Finite Group Databasebeta

← #44562268

Subgroups — #44562268

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#30632381C211yesmaximal, characteristic, complemented
2#30632344C211yesmaximal, characteristic, complemented
3#30632040C211yesmaximal, characteristic, complemented
4#30632028C211yesmaximal, characteristic, complemented
5#30632004C211yesmaximal, characteristic, complemented
6#30631992C211yesmaximal, characteristic, complemented
7#30631922C211yesmaximal, characteristic, complemented
8#30631916C211yesmaximal, characteristic, complemented
9#30631901C211yesmaximal, characteristic, complemented
10#30631832C211yesmaximal, characteristic, complemented
11#30631827C211yesmaximal, characteristic, complemented
12#30631772C211yesmaximal, characteristic, complemented
13C8×C22⋊D251C211yesmaximal, characteristic, complemented
14#103×D251C211yesmaximal, characteristic, complemented
15#416×C251C211yesmaximal, characteristic, complemented
16C2×Q8×D251C2211yescharacteristic
17C2×#16860799C2211yescharacteristic
18C2×#16860797C2211yescharacteristic
19C2×#16860795C2211yescharacteristic
20C2×C8×D251C2211yescharacteristic
21#22548709—21no—
22#22548706C2211yescharacteristic
23#22548702—21no—
24#22548702C2211yescharacteristic
25#22548697—21no—
26#22548694—21no—
27#22548694C2211yescharacteristic
28#22548690C2211yescharacteristic
29#22548689C2211yescharacteristic
30#22548688—21no—
31#22548686—21no—
32#22548686C2211yescharacteristic
33Q8×Dic251C2211yescharacteristic
34C4×D1004—21no—
35#22548677C2211yescharacteristic
36C4×C22⋊D251—21no—
37C4×C22⋊D251C2211yescharacteristic
38C4⋊C4×D251C2211yescharacteristic
39#22548648C2211yescharacteristic
40#22548647C2211yescharacteristic
41#22548646C2211yescharacteristic
42Dic251⋊C8C2211yescharacteristic
43#22548640C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548638C2211yescharacteristic
46#22548638C2211yescharacteristic
47#22548637C2211yescharacteristic
48#22548636C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C2×#41×C251C2211yescharacteristic
53C22×C8×C251C2211yescharacteristic
54#117×C251C2211yescharacteristic
55#103×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59C2×Dic502—21no—
60C2×Dic502C2311yescharacteristic
61Q8×D251—22no—
62Q8×D251—41no—
63C2×D1004—21no—
64C2×D1004C2311yescharacteristic
65C2×C22⋊D251—21no—
66C2×C22⋊D251—21no—
67C2×C22⋊D251D411yescharacteristic
68C2×C4×D251—21no—
69C2×C4×D251—21no—
70C2×C4×D251—21no—
71C2×C4×D251C2311yescharacteristic
72#16860799—41no—
73#16860797—22no—
74#16860795—41no—
75C2×C251⋊C8C2311yescharacteristic
76C2×C251⋊C8C2311yescharacteristic
77C8×D251—41no—
78C4⋊Dic251—21no—
79C4⋊Dic251—21no—
80C4⋊Dic251C2311yescharacteristic
81C4⋊Dic251C2311yescharacteristic
82Dic251⋊C4—21no—
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4D411yescharacteristic
86C22⋊Dic251—21no—
87C22⋊Dic251—21no—
88C22⋊Dic251—21no—
89C22⋊Dic251—21no—
90C22⋊Dic251D411yescharacteristic
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4D411yescharacteristic
