Finite Group Databasebeta

← #44560550

Subgroups — #44560550

Order 32128.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632310C211yesmaximal, characteristic, complemented
2#30632162C211yesmaximal, characteristic, complemented
3#30632025C211yesmaximal, characteristic, complemented
4#30632012C211yesmaximal, characteristic, complemented
5#30631975C211yesmaximal, characteristic, complemented
6#30631965C211yesmaximal, characteristic, complemented
7#30631904C211yesmaximal, characteristic, complemented
8#30631904C211yesmaximal, characteristic, complemented
9#30631821C211yesmaximal, characteristic
10#30631796C211yesmaximal, characteristic, complemented
11#30631788C211yesmaximal, characteristic, complemented
12SD16×Dic251C211yesmaximal, characteristic, complemented
13#30631767C211yesmaximal, characteristic, complemented
14#30631745C211yesmaximal, characteristic, complemented
15#450×C251C211yesmaximal, characteristic, complemented
16C2×#16860795C2211yescharacteristic
17C2×#16860795C2211yescharacteristic
18#22548714C2211yescharacteristic, complemented
19#22548710—21no—
20#22548704C2211yescharacteristic, complemented
21#22548701—21no—
22#22548697—21no—
23#22548695—21no—
24#22548689C2211yescharacteristic
25Q8×Dic251C2211yescharacteristic, complemented
26D4×Dic251—21no—
27D4×Dic251—21no—
28D4×Dic251C2211yescharacteristic
29D4×Dic251C2211yescharacteristic
30D4×Dic251C2211yescharacteristic
31D4×Dic251C2211yescharacteristic
32#22548677C2211yescharacteristic, complemented
33#22548674C2211yescharacteristic
34#22548674C2211yescharacteristic
35C2×C22⋊Dic251—21no—
36C2×C22⋊Dic251—21no—
37#22548649C2211yescharacteristic, complemented
38#22548648C2211yescharacteristic, complemented
39#22548647C2211yescharacteristic, complemented
40#22548646C2211yescharacteristic, complemented
41Dic251⋊C8C2211yescharacteristic, complemented
42Dic251⋊C8C2211yescharacteristic, complemented
43#22548639C2211yescharacteristic, complemented
44#22548639C2211yescharacteristic, complemented
45#22548637C2211yescharacteristic, complemented
46#22548637C2211yescharacteristic, complemented
47#22548636C2211yescharacteristic
48#22548636C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic
50#22548625C2211yescharacteristic, complemented
51C8×Dic251C2211yescharacteristic, complemented
52C22×D4×C251C2211yescharacteristic
53C2×SD16×C251C2211yescharacteristic
54C2×SD16×C251C2211yescharacteristic
55C22⋊Q8×C251C2211yescharacteristic
56#102×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic
59C2×Dic502—21no—
60C2×D1004C2311yescharacteristic
61C2×C22⋊D251—21no—
62C22×Dic251—21no—
63C22×Dic251—21no—
64C22×Dic251—21no—
65C22×Dic251—21no—
66C2×C4×D251—21no—
67#16860795—22no—
68#16860795—22no—
69C2×C251⋊C8C2311yescharacteristic
70C2×C251⋊C8C2311yescharacteristic
71C4⋊Dic251—21no—
72C4⋊Dic251D411yescharacteristic
73C4⋊Dic251D411yescharacteristic
74C4⋊Dic251C2311yescharacteristic
75C4⋊Dic251C2311yescharacteristic
76C4⋊Dic251C2311yescharacteristic
77Dic251⋊C4—21no—
78C22⋊Dic251—21no—
79C22⋊Dic251—21no—
80C22⋊Dic251—21no—
81C22⋊Dic251—21no—
82C22⋊Dic251—21no—
83C22⋊Dic251D411yescharacteristic
84C22⋊Dic251D411yescharacteristic
85(C2×D251)⋊C4—21no—
86(C2×D251)⋊C4—21no—
87(C2×D251)⋊C4—21no—
88(C2×D251)⋊C4—21no—
89C4×Dic251—21no—
90C4×Dic251—21no—
