Finite Group Databasebeta

← #49133276

Subgroups — #49133276

Order 36224.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D1132C211yesmaximal, characteristic
2#31826831C211yesmaximal, characteristic, complemented
3#31826789C211yesmaximal, characteristic, complemented
4#31826778C211yesmaximal, characteristic, complemented
5#31826762C211yesmaximal, characteristic, complemented
6#31826713C211yesmaximal, characteristic, complemented
7#31826696C211yesmaximal, characteristic, complemented
8#31826665C211yesmaximal, characteristic, complemented
9#31826615C211yesmaximal, characteristic, complemented
10#31826615C211yesmaximal, characteristic, complemented
11#31826563C211yesmaximal, characteristic, complemented
12#31826537C211yesmaximal, characteristic, complemented
13#31826469C211yesmaximal, characteristic, complemented
14C2×#22583530C211yesmaximal, characteristic, complemented
15#552×C283C211yesmaximal, characteristic, complemented
16C22×D1132C2211yescharacteristic
17C22×D1132C2211yescharacteristic
18C2×D4×D283—21no—
19C2×#17985379C2211yescharacteristic
20C2×#17985379C2211yescharacteristic, complemented
21C2×#17985379C2211yescharacteristic, complemented
22C2×#17985376C2211yescharacteristic, complemented
23C22×C283⋊C8C2211yescharacteristic, complemented
24#22583607C2211yescharacteristic
25#22583585—21no—
26#22583584C2211yescharacteristic
27#22583584C2211yescharacteristic
28#22583580—21no—
29#22583580—21no—
30C4×D1132C2211yescharacteristic
31#22583543C2211yescharacteristic, complemented
32#22583543C2211yescharacteristic, complemented
33#22583541C2211yescharacteristic, complemented
34C4⋊(C283⋊C8)C2211yescharacteristic, complemented
35#22583534C2211yescharacteristic, complemented
36#22583530C2211yescharacteristic
37#22583530C2211yescharacteristic
38#22583530C2211yescharacteristic
39#22583530C2211yescharacteristic, complemented
40#22583530C2211yescharacteristic, complemented
41#22583530C2211yescharacteristic, complemented
42#22583530C2211yescharacteristic, complemented
43#22583530C2211yescharacteristic, complemented
44#22583530C2211yescharacteristic, complemented
45C22⋊(C283⋊C8)C2211yescharacteristic, complemented
46C22⋊(C283⋊C8)C2211yescharacteristic, complemented
47#22583521C2211yescharacteristic, complemented
48#126×C283—21no—
49#125×C283—21no—
50#125×C283C2211yescharacteristic, complemented
51#123×C283—21no—
52C22⋊Q8×C283C2211yescharacteristic, complemented
53C22⋊Q8×C283C2211yescharacteristic, complemented
54C4×Q8×C283C2211yescharacteristic, complemented
55C4×D4×C283C2211yescharacteristic
56#117×C283—21no—
57#117×C283C2211yescharacteristic, complemented
58C2×C4⋊C4×C283C2211yescharacteristic, complemented
59C23×D283—21no—
60C23×D283—21no—
61C2×D1132—22no—
62C2×D1132—22no—
63C2×D1132—21no—
64C2×D1132—21no—
65C2×D1132—21no—
66C2×D1132—21no—
67C2×D1132—41no—
68C2×D1132—41no—
69C2×D1132C2311yescharacteristic
70C2×D1132C2311yescharacteristic
71C2×D1132C2311yescharacteristic
72C2×C22⋊D283—21no—
73C2×C22⋊D283—21no—
74D4×D283—44no—
75C2×C4×D283—21no—
76#17985379—22no—
77#17985379—41no—
78#17985379—41no—
79#17985376—41no—
80C2×C283⋊C8—41no—
81C2×C283⋊C8C2311yescharacteristic
82C2×C283⋊C8C2311yescharacteristic
83C2×C283⋊C8C2311yescharacteristic
84C2×C283⋊C8C2311yescharacteristic
85C4⋊Dic283C2311yescharacteristic
86(C2×D283)⋊C4—21no—
87(C2×D283)⋊C4—21no—
88(C2×D283)⋊C4—21no—
89C2×Q8×C283C2311yescharacteristic
90C2×D4×C283D411yescharacteristic
91C22×C4×C283—21no—
92C22×C4×C283C2311yescharacteristic
93C22×C4×C283C2311yescharacteristic
94C4⋊C4×C283—21no—
95C4⋊C4×C283—21no—
96C4⋊C4×C283—21no—
97C4⋊C4×C283—21no—
98C4⋊C4×C283—21no—
99C4⋊C4×C283—21no—
100C4⋊C4×C283D411yescharacteristic
101C4⋊C4×C283C2311yescharacteristic
