Finite Group Databasebeta

← #44560552

Subgroups — #44560552

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#30632161C211yesmaximal, characteristic, complemented
3#30632011C211yesmaximal, characteristic, complemented
4#30631976C211yesmaximal, characteristic, complemented
5#30631966C211yesmaximal, characteristic, complemented
6#30631905C211yesmaximal, characteristic, complemented
7#30631905C211yesmaximal, characteristic, complemented
8#30631837C211yesmaximal, characteristic, complemented
9#30631823C211yesmaximal, characteristic
10#30631798C211yesmaximal, characteristic, complemented
11#30631790C211yesmaximal, characteristic, complemented
12#30631772C211yesmaximal, characteristic, complemented
13#30631769C211yesmaximal, characteristic, complemented
14#30631745C211yesmaximal, characteristic, complemented
15#485×C251C211yesmaximal, characteristic, complemented
16C22×D1004C2211yescharacteristic
17C2×#16860797C2211yescharacteristic
18C2×#16860797C2211yescharacteristic
19C2×D2008C2211yescharacteristic
20C2×D2008C2211yescharacteristic
21#22548714C2211yescharacteristic, complemented
22#22548710—21no—
23#22548704C2211yescharacteristic, complemented
24#22548701—21no—
25#22548697—21no—
26#22548695—21no—
27#22548689C2211yescharacteristic
28Q8×Dic251C2211yescharacteristic, complemented
29C4×D1004—21no—
30C4×D1004—21no—
31#22548677C2211yescharacteristic
32#22548677C2211yescharacteristic
33#22548677C2211yescharacteristic
34#22548677C2211yescharacteristic
35#22548677C2211yescharacteristic, complemented
36#22548674C2211yescharacteristic
37C22⋊C4×D251—21no—
38C22⋊C4×D251—21no—
39#22548648C2211yescharacteristic, complemented
40#22548646C2211yescharacteristic, complemented
41Dic251⋊C8C2211yescharacteristic, complemented
42Dic251⋊C8C2211yescharacteristic, complemented
43#22548637C2211yescharacteristic
44#22548637C2211yescharacteristic
45#22548635C2211yescharacteristic
46#22548635C2211yescharacteristic
47#22548635C2211yescharacteristic, complemented
48#22548635C2211yescharacteristic, complemented
49C22⋊(C251⋊C8)C2211yescharacteristic
50#22548625C2211yescharacteristic, complemented
51C8×Dic251C2211yescharacteristic, complemented
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×D251—21no—
60C2×Dic502—21no—
61C2×D1004—22no—
62C2×D1004—22no—
63C2×D1004—21no—
64C2×D1004C2311yescharacteristic
65C2×D1004C2311yescharacteristic
66C2×D1004C2311yescharacteristic
67C2×C22⋊D251—21no—
68C2×C4×D251—21no—
69C2×C4×D251—21no—
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72C2×C4×D251—21no—
73#16860797—22no—
74#16860797—22no—
75D2008—22no—
76D2008—22no—
77C2×C251⋊C8C2311yescharacteristic
78C2×C251⋊C8C2311yescharacteristic
79C4⋊Dic251—21no—
80C4⋊Dic251D411yescharacteristic
81C4⋊Dic251D411yescharacteristic
82C4⋊Dic251C2311yescharacteristic
83Dic251⋊C4—21no—
84C22⋊Dic251D411yescharacteristic
85C22⋊Dic251D411yescharacteristic
86(C2×D251)⋊C4—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—
93(C2×D251)⋊C4—21no—
94C4×Dic251—21no—
95C4×Dic251C2311yescharacteristic
96C4×Dic251C2311yescharacteristic
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—21no—
109C22×D251—21no—
110C22×D251—24no—
111C22×D251—21no—
112C22×D251—81no—
113C22×D251—81no—
114Dic502—81no—
115D1004—24no—
116D1004—21no—
117D1004—21no—
118D1004—21no—
119D1004—21no—
120D1004#4112yes—
121D1004#4112yes—
122C22⋊D251—81no—
123C2×Dic251—21no—
124C2×Dic251C2×D411yescharacteristic
125C2×Dic251C2×D411yescharacteristic
126C2×Dic251C2×D411yescharacteristic
127C2×Dic251C2×D411yescharacteristic
128C4×D251—81no—
129C4×D251—81no—
130C4×D251—81no—
131C4×D251—81no—
132C4×D251—81no—
133C251⋊C8—21no—
134C251⋊C8—21no—
135C23×C251C2×D411yescharacteristic
136Q8×C251—21no—
137Q8×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251—21no—
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—21no—
150C22×C251—21no—
151C22×C251C22×D411yescharacteristic
152C2×D251—22no—
153C2×D251—22no—
154C2×D251—41no—
155C2×D251—41no—
156C2×D251—41no—
157C2×D251—41no—
158C2×D251—41no—
159C2×D251—41no—
160C2×D251—41no—
161C2×D251—84no—
162C4×C251—21no—
163C4×C251—41no—
164C4×C251—41no—
165C4×C251—41no—
166C4×C251—41no—
167C4×C251C2×#4111yescharacteristic
168C4×C251C2×#4111yescharacteristic
