Finite Group Databasebeta

← #49132872

Subgroups — #49132872

Order 36224.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583636C211yesmaximal, characteristic, complemented
2C2×#22583634C211yesmaximal, characteristic, complemented
3#31827112C211yesmaximal, characteristic, complemented
4#31827109C211yesmaximal, characteristic, complemented
5#31826848C211yesmaximal, characteristic, complemented
6#31826710C211yesmaximal, characteristic, complemented
7#31826709C211yesmaximal, characteristic, complemented
8#31826679C211yesmaximal, characteristic, complemented
9#31826678C211yesmaximal, characteristic, complemented
10#31826629C211yesmaximal, characteristic, complemented
11#31826628C211yesmaximal, characteristic, complemented
12#31826597C211yesmaximal, characteristic, complemented
13#31826545C211yesmaximal, characteristic, complemented
14#31826540C211yesmaximal, characteristic, complemented
15#447×C283C211yesmaximal, characteristic, complemented
16C2×#17985392C2211yescharacteristic
17C2×#17985391C2211yescharacteristic
18C2×D4×D283C2211yescharacteristic, complemented
19#22583636—24no—
20#22583634—24no—
21C2×Dic1132C2211yescharacteristic, complemented
22C2×#17985381C2211yescharacteristic, complemented
23C2×#17985380C2211yescharacteristic, complemented
24C2×#17985380C2211yescharacteristic, complemented
25C2×#17985380C2211yescharacteristic, complemented
26C2×#17985377C2211yescharacteristic
27C2×#17985377C2211yescharacteristic
28C2×#17985376C2211yescharacteristic, complemented
29C2×#17985375C2211yescharacteristic, complemented
30#22583597C2211yescharacteristic, complemented
31#22583597C2211yescharacteristic, complemented
32#22583592—21no—
33#22583591—21no—
34#22583590C2211yescharacteristic, complemented
35#22583589C2211yescharacteristic, complemented
36#22583582—21no—
37#22583581—21no—
38#22583581C2211yescharacteristic
39#22583579—21no—
40#22583578—21no—
41D4×Dic283C2211yescharacteristic, complemented
42#22583572C2211yescharacteristic, complemented
43C4×C22⋊D283C2211yescharacteristic
44C2×C22⋊Dic283—21no—
45C22⋊C4×D283—21no—
46#22583543C2211yescharacteristic, complemented
47Dic283⋊C8C2211yescharacteristic, complemented
48#22583535C2211yescharacteristic, complemented
49#22583533C2211yescharacteristic, complemented
50#22583532C2211yescharacteristic, complemented
51#22583531C2211yescharacteristic, complemented
52(C2×D283)⋊C8C2211yescharacteristic, complemented
53C8×Dic283C2211yescharacteristic, complemented
54C22×D4×C283C2211yescharacteristic, complemented
55C2×D8×C283C2211yescharacteristic
56C2×D8×C283C2211yescharacteristic
57#121×C283C2211yescharacteristic
58#102×C283C2211yescharacteristic, complemented
59#102×C283C2211yescharacteristic, complemented
60C22⋊C8×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62#17985392—22no—
63#17985392—41no—
64#17985391—22no—
65#17985391—41no—
66C2×Dic566C2311yescharacteristic
67C2×Dic566C2311yescharacteristic
68C2×D1132C2311yescharacteristic
69C2×C22⋊D283—21no—
70C2×C22⋊D283—21no—
71C2×C22⋊D283—21no—
72C2×C22⋊D283—21no—
73C2×C22⋊D283D411yescharacteristic
74D4×D283—22no—
75D4×D283—41no—
76C22×Dic283—21no—
77C22×Dic283—21no—
78C2×C4×D283—21no—
79C2×C4×D283—21no—
80C2×C4×D283C2311yescharacteristic
81Dic1132—41no—
82#17985381—22no—
83#17985380—22no—
84#17985380—22no—
85#17985380—22no—
86#17985377—22no—
87#17985377D414yescomplemented
88#17985376—22no—
89#17985375—22no—
90C2×C283⋊C8C2311yescharacteristic
91C2×C283⋊C8C2311yescharacteristic
92C4⋊Dic283—21no—
93C4⋊Dic283C2311yescharacteristic
94Dic283⋊C4—21no—
95Dic283⋊C4D411yescharacteristic
96C22⋊Dic283—21no—
97C22⋊Dic283—21no—
98C22⋊Dic283—21no—
99C22⋊Dic283—21no—
100C22⋊Dic283—21no—
101C22⋊Dic283—21no—
102C22⋊Dic283—21no—
103C22⋊Dic283D411yescharacteristic
104(C2×D283)⋊C4—21no—
105(C2×D283)⋊C4—21no—
106(C2×D283)⋊C4D411yescharacteristic
