85#17985381—41no—
86#17985380—22no—
87#17985380—22no—
88#17985379—22no—
89D2264—22no—
90#17985377—22no—
91#17985377D414yescomplemented
92#17985376—22no—
93#17985375—22no—
94C2×C283⋊C8C2311yescharacteristic
95C2×C283⋊C8C2311yescharacteristic
96C4⋊Dic283—21no—
97C4⋊Dic283C2311yescharacteristic
98Dic283⋊C4—21no—
99Dic283⋊C4—21no—
100Dic283⋊C4D411yescharacteristic
101C22⋊Dic283—21no—
102C22⋊Dic283—21no—
103C22⋊Dic283D411yescharacteristic
104(C2×D283)⋊C4—21no—
105(C2×D283)⋊C4—21no—
106(C2×D283)⋊C4—21no—
107(C2×D283)⋊C4D411yescharacteristic
108C4×Dic283C2311yescharacteristic
109C24×C283—21no—
110C2×Q8×C283C2311yescharacteristic
111C2×D4×C283—22no—
112C2×D4×C283—21no—
113C2×D4×C283—41no—
114C2×D4×C283C2311yescharacteristic
115C2×D4×C283C2311yescharacteristic
116C22×C4×C283C2311yescharacteristic
117SD16×C283—22no—
118SD16×C283—41no—
119C2×C8×C283C2311yescharacteristic
120C2×C8×C283C2311yescharacteristic
121C4⋊C4×C283—21no—
122C4⋊C4×C283C2311yescharacteristic
123C22⋊C4×C283—21no—
124C22×D283—21no—
125C22×D283—21no—
126C22×D283—42no—
127C22×D283—42no—
128C22×D283—81no—
129C22×D283C2×D411yescharacteristic
130Dic566—21no—
131Dic566—21no—
132D1132—21no—
133D1132—21no—
134D1132—21no—
135D1132—42no—
136D1132C2×D412yescomplemented
137C22⋊D283—22no—
138C22⋊D283—42no—
139C22⋊D283—42no—
140C22⋊D283—42no—
141C22⋊D283—42no—
142C22⋊D283—42no—
143C22⋊D283—81no—
144C2×Dic283—21no—
145C2×Dic283—21no—
146C2×Dic283—42no—
147C2×Dic283C2×D411yescharacteristic
148C2×Dic283C2×D411yescharacteristic
149C2×Dic283C2×D411yescharacteristic
150C4×D283—21no—
151C4×D283—21no—
152C4×D283—42no—
153C4×D283—42no—
154C4×D283—42no—
155C283⋊C8—21no—
156C283⋊C8C2×D412yescomplemented
157C23×C283—21no—
158C23×C283—21no—
159C23×C283—42no—
160C23×C283—42no—
161C23×C283—81no—
162C23×C283C2×D411yescharacteristic
163Q8×C283—21no—
164Q8×C283—21no—
165D4×C283—21no—
166D4×C283—21no—
167D4×C283—21no—
168D4×C283—42no—
169D4×C283C2×D412yes—
170C2×C4×C283—21no—
171C2×C4×C283—21no—
172C2×C4×C283—21no—
173C2×C4×C283—21no—
174C2×C4×C283C2×D411yescharacteristic
175C2×C4×C283C2411yescharacteristic
176C8×C283—21no—
177C8×C283—21no—
178C22×C283—21no—
179C22×C283—22no—
180C22×C283—21no—
181C22×C283—21no—
182C22×C283—41no—
183C22×C283—41no—
184C22×C283—41no—
185C22×C283—41no—
186C22×C283—41no—
187C22×C283—84no—
188C22×C283C22×D411yescharacteristic
189C2×D283—21no—
190C2×D283—22no—
191C2×D283—21no—
192C2×D283—21no—
193C2×D283—41no—
194C2×D283—41no—
195C2×D283—41no—
196C2×D283—41no—
197C2×D283—41no—
198C2×D283—84no—
199C4×C283—21no—
200C4×C283—41no—
201C4×C283—41no—
202C4×C283C22×D411yescharacteristic
203C4×C283H(5,𝔽2)11yescharacteristic
204Dic283—21no—
205Dic283—21no—
206Dic283—41no—
207Dic283—41no—
208Dic283AGL(1,ℤ/8)12yescomplemented
209C2×C283—42no—
210C2×C283—41no—
211C2×C283—81no—
212C2×C283D4211yescharacteristic
213C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C283#57711yescharacteristic
215D283—41no—
