Finite Group Databasebeta

← #44560283

Subgroups — #44560283

Order 32128.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632200C211yesmaximal, characteristic
2#117×D251C211yesmaximal, characteristic, complemented
3#30632025C211yesmaximal, characteristic, complemented
4#30632021C211yesmaximal, characteristic, complemented
5#30632016C211yesmaximal, characteristic, complemented
6#30632012C211yesmaximal, characteristic, complemented
7#30632011C211yesmaximal, characteristic, complemented
8#30632007C211yesmaximal, characteristic, complemented
9#30632002C211yesmaximal, characteristic, complemented
10#30631724C211yesmaximal, characteristic, complemented
11#30631722C211yesmaximal, characteristic, complemented
12#30631682C211yesmaximal, characteristic, complemented
13#30631682C211yesmaximal, characteristic, complemented
14C22⋊C8×D251C211yesmaximal, characteristic, complemented
15#485×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C22×C4×D251C2211yescharacteristic
18C2×C8×D251C2211yescharacteristic
19C2×C8×D251C2211yescharacteristic
20#22548700—21no—
21#22548691C2211yescharacteristic
22#22548690C2211yescharacteristic
23#22548689C2211yescharacteristic
24#22548685—21no—
25#22548677—21no—
26(C22⋊D251)⋊C4—21no—
27#22548674C2211yescharacteristic
28#22548673C2211yescharacteristic
29#22548673C2211yescharacteristic
30#22548673C2211yescharacteristic
31#22548672—21no—
32#22548671—21no—
33C4⋊C4×D251C2211yescharacteristic
34C4⋊C4×D251C2211yescharacteristic
35C22⋊C4×D251—21no—
36C42×D251—21no—
37#22548649C2211yescharacteristic, complemented
38#22548648C2211yescharacteristic, complemented
39#22548648C2211yescharacteristic, complemented
40#22548647C2211yescharacteristic, complemented
41#22548646C2211yescharacteristic, complemented
42#22548646C2211yescharacteristic, complemented
43#22548639C2211yescharacteristic, complemented
44#22548639C2211yescharacteristic, complemented
45#22548637C2211yescharacteristic, complemented
46#22548637C2211yescharacteristic, complemented
47#22548635C2211yescharacteristic, complemented
48#22548635C2211yescharacteristic, complemented
49C22⋊(C251⋊C8)C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51(C2×D251)⋊C8C2211yescharacteristic
52C22⋊Q8×C251C2211yescharacteristic
53#117×C251C2211yescharacteristic
54#107×C251C2211yescharacteristic, complemented
55#106×C251C2211yescharacteristic, complemented
56#103×C251C2211yescharacteristic, complemented
57#103×C251C2211yescharacteristic, complemented
58C22⋊C8×C251C2211yescharacteristic
59C23×D251D411yescharacteristic
60#16860811—22no—
61#16860811—22no—
62C2×D1004—21no—
63C2×D1004—21no—
64C2×D1004C2311yescharacteristic
65C2×C22⋊D251—21no—
66C22×Dic251D411yescharacteristic
67C2×C4×D251—22no—
68C2×C4×D251—22no—
69C2×C4×D251—21no—
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72C2×C4×D251D411yescharacteristic
73C2×C4×D251D411yescharacteristic
74C2×C4×D251C2311yescharacteristic
75C2×C4×D251C2311yescharacteristic
76C2×C251⋊C8C2311yescharacteristic
77C2×C251⋊C8C2311yescharacteristic
78C8×D251—22no—
79C8×D251—22no—
80C4⋊Dic251C2311yescharacteristic
81C4⋊Dic251C2311yescharacteristic
82C4⋊Dic251C2311yescharacteristic
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86C22⋊Dic251—21no—
87(C2×D251)⋊C4—21no—
88(C2×D251)⋊C4—21no—
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93C4×Dic251—21no—