107C4×Dic283C2311yescharacteristic
108C24×C283—21no—
109C2×D4×C283—22no—
110C2×D4×C283—21no—
111C2×D4×C283—21no—
112C2×D4×C283—41no—
113C2×D4×C283C2311yescharacteristic
114C2×D4×C283C2311yescharacteristic
115C2×D4×C283C2311yescharacteristic
116C22×C4×C283C2311yescharacteristic
117D8×C283—22no—
118D8×C283—41no—
119C2×C8×C283C2311yescharacteristic
120C2×C8×C283C2311yescharacteristic
121C4⋊C4×C283C2311yescharacteristic
122C22⋊C4×C283—21no—
123C22×D283—21no—
124C22×D283—42no—
125C22×D283—42no—
126C22×D283—81no—
127C22×D283C2×D411yescharacteristic
128Dic566—21no—
129Dic566—21no—
130Dic566—21no—
131Dic566—21no—
132D1132—21no—
133D1132C2×D412yes—
134C22⋊D283—41no—
135C22⋊D283—42no—
136C22⋊D283—42no—
137C22⋊D283—42no—
138C22⋊D283—81no—
139C2×Dic283—21no—
140C2×Dic283—21no—
141C2×Dic283—21no—
142C2×Dic283—42no—
143C2×Dic283—81no—
144C2×Dic283C2×D411yescharacteristic
145C2×Dic283C2×D411yescharacteristic
146C2×Dic283C2×D411yescharacteristic
147C4×D283—21no—
148C4×D283—21no—
149C4×D283—42no—
150C4×D283—81no—
151C283⋊C8—21no—
152C283⋊C8C2×D412yes—
153C23×C283—21no—
154C23×C283—21no—
155C23×C283—21no—
156C23×C283—42no—
157C23×C283—42no—
158C23×C283—81no—
159C23×C283C2×D411yescharacteristic
160D4×C283—21no—
161D4×C283—21no—
162D4×C283—21no—
163D4×C283—21no—
164D4×C283—21no—
165D4×C283—42no—
166D4×C283—81no—
167D4×C283C2×D412yes—
168C2×C4×C283—21no—
169C2×C4×C283—21no—
170C2×C4×C283—21no—
171C2×C4×C283C2×D411yescharacteristic
172C2×C4×C283C2411yescharacteristic
173C8×C283—21no—
174C8×C283—21no—
175C22×C283—21no—
176C22×C283—22no—
177C22×C283—21no—
178C22×C283—21no—
179C22×C283—41no—
180C22×C283—41no—
181C22×C283—41no—
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—84no—
196C4×C283—21no—
197C4×C283—41no—
198C4×C283C22×D411yescharacteristic
199C4×C283H(5,𝔽2)11yescharacteristic
200Dic283—21no—
201Dic283—21no—
202Dic283—21no—
203Dic283—41no—
204Dic283—41no—
205Dic283—41no—
206C2×C283—41no—
207C2×C283—42no—
208C2×C283—81no—
209C2×C283—81no—
210C2×C283D4211yescharacteristic
211C2×C283#57511yescharacteristic
212C2×C283#57611yescharacteristic
213D283—41no—
214D283—42no—
215C283#327311yescharacteristic, complemented, Sylow
216#3273—2831nomaximal, Sylow
217C2×AGL(1,ℤ/8)—2831no—
218C2×AGL(1,ℤ/8)—2831no—
219#546—2831no—
220#546—2831no—
221#495—2831no—
222#468—2831no—
223#468—2831no—
224#463—2831no—
225#463—2831no—
226#452—2831no—
227#452—2831no—
228#447—2831no—
229#447D28311yescharacteristic, complemented
230C4×D8—2831no—
231#436—2831no—
232C2×#41—2831no—
233C2×#41—2831no—
234C22×D4—2831no—
235C22×D4C2×D28311yescharacteristic, complemented
236AGL(1,ℤ/8)—5664no—
237AGL(1,ℤ/8)—5664no—
238C2×Q16—2831no—
239C2×SD16—2831no—
240C2×SD16—2831no—
241C2×SD16—2831no—
242C2×SD16—2831no—
243C2×D8—2831no—
244C2×D8—2831no—
245C2×D8C2×D28311yescharacteristic
246C2×D8C2×D28311yescharacteristic
247C2×M16—2831no—
248C2×M16—2831no—
249#124—2831no—
250#124—2831no—
251#123—5661no—
252#123—5661no—
253C22⋊Q8—2831no—
254C22⋊Q8—2831no—
255#121—2831no—
256#121—5661no—
257#121—5661no—
258#121C2×D28311yescharacteristic
259#120—5661no—
260#120—5661no—
261C4×D4—2831no—
262C4×D4—2831no—
263C4×D4—2831no—
264C2×C22⋊C4—5661no—
265C2×C22⋊C4—5661no—
266#107—2831no—
267C4⋊C8—2831no—
268#103—2831no—
269#103—2831no—
270#102—2831no—
271#102—2831no—
272#102C2×D28311yescharacteristic, complemented
273#102C2×D28311yescharacteristic, complemented
274C22⋊C8—2831no—
275C22⋊C8C2×D28311yescharacteristic, complemented
276C4×C8—2831no—
277C24—21no—
278C24—5661no—
279#41—5662no—
280#41—5662no—
281#41—11321no—
282#41—11321no—
283C2×Q8—2831no—
284C2×Q8—2831no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—2831no—
290C2×D4—2831no—
291C2×D4—5661no—
292C2×D4—5662no—
293C2×D4—5661no—
294C2×D4—5661no—
295C2×D4—5661no—
296C2×D4—11321no—
297C2×D4C22×D28311yescharacteristic
298C2×D4C22×D28311yescharacteristic
