Finite Group Databasebeta

← #44562965

Subgroups — #44562965

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#30632383C211yesmaximal, characteristic, complemented
2#30632086C211yesmaximal, characteristic, complemented
3#30632011C211yesmaximal, characteristic, complemented
4#30632008C211yesmaximal, characteristic, complemented
5#30632000C211yesmaximal, characteristic, complemented
6#30631998C211yesmaximal, characteristic, complemented
7#30631987C211yesmaximal, characteristic, complemented
8#30631984C211yesmaximal, characteristic, complemented
9#30631938C211yesmaximal, characteristic, complemented
10#30631938C211yesmaximal, characteristic, complemented
11C4×#16860795C211yesmaximal, characteristic, complemented
12#30631765C211yesmaximal, characteristic, complemented
13#30631730C211yesmaximal, characteristic, complemented
14#30631727C211yesmaximal, characteristic, complemented
15#553×C251C211yesmaximal, characteristic, complemented
16C2×#16860795C2211yescharacteristic
17C2×#16860794C2211yescharacteristic
18C2×#16860794C2211yescharacteristic
19#22548710—21no—
20#22548707C2211yescharacteristic
21#22548697—21no—
22#22548689C2211yescharacteristic
23#22548689C2211yescharacteristic
24#22548687—21no—
25C4×D1004C2211yescharacteristic
26(C22⋊D251)⋊C4—21no—
27#22548674C2211yescharacteristic
28C2×C4⋊Dic251C2211yescharacteristic
29#22548648C2211yescharacteristic
30#22548648C2211yescharacteristic
31#22548648C2211yescharacteristic
32#22548648C2211yescharacteristic
33#22548648C2211yescharacteristic
34#22548646C2211yescharacteristic
35#22548646C2211yescharacteristic
36#22548646C2211yescharacteristic
37#22548646C2211yescharacteristic
38C4⋊(C251⋊C8)C2211yescharacteristic
39#22548636C2211yescharacteristic
40#22548636C2211yescharacteristic
41#22548636C2211yescharacteristic
42#22548635C2211yescharacteristic
43#22548635C2211yescharacteristic
44#22548635C2211yescharacteristic
45C22⋊(C251⋊C8)C2211yescharacteristic
46C22⋊(C251⋊C8)C2211yescharacteristic
47C4×C251⋊C8C2211yescharacteristic
48#126×C251—21no—
49#125×C251C2211yescharacteristic
50#123×C251—21no—
51#121×C251—21no—
52#121×C251C2211yescharacteristic
53#121×C251C2211yescharacteristic
54C4×D4×C251—21no—
55C4×D4×C251C2211yescharacteristic
56C4×D4×C251C2211yescharacteristic
57#117×C251C2211yescharacteristic
58C2×C4⋊C4×C251C2211yescharacteristic
59C2×D1004C2311yescharacteristic
60C2×C22⋊D251—21no—
61C2×C22⋊D251—21no—
62C22×Dic251—21no—
63C2×C4×D251—21no—
64#16860795—22no—
65#16860794—22no—
66#16860794—22no—
67C2×C251⋊C8C2311yescharacteristic
68C2×C251⋊C8C2311yescharacteristic
69C2×C251⋊C8C2311yescharacteristic
70C2×C251⋊C8C2311yescharacteristic
71C4⋊Dic251—21no—
72C4⋊Dic251C2311yescharacteristic
73C4⋊Dic251C2311yescharacteristic
74C4⋊Dic251C2311yescharacteristic
75Dic251⋊C4—21no—
76Dic251⋊C4—21no—
77C22⋊Dic251—21no—
78(C2×D251)⋊C4—21no—
79(C2×D251)⋊C4—21no—
80(C2×D251)⋊C4—21no—
81C4×Dic251—21no—
