Finite Group Databasebeta

← #44562209

Subgroups — #44562209

Order 32128.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632333C211yesmaximal, characteristic, complemented
2#30632044C211yesmaximal, characteristic, complemented
3#30632022C211yesmaximal, characteristic, complemented
4#30632011C211yesmaximal, characteristic, complemented
5#30631984C211yesmaximal, characteristic, complemented
6#30631921C211yesmaximal, characteristic, complemented
7#30631914C211yesmaximal, characteristic, complemented
8#30631902C211yesmaximal, characteristic, complemented
9#30631830C211yesmaximal, characteristic, complemented
10#30631820C211yesmaximal, characteristic, complemented
11C4×#16860795C211yesmaximal, characteristic, complemented
12D4×C251⋊C8C211yesmaximal, characteristic, complemented
13#30631681C211yesmaximal, characteristic, complemented
14C2×#22548635C211yesmaximal, characteristic, complemented
15#553×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18C2×#16860798C2211yescharacteristic, complemented
19C2×#16860795C2211yescharacteristic
20C2×#16860795C2211yescharacteristic
21C22×C251⋊C8C2211yescharacteristic
22C22×C251⋊C8C2211yescharacteristic, complemented
23#22548703C2211yescharacteristic
24#22548700—21no—
25#22548694C2211yescharacteristic
26#22548689C2211yescharacteristic
27#22548688—21no—
28#22548683—21no—
29C4×D1004C2211yescharacteristic
30C22⋊C4×D251—21no—
31#22548648C2211yescharacteristic, complemented
32#22548648C2211yescharacteristic, complemented
33#22548646C2211yescharacteristic, complemented
34C4⋊(C251⋊C8)C2211yescharacteristic, complemented
35#22548638C2211yescharacteristic, complemented
36#22548638C2211yescharacteristic, complemented
37#22548638C2211yescharacteristic, complemented
38#22548636C2211yescharacteristic, complemented
39#22548635C2211yescharacteristic
40#22548635C2211yescharacteristic
41#22548635C2211yescharacteristic
42#22548635C2211yescharacteristic
43#22548635C2211yescharacteristic
44#22548635C2211yescharacteristic
45C22⋊(C251⋊C8)C2211yescharacteristic
46C22⋊(C251⋊C8)C2211yescharacteristic, complemented
47C4×C251⋊C8C2211yescharacteristic, complemented
48#126×C251—21no—
49#125×C251C2211yescharacteristic, complemented
50#123×C251—21no—
51#121×C251—21no—
52#121×C251C2211yescharacteristic
53#121×C251C2211yescharacteristic, complemented
54C4×D4×C251—21no—
55C4×D4×C251C2211yescharacteristic
56C4×D4×C251C2211yescharacteristic, complemented
57#117×C251C2211yescharacteristic, complemented
58C2×C4⋊C4×C251C2211yescharacteristic
59C23×D251—21no—
60#16860810—22no—
61#16860810—41no—
62C2×Dic502C2311yescharacteristic
63C2×D1004—22no—
64C2×D1004—21no—
65C2×D1004—41no—
66C2×D1004C2311yescharacteristic
67C2×D1004C2311yescharacteristic
68C2×C22⋊D251—21no—
69C2×C22⋊D251—21no—
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72#16860798—41no—
73#16860795—22no—
74#16860795—41no—
75C2×C251⋊C8—41no—
76C2×C251⋊C8—41no—
77C2×C251⋊C8C2311yescharacteristic
78C2×C251⋊C8C2311yescharacteristic
79C2×C251⋊C8C2311yescharacteristic
80C2×C251⋊C8C2311yescharacteristic
81C4⋊Dic251C2311yescharacteristic
82Dic251⋊C4—21no—
83C22⋊Dic251—21no—
84(C2×D251)⋊C4—21no—
85(C2×D251)⋊C4—21no—
86(C2×D251)⋊C4—21no—
87(C2×D251)⋊C4—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—