95C4×Dic251—21no—
96C4×Dic251C2311yescharacteristic
97#41×C251—22no—
98#41×C251—41no—
99C2×Q8×C251C2311yescharacteristic
100C2×D4×C251—21no—
101C2×D4×C251C2311yescharacteristic
102C22×C4×C251—21no—
103C22×C4×C251C2311yescharacteristic
104C2×C8×C251—41no—
105C2×C8×C251C2311yescharacteristic
106C2×C8×C251C2311yescharacteristic
107C4⋊C4×C251C2311yescharacteristic
108C4⋊C4×C251C2311yescharacteristic
109C22⋊C4×C251—21no—
110C42×C251—21no—
111C22×D251—21no—
112C22×D251C2×D411yescharacteristic
113Dic502—21no—
114Dic502—21no—
115Dic502—42no—
116D1004—21no—
117D1004—21no—
118D1004—42no—
119C22⋊D251—22no—
120C22⋊D251—42no—
121C22⋊D251—81no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251C2×D411yescharacteristic
126C2×Dic251C2×D411yescharacteristic
127C2×Dic251C2×D411yescharacteristic
128C4×D251—21no—
129C4×D251—21no—
130C4×D251—42no—
131C4×D251—81no—
132C4×D251—81no—
133C251⋊C8—21no—
134C251⋊C8—21no—
135C23×C251—21no—
136C23×C251C2×D411yescharacteristic
137Q8×C251—21no—
138Q8×C251#4112yes—
139D4×C251—21no—
140D4×C251—21no—
141D4×C251—42no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—42no—
148C2×C4×C251C2×D411yescharacteristic
149C2×C4×C251C2411yescharacteristic
150C8×C251—21no—
151C8×C251—21no—
152C22×C251—21no—
153C22×C251—21no—
154C22×C251—21no—
155C22×C251—41no—
156C22×C251—41no—
157C22×C251—41no—
158C22×C251C22×D411yescharacteristic
159C2×D251—21no—
160C2×D251—21no—
161C2×D251—41no—
162C2×D251—41no—
163C2×D251—41no—
164C2×D251#13512yes—
165C4×C251—21no—
166C4×C251—22no—
167C4×C251—41no—
168C4×C251—41no—
169C4×C251C2×#4111yescharacteristic
170C4×C251H(5,𝔽2)11yescharacteristic
171Dic251—21no—
172Dic251—21no—
173Dic251—21no—
174Dic251—41no—
175Dic251—41no—
176Dic251—41no—
177C2×C251—41no—
178C2×C251—81no—
179C2×C251#54611yescharacteristic
180C2×C251C2×#13511yescharacteristic
181C2×C251#57711yescharacteristic
182D251—22no—
183D251—81no—
184C251#330111yescharacteristic, complemented, Sylow
185#3301—2511nomaximal, Sylow
186#553—2511no—
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#416D25111yescharacteristic, complemented
200C2×#103—2511no—
201C2×#41C2×D25111yescharacteristic
202C22×Q8—2511no—
203C2×SD16—2511no—
204C2×SD16—2511no—
205C2×D8—2511no—
206C22×C8—2511no—
207C22×C8C2×D25111yescharacteristic
208#126—5021no—
209#125—2511no—
210#124—5021no—
211#124—2511no—
212#123—5021no—
213C22⋊Q8—5021no—
214C22⋊Q8—2511no—
215#121—5021no—
216#121—5021no—
217#121—2511no—
218#121—2511no—
219#121—2511no—
220C4×Q8—2511no—
221C4×D4—5021no—
222C4×D4—5021no—
223C4×D4—2511no—
224C4×D4—2511no—
225#117C2×D25111yescharacteristic
226C2×C4⋊C4—2511no—
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#103C2×D25111yescharacteristic
236#103C2×D25111yescharacteristic
237#102—2511no—
238#102—2511no—
239#102C2×D25111yescharacteristic
240#102C2×D25111yescharacteristic
241C22⋊C8—2511no—
242C22⋊C8—2511no—
243C4×C8—2511no—
244#41—22no—
245#41—41no—
246C2×Q8—10041no—
247C2×Q8—5021no—
248C2×Q8—5022no—
249C2×Q8—2511no—
