Finite Group Databasebeta

← #44562263

Subgroups — #44562263

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#30632385C211yesmaximal, characteristic, complemented
2#30632348C211yesmaximal, characteristic, complemented
3#30632043C211yesmaximal, characteristic, complemented
4#30632027C211yesmaximal, characteristic, complemented
5#30632005C211yesmaximal, characteristic, complemented
6#30631996C211yesmaximal, characteristic, complemented
7#30631923C211yesmaximal, characteristic, complemented
8#30631919C211yesmaximal, characteristic, complemented
9#30631905C211yesmaximal, characteristic, complemented
10#30631835C211yesmaximal, characteristic, complemented
11#30631826C211yesmaximal, characteristic, complemented
12#30631772C211yesmaximal, characteristic, complemented
13C8×C22⋊D251C211yesmaximal, characteristic, complemented
14#30631682C211yesmaximal, characteristic, complemented
15C2×#103×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×#16860797C2211yescharacteristic
18C2×#16860797C2211yescharacteristic
19C2×D2008C2211yescharacteristic
20C2×C8×D251C2211yescharacteristic
21#22548710—21no—
22#22548706C2211yescharacteristic
23#22548704—21no—
24#22548704C2211yescharacteristic
25#22548700—21no—
26#22548691—21no—
27#22548691C2211yescharacteristic
28#22548690C2211yescharacteristic
29#22548689—21no—
30#22548689—21no—
31#22548689C2211yescharacteristic
32#22548689C2211yescharacteristic
33Q8×Dic251C2211yescharacteristic
34#22548677C2211yescharacteristic
35C4×C22⋊D251—21no—
36C4×C22⋊D251—21no—
37C4×C22⋊D251C2211yescharacteristic
38#22548673C2211yescharacteristic
39#22548649C2211yescharacteristic
40#22548646C2211yescharacteristic
41#22548646C2211yescharacteristic
42Dic251⋊C8C2211yescharacteristic
43#22548639C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548637C2211yescharacteristic
46#22548637C2211yescharacteristic
47#22548635C2211yescharacteristic
48#22548635C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C22×Q8×C251C2211yescharacteristic
53C22×C8×C251C2211yescharacteristic
54C2×C4⋊C4×C251C2211yescharacteristic
55#103×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#103×C251C2211yescharacteristic
58#103×C251C2211yescharacteristic
59#16860811—22no—
60#16860811—41no—
61C2×D1004—21no—
62C2×D1004—21no—
63C2×D1004C2311yescharacteristic
64C2×D1004C2311yescharacteristic
65C2×C22⋊D251—21no—
66C2×C22⋊D251—21no—
67C2×C22⋊D251D411yescharacteristic
68C2×C4×D251—21no—
69C2×C4×D251—21no—
70C2×C4×D251C2311yescharacteristic
71#16860797—22no—
72#16860797—41no—
73D2008—41no—
74C2×C251⋊C8C2311yescharacteristic
75C2×C251⋊C8C2311yescharacteristic
76C8×D251—41no—
77C4⋊Dic251—21no—
78C4⋊Dic251—21no—
79C4⋊Dic251C2311yescharacteristic
80C4⋊Dic251C2311yescharacteristic
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83Dic251⋊C4D411yescharacteristic
84C22⋊Dic251D411yescharacteristic
85(C2×D251)⋊C4—21no—
86(C2×D251)⋊C4—21no—
87(C2×D251)⋊C4—21no—
88(C2×D251)⋊C4—21no—
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4D411yescharacteristic
94C4×Dic251—21no—
95C4×Dic251—21no—
96C4×Dic251C2311yescharacteristic
97C2×Q8×C251—21no—
98C2×Q8×C251—22no—
99C2×Q8×C251—41no—
100C2×Q8×C251C2311yescharacteristic
101C2×Q8×C251C2311yescharacteristic
102C22×C4×C251—21no—
103C22×C4×C251—21no—
104C22×C4×C251C2311yescharacteristic
105C2×C8×C251—41no—
106C2×C8×C251C2311yescharacteristic
107C2×C8×C251C2311yescharacteristic
108C4⋊C4×C251—21no—
109C4⋊C4×C251C2311yescharacteristic
110C4⋊C4×C251C2311yescharacteristic
111C22×D251—21no—
112C22×D251—21no—
113C22×D251C2×D411yescharacteristic
114D1004—21no—
115D1004—21no—
116D1004—21no—
117D1004—21no—
118D1004—42no—
119D1004—81no—
120C22⋊D251—22no—
121C22⋊D251—42no—
122C22⋊D251—42no—
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—
132C251⋊C8—21no—
133C251⋊C8—21no—
134C23×C251C2×D411yescharacteristic
135Q8×C251—21no—
136Q8×C251—21no—
