Finite Group Databasebeta

← #44562261

Subgroups — #44562261

Order 32128.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632346C211yesmaximal, characteristic, complemented
2#30632044C211yesmaximal, characteristic, complemented
3#30632028C211yesmaximal, characteristic, complemented
4#30632005C211yesmaximal, characteristic, complemented
5#30631991C211yesmaximal, characteristic, complemented
6#30631921C211yesmaximal, characteristic, complemented
7#30631914C211yesmaximal, characteristic, complemented
8#30631902C211yesmaximal, characteristic, complemented
9#30631830C211yesmaximal, characteristic, complemented
10#30631825C211yesmaximal, characteristic, complemented
11C4×#16860798C211yesmaximal, characteristic, complemented
12D4×C251⋊C8C211yesmaximal, characteristic, complemented
13#30631681C211yesmaximal, characteristic, complemented
14C2×#22548638C211yesmaximal, characteristic, complemented
15#553×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C22×Dic502C2211yescharacteristic
18C2×#16860798C2211yescharacteristic
19C2×#16860798C2211yescharacteristic
20C2×#16860795C2211yescharacteristic
21C22×C251⋊C8C2211yescharacteristic
22C22×C251⋊C8C2211yescharacteristic
23#22548703C2211yescharacteristic
24#22548701—21no—
25#22548694C2211yescharacteristic
26#22548693—21no—
27#22548689C2211yescharacteristic
28#22548687—21no—
29C4×Dic502C2211yescharacteristic
30(C22⋊D251)⋊C4—21no—
31#22548648C2211yescharacteristic
32#22548646C2211yescharacteristic
33#22548646C2211yescharacteristic
34C4⋊(C251⋊C8)C2211yescharacteristic
35#22548638C2211yescharacteristic
36#22548638C2211yescharacteristic
37#22548638C2211yescharacteristic
38#22548638C2211yescharacteristic
39#22548638C2211yescharacteristic
40#22548638C2211yescharacteristic
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
59#16860810—22no—
60#16860810—41no—
61C2×Dic502—22no—
62C2×Dic502—21no—
63C2×Dic502—41no—
64C2×Dic502C2311yescharacteristic
65C2×Dic502C2311yescharacteristic
66C2×D1004C2311yescharacteristic
67C2×C22⋊D251—21no—
68C2×C22⋊D251—21no—
69C22×Dic251—21no—
70C2×C4×D251—21no—
71#16860798—22no—
72#16860798—41no—
73#16860795—41no—
74C2×C251⋊C8—41no—
75C2×C251⋊C8—41no—
76C2×C251⋊C8C2311yescharacteristic
77C2×C251⋊C8C2311yescharacteristic
78C2×C251⋊C8C2311yescharacteristic
79C2×C251⋊C8C2311yescharacteristic
80C4⋊Dic251C2311yescharacteristic
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83C22⋊Dic251—21no—
84(C2×D251)⋊C4—21no—
85(C2×D251)⋊C4—21no—
86(C2×D251)⋊C4—21no—
87C4×Dic251—21no—
88C2×D4×C251—21no—
89C2×D4×C251—21no—
90C2×D4×C251D411yescharacteristic
91C2×D4×C251C2311yescharacteristic
92C22×C4×C251—21no—
93C22×C4×C251—21no—
94C22×C4×C251C2311yescharacteristic
95C22×C4×C251C2311yescharacteristic
96C4⋊C4×C251—21no—
97C4⋊C4×C251—21no—
98C4⋊C4×C251—21no—
99C4⋊C4×C251D411yescharacteristic
100C4⋊C4×C251C2311yescharacteristic
101C4⋊C4×C251C2311yescharacteristic
102C4⋊C4×C251C2311yescharacteristic
103C22⋊C4×C251—21no—
104C22⋊C4×C251—21no—
105C22⋊C4×C251—21no—
106C22⋊C4×C251—21no—
107C22⋊C4×C251D411yescharacteristic
108C22⋊C4×C251D411yescharacteristic
109C42×C251—21no—
110C42×C251C2311yescharacteristic
111C22×D251—21no—
112Dic502—21no—
113Dic502—21no—
114Dic502—21no—
115Dic502—42no—
116Dic502#4112yes—
117D1004—21no—
118D1004—21no—
119C22⋊D251—42no—
120C22⋊D251—42no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—42no—
125C4×D251—42no—
126C251⋊C8—21no—
127C251⋊C8—21no—
128C251⋊C8—21no—
129C251⋊C8—21no—
130C23×C251—21no—
131C23×C251C2×D411yescharacteristic
132C23×C251C2×D411yescharacteristic
133D4×C251—22no—
134D4×C251—21no—
135D4×C251—21no—
136D4×C251—42no—
137D4×C251—81no—
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×C251—21no—
145C2×C4×C251—81no—
146C2×C4×C251—81no—
147C2×C4×C251C2×D411yescharacteristic
148C2×C4×C251C2×D411yescharacteristic
149C2×C4×C251C2×D411yescharacteristic
150C2×C4×C251C2×D411yescharacteristic
151C2×C4×C251C2411yescharacteristic
152C22×C251—21no—
153C22×C251—21no—
154C22×C251—21no—
155C22×C251—21no—
156C22×C251—21no—
157C22×C251—41no—
158C22×C251—41no—
159C22×C251—41no—
160C22×C251#13512yes—
161C22×C251C22×D411yescharacteristic
162C2×D251—41no—
163C2×D251—41no—
164C2×D251—41no—
165C4×C251—21no—
166C4×C251—21no—
167C4×C251—21no—
168C4×C251—21no—
169C4×C251—41no—
170C4×C251—41no—
171C4×C251—41no—
172C4×C251C2×#4111yescharacteristic
173C4×C251H(5,𝔽2)11yescharacteristic
174Dic251—22no—
175Dic251—41no—
176Dic251—41no—
