Finite Group Databasebeta

← #44560716

Subgroups — #44560716

Order 32128.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632314C211yesmaximal, characteristic, complemented
2#30632255C211yesmaximal, characteristic
3#30632052C211yesmaximal, characteristic, complemented
4#30632050C211yesmaximal, characteristic, complemented
5#30632048C211yesmaximal, characteristic, complemented
6#30632043C211yesmaximal, characteristic, complemented
7#30632041C211yesmaximal, characteristic, complemented
8#30632035C211yesmaximal, characteristic, complemented
9#30632032C211yesmaximal, characteristic, complemented
10#30631690C211yesmaximal, characteristic, complemented
11#30631687C211yesmaximal, characteristic, complemented
12#30631682C211yesmaximal, characteristic, complemented
13#102×D251C211yesmaximal, characteristic, complemented
14#30631644C211yesmaximal, characteristic, complemented
15#489×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×D4×D251C2211yescharacteristic
18C2×#16860793C2211yescharacteristic
19C2×C8×D251C2211yescharacteristic
20#22548714C2211yescharacteristic, complemented
21#22548713—21no—
22#22548712C2211yescharacteristic
23#22548711C2211yescharacteristic
24#22548707C2211yescharacteristic, complemented
25#22548706C2211yescharacteristic, complemented
26#22548705—21no—
27#22548704C2211yescharacteristic
28#22548693—21no—
29#22548691—21no—
30#22548688—21no—
31#22548688—21no—
32#22548685—21no—
33#22548685—21no—
34#22548673C2211yescharacteristic
35#22548672C2211yescharacteristic
36C4⋊C4×D251C2211yescharacteristic
37#22548639C2211yescharacteristic, complemented
38#22548639C2211yescharacteristic, complemented
39#22548637C2211yescharacteristic
40#22548637C2211yescharacteristic
41#22548637C2211yescharacteristic, complemented
42#22548637C2211yescharacteristic, complemented
43#22548636C2211yescharacteristic
44#22548636C2211yescharacteristic
45#22548635C2211yescharacteristic
46#22548635C2211yescharacteristic
47#22548635C2211yescharacteristic, complemented
48#22548635C2211yescharacteristic, complemented
49#22548625C2211yescharacteristic, complemented
50C4×C251⋊C8C2211yescharacteristic, complemented
51C8×Dic251C2211yescharacteristic, complemented
52C4⋊Q8×C251C2211yescharacteristic, complemented
53#124×C251C2211yescharacteristic
54#103×C251C2211yescharacteristic, complemented
55#103×C251C2211yescharacteristic, complemented
56#102×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58#97×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60#16860811—22no—
61#16860811—22no—
62C2×Dic502—21no—
63C2×D1004—21no—
64C2×D1004—21no—
65C2×D1004—21no—
66C2×D1004C2311yescharacteristic
67C2×D1004C2311yescharacteristic
68C2×C22⋊D251—21no—
69C2×C22⋊D251—21no—
70C2×C22⋊D251—21no—
71D4×D251—22no—
72D4×D251—22no—
73C2×C4×D251—21no—
74C2×C4×D251—21no—
75C2×C4×D251C2311yescharacteristic
76#16860793—22no—
77C2×C251⋊C8C2311yescharacteristic
78C2×C251⋊C8C2311yescharacteristic
79C8×D251—22no—
80C4⋊Dic251—21no—
81C4⋊Dic251C2311yescharacteristic
82C4⋊Dic251C2311yescharacteristic
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88Dic251⋊C4D411yescharacteristic
89Dic251⋊C4D411yescharacteristic
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4D411yescharacteristic
94(C2×D251)⋊C4D411yescharacteristic
95C4×Dic251—21no—
96C4×Dic251C2311yescharacteristic