250C2×Q8C22×D25111yescharacteristic
251C2×D4—21no—
252C2×D4—5021no—
253C2×D4—5021no—
254C2×D4—5021no—
255C2×D4—2511no—
256C2×D4—2511no—
257C2×D4C22×D25111yescharacteristic
258C22×C4—21no—
259C22×C4—5021no—
260C22×C4—5021no—
261C22×C4—5021no—
262C22×C4—2511no—
263C22×C4C22×D25111yescharacteristic
264SD16—10041no—
265SD16—5022no—
266D8—10041no—
267C2×C8—41no—
268C2×C8—10041no—
269C2×C8—2511no—
270C2×C8—2511no—
271C2×C8C22×D25111yescharacteristic
272C2×C8C22×D25111yescharacteristic
273C4⋊C4—5021no—
274C4⋊C4—5021no—
275C4⋊C4—5021no—
276C4⋊C4—5021no—
277C4⋊C4—5021no—
278C4⋊C4—2511no—
279C4⋊C4—2511no—
280C4⋊C4—2511no—
281C4⋊C4C22×D25111yescharacteristic
282C4⋊C4C22×D25111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—5021no—
285C22⋊C4—5021no—
286C22⋊C4—5021no—
287C22⋊C4—5021no—
288C22⋊C4—5021no—
289C22⋊C4—5021no—
290C22⋊C4—5021no—
291C22⋊C4—2511no—
292C22⋊C4—2511no—
293C42—21no—
294C42—5021no—
295C42—2511no—
296C23—21no—
297C23—5021no—
298C23—2511no—
299C23D4×D25111yescharacteristic
300Q8—21no—
301Q8—10042no—
302Q8—5021no—
303Q8—5021no—
304Q8#1686081012yes—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—20081no—
309D4—10042no—
310D4—10042no—
311D4—5022no—
312D4—5021no—
313D4—5021no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—20081no—
321C2×C4—20081no—
322C2×C4—10042no—
323C2×C4—5021no—
324C2×C4—5021no—
325C2×C4—5021no—
326C2×C4—5021no—
327C2×C4—5021no—
328C2×C4—2511no—
329C2×C4—2511no—
330C2×C4—2511no—
331C2×C4D4×D25111yescharacteristic
332C2×C4C23×D25111yescharacteristic
333C8—21no—
334C8—21no—
335C8—5021no—
336C8—5021no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—10041no—
344C22—10041no—
345C22—10041no—
346C22—5021no—
347C22—5021no—
348C22—2512no—
349C22C2×D4×D25111yescharacteristic
350C4—21no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—10041no—
355C4—10041no—
356C4—10041no—
357C4—5021no—
358C4—5021no—
359C4—5021no—
360C4C2×#1686081011yescharacteristic
361C4#2254876011yescharacteristic
362C2—41no—
363C2—81no—
364C2—20081no—
365C2—5022no—
366C2#3063232711yescharacteristic
367C2#135×D25111yescharacteristic
368C2#3063249411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#416×C251C2
socleC22×C251C22×D4
hypercenter#416D251
hypocenterC251#3301
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #30632381C2semidirect
2. #30632344C2semidirect
3. #30632040C2semidirect
4. #30632028C2semidirect
5. #30632004C2semidirect
6. #30631992C2semidirect
7. #30631922C2semidirect
8. #30631916C2semidirect
9. #30631901C2semidirect
10. #30631832C2semidirect
11. #30631827C2semidirect
12. #30631772C2semidirect
13. C8×C22⋊D251C2semidirect
14. #103×D251C2semidirect
15. #416×C251C2semidirect
184. C251#3301semidirect
199. #416D251semidirect

Maximal subgroups

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

subgroupindex
#306323812
#306323442
#306320402
#306320282
#306320042
#306319922
#306319222
#306319162
#306319012
#306318322
#306318272
#306317722
C8×C22⋊D2512
#103×D2512
#416×C2512
#3301251