Finite Group Databasebeta

← #44560602

Subgroups — #44560602

Order 32128.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632290C211yesmaximal, characteristic, complemented
2#30632162C211yesmaximal, characteristic, complemented
3#30632012C211yesmaximal, characteristic, complemented
4#30632008C211yesmaximal, characteristic, complemented
5#30631989C211yesmaximal, characteristic, complemented
6#30631985C211yesmaximal, characteristic, complemented
7#30631880C211yesmaximal, characteristic, complemented
8#30631880C211yesmaximal, characteristic, complemented
9#30631821C211yesmaximal, characteristic, complemented
10#30631799C211yesmaximal, characteristic, complemented
11#30631796C211yesmaximal, characteristic, complemented
12D8×Dic251C211yesmaximal, characteristic, complemented
13#30631767C211yesmaximal, characteristic, complemented
14#30631745C211yesmaximal, characteristic, complemented
15#447×C251C211yesmaximal, characteristic, complemented
16C2×#16860795C2211yescharacteristic
17C2×#16860795C2211yescharacteristic
18#22548711C2211yescharacteristic, complemented
19#22548706C2211yescharacteristic, complemented
20#22548699—21no—
21#22548697—21no—
22#22548689C2211yescharacteristic
23#22548687—21no—
24#22548686—21no—
25D4×Dic251—21no—
26D4×Dic251—21no—
27D4×Dic251C2211yescharacteristic
28D4×Dic251C2211yescharacteristic
29D4×Dic251C2211yescharacteristic
30D4×Dic251C2211yescharacteristic
31D4×Dic251C2211yescharacteristic, complemented
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#22548637C2211yescharacteristic, complemented
44#22548637C2211yescharacteristic, 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×D1004C2311yescharacteristic
60C2×C22⋊D251—21no—
61C2×C22⋊D251—21no—
62C2×C22⋊D251—21no—
63C22×Dic251—21no—
64C22×Dic251—21no—
65C22×Dic251—21no—
66C22×Dic251—21no—
67C22×Dic251—21no—
68C2×C4×D251—21no—
69#16860795—22no—
70#16860795—22no—
71C2×C251⋊C8C2311yescharacteristic
72C2×C251⋊C8C2311yescharacteristic
73C4⋊Dic251—21no—
74C4⋊Dic251D411yescharacteristic
75C4⋊Dic251D411yescharacteristic
76C4⋊Dic251C2311yescharacteristic
77C4⋊Dic251C2311yescharacteristic
78C4⋊Dic251C2311yescharacteristic
79Dic251⋊C4—21no—
80C22⋊Dic251—21no—
81C22⋊Dic251—21no—
82C22⋊Dic251—21no—
83C22⋊Dic251—21no—
84C22⋊Dic251—21no—
85C22⋊Dic251—21no—
86C22⋊Dic251D411yescharacteristic
87C22⋊Dic251D411yescharacteristic
88(C2×D251)⋊C4—21no—
89(C2×D251)⋊C4—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—
108C22×D251—21no—
109D1004—21no—
110D1004—21no—
111C22⋊D251—81no—
112C22⋊D251—81no—
113C22⋊D251—81no—
114C2×Dic251—21no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251—81no—
118C2×Dic251—81no—
119C2×Dic251—81no—
120C2×Dic251—81no—
121C2×Dic251—81no—
122C2×Dic251C2×D411yescharacteristic
123C2×Dic251C2×D411yescharacteristic
124C2×Dic251C2×D411yescharacteristic
125C2×Dic251C2×D411yescharacteristic
126C4×D251—81no—
127C251⋊C8—21no—
128C251⋊C8—21no—
129C23×C251—24no—
130C23×C251—21no—
131C23×C251—21no—
132C23×C251—21no—
133C23×C251—81no—
134C23×C251—81no—
135C23×C251C2×D411yescharacteristic
136D4×C251—24no—
137D4×C251—21no—
138D4×C251—21no—
139D4×C251—21no—
140D4×C251—21no—
141D4×C251—81no—
142D4×C251#4112yes—
143D4×C251#4112yes—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251C2×D411yescharacteristic
148C2×C4×C251C2411yescharacteristic
149C8×C251—21no—
150C8×C251—21no—
151C22×C251—21no—
152C22×C251—22no—
153C22×C251—22no—
154C22×C251—21no—
155C22×C251—21no—
156C22×C251—41no—
157C22×C251—41no—
158C22×C251—41no—
159C22×C251—41no—
160C22×C251—41no—
161C22×C251—41no—
162C22×C251—41no—
163C22×C251—84no—
164C22×C251C22×D411yescharacteristic
165C2×D251—41no—
166C2×D251—41no—
167C2×D251—41no—
168C4×C251—21no—
169C4×C251—41no—
170C4×C251C2×#4111yescharacteristic
171C4×C251C2×#4111yescharacteristic
172Dic251—21no—
173Dic251—21no—
174Dic251—21no—
175Dic251—21no—
176Dic251—41no—
177Dic251—41no—
178Dic251—41no—
179C2×C251—41no—
180C2×C251—42no—
181C2×C251—42no—
182C2×C251—81no—
183C2×C251#52511yescharacteristic
184C2×C251#57611yescharacteristic
185C2×C251#57611yescharacteristic
186D251—81no—
187C251#312211yescharacteristic, complemented, Sylow
188#3122—2511nomaximal, Sylow
189#540—2511no—
190#518—2511no—
191#482—2511no—
192#482—2511no—
193#478—2511no—
194#478—2511no—
195#459—2511no—
196#459—2511no—
197#447—2511no—
198#447D25111yescharacteristic, complemented
199#442—2511no—
200#442—2511no—