82C2×D4×C251—21no—
83C2×D4×C251—21no—
84C2×D4×C251D411yescharacteristic
85C2×D4×C251C2311yescharacteristic
86C22×C4×C251—21no—
87C22×C4×C251—21no—
88C22×C4×C251C2311yescharacteristic
89C22×C4×C251C2311yescharacteristic
90C4⋊C4×C251—21no—
91C4⋊C4×C251—21no—
92C4⋊C4×C251—21no—
93C4⋊C4×C251D411yescharacteristic
94C4⋊C4×C251C2311yescharacteristic
95C4⋊C4×C251C2311yescharacteristic
96C4⋊C4×C251C2311yescharacteristic
97C22⋊C4×C251—21no—
98C22⋊C4×C251—21no—
99C22⋊C4×C251—21no—
100C22⋊C4×C251—21no—
101C22⋊C4×C251D411yescharacteristic
102C22⋊C4×C251D411yescharacteristic
103C42×C251—21no—
104C42×C251C2311yescharacteristic
105C22×D251—21no—
106D1004—21no—
107D1004—21no—
108C22⋊D251—42no—
109C22⋊D251—81no—
110C2×Dic251—21no—
111C2×Dic251—21no—
112C2×Dic251—21no—
113C2×Dic251—81no—
114C4×D251—81no—
115C251⋊C8—21no—
116C251⋊C8—21no—
117C251⋊C8—21no—
118C251⋊C8#4112yes—
119C23×C251—21no—
120C23×C251C2×D411yescharacteristic
121C23×C251C2×D411yescharacteristic
122D4×C251—21no—
123D4×C251—21no—
124D4×C251—41no—
125D4×C251—42no—
126D4×C251—81no—
127C2×C4×C251—21no—
128C2×C4×C251—21no—
129C2×C4×C251—21no—
130C2×C4×C251—21no—
131C2×C4×C251—21no—
132C2×C4×C251—21no—
133C2×C4×C251—21no—
134C2×C4×C251—81no—
135C2×C4×C251—81no—
136C2×C4×C251C2×D411yescharacteristic
137C2×C4×C251C2×D411yescharacteristic
138C2×C4×C251C2×D411yescharacteristic
139C2×C4×C251C2×D411yescharacteristic
140C2×C4×C251C2411yescharacteristic
141C22×C251—21no—
142C22×C251—21no—
143C22×C251—21no—
144C22×C251—21no—
145C22×C251—21no—
146C22×C251—21no—
147C22×C251—41no—
148C22×C251—41no—
149C22×C251—41no—
150C22×C251C22×D411yescharacteristic
151C2×D251—41no—
152C2×D251—41no—
153C2×D251—41no—
154C4×C251—21no—
155C4×C251—21no—
156C4×C251—21no—
157C4×C251—21no—
158C4×C251—41no—
159C4×C251—41no—
160C4×C251—41no—
161C4×C251C2×#4111yescharacteristic
162C4×C2512−1+411yescharacteristic
163Dic251—41no—
164Dic251—41no—
165Dic251—41no—
166C2×C251—41no—
167C2×C251—41no—
168C2×C251—81no—
169C2×C251#54711yescharacteristic
170C2×C251#57511yescharacteristic
171C2×C251#57611yescharacteristic
172D251—81no—
173C251#335511yescharacteristic, complemented, Sylow
174#3355—2511nomaximal, Sylow
175#553—2511no—
176#553D25111yescharacteristic, complemented
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#469—2511no—
186C4×D8—2511no—
187#436—2511no—
188#428—2511no—
189#428—2511no—
190C2×D8—2511no—
191C2×M16—2511no—
192C2×M16—2511no—
193#126—21no—
194#126—5021no—
195#125—2511no—
196#125C2×D25111yescharacteristic
197#123—21no—
198#123—5021no—
199#121—21no—
200#121—5021no—
201#121—2511no—
202#121—2511no—
203#121C2×D25111yescharacteristic
204#121C2×D25111yescharacteristic
205C4×D4—21no—
206C4×D4—5021no—
207C4×D4—2511no—
208C4×D4C2×D25111yescharacteristic
209C4×D4C2×D25111yescharacteristic
210#117—2511no—
211#117C2×D25111yescharacteristic
212C2×C4⋊C4—2511no—
213C2×C4⋊C4C2×D25111yescharacteristic