91C4×Dic251C2311yescharacteristic
92C4×Dic251C2311yescharacteristic
93C24×C251—21no—
94C2×Q8×C251C2311yescharacteristic
95C2×D4×C251—22no—
96C2×D4×C251—22no—
97C2×D4×C251—21no—
98C2×D4×C251C2311yescharacteristic
99C2×D4×C251C2311yescharacteristic
100C22×C4×C251C2311yescharacteristic
101SD16×C251—22no—
102SD16×C251—22no—
103C2×C8×C251C2311yescharacteristic
104C2×C8×C251C2311yescharacteristic
105C4⋊C4×C251—21no—
106C4⋊C4×C251C2311yescharacteristic
107C22⋊C4×C251—21no—
108C22×D251—21no—
109Dic502—81no—
110D1004—21no—
111D1004—21no—
112C22⋊D251—81no—
113C2×Dic251—21no—
114C2×Dic251—21no—
115C2×Dic251—21no—
116C2×Dic251—81no—
117C2×Dic251—81no—
118C2×Dic251—81no—
119C2×Dic251—81no—
120C2×Dic251C2×D411yescharacteristic
121C2×Dic251C2×D411yescharacteristic
122C2×Dic251C2×D411yescharacteristic
123C2×Dic251C2×D411yescharacteristic
124C4×D251—81no—
125C251⋊C8—21no—
126C251⋊C8—21no—
127C23×C251—24no—
128C23×C251—21no—
129C23×C251—21no—
130C23×C251—81no—
131C23×C251—81no—
132C23×C251C2×D411yescharacteristic
133Q8×C251—21no—
134Q8×C251—21no—
135D4×C251—24no—
136D4×C251—21no—
137D4×C251—21no—
138D4×C251#4112yes—
139D4×C251#4112yes—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251C2×D411yescharacteristic
145C2×C4×C251C2411yescharacteristic
146C8×C251—21no—
147C8×C251—21no—
148C22×C251—21no—
149C22×C251—22no—
150C22×C251—22no—
151C22×C251—21no—
152C22×C251—21no—
153C22×C251—41no—
154C22×C251—41no—
155C22×C251—41no—
156C22×C251—41no—
157C22×C251—84no—
158C22×C251C22×D411yescharacteristic
159C2×D251—41no—
160C2×D251—41no—
161C2×D251—41no—
162C4×C251—21no—
163C4×C251—41no—
164C4×C251—41no—
165C4×C251C2×#4111yescharacteristic
166C4×C251C2×#4111yescharacteristic
167Dic251—21no—
168Dic251—21no—
169Dic251—21no—
170Dic251—21no—
171Dic251—41no—
172Dic251—41no—
173Dic251—41no—
174C2×C251—41no—
175C2×C251—42no—
176C2×C251—42no—
177C2×C251#52511yescharacteristic
178C2×C251#57611yescharacteristic
179C2×C251#57711yescharacteristic
180D251—81no—
181C251#311811yescharacteristic, complemented, Sylow
182#3118—2511nomaximal, Sylow
183#542—2511no—
184#518—2511no—
185#485—2511no—
186#482—2511no—
187#476—2511no—
188#474—2511no—
189#463—2511no—
190#463—2511no—
191#450D25111yescharacteristic, complemented
192#447—2511no—
193#442—2511no—
194#440—2511no—
195C4×SD16—2511no—
196C4×D8—2511no—
197#433—2511no—
198C22×D4C2×D25111yescharacteristic
199C2×SD16C2×D25111yescharacteristic
200C2×SD16C2×D25111yescharacteristic
201C2×D8—2511no—
202C2×D8—2511no—
203C4⋊Q8—2511no—
204#126—5021no—
205#124—5021no—
206#124—2511no—
207#123—5021no—
208C22⋊Q8—5021no—
209C22⋊Q8C2×D25111yescharacteristic
210#121—2511no—
211C4×Q8—2511no—
212C4×D4—5021no—
213C4×D4—5021no—
214C4×D4—2511no—
215C4×D4—2511no—
216C4×D4—2511no—
217C4×D4—2511no—
218C4×D4—2511no—
219#117—2511no—
220#117—2511no—
221C2×C22⋊C4—5021no—
222C2×C22⋊C4—5021no—
223#107—2511no—
224#107—2511no—
225#106—2511no—
226#106—2511no—
227C4⋊C8—2511no—
228C4⋊C8—2511no—
229#103—2511no—
230#103—2511no—
231#102—2511no—
232#102—2511no—
233#102—2511no—
234#102—2511no—
235#102C2×D25111yescharacteristic
236#102C2×D25111yescharacteristic
237C22⋊C8—2511no—
238C22⋊C8C2×D25111yescharacteristic
239#97—2511no—
240C4×C8—2511no—
241C24—21no—
242C2×Q8—5021no—
243C2×Q8C22×D25111yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—5021no—