97C2×Q8×C251—21no—
98C2×Q8×C251C2311yescharacteristic
99C2×D4×C251C2311yescharacteristic
100C2×C8×C251C2311yescharacteristic
101C2×C8×C251C2311yescharacteristic
102C4⋊C4×C251—21no—
103C4⋊C4×C251C2311yescharacteristic
104C4⋊C4×C251C2311yescharacteristic
105C22⋊C4×C251—21no—
106C22⋊C4×C251—21no—
107C42×C251C2311yescharacteristic
108C22×D251—21no—
109C22×D251—21no—
110C22×D251—42no—
111C22×D251—81no—
112C22×D251—81no—
113C22×D251C2×D411yescharacteristic
114Dic502—81no—
115D1004—21no—
116D1004—21no—
117D1004—21no—
118D1004—21no—
119D1004—42no—
120D1004—81no—
121D1004—81no—
122C22⋊D251—42no—
123C22⋊D251—81no—
124C22⋊D251—81no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251C2×D411yescharacteristic
128C2×Dic251C2×D411yescharacteristic
129C2×Dic251C2×D411yescharacteristic
130C4×D251—42no—
131C4×D251—81no—
132C4×D251#4112yes—
133C4×D251#4112yes—
134C251⋊C8—21no—
135C251⋊C8—21no—
136C23×C251—21no—
137Q8×C251—21no—
138Q8×C251—21no—
139Q8×C251—81no—
140D4×C251—21no—
141D4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2×D411yescharacteristic
147C2×C4×C251C2411yescharacteristic
148C8×C251—21no—
149C8×C251—21no—
150C22×C251—41no—
151C22×C251—41no—
152C22×C251—41no—
153C22×C251C22×D411yescharacteristic
154C2×D251—21no—
155C2×D251—21no—
156C2×D251—41no—
157C2×D251—41no—
158C2×D251—41no—
159C2×D251—41no—
160C2×D251—41no—
161C2×D251—41no—
162C2×D251—84no—
163C2×D251#12412yes—
164C4×C251—21no—
165C4×C251—21no—
166C4×C251—41no—
167C4×C251—41no—
168C4×C251—41no—
169C4×C251C2×#4111yescharacteristic
170C4×C251C2×#4111yescharacteristic
171Dic251—21no—
172Dic251—21no—
173Dic251—41no—
174Dic251—41no—
175Dic251#12412yes—
176C2×C251—81no—
177C2×C251C2×#12411yescharacteristic
178C2×C251#57611yescharacteristic
179C2×C251#57711yescharacteristic
180D251—22no—
181D251—81no—
182D251—81no—
183C251#313611yescharacteristic, complemented, Sylow
184#3136—2511nomaximal, Sylow
185#544—2511no—
186#535—2511no—
187#490—2511no—
188#490—2511no—
189#489—2511no—
190#489D25111yescharacteristic, complemented
191#488—2511no—
192#488—2511no—
193#486—2511no—
194#486—2511no—
195#418—2511no—
196#417—2511no—
197#416—2511no—
198C2×#102—2511no—
199#405—2511no—
200C2×#41—2511no—
201C22×D4—2511no—
202C2×M16—2511no—
203C22×C8—2511no—
204C4⋊Q8—5021no—
205C4⋊Q8—2511no—
206C4⋊Q8C2×D25111yescharacteristic, complemented
207#127—2511no—
208#127—2511no—
209#125—5021no—
210#125—2511no—
211#125—2511no—
212#124—2511no—
213#124C2×D25111yescharacteristic
214C22⋊Q8—5021no—
215C22⋊Q8—5021no—
216#121—5021no—
217#121—5021no—
218#120—5021no—
219#120—5021no—
220#117—2511no—
221#117—2511no—
222C2×C4⋊C4—2511no—
223#103—2511no—
224#103—2511no—
225#103C2×D25111yescharacteristic, complemented
226#103C2×D25111yescharacteristic, complemented
227#102—2511no—
228#102—2511no—
229#102—2511no—
230#102—2511no—
231#102—2511no—
232#102—2511no—
233#102—2511no—
234#102—2511no—
235#102—2511no—
236#102—2511no—
237#102C2×D25111yescharacteristic
238#102C2×D25111yescharacteristic
239#97—2511no—
240#97C2×D25111yescharacteristic, complemented
241C4×C8—2511no—
242C4×C8—2511no—
243C24—5021no—
244#41—5022no—
245#41—5022no—
246C2×Q8—21no—
247C2×Q8—5021no—
248C2×Q8C22×D25111yescharacteristic
249C2×D4—5021no—
250C2×D4—5022no—
251C2×D4—5022no—
252C2×D4—5021no—
253C2×D4—5021no—
254C2×D4—5021no—
255C2×D4—5021no—
256C2×D4—5021no—