169Dic251—21no—
170Dic251—21no—
171Dic251—21no—
172Dic251—21no—
173Dic251—41no—
174C2×C251—41no—
175C2×C251#52511yescharacteristic
176C2×C251#57611yescharacteristic
177C2×C251#57711yescharacteristic
178D251—42no—
179D251—42no—
180D251—81no—
181C251#311811yescharacteristic, complemented, Sylow
182#3118—2511nomaximal, Sylow
183#542—2511no—
184#518—2511no—
185#485D25111yescharacteristic, complemented
186#482—2511no—
187#476—2511no—
188#474—2511no—
189#463—2511no—
190#463—2511no—
191#450—2511no—
192#447—2511no—
193#442—2511no—
194#440—2511no—
195C4×SD16—2511no—
196C4×D8—2511no—
197#433—2511no—
198C22×D4—2511no—
199C2×SD16—2511no—
200C2×SD16—2511no—
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#117C2×D25111yescharacteristic
221C2×C22⋊C4—5021no—
222C2×C22⋊C4—5021no—
223#107—2511no—
224#107C2×D25111yescharacteristic, complemented
225#106—2511no—
226#106C2×D25111yescharacteristic, complemented
227C4⋊C8—2511no—
228C4⋊C8—2511no—
229#103C2×D25111yescharacteristic, complemented
230#103C2×D25111yescharacteristic, complemented
231#102—2511no—
232#102—2511no—
233#102—2511no—
234#102—2511no—
235#102—2511no—
236#102—2511no—
237C22⋊C8—2511no—
238C22⋊C8C2×D25111yescharacteristic
239#97—2511no—
240C4×C8—2511no—
241C24—5021no—
242C2×Q8—5021no—
243C2×Q8C22×D25111yescharacteristic
244C2×D4—5022no—
245C2×D4—5022no—
246C2×D4—5021no—
247C2×D4—5021no—
248C2×D4—2511no—
249C2×D4—2511no—
250C2×D4—2511no—
251C22×C4—5021no—
252C22×C4—5021no—
253C22×C4—5021no—
254C22×C4—5021no—
255C22×C4—5021no—
256C22×C4C22×D25111yescharacteristic
257SD16—5022no—
258SD16—5022no—
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⋊C4C22×D25111yescharacteristic
272C4⋊C4C22×D25111yescharacteristic
273C4⋊C4C22×D25111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
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—21no—
287C42—5021no—
288C42—2511no—
289C42—2511no—
290C23—20081no—
291C23—20081no—
292C23—5024no—
293C23—5021no—
294C23—5021no—
295C23—5021no—
296C23D4×D25111yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—20081no—
300D4—20081no—
301D4—5024no—
302D4—5021no—
303D4—5021no—
304D4—5021no—
305D4—5021no—
306D4—2512no—
307D4—2512no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—20081no—
315C2×C4—20081no—
316C2×C4—20081no—
317C2×C4—20081no—
318C2×C4—20081no—
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—21no—
332C22—21no—
333C22—20084no—
334C22—10041no—
335C22—10041no—
336C22—10041no—
337C22—10041no—
338C22—10041no—
339C22—10041no—
340C22—10041no—
341C22—5022no—
342C22—5022no—
343C22C2×D4×D25111yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—10041no—
350C4—5021no—
351C4—5021no—
352C4—5021no—
353C4—5021no—
354C4#41×D25111yescharacteristic
355C4#41×D25111yescharacteristic
356C2—41no—
357C2—20081no—
358C2—10042no—
359C2—10042no—
360C2#3063219511yescharacteristic
361C2#3063248911yescharacteristic
362C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#485×C251C2
socleC22×C251C22×D4
prime radical#30631823C2
hypercenter#485D251
hypocenterC251#3118
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
1. #30632310C2semidirect
2. #30632161C2semidirect
3. #30632011C2semidirect
4. #30631976C2semidirect
5. #30631966C2semidirect
6. #30631905C2semidirect
7. #30631905C2semidirect
8. #30631837C2semidirect
10. #30631798C2semidirect
11. #30631790C2semidirect
12. #30631772C2semidirect
13. #30631769C2semidirect
14. #30631745C2semidirect
15. #485×C251C2semidirect
21. #22548714C22semidirect
23. #22548704C22semidirect
28. Q8×Dic251C22semidirect
35. #22548677C22semidirect
39. #22548648C22semidirect
40. #22548646C22semidirect
41. Dic251⋊C8C22semidirect
42. Dic251⋊C8C22semidirect
47. #22548635C22semidirect
48. #22548635C22semidirect
50. #22548625C22semidirect
51. C8×Dic251C22semidirect
54. #107×C251C22semidirect
55. #106×C251C22semidirect
56. #103×C251C22semidirect
57. #103×C251C22semidirect
181. C251#3118semidirect
185. #485D251semidirect
224. #107C2×D251semidirect
226. #106C2×D251semidirect
229. #103C2×D251semidirect
230. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323102
#306321612
#306320112
#306319762
#306319662
#306319052
#306318372
#306318232
#306317982
#306317902
#306317722
#306317692
#306317452
#485×C2512
#3118251