Finite Group Databasebeta

← #44562966

Subgroups — #44562966

Order 32128.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632389C211yesmaximal, characteristic, complemented
2#30632384C211yesmaximal, characteristic, complemented
3#30632087C211yesmaximal, characteristic, complemented
4#30632010C211yesmaximal, characteristic, complemented
5#30632008C211yesmaximal, characteristic, complemented
6#30631999C211yesmaximal, characteristic, complemented
7#30631998C211yesmaximal, characteristic, complemented
8#30631989C211yesmaximal, characteristic, complemented
9#30631985C211yesmaximal, characteristic, complemented
10#30631937C211yesmaximal, characteristic, complemented
11D8×Dic251C211yesmaximal, characteristic, complemented
12#30631763C211yesmaximal, characteristic, complemented
13#30631729C211yesmaximal, characteristic, complemented
14#30631726C211yesmaximal, characteristic, complemented
15#469×C251C211yesmaximal, characteristic, complemented
16C2×#16860793C2211yescharacteristic
17#22548708—21no—
18#22548708—21no—
19#22548706C2211yescharacteristic
20#22548706C2211yescharacteristic
21#22548699—21no—
22#22548696—21no—
23#22548687—21no—
24#22548686—21no—
25#22548686C2211yescharacteristic
26#22548686C2211yescharacteristic
27D4×Dic251—21no—
28D4×Dic251C2211yescharacteristic
29D4×Dic251C2211yescharacteristic
30C4×C22⋊D251C2211yescharacteristic
31(C22⋊D251)⋊C4—21no—
32#22548674C2211yescharacteristic
33#22548673C2211yescharacteristic
34C2×C4⋊Dic251C2211yescharacteristic
35C4⋊C4×D251C2211yescharacteristic
36#22548649C2211yescharacteristic
37#22548649C2211yescharacteristic
38#22548648C2211yescharacteristic
39#22548648C2211yescharacteristic
40#22548648C2211yescharacteristic
41#22548647C2211yescharacteristic
42#22548647C2211yescharacteristic
43#22548646C2211yescharacteristic
44Dic251⋊C8C2211yescharacteristic
45#22548636C2211yescharacteristic
46#22548636C2211yescharacteristic
47#22548636C2211yescharacteristic
48#22548636C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C2×D8×C251C2211yescharacteristic
53C2×M16×C251C2211yescharacteristic
54#121×C251C2211yescharacteristic
55#121×C251C2211yescharacteristic
56#106×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59C2×C22⋊D251—21no—
60C2×C22⋊D251—21no—
61C2×C22⋊D251D411yescharacteristic
62C22×Dic251—21no—
63C22×Dic251—21no—
64C22×Dic251—21no—
65C2×C4×D251—21no—
66C2×C4×D251C2311yescharacteristic
67#16860793—22no—
68C2×C251⋊C8C2311yescharacteristic
69C2×C251⋊C8C2311yescharacteristic
70C4⋊Dic251—21no—
71C4⋊Dic251—21no—
72C4⋊Dic251—21no—
73C4⋊Dic251C2311yescharacteristic
74C4⋊Dic251C2311yescharacteristic
75C4⋊Dic251C2311yescharacteristic
76C4⋊Dic251C2311yescharacteristic
77Dic251⋊C4—21no—
78Dic251⋊C4—21no—
79Dic251⋊C4—21no—
80Dic251⋊C4D411yescharacteristic
81C22⋊Dic251—21no—
82C22⋊Dic251—21no—
83C22⋊Dic251—21no—
84C22⋊Dic251—21no—
85C22⋊Dic251—21no—
86C22⋊Dic251D411yescharacteristic
87(C2×D251)⋊C4—21no—
88(C2×D251)⋊C4D411yescharacteristic
89C4×Dic251—21no—
