Finite Group Databasebeta

← #44560557

Subgroups — #44560557

Order 32128.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632290C211yesmaximal, characteristic, complemented
2#30632170C211yesmaximal, characteristic, complemented
3#30632003C211yesmaximal, characteristic, complemented
4#30632001C211yesmaximal, characteristic, complemented
5#30631996C211yesmaximal, characteristic, complemented
6#30631985C211yesmaximal, characteristic, complemented
7#30631885C211yesmaximal, characteristic, complemented
8#30631880C211yesmaximal, characteristic, complemented
9#30631832C211yesmaximal, characteristic, complemented
10#30631796C211yesmaximal, characteristic, complemented
11#30631788C211yesmaximal, characteristic, complemented
12#30631772C211yesmaximal, characteristic, complemented
13D8×Dic251C211yesmaximal, characteristic, complemented
14#30631745C211yesmaximal, characteristic, complemented
15#449×C251C211yesmaximal, characteristic, complemented
16C2×#16860797C2211yescharacteristic
17C2×#16860795C2211yescharacteristic
18#22548711C2211yescharacteristic
19#22548706C2211yescharacteristic
20#22548699—21no—
21#22548697—21no—
22#22548689C2211yescharacteristic
23#22548687—21no—
24#22548686—21no—
25Q8×Dic251C2211yescharacteristic
26Q8×Dic251C2211yescharacteristic
27D4×Dic251—21no—
28D4×Dic251—21no—
29D4×Dic251C2211yescharacteristic
30D4×Dic251C2211yescharacteristic
31D4×Dic251C2211yescharacteristic
32#22548677C2211yescharacteristic
33#22548674—21no—
34#22548674C2211yescharacteristic
35C2×C4⋊Dic251—21no—
36C2×C4⋊Dic251C2211yescharacteristic
37#22548649C2211yescharacteristic
38#22548649C2211yescharacteristic
39#22548646C2211yescharacteristic
40#22548646C2211yescharacteristic
41Dic251⋊C8C2211yescharacteristic
42Dic251⋊C8C2211yescharacteristic
43#22548639C2211yescharacteristic
44#22548637C2211yescharacteristic
45#22548637C2211yescharacteristic
46#22548636C2211yescharacteristic
47#22548636C2211yescharacteristic
48#22548636C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic
50#22548625C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C2×#41×C251C2211yescharacteristic
53C2×SD16×C251C2211yescharacteristic
54C2×D8×C251C2211yescharacteristic
55#121×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic
59C2×D1004C2311yescharacteristic
60C2×C22⋊D251—21no—
61C2×C22⋊D251—21no—
62C2×C22⋊D251—21no—
63C22×Dic251—21no—
64C22×Dic251—21no—
65C22×Dic251—21no—
66C22×Dic251—21no—
67C2×C4×D251—21no—
68#16860797—22no—
69#16860795—22no—
70C2×C251⋊C8C2311yescharacteristic
71C2×C251⋊C8C2311yescharacteristic
72C4⋊Dic251—21no—
73C4⋊Dic251—21no—
74C4⋊Dic251—21no—
75C4⋊Dic251—21no—
76C4⋊Dic251D411yescharacteristic
77C4⋊Dic251D411yescharacteristic
78C4⋊Dic251C2311yescharacteristic
79C4⋊Dic251C2311yescharacteristic
80C4⋊Dic251C2311yescharacteristic
81Dic251⋊C4—21no—
82C22⋊Dic251—21no—
83C22⋊Dic251—21no—
84C22⋊Dic251—21no—
85C22⋊Dic251D411yescharacteristic
86C22⋊Dic251D411yescharacteristic
87(C2×D251)⋊C4—21no—
88(C2×D251)⋊C4—21no—
89C4×Dic251—21no—
90C4×Dic251—21no—
91C4×Dic251C2311yescharacteristic