216D283—42no—
217D283—81no—
218C283#327211yescharacteristic, complemented, Sylow
219#3272—2831nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2831no—
221C2×AGL(1,ℤ/8)—2831no—
222#547—2831no—
223D42—2831no—
224#492—2831no—
225#468—2831no—
226#468—2831no—
227#461—2831no—
228#459—2831no—
229#450—2831no—
230#450D28311yescharacteristic, complemented
231#449—2831no—
232#449—2831no—
233C4×D8—2831no—
234#436—2831no—
235C2×#41—2831no—
236C2×#41—2831no—
237C22×D4—2831no—
238C22×D4—5661no—
239C22×D4C2×D28311yescharacteristic
240AGL(1,ℤ/8)—5664no—
241AGL(1,ℤ/8)—5664no—
242C2×SD16—2831no—
243C2×SD16—2831no—
244C2×SD16—2831no—
245C2×SD16—2831no—
246C2×SD16C2×D28311yescharacteristic
247C2×SD16C2×D28311yescharacteristic, complemented
248C2×D8—2831no—
249C2×D8—2831no—
250C2×D8—2831no—
251C2×M16—2831no—
252C2×M16—2831no—
253C4⋊Q8—2831no—
254#127—2831no—
255#123—5661no—
256#123—5661no—
257C22⋊Q8—2831no—
258C22⋊Q8C2×D28311yescharacteristic, complemented
259#121—2831no—
260#121—2831no—
261#121—5661no—
262#121—5661no—
263#120—5661no—
264#120—5661no—
265C4×D4—2831no—
266C4×D4—2831no—
267C4×D4—2831no—
268C2×C4⋊C4—5661no—
269#107—2831no—
270C4⋊C8—2831no—
271#103—2831no—
272#103—2831no—
273#102—2831no—
274#102—2831no—
275#102C2×D28311yescharacteristic
276#102C2×D28311yescharacteristic, complemented
277C22⋊C8—2831no—
278C22⋊C8C2×D28311yescharacteristic, complemented
279C4×C8—2831no—
280C24—21no—
281C24—5661no—
282#41—5662no—
283#41—5662no—
284#41—11321no—
285#41—11321no—
286C2×Q8—2831no—
287C2×Q8C22×D28311yescharacteristic
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—41no—
291C2×D4—2831no—
292C2×D4—2831no—
293C2×D4—2831no—
294C2×D4—5661no—
295C2×D4—5661no—
296C2×D4—5661no—
297C2×D4—5661no—
298C2×D4—5662no—
299C2×D4—5661no—
300C2×D4—5661no—
301C2×D4—5661no—
302C2×D4—11321no—
303C2×D4—11324no—
304C2×D4C22×D28311yescharacteristic
305C2×D4C22×D28311yescharacteristic
306C22×C4—2831no—
307C22×C4—5661no—
308C22×C4—5661no—
309C22×C4—5661no—
310C22×C4—5661no—
311C22×C4C22×D28311yescharacteristic
312SD16—22no—
313SD16—41no—
314SD16—5662no—
315SD16—5662no—
316SD16—5662no—
317SD16—11321no—
318D8—2834no—
319D8—5662no—
320D8—5662no—
321M16—5662no—
322M16—5662no—
323C2×C8—2831no—
324C2×C8—2831no—
325C2×C8C22×D28311yescharacteristic
326C2×C8C22×D28311yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—2831no—
329C4⋊C4—2831no—
330C4⋊C4—5661no—
331C4⋊C4—5661no—
332C4⋊C4—5661no—
333C4⋊C4C22×D28311yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—2831no—
336C22⋊C4—2831no—
337C22⋊C4—5661no—
338C22⋊C4—5661no—
339C22⋊C4—5661no—
340C22⋊C4—5661no—
341C22⋊C4—5661no—
342C42—2831no—
343C23—21no—
344C23—21no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—2831no—
349C23—5661no—
350C23—5661no—
351C23—11322no—
352C23—11322no—
353C23—22641no—
354C23D4×D28311yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—5661no—
358Q8—5661no—
359D4—21no—
360D4—21no—
361D4—21no—
