Finite Group Databasebeta

← #44562215

Subgroups — #44562215

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#30632383C211yesmaximal, characteristic, complemented
2#30632041C211yesmaximal, characteristic, complemented
3#30632020C211yesmaximal, characteristic, complemented
4#30632008C211yesmaximal, characteristic, complemented
5#30631987C211yesmaximal, characteristic, complemented
6#30631921C211yesmaximal, characteristic, complemented
7#30631918C211yesmaximal, characteristic, complemented
8#30631899C211yesmaximal, characteristic, complemented
9#30631832C211yesmaximal, characteristic, complemented
10#30631821C211yesmaximal, characteristic, complemented
11C4×#16860795C211yesmaximal, characteristic, complemented
12D4×C251⋊C8C211yesmaximal, characteristic, complemented
13#30631683C211yesmaximal, characteristic, complemented
14C2×#22548636C211yesmaximal, characteristic, complemented
15#546×C251C211yesmaximal, characteristic, complemented
16C2×#16860797C2211yescharacteristic, complemented
17C2×#16860795C2211yescharacteristic
18C2×#16860795C2211yescharacteristic
19C22×C251⋊C8C2211yescharacteristic
20C22×C251⋊C8C2211yescharacteristic, complemented
21#22548710—21no—
22#22548707C2211yescharacteristic, complemented
23#22548697—21no—
24#22548689C2211yescharacteristic
25#22548689C2211yescharacteristic, complemented
26#22548687—21no—
27C4×D1004C2211yescharacteristic, complemented
28(C22⋊D251)⋊C4—21no—
29#22548674C2211yescharacteristic, complemented
30C2×C4⋊Dic251C2211yescharacteristic
31#22548648C2211yescharacteristic, complemented
32#22548648C2211yescharacteristic, complemented
33#22548646C2211yescharacteristic, complemented
34C4⋊(C251⋊C8)C2211yescharacteristic, complemented
35#22548639C2211yescharacteristic, complemented
36#22548639C2211yescharacteristic, complemented
37#22548639C2211yescharacteristic, complemented
38#22548636C2211yescharacteristic
39#22548636C2211yescharacteristic
40#22548636C2211yescharacteristic
41#22548636C2211yescharacteristic
42#22548636C2211yescharacteristic
43#22548636C2211yescharacteristic
44#22548635C2211yescharacteristic, complemented
45C22⋊(C251⋊C8)C2211yescharacteristic
46C22⋊(C251⋊C8)C2211yescharacteristic, complemented
47C4×C251⋊C8C2211yescharacteristic, complemented
48C2×#41×C251C2211yescharacteristic
49C22×D4×C251C2211yescharacteristic
50#124×C251C2211yescharacteristic
51#123×C251—21no—
52C22⋊Q8×C251C2211yescharacteristic
53#121×C251—21no—
54#121×C251C2211yescharacteristic
55#120×C251—21no—
56C4×D4×C251C2211yescharacteristic
57C4×D4×C251C2211yescharacteristic
58C2×C22⋊C4×C251—21no—
59C2×D1004C2311yescharacteristic
60C2×C22⋊D251—21no—
61C2×C22⋊D251—21no—
62C22×Dic251—21no—
63C2×C4×D251—21no—
64#16860797—41no—
65#16860795—22no—
66#16860795—41no—
67C2×C251⋊C8—41no—
68C2×C251⋊C8—41no—
69C2×C251⋊C8C2311yescharacteristic
70C2×C251⋊C8C2311yescharacteristic
71C2×C251⋊C8C2311yescharacteristic
72C2×C251⋊C8C2311yescharacteristic
73C4⋊Dic251—21no—
74C4⋊Dic251C2311yescharacteristic
75C4⋊Dic251C2311yescharacteristic
76C4⋊Dic251C2311yescharacteristic
77Dic251⋊C4—21no—
78Dic251⋊C4—21no—
79C22⋊Dic251—21no—
80(C2×D251)⋊C4—21no—
81(C2×D251)⋊C4—21no—
82(C2×D251)⋊C4—21no—
83C4×Dic251—21no—
84C24×C251—21no—
85#41×C251—22no—
86#41×C251—41no—
87C2×Q8×C251C2311yescharacteristic
88C2×D4×C251—22no—
89C2×D4×C251—21no—
90C2×D4×C251—21no—
91C2×D4×C251—21no—
92C2×D4×C251—41no—
93C2×D4×C251D411yescharacteristic
94C2×D4×C251C2311yescharacteristic
95C2×D4×C251C2311yescharacteristic
96C22×C4×C251—21no—
97C22×C4×C251—21no—
98C22×C4×C251C2311yescharacteristic
99C22×C4×C251C2311yescharacteristic
100C4⋊C4×C251—21no—
101C4⋊C4×C251D411yescharacteristic
102C4⋊C4×C251C2311yescharacteristic
103C22⋊C4×C251—21no—
104C22⋊C4×C251—21no—
105C22⋊C4×C251—21no—
106C22⋊C4×C251—21no—
107C22⋊C4×C251—21no—
108C22⋊C4×C251D411yescharacteristic
109C22⋊C4×C251D411yescharacteristic
110C42×C251C2311yescharacteristic
111C22×D251—21no—
112D1004—21no—
113D1004—21no—
114C22⋊D251—42no—
115C22⋊D251—81no—
116C2×Dic251—21no—
117C2×Dic251—21no—
118C2×Dic251—21no—
119C2×Dic251—81no—
120C4×D251—81no—
121C251⋊C8—21no—
122C251⋊C8—21no—
123C251⋊C8—21no—
124C251⋊C8—21no—
125C23×C251—21no—
126C23×C251—21no—
127C23×C251—42no—
128C23×C251—42no—
129C23×C251—81no—
130C23×C251C2×D411yescharacteristic
131C23×C251C2×D411yescharacteristic
132Q8×C251—21no—
133Q8×C251—21no—
134D4×C251—22no—
135D4×C251—21no—
136D4×C251—21no—
137D4×C251—21no—
138D4×C251—42no—
139D4×C251—42no—
140D4×C251—81no—
141D4×C251#4112yes—
