Finite Group Databasebeta

← #44560542

Subgroups — #44560542

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#30632304C211yesmaximal, characteristic, complemented
2#30632162C211yesmaximal, characteristic, complemented
3#30632029C211yesmaximal, characteristic, complemented
4#30632008C211yesmaximal, characteristic, complemented
5#30631975C211yesmaximal, characteristic, complemented
6#30631965C211yesmaximal, characteristic, complemented
7#30631900C211yesmaximal, characteristic, complemented
8#30631900C211yesmaximal, characteristic, complemented
9#30631838C211yesmaximal, characteristic, complemented
10#30631799C211yesmaximal, characteristic, complemented
11#30631784C211yesmaximal, characteristic, complemented
12#30631778C211yesmaximal, characteristic, complemented
13D8×Dic251C211yesmaximal, characteristic, complemented
14#30631745C211yesmaximal, characteristic, complemented
15#447×C251C211yesmaximal, characteristic
16C2×#16860798C2211yescharacteristic
17C2×#16860798C2211yescharacteristic
18#22548714C2211yescharacteristic, complemented
19#22548708—21no—
20#22548702—21no—
21#22548702C2211yescharacteristic, complemented
22#22548696—21no—
23#22548694C2211yescharacteristic
24#22548692—21no—
25#22548680C2211yescharacteristic, complemented
26D4×Dic251—21no—
27D4×Dic251—21no—
28D4×Dic251C2211yescharacteristic
29D4×Dic251C2211yescharacteristic
30D4×Dic251C2211yescharacteristic
31D4×Dic251C2211yescharacteristic
32D4×Dic251C2211yescharacteristic, 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#22548640C2211yescharacteristic, complemented
44#22548640C2211yescharacteristic, complemented
45#22548636C2211yescharacteristic
46#22548636C2211yescharacteristic
47#22548636C2211yescharacteristic, complemented
48#22548636C2211yescharacteristic, complemented
49C22⋊(C251⋊C8)C2211yescharacteristic
50#22548625C2211yescharacteristic, complemented
51C8×Dic251C2211yescharacteristic, complemented
52C22×D4×C251C2211yescharacteristic
53C2×D8×C251C2211yescharacteristic
54C2×D8×C251C2211yescharacteristic
55#121×C251C2211yescharacteristic
56#102×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic
59C2×Dic502—21no—
60C2×Dic502C2311yescharacteristic
61C22×Dic251—21no—
62C22×Dic251—21no—
63C22×Dic251—21no—
64C22×Dic251—21no—
65C22×Dic251—21no—
66#16860798—22no—
67#16860798—22no—
68C2×C251⋊C8C2311yescharacteristic
69C2×C251⋊C8C2311yescharacteristic
70C4⋊Dic251D411yescharacteristic
71C4⋊Dic251D411yescharacteristic
72C4⋊Dic251C2311yescharacteristic
73C4⋊Dic251C2311yescharacteristic
74C4⋊Dic251C2311yescharacteristic
75Dic251⋊C4—21no—
76Dic251⋊C4—21no—
77Dic251⋊C4—21no—
78C22⋊Dic251—21no—
79C22⋊Dic251—21no—
80C22⋊Dic251—21no—
81C22⋊Dic251—21no—
82C22⋊Dic251—21no—
83C22⋊Dic251—21no—
84C22⋊Dic251—21no—
85C22⋊Dic251—21no—
86C22⋊Dic251—21no—
87C22⋊Dic251D411yescharacteristic
88C22⋊Dic251D411yescharacteristic
89C4×Dic251—21no—
90C4×Dic251—21no—
91C4×Dic251C2311yescharacteristic
92C4×Dic251C2311yescharacteristic
93C24×C251—21no—
94C2×D4×C251—22no—
95C2×D4×C251—22no—
96C2×D4×C251—21no—
97C2×D4×C251—21no—
98C2×D4×C251C2311yescharacteristic
99C2×D4×C251C2311yescharacteristic
100C2×D4×C251C2311yescharacteristic
101C22×C4×C251C2311yescharacteristic
102D8×C251—22no—
103D8×C251—22no—
104C2×C8×C251C2311yescharacteristic
105C2×C8×C251C2311yescharacteristic
106C4⋊C4×C251C2311yescharacteristic
107C22⋊C4×C251—21no—
108Dic502—21no—
109Dic502—21no—
110Dic502—81no—
111C2×Dic251—21no—
112C2×Dic251—21no—
113C2×Dic251—21no—
114C2×Dic251—21no—
115C2×Dic251—81no—
116C2×Dic251—81no—
117C2×Dic251—81no—
118C2×Dic251—81no—
119C2×Dic251—81no—
120C2×Dic251C2×D411yescharacteristic
121C2×Dic251C2×D411yescharacteristic
122C2×Dic251C2×D411yescharacteristic
123C2×Dic251C2×D411yescharacteristic