90C4×Dic251—21no—
91C4×Dic251C2311yescharacteristic
92C2×D4×C251—21no—
93C2×D4×C251—21no—
94C2×D4×C251C2311yescharacteristic
95C2×D4×C251C2311yescharacteristic
96C22×C4×C251C2311yescharacteristic
97D8×C251—22no—
98M16×C251—22no—
99C2×C8×C251C2311yescharacteristic
100C2×C8×C251C2311yescharacteristic
101C4⋊C4×C251C2311yescharacteristic
102C4⋊C4×C251C2311yescharacteristic
103C22⋊C4×C251—21no—
104C22⋊C4×C251—21no—
105C22×D251C2×D411yescharacteristic
106C22⋊D251—41no—
107C22⋊D251—42no—
108C22⋊D251—81no—
109C2×Dic251—21no—
110C2×Dic251—21no—
111C2×Dic251—21no—
112C2×Dic251—21no—
113C2×Dic251—81no—
114C2×Dic251—81no—
115C2×Dic251—81no—
116C2×Dic251C2×D411yescharacteristic
117C2×Dic251C2×D411yescharacteristic
118C2×Dic251C2×D411yescharacteristic
119C4×D251—21no—
120C4×D251—21no—
121C4×D251—81no—
122C251⋊C8—21no—
123C251⋊C8—21no—
124C23×C251—21no—
125C23×C251—21no—
126C23×C251C2×D411yescharacteristic
127D4×C251—21no—
128D4×C251—21no—
129D4×C251—21no—
130D4×C251—21no—
131D4×C251—42no—
132D4×C251—81no—
133C2×C4×C251—21no—
134C2×C4×C251—21no—
135C2×C4×C251—21no—
136C2×C4×C251—21no—
137C2×C4×C251C2×D411yescharacteristic
138C2×C4×C251C2411yescharacteristic
139C8×C251—21no—
140C8×C251#4112yes—
141C22×C251—21no—
142C22×C251—21no—
143C22×C251—21no—
144C22×C251—41no—
145C22×C251—41no—
146C22×C251—41no—
147C22×C251—41no—
148C22×C251—41no—
149C22×C251—41no—
150C22×C251C22×D411yescharacteristic
151C2×D251—21no—
152C2×D251—21no—
153C2×D251—21no—
154C4×C251—21no—
155C4×C251—41no—
156C4×C251—41no—
157C4×C251C2×#4111yescharacteristic
158C4×C2512−1+411yescharacteristic
159Dic251—21no—
160Dic251—21no—
161Dic251—21no—
162Dic251—41no—
163Dic251—41no—
164Dic251—41no—
165Dic251—41no—
166C2×C251—41no—
167C2×C251—81no—
168C2×C251—81no—
169C2×C251#54711yescharacteristic
170C2×C251#57511yescharacteristic
171C2×C251#57611yescharacteristic
172D251—41no—
173C251#335511yescharacteristic, complemented, Sylow
174#3355—2511nomaximal, Sylow
175#553—2511no—
176#553—2511no—
177#499—2511no—
178#482—2511no—
179#482—2511no—
180#480—2511no—
181#480—2511no—
182#478—2511no—
183#478—2511no—
184#469—2511no—
185#469D25111yescharacteristic, complemented
186C4×D8—2511no—
187#436—2511no—
188#428—2511no—
189#428—2511no—
190C2×D8C2×D25111yescharacteristic
191C2×M16—2511no—
192C2×M16C2×D25111yescharacteristic
193#126—5021no—
194#126—5021no—
195#125—2511no—
196#125—2511no—
197#123—5021no—
198#123—5021no—
199#121—5021no—
200#121—5021no—
201#121—2511no—
202#121—2511no—
203#121C2×D25111yescharacteristic
204#121C2×D25111yescharacteristic
205C4×D4—5021no—
206C4×D4—5021no—
207C4×D4—2511no—
208C4×D4—2511no—
209C4×D4—2511no—
210#117—2511no—
211#117—2511no—
212C2×C4⋊C4—2511no—
213C2×C4⋊C4—2511no—
214#107—2511no—
215#107—2511no—
216#107—2511no—
217#107—2511no—
218#107—2511no—
219#106—2511no—
220#106—2511no—
221#106—2511no—
222#106C2×D25111yescharacteristic
223C4⋊C8—2511no—
224#102—2511no—
225#102—2511no—
226#102—2511no—
227#102—2511no—
228#102C2×D25111yescharacteristic