137Q8×C251—21no—
138Q8×C251—42no—
139Q8×C251#4112yes—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—42no—
147C2×C4×C251—81no—
148C2×C4×C251C2×D411yescharacteristic
149C2×C4×C251C2411yescharacteristic
150C8×C251—21no—
151C8×C251—21no—
152C22×C251—21no—
153C22×C251—21no—
154C22×C251#13512yes—
155C22×C251C22×D411yescharacteristic
156C2×D251—21no—
157C2×D251—21no—
158C2×D251—21no—
159C2×D251—41no—
160C2×D251—41no—
161C2×D251—41no—
162C2×D251—41no—
163C2×D251—41no—
164C2×D251—41no—
165C4×C251—21no—
166C4×C251—22no—
167C4×C251—41no—
168C4×C251—41no—
169C4×C251—41no—
170C4×C251C2×#4111yescharacteristic
171C4×C251H(5,𝔽2)11yescharacteristic
172Dic251—21no—
173Dic251—21no—
174Dic251—21no—
175Dic251—41no—
176Dic251—41no—
177C2×C251—22no—
178C2×C251#54611yescharacteristic
179C2×C251C2×#13511yescharacteristic
180C2×C251#57711yescharacteristic
181D251—41no—
182D251—81no—
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#416—2511no—
200C2×#103D25111yescharacteristic, complemented
201C2×#41—2511no—
202C22×Q8C2×D25111yescharacteristic
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#117—2511no—
226C2×C4⋊C4C2×D25111yescharacteristic
227#107—2511no—
228#106—2511no—
229#106—2511no—
230C4⋊C8—2511no—
231#103—2511no—
232#103—2511no—
233#103C2×D25111yescharacteristic
234#103C2×D25111yescharacteristic
235#103C2×D25111yescharacteristic
236#103C2×D25111yescharacteristic
237#102—2511no—
238#102—2511no—
239#102—2511no—
240#102—2511no—
241C22⋊C8—2511no—
242C22⋊C8—2511no—
243C4×C8—2511no—
244#41—10041no—
245#41—5022no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D25111yescharacteristic
250C2×Q8C22×D25111yescharacteristic
251C2×D4—5021no—
252C2×D4—5021no—
253C2×D4—5021no—
254C2×D4—5021no—
255C2×D4—2511no—
256C2×D4—2511no—
257C2×D4—2511no—
258C22×C4—21no—
259C22×C4—21no—
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—21no—
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—5021no—
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—5021no—
294C42—5021no—
295C42—2511no—
296C23—5021no—
297C23—5021no—
298C23—2511no—
299C23D4×D25111yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1686081012yes—
305D4—20081no—
306D4—10042no—
307D4—10042no—
308D4—10042no—
309D4—5022no—
310D4—5021no—
311D4—5021no—
312D4—5021no—
313D4—5021no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—20081no—
323C2×C4—10042no—
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—10041no—
340C22—10041no—
341C22—10041no—
342C22—10041no—
343C22—10041no—
344C22—10041no—
345C22—5021no—
346C22—5021no—
347C22—5021no—
348C22#2254873812yes—
349C22C2×D4×D25111yescharacteristic
350C4—21no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—10041no—
356C4—10041no—
357C4—5021no—
358C4—5021no—
359C4—5021no—
360C4C2×#1686081011yescharacteristic
361C4#2254876011yescharacteristic
362C2—22no—
363C2—20081no—
364C2—20081no—
365C2—10041no—
366C2#3063232711yescharacteristic
367C2C2×#2254873811yescharacteristic
368C2#3063249311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#103×C251C2
socleC22×C251C22×D4
hypercenterC2×#103D251
hypocenterC251#3301
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. #30632385C2semidirect
2. #30632348C2semidirect
3. #30632043C2semidirect
4. #30632027C2semidirect
5. #30632005C2semidirect
6. #30631996C2semidirect
7. #30631923C2semidirect
8. #30631919C2semidirect
9. #30631905C2semidirect
10. #30631835C2semidirect
11. #30631826C2semidirect
12. #30631772C2semidirect
13. C8×C22⋊D251C2semidirect
14. #30631682C2semidirect
15. C2×#103×C251C2semidirect
184. C251#3301semidirect
200. C2×#103D251semidirect

Maximal subgroups

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

subgroupindex
#306323852
#306323482
#306320432
#306320272
#306320052
#306319962
#306319232
#306319192
#306319052
#306318352
#306318262
#306317722
C8×C22⋊D2512
#306316822
C2×#103×C2512
#3301251