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—42no—
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—22no—
159C22×C251—21no—
160C22×C251—21no—
161C22×C251—41no—
162C22×C251—41no—
163C22×C251—41no—
164C22×C251—41no—
165C22×C251—41no—
166C22×C251—84no—
167C22×C251#13512yes—
168C22×C251C22×D411yescharacteristic
169C2×D251—41no—
170C2×D251—41no—
171C2×D251—41no—
172C4×C251—21no—
173C4×C251—21no—
174C4×C251—21no—
175C4×C251—21no—
176C4×C251—41no—
177C4×C251—41no—
178C4×C251C2×#4111yescharacteristic
179C4×C251H(5,𝔽2)11yescharacteristic
180Dic251—41no—
181Dic251—41no—
182Dic251—41no—
183C2×C251—22no—
184C2×C251—41no—
185C2×C251—42no—
186C2×C251—81no—
187C2×C251#54611yescharacteristic
188C2×C251C2×#13511yescharacteristic
189C2×C251#57611yescharacteristic
190D251—81no—
191C251#329711yescharacteristic, complemented, Sylow
192#3297—2511nomaximal, Sylow
193#553—2511no—
194#546D25111yescharacteristic, complemented
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×#41C2×D25111yescharacteristic
209C22×D4C2×D25111yescharacteristic
210C2×SD16—2511no—
211C2×D8—2511no—
212C2×D8—2511no—
213C22×C8—2511no—
214C22×C8—2511no—
215#126—5021no—
216#125—2511no—
217#124C2×D25111yescharacteristic
218#123—21no—
219#123—5021no—
220C22⋊Q8C2×D25111yescharacteristic
221#121—21no—
222#121—5021no—
223#121—2511no—
224#121—2511no—
225#121C2×D25111yescharacteristic
226#120—21no—
227C4×D4—5021no—
228C4×D4—2511no—
229C4×D4C2×D25111yescharacteristic
230C4×D4C2×D25111yescharacteristic
231#117—2511no—
232C2×C4⋊C4—2511no—
233C2×C22⋊C4—21no—
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—21no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D25111yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—5021no—
261C2×D4—5021no—
262C2×D4—2511no—
263C2×D4C22⋊D25111yescharacteristic
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—5021no—
283C4⋊C4—5021no—
284C4⋊C4—5021no—
285C4⋊C4—2511no—
286C4⋊C4—2511no—
287C4⋊C4—2511no—
288C4⋊C4C22⋊D25111yescharacteristic
289C4⋊C4C22×D25111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—5021no—
296C22⋊C4—5021no—
297C22⋊C4—5021no—
298C22⋊C4—5021no—
299C22⋊C4C22⋊D25111yescharacteristic
300C22⋊C4C22⋊D25111yescharacteristic
301C42—5021no—
302C42C22×D25111yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—5021no—
309C23C2×C22⋊D25111yescharacteristic
310C23C2×C22⋊D25111yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—42no—
319D4—81no—
320D4—20081no—
321D4—10042no—
322D4—5021no—
323D4—5021no—
324D4#1686080912yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—20081no—
334C2×C4—20081no—
335C2×C4—5021no—
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—22no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—10041no—
360C22—10041no—
361C22—10041no—
362C22#2254873512yes—
363C22C22×C22⋊D25111yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—10041no—
371C4—10041no—
372C4—10041no—
373C4C2×#1686080911yescharacteristic
374C4#2254876011yescharacteristic
375C2—22no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—20081no—
380C2#3063233111yescharacteristic
381C2C2×#2254873511yescharacteristic
382C2#3063249011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#546×C251C2
socleC22×C251C22×D4
hypercenter#546D251
hypocenterC251#3297
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #30632383C2semidirect
2. #30632041C2semidirect
3. #30632020C2semidirect
4. #30632008C2semidirect
5. #30631987C2semidirect
6. #30631921C2semidirect
7. #30631918C2semidirect
8. #30631899C2semidirect
9. #30631832C2semidirect
10. #30631821C2semidirect
11. C4×#16860795C2semidirect
12. D4×C251⋊C8C2semidirect
13. #30631683C2semidirect
14. C2×#22548636C2semidirect
15. #546×C251C2semidirect
16. C2×#16860797C22semidirect
20. C22×C251⋊C8C22semidirect
22. #22548707C22semidirect
25. #22548689C22semidirect
27. C4×D1004C22semidirect
29. #22548674C22semidirect
31. #22548648C22semidirect
32. #22548648C22semidirect
33. #22548646C22semidirect
34. C4⋊(C251⋊C8)C22semidirect
35. #22548639C22semidirect
36. #22548639C22semidirect
37. #22548639C22semidirect
44. #22548635C22semidirect
46. C22⋊(C251⋊C8)C22semidirect
47. C4×C251⋊C8C22semidirect
191. C251#3297semidirect
194. #546D251semidirect

Maximal subgroups

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

subgroupindex
#306323832
#306320412
#306320202
#306320082
#306319872
#306319212
#306319182
#306318992
#306318322
#306318212
C4×#168607952
D4×C251⋊C82
#306316832
C2×#225486362
#546×C2512
#3297251