102C4⋊C4×C283C2311yescharacteristic
103C4⋊C4×C283C2311yescharacteristic
104C22⋊C4×C283—21no—
105C22⋊C4×C283—21no—
106C22⋊C4×C283—21no—
107C22⋊C4×C283D411yescharacteristic
108C22⋊C4×C283D411yescharacteristic
109C42×C283—21no—
110C42×C283—21no—
111C42×C283C2311yescharacteristic
112C22×D283—21no—
113C22×D283—21no—
114C22×D283—21no—
115C22×D283—42no—
116C22×D283—42no—
117C22×D283—42no—
118C22×D283—42no—
119C22×D283—81no—
120C22×D283—81no—
121D1132—21no—
122D1132—21no—
123D1132—21no—
124D1132—21no—
125D1132—21no—
126D1132—42no—
127D1132—42no—
128D1132—42no—
129D1132—81no—
130D1132#4112yes—
131C22⋊D283—42no—
132C22⋊D283—42no—
133C2×Dic283—21no—
134C4×D283—42no—
135C283⋊C8—21no—
136C283⋊C8—21no—
137C283⋊C8—21no—
138C283⋊C8—21no—
139C23×C283C2×D411yescharacteristic
140C23×C283C2×D411yescharacteristic
141Q8×C283—21no—
142Q8×C283—21no—
143D4×C283—22no—
144C2×C4×C283—21no—
145C2×C4×C283—21no—
146C2×C4×C283—21no—
147C2×C4×C283—21no—
148C2×C4×C283—21no—
149C2×C4×C283—21no—
150C2×C4×C283—21no—
151C2×C4×C283—21no—
152C2×C4×C283—42no—
153C2×C4×C283C2×D411yescharacteristic
154C2×C4×C283C2×D411yescharacteristic
155C2×C4×C283C2×D411yescharacteristic
156C2×C4×C283C2×D411yescharacteristic
157C2×C4×C283C2411yescharacteristic
158C22×C283—21no—
159C22×C283—21no—
160C22×C283—21no—
161C22×C283—21no—
162C22×C283—21no—
163C22×C283AGL(1,ℤ/8)12yes—
164C22×C283C22×D411yescharacteristic
165C2×D283—22no—
166C2×D283—41no—
167C2×D283—41no—
168C2×D283—41no—
169C2×D283—41no—
170C2×D283—41no—
171C2×D283—41no—
172C2×D283—41no—
173C2×D283—41no—
174C2×D283—84no—
175C2×D283—84no—
176C4×C283—21no—
177C4×C283—21no—
178C4×C283—21no—
179C4×C283—21no—
180C4×C283—41no—
181C4×C283—41no—
182C4×C283—41no—
183C4×C283—41no—
184C4×C283C2×#4111yescharacteristic
185C4×C283H(5,𝔽2)11yescharacteristic
186Dic283—41no—
187C2×C283—22no—
188C2×C283—41no—
189C2×C283#54611yescharacteristic
190C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C283#57711yescharacteristic
192D283—42no—
193D283—81no—
194D283—81no—
195C283#331011yescharacteristic, complemented, Sylow
196#3310—2831nomaximal, Sylow
197#552D28311yescharacteristic, complemented
198D42—2831no—
199#490—2831no—
200#482—2831no—
201#480—2831no—
202#478—2831no—
203#468—2831no—
204#465—2831no—
205#460—2831no—
206#450—2831no—
207#450—2831no—
208#440—2831no—
209#435—2831no—
210#418—2831no—
211C2×#102—2831no—
212C22×D4—2831no—
213C22×D4—2831no—
214C22×D4—5661no—
215C2×SD16—2831no—
216C2×SD16—2831no—
217C2×SD16—2831no—
218C2×M16—2831no—
219C22×C8—2831no—
220#127—2831no—
221#126—21no—
222#125—21no—
223#125C2×D28311yescharacteristic, complemented
224#123—21no—
225C22⋊Q8C2×D28311yescharacteristic, complemented
226C22⋊Q8C2×D28311yescharacteristic, complemented
227#121—2831no—
228#121—2831no—
229#121—5661no—
230#120—5661no—
231#120—5661no—
232C4×Q8C2×D28311yescharacteristic, complemented
233C4×D4—2831no—
234C4×D4C2×D28311yescharacteristic
235#117—21no—
236#117C2×D28311yescharacteristic, complemented
237C2×C4⋊C4C2×D28311yescharacteristic, complemented
238#107—2831no—
239#107—2831no—
240#106—2831no—
241C4⋊C8—2831no—
242#103—2831no—
243#102—2831no—
244#102—2831no—
245#102—2831no—
246#102—2831no—
247#102—2831no—
248#102—2831no—
249#102—2831no—
250#102—2831no—
251#102—2831no—
252C22⋊C8—2831no—
253C22⋊C8—2831no—
254#97—2831no—
255C24—5661no—
256C24—5661no—
257C2×Q8C22×D28311yescharacteristic
258C2×D4—2831no—
259C2×D4—2831no—
260C2×D4—2831no—
261C2×D4—5662no—
262C2×D4—5662no—
263C2×D4—5661no—
264C2×D4—5661no—
265C2×D4—5661no—
266C2×D4—5661no—
267C2×D4—5661no—
268C2×D4—5661no—
269C2×D4—11321no—
270C2×D4—11321no—
271C2×D4—11324no—
272C2×D4C22⋊D28311yescharacteristic
273C22×C4—21no—
274C22×C4—5661no—