112C22×D251—21no—
113C22×D251—42no—
114C22×D251—42no—
115C22×D251—81no—
116Dic502—21no—
117Dic502—21no—
118D1004—21no—
119D1004—21no—
120D1004—21no—
121D1004—42no—
122D1004#4112yes—
123C22⋊D251—42no—
124C22⋊D251—81no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C4×D251—42no—
128C4×D251—81no—
129C251⋊C8—21no—
130C251⋊C8—21no—
131C251⋊C8—21no—
132C251⋊C8—21no—
133C23×C251—21no—
134C23×C251C2×D411yescharacteristic
135C23×C251C2×D411yescharacteristic
136D4×C251—22no—
137D4×C251—21no—
138D4×C251—21no—
139D4×C251—42no—
140D4×C251—81no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—81no—
149C2×C4×C251—81no—
150C2×C4×C251C2×D411yescharacteristic
151C2×C4×C251C2×D411yescharacteristic
152C2×C4×C251C2×D411yescharacteristic
153C2×C4×C251C2×D411yescharacteristic
154C2×C4×C251C2411yescharacteristic
155C22×C251—21no—
156C22×C251—21no—
157C22×C251—21no—
158C22×C251—21no—
159C22×C251—21no—
160C22×C251—41no—
161C22×C251—41no—
162C22×C251—41no—
163C22×C251#13512yes—
164C22×C251C22×D411yescharacteristic
165C2×D251—22no—
166C2×D251—41no—
167C2×D251—41no—
168C2×D251—41no—
169C2×D251—41no—
170C2×D251—41no—
171C2×D251—84no—
172C4×C251—21no—
173C4×C251—21no—
174C4×C251—21no—
175C4×C251—21no—
176C4×C251—41no—
177C4×C251—41no—
178C4×C251—41no—
179C4×C251C2×#4111yescharacteristic
180C4×C251H(5,𝔽2)11yescharacteristic
181Dic251—41no—
182Dic251—41no—
183C2×C251—22no—
184C2×C251—41no—
185C2×C251—81no—
186C2×C251#54611yescharacteristic
187C2×C251C2×#13511yescharacteristic
188C2×C251#57611yescharacteristic
189D251—42no—
190D251—81no—
191C251#329711yescharacteristic, complemented, Sylow
192#3297—2511nomaximal, Sylow
193#553D25111yescharacteristic, complemented
194#546—2511no—
195#488—2511no—
196#484—2511no—
197#482—2511no—
198#478—2511no—
199#466—2511no—
200#465—2511no—
201#463—2511no—
202#449—2511no—
203#447—2511no—
204C4×D8—2511no—
205C8×D4—2511no—
206#416—2511no—
207C2×#102—2511no—
208C2×#41—2511no—
209C22×D4—2511no—
210C2×SD16—2511no—
211C2×D8—2511no—
212C2×D8—2511no—
213C22×C8—2511no—
214C22×C8—2511no—
215#126—21no—
216#125C2×D25111yescharacteristic, complemented
217#124—2511no—
218#123—21no—
219#123—5021no—
220C22⋊Q8—2511no—
221#121—21no—
222#121—5021no—
223#121—2511no—
224#121C2×D25111yescharacteristic
225#121C2×D25111yescharacteristic, complemented
226#120—5021no—
227C4×D4—21no—
228C4×D4—2511no—
229C4×D4C2×D25111yescharacteristic
230C4×D4C2×D25111yescharacteristic, complemented
231#117C2×D25111yescharacteristic, complemented
232C2×C4⋊C4C2×D25111yescharacteristic
233C2×C22⋊C4—5021no—
234#107—2511no—
235#107—2511no—
236#106—2511no—
237C4⋊C8—2511no—
238#103—2511no—
239#103—2511no—
240#103—2511no—
241#102—2511no—
242#102—2511no—
243#102—2511no—
244#102—2511no—
245#102—2511no—
246#102—2511no—
247#102—2511no—
248C22⋊C8—2511no—
249C22⋊C8—2511no—
250C4×C8—2511no—
251C24—5021no—
252#41—10041no—
253#41—5022no—
254C2×Q8—2511no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—10041no—
258C2×D4—5022no—
259C2×D4—5021no—
260C2×D4—5021no—
261C2×D4—5021no—
262C2×D4—2511no—
263C2×D4—2511no—
264C2×D4C22⋊D25111yescharacteristic
265C2×D4C22×D25111yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—5021no—
269C22×C4—5021no—
270C22×C4C22×D25111yescharacteristic