94C4×Dic251—21no—
95C4×Dic251—21no—
96C4×Dic251—21no—
97C2×Q8×C251C2311yescharacteristic
98C22×C4×C251C2311yescharacteristic
99C2×C8×C251C2311yescharacteristic
100C2×C8×C251C2311yescharacteristic
101C4⋊C4×C251—21no—
102C4⋊C4×C251C2311yescharacteristic
103C4⋊C4×C251C2311yescharacteristic
104C4⋊C4×C251C2311yescharacteristic
105C22⋊C4×C251—21no—
106C22⋊C4×C251—21no—
107C42×C251—21no—
108C22×D251—22no—
109C22×D251—22no—
110C22×D251—21no—
111C22×D251—41no—
112C22×D251C2×D411yescharacteristic
113C22×D251C2×D411yescharacteristic
114D1004—21no—
115D1004—21no—
116D1004—42no—
117D1004—42no—
118C22⋊D251—42no—
119C2×Dic251—22no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251C2×D411yescharacteristic
124C2×Dic251C2×D411yescharacteristic
125C4×D251—22no—
126C4×D251—21no—
127C4×D251—21no—
128C4×D251—22no—
129C4×D251—42no—
130C4×D251—42no—
131C4×D251—42no—
132C4×D251#4112yes—
133C4×D251#4112yes—
134C251⋊C8—21no—
135C251⋊C8—21no—
136C23×C251C2×D411yescharacteristic
137Q8×C251—21no—
138Q8×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2411yescharacteristic
147C8×C251—21no—
148C8×C251—21no—
149C22×C251—21no—
150C22×C251—21no—
151C22×C251—21no—
152C22×C251C22×D411yescharacteristic
153C2×D251—21no—
154C2×D251—21no—
155C2×D251—21no—
156C2×D251—21no—
157C2×D251—44no—
158C2×D251—41no—
159C2×D251—41no—
160C2×D251—41no—
161C2×D251#12312yes—
162C2×D251#13512yes—
163C4×C251—21no—
164C4×C251—41no—
165C4×C251—41no—
166C4×C251—41no—
167C4×C251—41no—
168C4×C251C2×#4111yescharacteristic
169C4×C251C2×#4111yescharacteristic
170Dic251—21no—
171Dic251—41no—
172Dic251—41no—
173Dic251—41no—
174Dic251#12312yes—
175C2×C251—41no—
176C2×C251C2×#12311yescharacteristic
177C2×C251C2×#13511yescharacteristic
178C2×C251#57511yescharacteristic
179D251—22no—
180D251—22no—
181D251—81no—
182C251#309111yescharacteristic, complemented, Sylow
183#3091—2511nomaximal, Sylow
184#525—2511no—
185C2×#117—2511no—
186#485—2511no—
187#485D25111yescharacteristic, complemented
188#484—2511no—
189#483—2511no—
190#482—2511no—
191#482—2511no—
192#481—2511no—
193#480—2511no—
194#427—2511no—
195#427—2511no—
196#416—2511no—
197#416—2511no—
198C2×C22⋊C8—2511no—
199C2×#41—2511no—
200C23×C4—2511no—
201C22×C8—2511no—
202C22×C8—2511no—
203#123—5021no—
204C22⋊Q8—2511no—
205C22⋊Q8C2×D25111yescharacteristic
206#121—2511no—
207#121—2511no—
208#120—5021no—
209C4×D4—5021no—
210C4×D4—5021no—
211#117—5021no—
212#117—5021no—
213#117—2511no—
214#117—2511no—
215#117—2511no—
216#117—2511no—
217#117C2×D25111yescharacteristic
218C2×C4⋊C4—2511no—
219C2×C4⋊C4—2511no—
220C2×C22⋊C4—5021no—
221C2×C42—5021no—
222#107—2511no—
223#107—2511no—
224#107—2511no—
225#107C2×D25111yescharacteristic, complemented
226#106—2511no—
227#106—2511no—
228#106—2511no—
229#106C2×D25111yescharacteristic, complemented
230#103—2511no—
231#103—2511no—
232#103C2×D25111yescharacteristic, complemented
233#103C2×D25111yescharacteristic, complemented
234#102—2511no—
235#102—2511no—
236#102—2511no—
237#102—2511no—
238C22⋊C8—2511no—
239C22⋊C8—2511no—
240C22⋊C8—2511no—
241C22⋊C8C2×D25111yescharacteristic
242C24—2511no—
243#41—5022no—
244#41—5022no—
245C2×Q8C22×D25111yescharacteristic
246C2×D4—5021no—
247C2×D4—5021no—
248C2×D4—5021no—
249C2×D4—2511no—
250C22×C4—5021no—
251C22×C4—5021no—
252C22×C4—5021no—
253C22×C4—5022no—
