Finite Group Databasebeta

← #49132855

Subgroups — #49132855

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×#22583634C211yesmaximal, characteristic, complemented
2#31827114C211yesmaximal, characteristic, complemented
3D4×C22⋊D283C211yesmaximal, characteristic, complemented
4#31826836C211yesmaximal, characteristic, complemented
5#31826711C211yesmaximal, characteristic, complemented
6#31826708C211yesmaximal, characteristic, complemented
7#31826668C211yesmaximal, characteristic, complemented
8#31826658C211yesmaximal, characteristic, complemented
9#31826616C211yesmaximal, characteristic, complemented
10#31826614C211yesmaximal, characteristic, complemented
11#31826610C211yesmaximal, characteristic, complemented
12#31826607C211yesmaximal, characteristic, complemented
13D8×Dic283C211yesmaximal, characteristic, complemented
14#31826541C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic, complemented
17C22×C22⋊D283—21no—
18C2×D4×D283C2211yescharacteristic
19#22583634—24no—
20C2×#17985381C2211yescharacteristic, complemented
21C2×#17985380C2211yescharacteristic
22C2×#17985380C2211yescharacteristic, complemented
23C2×#17985379C2211yescharacteristic, complemented
24C2×#17985377C2211yescharacteristic, complemented
25C2×#17985375C2211yescharacteristic, complemented
26#22583608C2211yescharacteristic, complemented
27#22583606C2211yescharacteristic
28#22583593—21no—
29#22583591—21no—
30#22583589C2211yescharacteristic, complemented
31#22583586C2211yescharacteristic, complemented
32#22583584C2211yescharacteristic
33#22583582—21no—
34#22583582—21no—
35#22583581C2211yescharacteristic
36#22583579—21no—
37#22583578—21no—
38D4×Dic283C2211yescharacteristic
39D4×Dic283C2211yescharacteristic, complemented
40C4×C22⋊D283C2211yescharacteristic
41C2×Dic283⋊C4—21no—
42#22583544C2211yescharacteristic, complemented
43Dic283⋊C8C2211yescharacteristic, complemented
44#22583535C2211yescharacteristic, complemented
45#22583534C2211yescharacteristic, complemented
46#22583532C2211yescharacteristic, complemented
47#22583531C2211yescharacteristic
48#22583531C2211yescharacteristic, complemented
49#22583531C2211yescharacteristic, complemented
50C22⋊(C283⋊C8)C2211yescharacteristic, complemented
51(C2×D283)⋊C8C2211yescharacteristic, complemented
52C8×Dic283C2211yescharacteristic, complemented
53C2×#41×C283C2211yescharacteristic, complemented
54C22×D4×C283C2211yescharacteristic
55AGL(1,ℤ/8)×C283—24no—
56C2×SD16×C283C2211yescharacteristic
57C2×SD16×C283C2211yescharacteristic, complemented
58C2×D8×C283C2211yescharacteristic
59C2×D8×C283C2211yescharacteristic, complemented
60C2×M16×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62#17985391—22no—
63#17985391—41no—
64C2×Dic566C2311yescharacteristic
65C2×D1132C2311yescharacteristic
66C2×C22⋊D283—21no—
67C2×C22⋊D283—21no—
68C2×C22⋊D283—21no—
69C2×C22⋊D283—21no—