271C22×C4C22×D25111yescharacteristic
272SD16—10041no—
273D8—10041no—
274D8—5022no—
275C2×C8—10041no—
276C2×C8—10041no—
277C2×C8—2511no—
278C2×C8—2511no—
279C2×C8—2511no—
280C2×C8—2511no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—5021no—
285C4⋊C4—2511no—
286C4⋊C4C22⋊D25111yescharacteristic
287C4⋊C4C22×D25111yescharacteristic
288C4⋊C4C22×D25111yescharacteristic
289C4⋊C4C22×D25111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—5021no—
295C22⋊C4—5021no—
296C22⋊C4—5021no—
297C22⋊C4—5021no—
298C22⋊C4—5021no—
299C22⋊C4C22⋊D25111yescharacteristic
300C22⋊C4C22⋊D25111yescharacteristic
301C42—21no—
302C42C22×D25111yescharacteristic
303C23—21no—
304C23—20081no—
305C23—10042no—
306C23—10042no—
307C23—5021no—
308C23—5021no—
309C23C2×C22⋊D25111yescharacteristic
310C23C2×C22⋊D25111yescharacteristic
311Q8—5021no—
312Q8—5021no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—20081no—
319D4—10042no—
320D4—10042no—
321D4—5021no—
322D4—5021no—
323D4—5021no—
324D4—2512no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—20081no—
335C2×C4—10042no—
336C2×C4—5021no—
337C2×C4—5021no—
338C2×C4C2×C22⋊D25111yescharacteristic
339C2×C4C2×C22⋊D25111yescharacteristic
340C2×C4C2×C22⋊D25111yescharacteristic
341C2×C4C2×C22⋊D25111yescharacteristic
342C2×C4C23×D25111yescharacteristic
343C8—5021no—
344C8—5021no—
345C8—5021no—
346C8—5021no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—20084no—
356C22—10041no—
357C22—10041no—
358C22—10041no—
359C22—10041no—
360C22—10041no—
361C22—5022no—
362C22#2254873512yes—
363C22C22×C22⋊D25111yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—10041no—
372C4—10041no—
373C4#41×D25111yescharacteristic
374C4#2254876111yescharacteristic
375C2—22no—
376C2—41no—
377C2—81no—
378C2—20081no—
379C2—10042no—
380C2#3063232511yescharacteristic
381C2C2×#2254873511yescharacteristic
382C2#3063249011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#553×C251C2
socleC22×C251C22×D4
hypercenter#553D251
hypocenterC251#3297
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. #30632333C2semidirect
2. #30632044C2semidirect
3. #30632022C2semidirect
4. #30632011C2semidirect
5. #30631984C2semidirect
6. #30631921C2semidirect
7. #30631914C2semidirect
8. #30631902C2semidirect
9. #30631830C2semidirect
10. #30631820C2semidirect
11. C4×#16860795C2semidirect
12. D4×C251⋊C8C2semidirect
13. #30631681C2semidirect
14. C2×#22548635C2semidirect
15. #553×C251C2semidirect
18. C2×#16860798C22semidirect
22. C22×C251⋊C8C22semidirect
31. #22548648C22semidirect
32. #22548648C22semidirect
33. #22548646C22semidirect
34. C4⋊(C251⋊C8)C22semidirect
35. #22548638C22semidirect
36. #22548638C22semidirect
37. #22548638C22semidirect
38. #22548636C22semidirect
46. C22⋊(C251⋊C8)C22semidirect
47. C4×C251⋊C8C22semidirect
49. #125×C251C22semidirect
53. #121×C251C22semidirect
56. C4×D4×C251C22semidirect
57. #117×C251C22semidirect
191. C251#3297semidirect
193. #553D251semidirect
216. #125C2×D251semidirect
225. #121C2×D251semidirect
230. C4×D4C2×D251semidirect
231. #117C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323332
#306320442
#306320222
#306320112
#306319842
#306319212
#306319142
#306319022
#306318302
#306318202
C4×#168607952
D4×C251⋊C82
#306316812
C2×#225486352
#553×C2512
#3297251