254C22×C4—5022no—
255C22×C4—2511no—
256C22×C4—2511no—
257C22×C4—2511no—
258C22×C4—2511no—
259C22×C4—2511no—
260C22×C4C22×D25111yescharacteristic
261C2×C8—5022no—
262C2×C8—5022no—
263C2×C8—2511no—
264C2×C8—2511no—
265C2×C8C22×D25111yescharacteristic
266C2×C8C22×D25111yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—5021no—
269C4⋊C4—5021no—
270C4⋊C4—5021no—
271C4⋊C4—2511no—
272C4⋊C4—2511no—
273C4⋊C4—2511no—
274C4⋊C4C22×D25111yescharacteristic
275C4⋊C4C22×D25111yescharacteristic
276C4⋊C4C22×D25111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—5021no—
280C22⋊C4—5021no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4—5021no—
284C22⋊C4—5021no—
285C22⋊C4—5021no—
286C42—21no—
287C42—5021no—
288C42—5021no—
289C42—5021no—
290C42—5021no—
291C23—10041no—
292C23—5022no—
293C23—5022no—
294C23—5021no—
295C23—2511no—
296C23—2511no—
297C23D4×D25111yescharacteristic
298Q8—21no—
299Q8—21no—
300D4—10042no—
301D4—10042no—
302D4—10042no—
303D4—5021no—
304D4—5021no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—10042no—
312C2×C4—10042no—
313C2×C4—10042no—
314C2×C4—5022no—
315C2×C4—5022no—
316C2×C4—5021no—
317C2×C4—5021no—
318C2×C4—5021no—
319C2×C4—5022no—
320C2×C4—5021no—
321C2×C4—5021no—
322C2×C4—2511no—
323C2×C4—2511no—
324C2×C4—2512no—
325C2×C4—2512no—
326C2×C4D4×D25111yescharacteristic
327C2×C4C23×D25111yescharacteristic
328C8—21no—
329C8—21no—
330C8—5021no—
331C8—5021no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—10044no—
336C22—10041no—
337C22—10041no—
338C22—10041no—
339C22—5021no—
340C22—5021no—
341C22—5021no—
342C22—5021no—
343C22—2512no—
344C22—2512no—
345C22C2×D4×D25111yescharacteristic
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—10041no—
352C4—10041no—
353C4—10041no—
354C4—5021no—
355C4—2512no—
356C4#41×D25111yescharacteristic
357C4#41×D25111yescharacteristic
358C2—41no—
359C2—20081no—
360C2—5022no—
361C2—5022no—
362C2#123×D25111yescharacteristic
363C2#135×D25111yescharacteristic
364C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#485×C251C2
socleC22×C251C22×D4
prime radical#30632200C2
hypercenter#485D251
hypocenterC251#3091
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
2. #117×D251C2semidirect
3. #30632025C2semidirect
4. #30632021C2semidirect
5. #30632016C2semidirect
6. #30632012C2semidirect
7. #30632011C2semidirect
8. #30632007C2semidirect
9. #30632002C2semidirect
10. #30631724C2semidirect
11. #30631722C2semidirect
12. #30631682C2semidirect
13. #30631682C2semidirect
14. C22⋊C8×D251C2semidirect
15. #485×C251C2semidirect
37. #22548649C22semidirect
38. #22548648C22semidirect
39. #22548648C22semidirect
40. #22548647C22semidirect
41. #22548646C22semidirect
42. #22548646C22semidirect
43. #22548639C22semidirect
44. #22548639C22semidirect
45. #22548637C22semidirect
46. #22548637C22semidirect
47. #22548635C22semidirect
48. #22548635C22semidirect
54. #107×C251C22semidirect
55. #106×C251C22semidirect
56. #103×C251C22semidirect
57. #103×C251C22semidirect
182. C251#3091semidirect
187. #485D251semidirect
225. #107C2×D251semidirect
229. #106C2×D251semidirect
232. #103C2×D251semidirect
233. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322002
#117×D2512
#306320252
#306320212
#306320162
#306320122
#306320112
#306320072
#306320022
#306317242
#306317222
#306316822
C22⋊C8×D2512
#485×C2512
#3091251