201C4×D8—2511no—
202C4×D8—2511no—
203#433—2511no—
204C22×D4C2×D25111yescharacteristic
205C2×D8—2511no—
206C2×D8—2511no—
207C2×D8C2×D25111yescharacteristic
208C2×D8C2×D25111yescharacteristic
209#127—2511no—
210#125—2511no—
211#123—5021no—
212#123—5021no—
213#121—5021no—
214#121—5021no—
215#121—2511no—
216#121C2×D25111yescharacteristic
217C4×D4—5021no—
218C4×D4—5021no—
219C4×D4—2511no—
220C4×D4—2511no—
221C4×D4—2511no—
222C4×D4—2511no—
223C4×D4—2511no—
224C4×D4—2511no—
225#117—2511no—
226#117—2511no—
227C2×C22⋊C4—5021no—
228C2×C22⋊C4—5021no—
229#107—2511no—
230#107—2511no—
231#106—2511no—
232#106—2511no—
233C4⋊C8—2511no—
234C4⋊C8—2511no—
235#102—2511no—
236#102—2511no—
237#102—2511no—
238#102—2511no—
239#102—2511no—
240#102—2511no—
241#102C2×D25111yescharacteristic
242#102C2×D25111yescharacteristic
243C22⋊C8—2511no—
244C22⋊C8C2×D25111yescharacteristic
245#97—2511no—
246C4×C8—2511no—
247C24—21no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—5021no—
253C2×D4—5021no—
254C2×D4—5021no—
255C2×D4—2511no—
256C2×D4C22×D25111yescharacteristic
257C2×D4C22×D25111yescharacteristic
258C2×D4C22×D25111yescharacteristic
259C22×C4—5021no—
260C22×C4—5021no—
261C22×C4—5021no—
262C22×C4—5021no—
263C22×C4—5021no—
264C22×C4—5021no—
265C22×C4C22×D25111yescharacteristic
266D8—22no—
267D8—22no—
268D8—5022no—
269D8—5022no—
270C2×C8—2511no—
271C2×C8—2511no—
272C2×C8C22×D25111yescharacteristic
273C2×C8C22×D25111yescharacteristic
274C4⋊C4—5021no—
275C4⋊C4—5021no—
276C4⋊C4—2511no—
277C4⋊C4—2511no—
278C4⋊C4—2511no—
279C4⋊C4—2511no—
280C4⋊C4—2511no—
281C4⋊C4C22×D25111yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—5021no—
284C22⋊C4—5021no—
285C22⋊C4—5021no—
286C22⋊C4—5021no—
287C22⋊C4—5021no—
288C22⋊C4—5021no—
289C22⋊C4—5021no—
290C22⋊C4—5021no—
291C22⋊C4—2511no—
292C22⋊C4—2511no—
293C42—5021no—
294C42—2511no—
295C42—2511no—
296C23—24no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—81no—
301C23—81no—
302C23—5021no—
303C23D4×D25111yescharacteristic
304D4—24no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—81no—
310D4—20081no—
311D4—20081no—
312D4—20081no—
313D4—5021no—
314D4—5021no—
315D4#1686080912yes—
316D4#1686080912yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—20081no—
321C2×C4—20081no—
322C2×C4—20081no—
323C2×C4—20081no—
324C2×C4—20081no—
325C2×C4—20081no—
326C2×C4—5021no—
327C2×C4—5021no—
328C2×C4—5021no—
329C2×C4—2511no—
330C2×C4—2511no—
331C2×C4—2511no—
332C2×C4—2511no—
333C2×C4D4×D25111yescharacteristic
334C2×C4C23×D25111yescharacteristic
335C8—21no—
336C8—21no—
337C8—5021no—
338C8—5021no—
339C22—21no—
340C22—22no—
341C22—22no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—84no—
352C22—10041no—
353C22—10041no—
354C22—10041no—
355C22C2×D4×D25111yescharacteristic
356C4—21no—
357C4—41no—
358C4—10041no—
359C4—10041no—
360C4—10041no—
361C4—5021no—
362C4—5021no—
363C4—5021no—
364C4—5021no—
365C4C2×#1686080911yescharacteristic
366C4C2×#1686080911yescharacteristic
367C2—41no—
368C2—42no—
369C2—42no—
370C2—81no—
371C2—20081no—
372C2#3063219311yescharacteristic
373C2#3063248811yescharacteristic
374C2#3063249111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#447×C251C2
socleC22×C251C22×D4
hypercenter#447D251
hypocenterC251#3122
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. #30632290C2semidirect
2. #30632162C2semidirect
3. #30632012C2semidirect
4. #30632008C2semidirect
5. #30631989C2semidirect
6. #30631985C2semidirect
7. #30631880C2semidirect
8. #30631880C2semidirect
9. #30631821C2semidirect
10. #30631799C2semidirect
11. #30631796C2semidirect
12. D8×Dic251C2semidirect
13. #30631767C2semidirect
14. #30631745C2semidirect
15. #447×C251C2semidirect
18. #22548711C22semidirect
19. #22548706C22semidirect
31. D4×Dic251C22semidirect
32. #22548677C22semidirect
37. #22548649C22semidirect
38. #22548648C22semidirect
39. #22548647C22semidirect
40. #22548646C22semidirect
41. Dic251⋊C8C22semidirect
42. Dic251⋊C8C22semidirect
43. #22548637C22semidirect
44. #22548637C22semidirect
47. #22548636C22semidirect
48. #22548636C22semidirect
50. #22548625C22semidirect
51. C8×Dic251C22semidirect
187. C251#3122semidirect
198. #447D251semidirect

Maximal subgroups

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

subgroupindex
#306322902
#306321622
#306320122
#306320082
#306319892
#306319852
#306318802
#306318212
#306317992
#306317962
D8×Dic2512
#306317672
#306317452
#447×C2512
#3122251