Finite Group Databasebeta

← #44562217

Subgroups — #44562217

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#30632333C211yesmaximal, characteristic, complemented
3#30632042C211yesmaximal, characteristic, complemented
4#30632023C211yesmaximal, characteristic, complemented
5#30632012C211yesmaximal, characteristic, complemented
6#30631988C211yesmaximal, characteristic, complemented
7#30631923C211yesmaximal, characteristic, complemented
8#30631916C211yesmaximal, characteristic, complemented
9#30631903C211yesmaximal, characteristic, complemented
10#30631834C211yesmaximal, characteristic, complemented
11#30631823C211yesmaximal, characteristic, complemented
12C4×D2008C211yesmaximal, characteristic, complemented
13#30631684C211yesmaximal, characteristic, complemented
14C2×#22548637C211yesmaximal, characteristic, complemented
15C8×D4×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18C2×#16860799C2211yescharacteristic, complemented
19C2×D2008C2211yescharacteristic
20C2×D2008C2211yescharacteristic
21#22548710—21no—
22#22548707C2211yescharacteristic, complemented
23#22548703C2211yescharacteristic
24#22548700—21no—
25#22548697—21no—
26#22548694C2211yescharacteristic
27#22548689C2211yescharacteristic
28#22548689C2211yescharacteristic
29#22548689C2211yescharacteristic, complemented
30#22548688—21no—
31#22548687—21no—
32#22548683—21no—
33C4×D1004C2211yescharacteristic
34C4×D1004C2211yescharacteristic, complemented
35(C22⋊D251)⋊C4—21no—
36#22548674C2211yescharacteristic, complemented
37C2×C4⋊Dic251C2211yescharacteristic
38C22⋊C4×D251—21no—
39#22548649C2211yescharacteristic, complemented
40#22548649C2211yescharacteristic, complemented
41#22548647C2211yescharacteristic, complemented
42#22548640C2211yescharacteristic, complemented
43#22548640C2211yescharacteristic, complemented
44#22548640C2211yescharacteristic, complemented
45#22548637C2211yescharacteristic
46#22548637C2211yescharacteristic
47#22548637C2211yescharacteristic
48#22548637C2211yescharacteristic
49#22548637C2211yescharacteristic
50#22548637C2211yescharacteristic
51#22548637C2211yescharacteristic, complemented
52C22×C8×C251C2211yescharacteristic
53C22×C8×C251C2211yescharacteristic, complemented
54C4×D4×C251C2211yescharacteristic
55C4⋊C8×C251C2211yescharacteristic, complemented
56C22⋊C8×C251C2211yescharacteristic
57C22⋊C8×C251C2211yescharacteristic, complemented
58C4×C8×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60#16860810—22no—
61#16860810—41no—
62C2×Dic502C2311yescharacteristic
63C2×D1004—22no—
64C2×D1004—21no—
65C2×D1004—41no—
66C2×D1004C2311yescharacteristic
67C2×D1004C2311yescharacteristic
68C2×D1004C2311yescharacteristic
69C2×C22⋊D251—21no—
70C2×C22⋊D251—21no—
71C2×C22⋊D251—21no—
72C2×C22⋊D251—21no—
73C22×Dic251—21no—
74C2×C4×D251—21no—
75C2×C4×D251—21no—
76C2×C4×D251—21no—
77#16860799—41no—
78D2008—22no—
79D2008—41no—
80C4⋊Dic251—21no—
81C4⋊Dic251C2311yescharacteristic
82C4⋊Dic251C2311yescharacteristic
83C4⋊Dic251C2311yescharacteristic
84C4⋊Dic251C2311yescharacteristic
85Dic251⋊C4—21no—
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88C22⋊Dic251—21no—
89C22⋊Dic251—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4—21no—
96(C2×D251)⋊C4—21no—
97C4×Dic251—21no—
98C2×D4×C251D411yescharacteristic
99C22×C4×C251C2311yescharacteristic
100C22×C4×C251C2311yescharacteristic
101C2×C8×C251—41no—
102C2×C8×C251—41no—