70C2×C22⋊D283—21no—
71C2×C22⋊D283—21no—
72C2×C22⋊D283—44no—
73C2×C22⋊D283D411yescharacteristic
74D4×D283—22no—
75D4×D283—41no—
76C22×Dic283—21no—
77C22×Dic283—21no—
78C22×Dic283—21no—
79C2×C4×D283C2311yescharacteristic
80#17985381—22no—
81#17985380—22no—
82#17985380—41no—
83#17985379—41no—
84#17985377—22no—
85#17985375—22no—
86C2×C283⋊C8C2311yescharacteristic
87C2×C283⋊C8C2311yescharacteristic
88C4⋊Dic283C2311yescharacteristic
89C4⋊Dic283C2311yescharacteristic
90Dic283⋊C4—21no—
91Dic283⋊C4—21no—
92Dic283⋊C4—21no—
93Dic283⋊C4—21no—
94Dic283⋊C4D411yescharacteristic
95C22⋊Dic283—21no—
96C22⋊Dic283—21no—
97C22⋊Dic283—21no—
98C22⋊Dic283—21no—
99C22⋊Dic283D411yescharacteristic
100(C2×D283)⋊C4—21no—
101(C2×D283)⋊C4—21no—
102(C2×D283)⋊C4D411yescharacteristic
103C4×Dic283C2311yescharacteristic
104C24×C283—21no—
105#41×C283—22no—
106#41×C283—41no—
107C2×Q8×C283C2311yescharacteristic
108C2×D4×C283—22no—
109C2×D4×C283—21no—
110C2×D4×C283—21no—
111C2×D4×C283—41no—
112C2×D4×C283C2311yescharacteristic
113C2×D4×C283C2311yescharacteristic
114C2×D4×C283C2311yescharacteristic
115C22×C4×C283—21no—
116C22×C4×C283C2311yescharacteristic
117SD16×C283—22no—
118SD16×C283—22no—
119D8×C283—22no—
120D8×C283D414yescomplemented
121M16×C283—22no—
122C2×C8×C283C2311yescharacteristic
123C2×C8×C283C2311yescharacteristic
124C22×D283—21no—
125C22×D283—42no—
126C22×D283—42no—
127C22×D283—81no—
128C22×D283C2×D411yescharacteristic
129Dic566—21no—
130Dic566—21no—
131D1132—21no—
132D1132—21no—
133C22⋊D283—22no—
134C22⋊D283—42no—
135C22⋊D283—42no—
136C22⋊D283—42no—
137C22⋊D283—42no—
138C22⋊D283—42no—
139C22⋊D283—81no—
140C2×Dic283—21no—
141C2×Dic283—21no—
142C2×Dic283—21no—
143C2×Dic283—42no—
144C2×Dic283—42no—
145C2×Dic283—42no—
146C2×Dic283C2×D411yescharacteristic
147C2×Dic283C2×D411yescharacteristic
148C2×Dic283C2×D411yescharacteristic
149C4×D283—21no—
150C4×D283—21no—
151C283⋊C8—21no—
152C283⋊C8—21no—
153C23×C283—21no—
154C23×C283—21no—
155C23×C283—21no—
156C23×C283—42no—
157C23×C283—42no—
158C23×C283—81no—
159C23×C283C2×D411yescharacteristic
160Q8×C283—21no—
161Q8×C283—21no—
162D4×C283—21no—
163D4×C283—21no—
164D4×C283—21no—
165D4×C283—21no—
166D4×C283—42no—
167D4×C283—42no—
168D4×C283C2×D412yes—
169D4×C283C2×D412yescomplemented
170C2×C4×C283—21no—
171C2×C4×C283—21no—
172C2×C4×C283—21no—
173C2×C4×C283—42no—
174C2×C4×C283C2×D411yescharacteristic
175C2×C4×C283C2411yescharacteristic
176C8×C283—21no—
177C8×C283C2×D412yescomplemented
178C22×C283—21no—
179C22×C283—22no—
180C22×C283—22no—
181C22×C283—21no—
182C22×C283—21no—
183C22×C283—41no—
184C22×C283—41no—
185C22×C283—41no—
186C22×C283—41no—
187C22×C283—41no—
188C22×C283—41no—
189C22×C283—41no—
190C22×C283—84no—
191C22×C283C22×D411yescharacteristic
192C2×D283—21no—
193C2×D283—21no—
194C2×D283—21no—
195C2×D283—41no—
196C2×D283—41no—
197C2×D283—41no—
198C2×D283—84no—
199C4×C283—21no—