248C2×D4—2511no—
249C2×D4C22×D25111yescharacteristic
250C2×D4C22×D25111yescharacteristic
251C22×C4—5021no—
252C22×C4—5021no—
253C22×C4—5021no—
254C22×C4—5021no—
255C22×C4—5021no—
256C22×C4C22×D25111yescharacteristic
257SD16—22no—
258SD16—22no—
259D8—5022no—
260D8—5022no—
261C2×C8—2511no—
262C2×C8—2511no—
263C2×C8C22×D25111yescharacteristic
264C2×C8C22×D25111yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—5021no—
267C4⋊C4—5021no—
268C4⋊C4—2511no—
269C4⋊C4—2511no—
270C4⋊C4—2511no—
271C4⋊C4—2511no—
272C4⋊C4—2511no—
273C4⋊C4C22×D25111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—5021no—
276C22⋊C4—5021no—
277C22⋊C4—5021no—
278C22⋊C4—5021no—
279C22⋊C4—5021no—
280C22⋊C4—5021no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4—5021no—
284C22⋊C4—2511no—
285C22⋊C4—2511no—
286C42—5021no—
287C42—5021no—
288C42—2511no—
289C42—2511no—
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—5021no—
296C23D4×D25111yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—20081no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—20081no—
304D4—5021no—
305D4—5021no—
306D4#1686080912yes—
307D4#1686080912yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—20081no—
313C2×C4—20081no—
314C2×C4—20081no—
315C2×C4—20081no—
316C2×C4—20081no—
317C2×C4—5021no—
318C2×C4—5021no—
319C2×C4—5021no—
320C2×C4—2511no—
321C2×C4—2511no—
322C2×C4—2511no—
323C2×C4—2511no—
324C2×C4D4×D25111yescharacteristic
325C2×C4C23×D25111yescharacteristic
326C8—21no—
327C8—21no—
328C8—5021no—
329C8—5021no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—10041no—
341C22—10041no—
342C22—10041no—
343C22C2×D4×D25111yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—10041no—
348C4—10041no—
349C4—10041no—
350C4—5021no—
351C4—5021no—
352C4—5021no—
353C4—5021no—
354C4C2×#1686080911yescharacteristic
355C4C2×#1686080911yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—20081no—
360C2#3063219311yescharacteristic
361C2#3063249111yescharacteristic
362C2#3063249311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#450×C251C2
socleC22×C251C22×D4
prime radical#30631821C2
hypercenter#450D251
hypocenterC251#3118
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #30632310C2semidirect
2. #30632162C2semidirect
3. #30632025C2semidirect
4. #30632012C2semidirect
5. #30631975C2semidirect
6. #30631965C2semidirect
7. #30631904C2semidirect
8. #30631904C2semidirect
10. #30631796C2semidirect
11. #30631788C2semidirect
12. SD16×Dic251C2semidirect
13. #30631767C2semidirect
14. #30631745C2semidirect
15. #450×C251C2semidirect
18. #22548714C22semidirect
20. #22548704C22semidirect
25. Q8×Dic251C22semidirect
32. #22548677C22semidirect
37. #22548649C22semidirect
38. #22548648C22semidirect
39. #22548647C22semidirect
40. #22548646C22semidirect
41. Dic251⋊C8C22semidirect
42. Dic251⋊C8C22semidirect
43. #22548639C22semidirect
44. #22548639C22semidirect
45. #22548637C22semidirect
46. #22548637C22semidirect
50. #22548625C22semidirect
51. C8×Dic251C22semidirect
181. C251#3118semidirect
191. #450D251semidirect

Maximal subgroups

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

subgroupindex
#306323102
#306321622
#306320252
#306320122
#306319752
#306319652
#306319042
#306318212
#306317962
#306317882
SD16×Dic2512
#306317672
#306317452
#450×C2512
#3118251