229#102C2×D25111yescharacteristic
230C22⋊C8—2511no—
231C22⋊C8—2511no—
232C4×C8—2511no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—5021no—
236C2×D4—5021no—
237C2×D4—2511no—
238C2×D4C22×D25111yescharacteristic
239C2×D4C22×D25111yescharacteristic
240C22×C4—5021no—
241C22×C4—5021no—
242C22×C4—5021no—
243C22×C4—5021no—
244C22×C4—2511no—
245C22×C4C22×D25111yescharacteristic
246D8—22no—
247M16—22no—
248M16—5022no—
249C2×C8—2511no—
250C2×C8—2511no—
251C2×C8C22×D25111yescharacteristic
252C2×C8C22×D25111yescharacteristic
253C4⋊C4—5021no—
254C4⋊C4—5021no—
255C4⋊C4—5021no—
256C4⋊C4—5021no—
257C4⋊C4—5021no—
258C4⋊C4—5021no—
259C4⋊C4—2511no—
260C4⋊C4—2511no—
261C4⋊C4—2511no—
262C4⋊C4—2511no—
263C4⋊C4—2511no—
264C4⋊C4C22×D25111yescharacteristic
265C4⋊C4C22×D25111yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—5021no—
269C22⋊C4—5021no—
270C22⋊C4—5021no—
271C22⋊C4—5021no—
272C22⋊C4—5021no—
273C22⋊C4—5021no—
274C22⋊C4—2511no—
275C22⋊C4—2511no—
276C42—5021no—
277C42—5021no—
278C42—2511no—
279C23—21no—
280C23—21no—
281C23—2511no—
282C23D4×D25111yescharacteristic
283D4—21no—
284D4—21no—
285D4—21no—
286D4—21no—
287D4—42no—
288D4—81no—
289D4—20081no—
290D4—10042no—
291D4—10041no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—20081no—
297C2×C4—20081no—
298C2×C4—20081no—
299C2×C4—20081no—
300C2×C4—5021no—
301C2×C4—5021no—
302C2×C4—5021no—
303C2×C4—5021no—
304C2×C4—5021no—
305C2×C4—5021no—
306C2×C4—2511no—
307C2×C4—2511no—
308C2×C4—2511no—
309C2×C4D4×D25111yescharacteristic
310C2×C4C23×D25111yescharacteristic
311C8—21no—
312C8—5021no—
313C8—5021no—
314C8#1686080912yes—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—5021no—
325C22—5021no—
326C22—5021no—
327C22C2×D4×D25111yescharacteristic
328C4—21no—
329C4—41no—
330C4—41no—
331C4—10041no—
332C4—10041no—
333C4—10041no—
334C4—10041no—
335C4—5021no—
336C4—5021no—
337C4—5021no—
338C4C2×#1686080911yescharacteristic
339C4#2254876211yescharacteristic
340C2—41no—
341C2—81no—
342C2—81no—
343C2—10041no—
344C2#3063233711yescharacteristic
345C2#3063248311yescharacteristic
346C2#3063249111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#469×C251C2
socleC22×C251C22×D4
hypercenter#469D251
hypocenterC251#3355
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #30632389C2semidirect
2. #30632384C2semidirect
3. #30632087C2semidirect
4. #30632010C2semidirect
5. #30632008C2semidirect
6. #30631999C2semidirect
7. #30631998C2semidirect
8. #30631989C2semidirect
9. #30631985C2semidirect
10. #30631937C2semidirect
11. D8×Dic251C2semidirect
12. #30631763C2semidirect
13. #30631729C2semidirect
14. #30631726C2semidirect
15. #469×C251C2semidirect
173. C251#3355semidirect
185. #469D251semidirect

Maximal subgroups

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

subgroupindex
#306323892
#306323842
#306320872
#306320102
#306320082
#306319992
#306319982
#306319892
#306319852
#306319372
D8×Dic2512
#306317632
#306317292
#306317262
#469×C2512
#3355251