257C2×D4—2511no—
258C2×D4—2511no—
259C2×D4C22×D25111yescharacteristic
260C22×C4—5021no—
261C22×C4—5021no—
262C22×C4—2511no—
263M16—5022no—
264C2×C8—5022no—
265C2×C8—2511no—
266C2×C8—2511no—
267C2×C8C22×D25111yescharacteristic
268C2×C8C22×D25111yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—5021no—
271C4⋊C4—5021no—
272C4⋊C4—5021no—
273C4⋊C4—5021no—
274C4⋊C4—5021no—
275C4⋊C4—5021no—
276C4⋊C4—2511no—
277C4⋊C4—2511no—
278C4⋊C4—2511no—
279C4⋊C4—2511no—
280C4⋊C4C22×D25111yescharacteristic
281C4⋊C4C22×D25111yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—5021no—
285C22⋊C4—5021no—
286C22⋊C4—5021no—
287C22⋊C4—2511no—
288C22⋊C4—2511no—
289C42—5021no—
290C42—2511no—
291C42C22×D25111yescharacteristic
292C23—21no—
293C23—20081no—
294C23—20081no—
295C23—10042no—
296C23—5021no—
297C23—5021no—
298C23—2511no—
299Q8—21no—
300Q8—21no—
301Q8—81no—
302Q8—20081no—
303D4—21no—
304D4—21no—
305D4—20081no—
306D4—20081no—
307D4—20081no—
308D4—20081no—
309D4—10042no—
310D4—10042no—
311D4—5021no—
312D4—5021no—
313D4—5021no—
314D4—5021no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—20081no—
319C2×C4—10042no—
320C2×C4—5021no—
321C2×C4—5021no—
322C2×C4—2511no—
323C2×C4—2511no—
324C2×C4—2511no—
325C2×C4—2512no—
326C2×C4—2512no—
327C2×C4D4×D25111yescharacteristic
328C2×C4D4×D25111yescharacteristic
329C2×C4C23×D25111yescharacteristic
330C8—21no—
331C8—21no—
332C8—5021no—
333C8—5021no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—20084no—
338C22—10041no—
339C22—10041no—
340C22—10041no—
341C22—10041no—
342C22—10041no—
343C22—10041no—
344C22—5021no—
345C22—5021no—
346C22—2512no—
347C22C2×D4×D25111yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—10041no—
354C4—10041no—
355C4—5021no—
356C4—5021no—
357C4—2512no—
358C4#41×D25111yescharacteristic
359C4#41×D25111yescharacteristic
360C2—81no—
361C2—20081no—
362C2—20081no—
363C2—5022no—
364C2#124×D25111yescharacteristic
365C2#3063249111yescharacteristic
366C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#489×C251C2
socleC22×C251C22×D4
prime radical#30632255C2
hypercenter#489D251
hypocenterC251#3136
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #30632314C2semidirect
3. #30632052C2semidirect
4. #30632050C2semidirect
5. #30632048C2semidirect
6. #30632043C2semidirect
7. #30632041C2semidirect
8. #30632035C2semidirect
9. #30632032C2semidirect
10. #30631690C2semidirect
11. #30631687C2semidirect
12. #30631682C2semidirect
13. #102×D251C2semidirect
14. #30631644C2semidirect
15. #489×C251C2semidirect
20. #22548714C22semidirect
24. #22548707C22semidirect
25. #22548706C22semidirect
37. #22548639C22semidirect
38. #22548639C22semidirect
41. #22548637C22semidirect
42. #22548637C22semidirect
47. #22548635C22semidirect
48. #22548635C22semidirect
49. #22548625C22semidirect
50. C4×C251⋊C8C22semidirect
51. C8×Dic251C22semidirect
52. C4⋊Q8×C251C22semidirect
54. #103×C251C22semidirect
55. #103×C251C22semidirect
58. #97×C251C22semidirect
183. C251#3136semidirect
190. #489D251semidirect
206. C4⋊Q8C2×D251semidirect
225. #103C2×D251semidirect
226. #103C2×D251semidirect
240. #97C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323142
#306322552
#306320522
#306320502
#306320482
#306320432
#306320412
#306320352
#306320322
#306316902
#306316872
#306316822
#102×D2512
#306316442
#489×C2512
#3136251