299C2×D4C22×D28311yescharacteristic
300C22×C4—2831no—
301C22×C4—5661no—
302C22×C4—5661no—
303C22×C4—5661no—
304C22×C4—5661no—
305C22×C4C22×D28311yescharacteristic
306Q16—11321no—
307SD16—5662no—
308SD16—5662no—
309SD16—5662no—
310SD16—5662no—
311D8—22no—
312D8—41no—
313D8—2834no—
314D8—5662no—
315M16—5662no—
316M16—5662no—
317C2×C8—2831no—
318C2×C8—2831no—
319C2×C8C22×D28311yescharacteristic
320C2×C8C22×D28311yescharacteristic
321C4⋊C4—2831no—
322C4⋊C4—2831no—
323C4⋊C4—5661no—
324C4⋊C4—5661no—
325C4⋊C4C22×D28311yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—2831no—
328C22⋊C4—2831no—
329C22⋊C4—5661no—
330C22⋊C4—5661no—
331C22⋊C4—5661no—
332C22⋊C4—5661no—
333C22⋊C4—5661no—
334C22⋊C4—5661no—
335C22⋊C4—5661no—
336C22⋊C4—5661no—
337C22⋊C4—5661no—
338C42—2831no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—2831no—
346C23—5661no—
347C23—11322no—
348C23—11322no—
349C23—22641no—
350C23D4×D28311yescharacteristic
351Q8—5661no—
352Q8—5661no—
353Q8—5661no—
354Q8—5661no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—21no—
360D4—42no—
361D4—81no—
362D4—2832no—
363D4—5661no—
364D4—11321no—
365D4—11322no—
366D4—11322no—
367D4—11322no—
368D4—22641no—
369D4D4×D28312yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—2831no—
374C2×C4—2831no—
375C2×C4—2831no—
376C2×C4—5661no—
377C2×C4—5661no—
378C2×C4—5661no—
379C2×C4—5661no—
380C2×C4—5661no—
381C2×C4—11322no—
382C2×C4—11322no—
383C2×C4—22641no—
384C2×C4—22641no—
385C2×C4D4×D28311yescharacteristic
386C2×C4C23×D28311yescharacteristic
387C8—21no—
388C8—21no—
389C8—2832no—
390C8—5661no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—5661no—
405C22—5661no—
406C22—5661no—
407C22—5662no—
408C22—11321no—
409C22—11321no—
410C22—22644no—
411C22C2×D4×D28311yescharacteristic
412C4—21no—
413C4—41no—
414C4—5661no—
415C4—5661no—
416C4—5661no—
417C4—11321no—
418C4—11321no—
419C4—11321no—
420C4C2×D4×D28311yescharacteristic
421C4#2258365511yescharacteristic
422C2—41no—
423C2—42no—
424C2—81no—
425C2—81no—
426C2—11321no—
427C2—11322no—
428C2#3182709711yescharacteristic
429C2#3182726111yescharacteristic
430C2#3182726611yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#447×C283C2
socleC22×C283C22×D4
hypercenter#447D283
hypocenterC283#3273
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#22583636C2semidirect
2. C2×#22583634C2semidirect
3. #31827112C2semidirect
4. #31827109C2semidirect
5. #31826848C2semidirect
6. #31826710C2semidirect
7. #31826709C2semidirect
8. #31826679C2semidirect
9. #31826678C2semidirect
10. #31826629C2semidirect
11. #31826628C2semidirect
12. #31826597C2semidirect
13. #31826545C2semidirect
14. #31826540C2semidirect
15. #447×C283C2semidirect
18. C2×D4×D283C22semidirect
21. C2×Dic1132C22semidirect
22. C2×#17985381C22semidirect
23. C2×#17985380C22semidirect
24. C2×#17985380C22semidirect
25. C2×#17985380C22semidirect
28. C2×#17985376C22semidirect
29. C2×#17985375C22semidirect
30. #22583597C22semidirect
31. #22583597C22semidirect
34. #22583590C22semidirect
35. #22583589C22semidirect
41. D4×Dic283C22semidirect
42. #22583572C22semidirect
46. #22583543C22semidirect
47. Dic283⋊C8C22semidirect
48. #22583535C22semidirect
49. #22583533C22semidirect
50. #22583532C22semidirect
51. #22583531C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. C8×Dic283C22semidirect
54. C22×D4×C283C22semidirect
58. #102×C283C22semidirect
59. #102×C283C22semidirect
60. C22⋊C8×C283C22semidirect
87. #17985377D4semidirect
215. C283#3273semidirect
229. #447D283semidirect
235. C22×D4C2×D283semidirect
272. #102C2×D283semidirect
273. #102C2×D283semidirect
275. C22⋊C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836362
C2×#225836342
#318271122
#318271092
#318268482
#318267102
#318267092
#318266792
#318266782
#318266292
#318266282
#318265972
#318265452
#318265402
#447×C2832
#3273283