200C4×C283—41no—
201C4×C283C22×D411yescharacteristic
202C4×C283H(5,𝔽2)11yescharacteristic
203Dic283—21no—
204Dic283—21no—
205Dic283—41no—
206Dic283—41no—
207Dic283—41no—
208Dic283AGL(1,ℤ/8)12yescomplemented
209C2×C283—42no—
210C2×C283—41no—
211C2×C283—42no—
212C2×C283—81no—
213C2×C283D4211yescharacteristic
214C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C283#57711yescharacteristic
216D283—41no—
217D283—81no—
218C283#327211yescharacteristic, complemented, Sylow
219#3272—2831nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2831no—
221C2×AGL(1,ℤ/8)D28311yescharacteristic, complemented
222#547—2831no—
223D42—2831no—
224#492—2831no—
225#468—2831no—
226#468—2831no—
227#461—2831no—
228#459—2831no—
229#450—2831no—
230#450—2831no—
231#449—2831no—
232#449—2831no—
233C4×D8—2831no—
234#436—2831no—
235C2×#41—2831no—
236C2×#41C2×D28311yescharacteristic, complemented
237C22×D4—2831no—
238C22×D4—5661no—
239C22×D4C2×D28311yescharacteristic
240AGL(1,ℤ/8)—24no—
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×D8C2×D28311yescharacteristic
250C2×D8C2×D28311yescharacteristic, complemented
251C2×M16—2831no—
252C2×M16C2×D28311yescharacteristic, complemented
253C4⋊Q8—2831no—
254#127—2831no—
255#123—5661no—
256#123—5661no—
257C22⋊Q8—2831no—
258C22⋊Q8—2831no—
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#102—2831no—
276#102—2831no—
277C22⋊C8—2831no—
278C22⋊C8—2831no—
279C4×C8—2831no—
280C24—21no—
281C24—5661no—
282#41—22no—
283#41—41no—
284#41—5662no—
285#41—11321no—
286C2×Q8—2831no—
287C2×Q8C22×D28311yescharacteristic
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—2831no—
293C2×D4—2831no—
294C2×D4—5661no—
295C2×D4—5661no—
296C2×D4—5661no—
297C2×D4—5662no—
298C2×D4—5661no—
299C2×D4—5661no—
300C2×D4—5661no—
301C2×D4—11321no—
302C2×D4—11324no—
303C2×D4C22×D28311yescharacteristic
304C2×D4C22×D28311yescharacteristic
305C2×D4C22×D28311yescharacteristic
306C22×C4—21no—
307C22×C4—2831no—
308C22×C4—5661no—
309C22×C4—5661no—
310C22×C4—5661no—
311C22×C4C22×D28311yescharacteristic
312SD16—22no—
313SD16—22no—
314SD16—5662no—
315SD16—5662no—
316SD16—11321no—
317SD16—11321no—
318D8—22no—
319D8—5662no—
320D8C22⋊D28314yescomplemented
321M16—22no—
322M16—5662no—
323C2×C8—2831no—
324C2×C8—2831no—
325C2×C8C22×D28311yescharacteristic
326C2×C8C22×D28311yescharacteristic
327C4⋊C4—2831no—
328C4⋊C4—2831no—
329C4⋊C4—2831no—
330C4⋊C4—5661no—
331C4⋊C4—5661no—
332C4⋊C4—5661no—
333C4⋊C4—5661no—
334C22⋊C4—2831no—
335C22⋊C4—2831no—
336C22⋊C4—5661no—
337C22⋊C4—5661no—
338C22⋊C4—5661no—
339C22⋊C4—5661no—
340C22⋊C4—5661no—
341C22⋊C4—5661no—
342C42—2831no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—2831no—
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—21no—
363D4—42no—
364D4—42no—
365D4—5662no—
366D4—5661no—
367D4—5661no—
368D4—11322no—
369D4—11322no—
370D4—11322no—
371D4—11322no—
372D4—11322no—