362D4—42no—
363D4—2832no—
364D4—5662no—
365D4—5661no—
366D4—5661no—
367D4—5661no—
368D4—11322no—
369D4—11322no—
370D4—11322no—
371D4—11322no—
372D4—11322no—
373D4—11322no—
374D4—22641no—
375D4D4×D28312yes—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—2831no—
381C2×C4—2831no—
382C2×C4—2831no—
383C2×C4—5661no—
384C2×C4—5661no—
385C2×C4—5661no—
386C2×C4—5661no—
387C2×C4—11322no—
388C2×C4—11322no—
389C2×C4—11322no—
390C2×C4—11322no—
391C2×C4D4×D28311yescharacteristic
392C2×C4C23×D28311yescharacteristic
393C8—21no—
394C8—21no—
395C8—2832no—
396C8—5661no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—84no—
407C22—5661no—
408C22—5661no—
409C22—5661no—
410C22—5662no—
411C22—11321no—
412C22—11321no—
413C22—11321no—
414C22—11321no—
415C22—11321no—
416C22—22644no—
417C22C2×D4×D28311yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—2832no—
422C4—5661no—
423C4—5661no—
424C4—11321no—
425C4—11321no—
426C4C2×D4×D28311yescharacteristic
427C4#2258365511yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—11321no—
432C2—11322no—
433C2—22641no—
434C2#3182709711yescharacteristic
435C2AGL(1,ℤ/8)×D28311yescharacteristic
436C2#3182727111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#450×C283C2
socleC22×C283C22×D4
hypercenter#450D283
hypocenterC283#3272
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#22583637C2semidirect
2. C2×#22583636C2semidirect
3. #31827117C2semidirect
4. D4×C22⋊D283C2semidirect
5. #31826836C2semidirect
6. #31826713C2semidirect
7. #31826710C2semidirect
8. #31826668C2semidirect
9. #31826659C2semidirect
10. #31826617C2semidirect
11. #31826612C2semidirect
12. #31826609C2semidirect
13. #31826545C2semidirect
14. #31826540C2semidirect
15. #450×C283C2semidirect
16. C2×#17985393C22semidirect
17. C2×#17985392C22semidirect
22. C2×#17985381C22semidirect
24. C2×#17985380C22semidirect
25. C2×#17985379C22semidirect
26. C2×D2264C22semidirect
28. C2×#17985377C22semidirect
29. C2×#17985376C22semidirect
30. C2×#17985375C22semidirect
31. #22583608C22semidirect
35. #22583590C22semidirect
43. #22583572C22semidirect
46. #22583543C22semidirect
47. Dic283⋊C8C22semidirect
48. #22583535C22semidirect
49. #22583534C22semidirect
50. #22583532C22semidirect
51. #22583530C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. C8×Dic283C22semidirect
56. C2×SD16×C283C22semidirect
57. C22⋊Q8×C283C22semidirect
59. #102×C283C22semidirect
60. C22⋊C8×C283C22semidirect
91. #17985377D4semidirect
136. D1132C2×D4semidirect
156. C283⋊C8C2×D4semidirect
208. Dic283AGL(1,ℤ/8)semidirect
218. C283#3272semidirect
230. #450D283semidirect
247. C2×SD16C2×D283semidirect
258. C22⋊Q8C2×D283semidirect
276. #102C2×D283semidirect
278. C22⋊C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836372
C2×#225836362
#318271172
D4×C22⋊D2832
#318268362
#318267132
#318267102
#318266682
#318266592
#318266172
#318266122
#318266092
#318265452
#318265402
#450×C2832
#3272283