92C4×Dic251C2311yescharacteristic
93#41×C251—22no—
94#41×C251—22no—
95C2×Q8×C251C2311yescharacteristic
96C2×D4×C251—21no—
97C2×D4×C251—21no—
98C2×D4×C251C2311yescharacteristic
99C2×D4×C251C2311yescharacteristic
100C22×C4×C251—21no—
101C22×C4×C251C2311yescharacteristic
102SD16×C251—22no—
103D8×C251—22no—
104C2×C8×C251C2311yescharacteristic
105C2×C8×C251C2311yescharacteristic
106C4⋊C4×C251C2311yescharacteristic
107C22⋊C4×C251—21no—
108C22×D251—21no—
109D1004—21no—
110D1004—21no—
111C22⋊D251—81no—
112C22⋊D251—81no—
113C22⋊D251—81no—
114C2×Dic251—21no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251—81no—
118C2×Dic251—81no—
119C2×Dic251—81no—
120C2×Dic251—81no—
121C2×Dic251C2×D411yescharacteristic
122C2×Dic251C2×D411yescharacteristic
123C2×Dic251C2×D411yescharacteristic
124C2×Dic251C2×D411yescharacteristic
125C4×D251—81no—
126C251⋊C8—21no—
127C251⋊C8—21no—
128C23×C251—21no—
129C23×C251—21no—
130C23×C251C2×D411yescharacteristic
131Q8×C251—21no—
132Q8×C251#4112yes—
133D4×C251—24no—
134D4×C251—21no—
135D4×C251—21no—
136D4×C251—21no—
137D4×C251—81no—
138D4×C251#4112yes—
139C2×C4×C251—24no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251C2×D411yescharacteristic
145C2×C4×C251C2411yescharacteristic
146C8×C251—21no—
147C8×C251—21no—
148C22×C251—21no—
149C22×C251—22no—
150C22×C251—21no—
151C22×C251—21no—
152C22×C251—41no—
153C22×C251—41no—
154C22×C251—41no—
155C22×C251—41no—
156C22×C251—41no—
157C22×C251C22×D411yescharacteristic
158C2×D251—41no—
159C2×D251—41no—
160C2×D251—41no—
161C4×C251—21no—
162C4×C251—22no—
163C4×C251—41no—
164C4×C251C2×#4111yescharacteristic
165C4×C251C2×#4111yescharacteristic
166Dic251—21no—
167Dic251—21no—
168Dic251—21no—
169Dic251—21no—
170Dic251—41no—
171Dic251—41no—
172Dic251—41no—
173C2×C251—41no—
174C2×C251—42no—
175C2×C251—81no—
176C2×C251#52511yescharacteristic
177C2×C251#57611yescharacteristic
178C2×C251#57711yescharacteristic
179D251—81no—
180C251#311911yescharacteristic, complemented, Sylow
181#3119—2511nomaximal, Sylow
182#540—2511no—
183#520—2511no—
184#481—2511no—
185#480—2511no—
186#479—2511no—
187#478—2511no—
188#460—2511no—
189#459—2511no—
190#449—2511no—
191#449D25111yescharacteristic, complemented
192#442—2511no—
193#440—2511no—
194C4×SD16—2511no—
195C4×D8—2511no—
196#433—2511no—
197C2×#41C2×D25111yescharacteristic
198C2×SD16—2511no—
199C2×SD16C2×D25111yescharacteristic
200C2×D8—2511no—
201C2×D8C2×D25111yescharacteristic
202#127—2511no—
203#125—2511no—
204#123—5021no—
205#123—5021no—
206#121—5021no—
207#121—5021no—
208#121—2511no—
209#121C2×D25111yescharacteristic
210C4×Q8—2511no—
211C4×Q8—2511no—
212C4×D4—5021no—
213C4×D4—5021no—
214C4×D4—2511no—
215C4×D4—2511no—
216C4×D4—2511no—
217C4×D4—2511no—
218#117—5021no—
219#117—2511no—
220C2×C4⋊C4—5021no—
221C2×C4⋊C4—2511no—
222#107—2511no—
223#107—2511no—
224#106—2511no—
225#106—2511no—
226C4⋊C8—2511no—
227C4⋊C8—2511no—
228#103—2511no—
229#103C2×D25111yescharacteristic
230#102—2511no—
231#102—2511no—
232#102—2511no—
233#102—2511no—
234#102—2511no—
235#102C2×D25111yescharacteristic