373D4—22641no—
374D4C2×C22⋊D28312yes—
375D4C2×C22⋊D28312yescomplemented
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—2831no—
381C2×C4—2831no—
382C2×C4—2831no—
383C2×C4—5661no—
384C2×C4—5661no—
385C2×C4—5661no—
386C2×C4—5661no—
387C2×C4—5661no—
388C2×C4—11322no—
389C2×C4—11322no—
390C2×C4—11322no—
391C2×C4D4×D28311yescharacteristic
392C2×C4C23×D28311yescharacteristic
393C8—21no—
394C8—5661no—
395C8—5661no—
396C8C2×C22⋊D28312yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—5661no—
411C22—5661no—
412C22—5661no—
413C22—11321no—
414C22—11321no—
415C22—11321no—
416C22—22644no—
417C22C2×D4×D28311yescharacteristic
418C4—21no—
419C4—41no—
420C4—2832no—
421C4—5661no—
422C4—5661no—
423C4—11321no—
424C4—11321no—
425C4—11321no—
426C4C22×C22⋊D28311yescharacteristic
427C4#2258365511yescharacteristic
428C2—41no—
429C2—42no—
430C2—42no—
431C2—81no—
432C2—11321no—
433C2—22641no—
434C2D4×C22⋊D28311yescharacteristic
435C2AGL(1,ℤ/8)×D28311yescharacteristic
436C2#3182727511yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×AGL(1,ℤ/8)×C283C2
socleC22×C283C22×D4
hypercenterC2×AGL(1,ℤ/8)D283
hypocenterC283#3272
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22583634C2semidirect
2. #31827114C2semidirect
3. D4×C22⋊D283C2semidirect
4. #31826836C2semidirect
5. #31826711C2semidirect
6. #31826708C2semidirect
7. #31826668C2semidirect
8. #31826658C2semidirect
9. #31826616C2semidirect
10. #31826614C2semidirect
11. #31826610C2semidirect
12. #31826607C2semidirect
13. D8×Dic283C2semidirect
14. #31826541C2semidirect
15. C2×AGL(1,ℤ/8)×C283C2semidirect
16. C2×#17985391C22semidirect
20. C2×#17985381C22semidirect
22. C2×#17985380C22semidirect
23. C2×#17985379C22semidirect
24. C2×#17985377C22semidirect
25. C2×#17985375C22semidirect
26. #22583608C22semidirect
30. #22583589C22semidirect
31. #22583586C22semidirect
39. D4×Dic283C22semidirect
42. #22583544C22semidirect
43. Dic283⋊C8C22semidirect
44. #22583535C22semidirect
45. #22583534C22semidirect
46. #22583532C22semidirect
48. #22583531C22semidirect
49. #22583531C22semidirect
50. C22⋊(C283⋊C8)C22semidirect
51. (C2×D283)⋊C8C22semidirect
52. C8×Dic283C22semidirect
53. C2×#41×C283C22semidirect
57. C2×SD16×C283C22semidirect
59. C2×D8×C283C22semidirect
60. C2×M16×C283C22semidirect
120. D8×C283D4semidirect
169. D4×C283C2×D4semidirect
177. C8×C283C2×D4semidirect
208. Dic283AGL(1,ℤ/8)semidirect
218. C283#3272semidirect
221. C2×AGL(1,ℤ/8)D283semidirect
236. C2×#41C2×D283semidirect
247. C2×SD16C2×D283semidirect
250. C2×D8C2×D283semidirect
252. C2×M16C2×D283semidirect
320. D8C22⋊D283semidirect
375. D4C2×C22⋊D283semidirect
396. C8C2×C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836342
#318271142
D4×C22⋊D2832
#318268362
#318267112
#318267082
#318266682
#318266582
#318266162
#318266142
#318266102
#318266072
D8×Dic2832
#318265412
C2×AGL(1,ℤ/8)×C2832
#3272283