124C251⋊C8—21no—
125C251⋊C8—21no—
126C23×C251—24no—
127C23×C251—21no—
128C23×C251—21no—
129C23×C251—21no—
130C23×C251—81no—
131C23×C251—81no—
132C23×C251C2×D411yescharacteristic
133D4×C251—24no—
134D4×C251—21no—
135D4×C251—21no—
136D4×C251—21no—
137D4×C251—21no—
138D4×C251—81no—
139D4×C251#4112yes—
140D4×C251#4112yes—
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—41no—
158C22×C251—41no—
159C22×C251—41no—
160C22×C251—84no—
161C22×C251C22×D411yescharacteristic
162C4×C251—21no—
163C4×C251—41no—
164C4×C251C2×#4111yescharacteristic
165C4×C251C2×#4111yescharacteristic
166Dic251—21no—
167Dic251—21no—
168Dic251—21no—
169Dic251—21no—
170Dic251—41no—
171Dic251—41no—
172Dic251—41no—
173Dic251—41no—
174C2×C251—41no—
175C2×C251—42no—
176C2×C251—42no—
177C2×C251—81no—
178C2×C251#52511yescharacteristic
179C2×C251#57611yescharacteristic
180C2×C251#57711yescharacteristic
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#450—2511no—
192#447D25111yescharacteristic
193#442—2511no—
194#440—2511no—
195C4×SD16—2511no—
196C4×D8—2511no—
197#433—2511no—
198C22×D4C2×D25111yescharacteristic
199C2×SD16—2511no—
200C2×SD16—2511no—
201C2×D8C2×D25111yescharacteristic
202C2×D8C2×D25111yescharacteristic
203C4⋊Q8—2511no—
204#126—5021no—
205#124—5021no—
206#124—2511no—
207#123—5021no—
208C22⋊Q8—5021no—
209C22⋊Q8—2511no—
210#121C2×D25111yescharacteristic
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×Q8—2511no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4C22×D25111yescharacteristic
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—5022no—
258SD16—5022no—
259D8—22no—
260D8—22no—
261C2×C8—2511no—
262C2×C8—2511no—
263C2×C8C22×D25111yescharacteristic
264C2×C8C22×D25111yescharacteristic
265C4⋊C4—5021no—
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—21no—
294C23—81no—
295C23—81no—
296C23D4×D25111yescharacteristic
297Q8—20081no—
298Q8—5021no—
299Q8—5021no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—21no—
305D4—81no—
306D4#1686080912yes—
307D4#1686080912yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—20081no—
312C2×C4—20081no—
313C2×C4—20081no—
314C2×C4—20081no—
315C2×C4—20081no—
316C2×C4—5021no—
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—41no—
340C22—41no—
341C22—41no—
342C22—84no—
343C22C2×D4×D25111yescharacteristic
344C4—21no—
345C4—41no—
346C4—10041no—
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—81no—
360C2#3063219311yescharacteristic
361C2#3063248811yescharacteristic
362C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting, prime radical#447×C251C2
socleC22×C251C22×D4
hypercenter#447D251
hypocenterC251#3118
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #30632304C2semidirect
2. #30632162C2semidirect
3. #30632029C2semidirect
4. #30632008C2semidirect
5. #30631975C2semidirect
6. #30631965C2semidirect
7. #30631900C2semidirect
8. #30631900C2semidirect
9. #30631838C2semidirect
10. #30631799C2semidirect
11. #30631784C2semidirect
12. #30631778C2semidirect
13. D8×Dic251C2semidirect
14. #30631745C2semidirect
18. #22548714C22semidirect
21. #22548702C22semidirect
25. #22548680C22semidirect
32. D4×Dic251C22semidirect
37. #22548649C22semidirect
38. #22548648C22semidirect
39. #22548647C22semidirect
40. #22548646C22semidirect
41. Dic251⋊C8C22semidirect
42. Dic251⋊C8C22semidirect
43. #22548640C22semidirect
44. #22548640C22semidirect
47. #22548636C22semidirect
48. #22548636C22semidirect
50. #22548625C22semidirect
51. C8×Dic251C22semidirect
181. C251#3118semidirect

Maximal subgroups

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

subgroupindex
#306323042
#306321622
#306320292
#306320082
#306319752
#306319652
#306319002
#306318382
#306317992
#306317842
#306317782
D8×Dic2512
#306317452
#447×C2512
#3118251