103C2×C8×C251C2311yescharacteristic
104C2×C8×C251C2311yescharacteristic
105C2×C8×C251C2311yescharacteristic
106C2×C8×C251C2311yescharacteristic
107C4⋊C4×C251D411yescharacteristic
108C22⋊C4×C251D411yescharacteristic
109C22⋊C4×C251D411yescharacteristic
110C42×C251C2311yescharacteristic
111C22×D251—21no—
112C22×D251—21no—
113C22×D251—21no—
114C22×D251—42no—
115C22×D251—42no—
116C22×D251—81no—
117Dic502—21no—
118Dic502—21no—
119D1004—21no—
120D1004—21no—
121D1004—21no—
122D1004—21no—
123D1004—21no—
124D1004—42no—
125D1004#4112yes—
126C22⋊D251—42no—
127C22⋊D251—42no—
128C22⋊D251—81no—
129C22⋊D251—81no—
130C2×Dic251—21no—
131C2×Dic251—21no—
132C2×Dic251—21no—
133C2×Dic251—21no—
134C2×Dic251—21no—
135C2×Dic251—81no—
136C4×D251—42no—
137C4×D251—81no—
138C4×D251—81no—
139C23×C251C2×D411yescharacteristic
140C23×C251C2×D411yescharacteristic
141D4×C251—22no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251C2×D411yescharacteristic
147C2×C4×C251C2×D411yescharacteristic
148C2×C4×C251C2×D411yescharacteristic
149C2×C4×C251C2×D411yescharacteristic
150C2×C4×C251C2411yescharacteristic
151C8×C251—21no—
152C8×C251—21no—
153C8×C251—21no—
154C8×C251—21no—
155C22×C251—21no—
156C22×C251—21no—
157C22×C251—21no—
158C22×C251—21no—
159C22×C251—21no—
160C22×C251#13512yes—
161C22×C251C22×D411yescharacteristic
162C2×D251—22no—
163C2×D251—41no—
164C2×D251—41no—
165C2×D251—41no—
166C2×D251—41no—
167C2×D251—41no—
168C2×D251—41no—
169C2×D251—41no—
170C2×D251—41no—
171C2×D251—84no—
172C4×C251—21no—
173C4×C251—21no—
174C4×C251—21no—
175C4×C251—21no—
176C4×C251C2×#4111yescharacteristic
177C4×C251H(5,𝔽2)11yescharacteristic
178Dic251—41no—
179Dic251—41no—
180Dic251—41no—
181Dic251—41no—
182Dic251—41no—
183C2×C251—22no—
184C2×C251—41no—
185C2×C251#54611yescharacteristic
186C2×C251C2×#13511yescharacteristic
187C2×C251#57611yescharacteristic
188D251—42no—
189D251—81no—
190D251—81no—
191C251#329711yescharacteristic, complemented, Sylow
192#3297—2511nomaximal, Sylow
193#553—2511no—
194#546—2511no—
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×D4D25111yescharacteristic, complemented
206#416—2511no—
207C2×#102—2511no—
208C2×#41—2511no—
209C22×D4—2511no—
210C2×SD16—2511no—
211C2×D8—2511no—
212C2×D8—2511no—
213C22×C8C2×D25111yescharacteristic
214C22×C8C2×D25111yescharacteristic, complemented
215#126—5021no—
216#125—2511no—
217#124—2511no—
218#123—5021no—
219#123—5021no—
220C22⋊Q8—2511no—
221#121—5021no—
222#121—5021no—
223#121—2511no—
224#121—2511no—
225#121—2511no—
226#120—5021no—
227C4×D4—5021no—
228C4×D4—2511no—
229C4×D4—2511no—
230C4×D4C2×D25111yescharacteristic
231#117—2511no—
232C2×C4⋊C4—2511no—
233C2×C22⋊C4—5021no—
234#107—2511no—
235#107—2511no—
236#106—2511no—
237C4⋊C8C2×D25111yescharacteristic, complemented
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⋊C8C2×D25111yescharacteristic
249C22⋊C8C2×D25111yescharacteristic, complemented
250C4×C8C2×D25111yescharacteristic, complemented
251C24—5021no—
252#41—10041no—
253#41—5022no—
254C2×Q8—2511no—
255C2×D4—10041no—
256C2×D4—5022no—
257C2×D4—5021no—
258C2×D4—5021no—
259C2×D4—5021no—
260C2×D4—5021no—
261C2×D4—5021no—
262C2×D4—2511no—
263C2×D4—2511no—
264C2×D4—2511no—
265C2×D4D100411yescharacteristic
266C22×C4—5021no—
267C22×C4—5021no—
268C22×C4—5021no—
269C22×C4—5021no—
270C22×C4C22×D25111yescharacteristic
271C22×C4C22×D25111yescharacteristic
272SD16—10041no—