177C2×C251—22no—
178C2×C251—41no—
179C2×C251—81no—
180C2×C251#54611yescharacteristic
181C2×C251C2×#13511yescharacteristic
182C2×C251#57711yescharacteristic
183D251—81no—
184C251#330111yescharacteristic, complemented, Sylow
185#3301—2511nomaximal, Sylow
186#553D25111yescharacteristic, complemented
187#548—2511no—
188#488—2511no—
189#485—2511no—
190#481—2511no—
191#479—2511no—
192#466—2511no—
193#465—2511no—
194#463—2511no—
195#449—2511no—
196#448—2511no—
197C4×SD16—2511no—
198C8×D4—2511no—
199#416—2511no—
200C2×#103—2511no—
201C2×#41—2511no—
202C22×Q8—2511no—
203C2×SD16—2511no—
204C2×SD16—2511no—
205C2×D8—2511no—
206C22×C8—2511no—
207C22×C8—2511no—
208#126—21no—
209#125C2×D25111yescharacteristic
210#124—5021no—
211#124—2511no—
212#123—21no—
213C22⋊Q8—5021no—
214C22⋊Q8—2511no—
215#121—21no—
216#121—5021no—
217#121—2511no—
218#121C2×D25111yescharacteristic
219#121C2×D25111yescharacteristic
220C4×Q8—2511no—
221C4×D4—21no—
222C4×D4—5021no—
223C4×D4C2×D25111yescharacteristic
224C4×D4C2×D25111yescharacteristic
225#117C2×D25111yescharacteristic
226C2×C4⋊C4C2×D25111yescharacteristic
227#107—2511no—
228#106—2511no—
229#106—2511no—
230C4⋊C8—2511no—
231#103—2511no—
232#103—2511no—
233#103—2511no—
234#103—2511no—
235#103—2511no—
236#103—2511no—
237#102—2511no—
238#102—2511no—
239#102—2511no—
240#102—2511no—
241C22⋊C8—2511no—
242C22⋊C8—2511no—
243C4×C8—2511no—
244#41—10041no—
245#41—5022no—
246C2×Q8—10041no—
247C2×Q8—5022no—
248C2×Q8—5021no—
249C2×Q8—2511no—
250C2×Q8—2511no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—5021no—
254C2×D4—5021no—
255C2×D4—2511no—
256C2×D4C22⋊D25111yescharacteristic
257C2×D4C22×D25111yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—5021no—
261C22×C4—5021no—
262C22×C4C22×D25111yescharacteristic
263C22×C4C22×D25111yescharacteristic
264SD16—10041no—
265SD16—5022no—
266D8—10041no—
267C2×C8—10041no—
268C2×C8—10041no—
269C2×C8—2511no—
270C2×C8—2511no—
271C2×C8—2511no—
272C2×C8—2511no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—5021no—
277C4⋊C4—5021no—
278C4⋊C4—2511no—
279C4⋊C4C22⋊D25111yescharacteristic
280C4⋊C4C22×D25111yescharacteristic
281C4⋊C4C22×D25111yescharacteristic
282C4⋊C4C22×D25111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—5021no—
288C22⋊C4—5021no—
289C22⋊C4—5021no—
290C22⋊C4—5021no—
291C22⋊C4C22⋊D25111yescharacteristic
292C22⋊C4C22⋊D25111yescharacteristic
293C42—21no—
294C42—5021no—
295C42C22×D25111yescharacteristic
296C23—21no—
297C23—5021no—
298C23C2×C22⋊D25111yescharacteristic
299C23C2×C22⋊D25111yescharacteristic
300Q8—10042no—
301Q8—5021no—
302Q8—5021no—
303Q8—5021no—
304Q8—2512no—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310D4—10042no—
311D4—10042no—
312D4—5021no—
313D4—5021no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—10042no—
324C2×C4—10042no—
325C2×C4—5021no—
326C2×C4—5021no—
327C2×C4—5021no—
328C2×C4C2×C22⋊D25111yescharacteristic
329C2×C4C2×C22⋊D25111yescharacteristic
330C2×C4C2×C22⋊D25111yescharacteristic
331C2×C4C2×C22⋊D25111yescharacteristic
332C2×C4C23×D25111yescharacteristic
333C8—5021no—
334C8—5021no—
335C8—5021no—
336C8—5021no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—10041no—
346C22—10041no—
347C22—10041no—
348C22#2254873512yes—
349C22C22×C22⋊D25111yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—10041no—
358C4—10041no—
359C4—5022no—
360C4#41×D25111yescharacteristic
361C4#2254876111yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—20081no—
366C2#3063232511yescharacteristic
367C2C2×#2254873511yescharacteristic
368C2#3063249811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#553×C251C2
socleC22×C251C22×D4
hypercenter#553D251
hypocenterC251#3301
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. #30632346C2semidirect
2. #30632044C2semidirect
3. #30632028C2semidirect
4. #30632005C2semidirect
5. #30631991C2semidirect
6. #30631921C2semidirect
7. #30631914C2semidirect
8. #30631902C2semidirect
9. #30631830C2semidirect
10. #30631825C2semidirect
11. C4×#16860798C2semidirect
12. D4×C251⋊C8C2semidirect
13. #30631681C2semidirect
14. C2×#22548638C2semidirect
15. #553×C251C2semidirect
184. C251#3301semidirect
186. #553D251semidirect

Maximal subgroups

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

subgroupindex
#306323462
#306320442
#306320282
#306320052
#306319912
#306319212
#306319142
#306319022
#306318302
#306318252
C4×#168607982
D4×C251⋊C82
#306316812
C2×#225486382
#553×C2512
#3301251