275C22×C4C22×D28311yescharacteristic
276C22×C4C22×D28311yescharacteristic
277SD16—5662no—
278SD16—11321no—
279SD16—11321no—
280M16—11321no—
281C2×C8—2831no—
282C2×C8—2831no—
283C2×C8—2831no—
284C2×C8—2831no—
285C2×C8—11321no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—2831no—
293C4⋊C4C22⋊D28311yescharacteristic
294C4⋊C4C22×D28311yescharacteristic
295C4⋊C4C22×D28311yescharacteristic
296C4⋊C4C22×D28311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—5661no—
301C22⋊C4—5661no—
302C22⋊C4—5661no—
303C22⋊C4C22⋊D28311yescharacteristic
304C22⋊C4C22⋊D28311yescharacteristic
305C42—21no—
306C42—21no—
307C42C22×D28311yescharacteristic
308C23—5661no—
309C23—5661no—
310C23—5661no—
311C23—11322no—
312C23—11322no—
313C23—11322no—
314C23—11322no—
315C23—22641no—
316C23—22641no—
317C23C2×C22⋊D28311yescharacteristic
318C23C2×C22⋊D28311yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—22no—
322D4—2832no—
323D4—5661no—
324D4—5661no—
325D4—5661no—
326D4—5661no—
327D4—5661no—
328D4—11322no—
329D4—11322no—
330D4—11322no—
331D4—11322no—
332D4—11322no—
333D4—22641no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—5661no—
344C2×C4—11322no—
345C2×C4C2×C22⋊D28311yescharacteristic
346C2×C4C2×C22⋊D28311yescharacteristic
347C2×C4C2×C22⋊D28311yescharacteristic
348C2×C4C2×C22⋊D28311yescharacteristic
349C2×C4C23×D28311yescharacteristic
350C8—5661no—
351C8—5661no—
352C8—5661no—
353C8—5661no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—5662no—
360C22—11321no—
361C22—11321no—
362C22—11321no—
363C22—11321no—
364C22—11321no—
365C22—11321no—
366C22—11321no—
367C22—11321no—
368C22—22644no—
369C22—22644no—
370C22#2258363512yes—
371C22C22×C22⋊D28311yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—11321no—
381C4#41×D28311yescharacteristic
382C4#2258365611yescharacteristic
383C2—22no—
384C2—41no—
385C2—11322no—
386C2—22641no—
387C2—22641no—
388C2#3182710311yescharacteristic
389C2C2×#2258363511yescharacteristic
390C2#3182727311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#552×C283C2
socleC22×C283C22×D4
prime radicalD4×D1132C2
hypercenter#552D283
hypocenterC283#3310
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #31826831C2semidirect
3. #31826789C2semidirect
4. #31826778C2semidirect
5. #31826762C2semidirect
6. #31826713C2semidirect
7. #31826696C2semidirect
8. #31826665C2semidirect
9. #31826615C2semidirect
10. #31826615C2semidirect
11. #31826563C2semidirect
12. #31826537C2semidirect
13. #31826469C2semidirect
14. C2×#22583530C2semidirect
15. #552×C283C2semidirect
20. C2×#17985379C22semidirect
21. C2×#17985379C22semidirect
22. C2×#17985376C22semidirect
23. C22×C283⋊C8C22semidirect
31. #22583543C22semidirect
32. #22583543C22semidirect
33. #22583541C22semidirect
34. C4⋊(C283⋊C8)C22semidirect
35. #22583534C22semidirect
39. #22583530C22semidirect
40. #22583530C22semidirect
41. #22583530C22semidirect
42. #22583530C22semidirect
43. #22583530C22semidirect
44. #22583530C22semidirect
45. C22⋊(C283⋊C8)C22semidirect
46. C22⋊(C283⋊C8)C22semidirect
47. #22583521C22semidirect
50. #125×C283C22semidirect
52. C22⋊Q8×C283C22semidirect
53. C22⋊Q8×C283C22semidirect
54. C4×Q8×C283C22semidirect
57. #117×C283C22semidirect
58. C2×C4⋊C4×C283C22semidirect
195. C283#3310semidirect
197. #552D283semidirect
223. #125C2×D283semidirect
225. C22⋊Q8C2×D283semidirect
226. C22⋊Q8C2×D283semidirect
232. C4×Q8C2×D283semidirect
236. #117C2×D283semidirect
237. C2×C4⋊C4C2×D283semidirect

Maximal subgroups

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

subgroupindex
D4×D11322
#318268312
#318267892
#318267782
#318267622
#318267132
#318266962
#318266652
#318266152
#318265632
#318265372
#318264692
C2×#225835302
#552×C2832
#3310283