273D8—10041no—
274D8—5022no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8C22×D25111yescharacteristic
278C2×C8C22×D25111yescharacteristic
279C2×C8C22×D25111yescharacteristic
280C2×C8C22×D25111yescharacteristic
281C4⋊C4—5021no—
282C4⋊C4—5021no—
283C4⋊C4—5021no—
284C4⋊C4—5021no—
285C4⋊C4—2511no—
286C4⋊C4—2511no—
287C4⋊C4—2511no—
288C4⋊C4—2511no—
289C4⋊C4D100411yescharacteristic
290C22⋊C4—5021no—
291C22⋊C4—5021no—
292C22⋊C4—5021no—
293C22⋊C4—5021no—
294C22⋊C4—5021no—
295C22⋊C4—5021no—
296C22⋊C4—5021no—
297C22⋊C4—5021no—
298C22⋊C4—5021no—
299C22⋊C4D100411yescharacteristic
300C22⋊C4D100411yescharacteristic
301C42—5021no—
302C42C22×D25111yescharacteristic
303C23—20081no—
304C23—10042no—
305C23—10042no—
306C23—5021no—
307C23—5021no—
308C23—5021no—
309C23C2×D100411yescharacteristic
310C23C2×D100411yescharacteristic
311Q8—5021no—
312Q8—5021no—
313D4—22no—
314D4—20081no—
315D4—20081no—
316D4—10042no—
317D4—10042no—
318D4—10042no—
319D4—5021no—
320D4—5021no—
321D4—5021no—
322D4—5021no—
323D4—5021no—
324D4—2512no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—20081no—
330C2×C4—20081no—
331C2×C4—20081no—
332C2×C4—10042no—
333C2×C4—5021no—
334C2×C4—5021no—
335C2×C4—5021no—
336C2×C4—5021no—
337C2×C4—5021no—
338C2×C4C2×D100411yescharacteristic
339C2×C4C2×D100411yescharacteristic
340C2×C4C2×D100411yescharacteristic
341C2×C4C2×D100411yescharacteristic
342C2×C4C23×D25111yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—20084no—
353C22—10041no—
354C22—10041no—
355C22—10041no—
356C22—10041no—
357C22—10041no—
358C22—10041no—
359C22—10041no—
360C22—10041no—
361C22—5022no—
362C22#2254873712yes—
363C22C22×D100411yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—10041no—
369C4—10041no—
370C4—10041no—
371C4—10041no—
372C4—10041no—
373C4#41×D25111yescharacteristic
374C4#2254876011yescharacteristic
375C2—22no—
376C2—41no—
377C2—20081no—
378C2—20081no—
379C2—10042no—
380C2#3063233211yescharacteristic
381C2C2×#2254873711yescharacteristic
382C2#3063249211yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC8×D4×C251C2
socleC22×C251C22×D4
hypercenterC8×D4D251
hypocenterC251#3297
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #30632383C2semidirect
2. #30632333C2semidirect
3. #30632042C2semidirect
4. #30632023C2semidirect
5. #30632012C2semidirect
6. #30631988C2semidirect
7. #30631923C2semidirect
8. #30631916C2semidirect
9. #30631903C2semidirect
10. #30631834C2semidirect
11. #30631823C2semidirect
12. C4×D2008C2semidirect
13. #30631684C2semidirect
14. C2×#22548637C2semidirect
15. C8×D4×C251C2semidirect
18. C2×#16860799C22semidirect
22. #22548707C22semidirect
29. #22548689C22semidirect
34. C4×D1004C22semidirect
36. #22548674C22semidirect
39. #22548649C22semidirect
40. #22548649C22semidirect
41. #22548647C22semidirect
42. #22548640C22semidirect
43. #22548640C22semidirect
44. #22548640C22semidirect
51. #22548637C22semidirect
53. C22×C8×C251C22semidirect
55. C4⋊C8×C251C22semidirect
57. C22⋊C8×C251C22semidirect
58. C4×C8×C251C22semidirect
191. C251#3297semidirect
205. C8×D4D251semidirect
214. C22×C8C2×D251semidirect
237. C4⋊C8C2×D251semidirect
249. C22⋊C8C2×D251semidirect
250. C4×C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323832
#306323332
#306320422
#306320232
#306320122
#306319882
#306319232
#306319162
#306319032
#306318342
#306318232
C4×D20082
#306316842
C2×#225486372
C8×D4×C2512
#3297251