236C22⋊C8—2511no—
237C22⋊C8C2×D25111yescharacteristic
238#97—2511no—
239C4×C8—2511no—
240#41—22no—
241#41—22no—
242C2×Q8C22×D25111yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—5021no—
246C2×D4—5021no—
247C2×D4—5021no—
248C2×D4—2511no—
249C2×D4C22×D25111yescharacteristic
250C2×D4C22×D25111yescharacteristic
251C22×C4—21no—
252C22×C4—5021no—
253C22×C4—5021no—
254C22×C4—5021no—
255C22×C4—5021no—
256C22×C4—5021no—
257C22×C4C22×D25111yescharacteristic
258SD16—22no—
259SD16—5022no—
260D8—22no—
261D8—5022no—
262C2×C8—2511no—
263C2×C8—2511no—
264C2×C8C22×D25111yescharacteristic
265C2×C8C22×D25111yescharacteristic
266C4⋊C4—5021no—
267C4⋊C4—5021no—
268C4⋊C4—5021no—
269C4⋊C4—5021no—
270C4⋊C4—5021no—
271C4⋊C4—2511no—
272C4⋊C4—2511no—
273C4⋊C4—2511no—
274C4⋊C4—2511no—
275C4⋊C4—2511no—
276C4⋊C4C22×D25111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—5021no—
279C22⋊C4—5021no—
280C22⋊C4—5021no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4—2511no—
284C22⋊C4—2511no—
285C42—5021no—
286C42—5021no—
287C42—2511no—
288C42—2511no—
289C23—21no—
290C23—21no—
291C23—5021no—
292C23D4×D25111yescharacteristic
293Q8—21no—
294Q8#1686080912yes—
295D4—24no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—20081no—
301D4—20081no—
302D4—20081no—
303D4—5021no—
304D4—5021no—
305D4#1686080912yes—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—20081no—
312C2×C4—20081no—
313C2×C4—20081no—
314C2×C4—20081no—
315C2×C4—20081no—
316C2×C4—5021no—
317C2×C4—5021no—
318C2×C4—5021no—
319C2×C4—2511no—
320C2×C4—2511no—
321C2×C4—2511no—
322C2×C4—2511no—
323C2×C4D4×D25111yescharacteristic
324C2×C4C23×D25111yescharacteristic
325C8—21no—
326C8—21no—
327C8—5021no—
328C8—5021no—
329C22—21no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—10041no—
339C22—10041no—
340C22—10041no—
341C22C2×D4×D25111yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—10041no—
346C4—10041no—
347C4—10041no—
348C4—5021no—
349C4—5021no—
350C4—5021no—
351C4—5021no—
352C4C2×#1686080911yescharacteristic
353C4C2×#1686080911yescharacteristic
354C2—41no—
355C2—42no—
356C2—81no—
357C2—20081no—
358C2#3063219311yescharacteristic
359C2#3063248911yescharacteristic
360C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#449×C251C2
socleC22×C251C22×D4
hypercenter#449D251
hypocenterC251#3119
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #30632290C2semidirect
2. #30632170C2semidirect
3. #30632003C2semidirect
4. #30632001C2semidirect
5. #30631996C2semidirect
6. #30631985C2semidirect
7. #30631885C2semidirect
8. #30631880C2semidirect
9. #30631832C2semidirect
10. #30631796C2semidirect
11. #30631788C2semidirect
12. #30631772C2semidirect
13. D8×Dic251C2semidirect
14. #30631745C2semidirect
15. #449×C251C2semidirect
180. C251#3119semidirect
191. #449D251semidirect

Maximal subgroups

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

subgroupindex
#306322902
#306321702
#306320032
#306320012
#306319962
#306319852
#306318852
#306318802
#306318322
#306317962
#306317882
#306317722
D8×Dic2512
#306317452
#449×C2512
#3119251