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#106—2511no—
223C4⋊C8—2511no—
224#102—2511no—
225#102—2511no—
226#102—2511no—
227#102—2511no—
228#102—2511no—
229#102—2511no—
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—21no—
241C22×C4—21no—
242C22×C4—5021no—
243C22×C4—5021no—
244C22×C4C22×D25111yescharacteristic
245C22×C4C22×D25111yescharacteristic
246D8—5022no—
247M16—5022no—
248M16—5022no—
249C2×C8—2511no—
250C2×C8—2511no—
251C2×C8—2511no—
252C2×C8—2511no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—5021no—
257C4⋊C4—5021no—
258C4⋊C4—5021no—
259C4⋊C4—2511no—
260C4⋊C4—2511no—
261C4⋊C4—2511no—
262C4⋊C4C22⋊D25111yescharacteristic
263C4⋊C4C22×D25111yescharacteristic
264C4⋊C4C22×D25111yescharacteristic
265C4⋊C4C22×D25111yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—5021no—
271C22⋊C4—5021no—
272C22⋊C4—5021no—
273C22⋊C4—5021no—
274C22⋊C4C22⋊D25111yescharacteristic
275C22⋊C4C22⋊D25111yescharacteristic
276C42—21no—
277C42—5021no—
278C42C22×D25111yescharacteristic
279C23—21no—
280C23—5021no—
281C23C2×C22⋊D25111yescharacteristic
282C23C2×C22⋊D25111yescharacteristic
283D4—21no—
284D4—21no—
285D4—41no—
286D4—42no—
287D4—81no—
288D4—20081no—
289D4—10042no—
290D4—5021no—
291D4—5021no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—20081no—
302C2×C4—20081no—
303C2×C4—5021no—
304C2×C4—5021no—
305C2×C4—5021no—
306C2×C4C2×C22⋊D25111yescharacteristic
307C2×C4C2×C22⋊D25111yescharacteristic
308C2×C4C2×C22⋊D25111yescharacteristic
309C2×C4C2×C22⋊D25111yescharacteristic
310C2×C4C23×D25111yescharacteristic
311C8—5021no—
312C8—5021no—
313C8—5021no—
314C8—2512no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—10041no—
325C22—10041no—
326C22—10041no—
327C22C22×C22⋊D25111yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—10041no—
336C4—10041no—
337C4—10041no—
338C4#41×D25111yescharacteristic
339C4#2254876211yescharacteristic
340C2—41no—
341C2—41no—
342C2—81no—
343C2—20081no—
344C2#3063233611yescharacteristic
345C2#3063248511yescharacteristic
346C2#3063249011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#553×C251C2
socleC22×C251C22×D4
hypercenter#553D251
hypocenterC251#3355
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #30632383C2semidirect
2. #30632086C2semidirect
3. #30632011C2semidirect
4. #30632008C2semidirect
5. #30632000C2semidirect
6. #30631998C2semidirect
7. #30631987C2semidirect
8. #30631984C2semidirect
9. #30631938C2semidirect
10. #30631938C2semidirect
11. C4×#16860795C2semidirect
12. #30631765C2semidirect
13. #30631730C2semidirect
14. #30631727C2semidirect
15. #553×C251C2semidirect
173. C251#3355semidirect
176. #553D251semidirect

Maximal subgroups

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

subgroupindex
#306323832
#306320862
#306320112
#306320082
#306320002
#306319982
#306319872
#306319842
#306319382
C4×#168607952
#306317652
#306317302
#306317272
#553×C2512
#3355251