Finite Group Databasebeta

← #44562328

Subgroups — #44562328

Order 32128.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632341C211yesmaximal, characteristic, complemented
2#30632333C211yesmaximal, characteristic, complemented
3#30632047C211yesmaximal, characteristic, complemented
4#30632018C211yesmaximal, characteristic, complemented
5#30632006C211yesmaximal, characteristic, complemented
6#30631978C211yesmaximal, characteristic, complemented
7#30631940C211yesmaximal, characteristic, complemented
8#30631916C211yesmaximal, characteristic, complemented
9#30631903C211yesmaximal, characteristic, complemented
10#30631834C211yesmaximal, characteristic, complemented
11#30631823C211yesmaximal, characteristic, complemented
12#30631800C211yesmaximal, characteristic, complemented
13#30631689C211yesmaximal, characteristic, complemented
14C2×#22548637C211yesmaximal, characteristic, complemented
15#435×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C2×#16860809—21no—
18C22×D1004C2211yescharacteristic
19C2×#16860799C2211yescharacteristic, complemented
20C2×D2008C2211yescharacteristic
21C2×D2008C2211yescharacteristic
22#22548715C2211yescharacteristic, complemented
23#22548703C2211yescharacteristic
24#22548700—21no—
25#22548699—21no—
26#22548699—21no—
27#22548695—21no—
28#22548694C2211yescharacteristic
29#22548689C2211yescharacteristic
30#22548689C2211yescharacteristic
31#22548689C2211yescharacteristic, complemented
32#22548688—21no—
33#22548683—21no—
34C4×D1004C2211yescharacteristic
35C4×D1004C2211yescharacteristic, complemented
36C2×C4⋊Dic251C2211yescharacteristic
37C2×C4⋊Dic251C2211yescharacteristic, complemented
38C22⋊C4×D251—21no—
39#22548647C2211yescharacteristic, complemented
40#22548647C2211yescharacteristic, 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
52C2×M16×C251C2211yescharacteristic, complemented
53C22×C8×C251C2211yescharacteristic
54C4×D4×C251C2211yescharacteristic
55C4⋊C8×C251C2211yescharacteristic, complemented
56C22⋊C8×C251C2211yescharacteristic
57C22⋊C8×C251C2211yescharacteristic, complemented
58#97×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60#16860810—22no—
61#16860810—41no—
62#16860809—44no—
63C2×Dic502—21no—
64C2×Dic502C2311yescharacteristic
65C2×D1004—22no—
66C2×D1004—21no—
67C2×D1004—41no—
68C2×D1004C2311yescharacteristic
69C2×D1004C2311yescharacteristic
70C2×D1004C2311yescharacteristic
71C2×C22⋊D251—21no—
72C2×C22⋊D251—21no—
73C2×C22⋊D251—21no—
74C2×C22⋊D251—21no—
75C22×Dic251—21no—
76C22×Dic251—21no—
77C2×C4×D251—21no—
78C2×C4×D251—21no—
79C2×C4×D251—21no—
80#16860799—41no—
81D2008—22no—
82D2008—41no—
83C4⋊Dic251—21no—
84C4⋊Dic251—21no—
85C4⋊Dic251—21no—
86C4⋊Dic251C2311yescharacteristic
87C4⋊Dic251C2311yescharacteristic
88C4⋊Dic251C2311yescharacteristic
89C4⋊Dic251C2311yescharacteristic
90Dic251⋊C4—21no—
91C22⋊Dic251—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—
97(C2×D251)⋊C4—21no—
98(C2×D251)⋊C4—21no—
99C2×D4×C251D411yescharacteristic
100C22×C4×C251C2311yescharacteristic
101C22×C4×C251C2311yescharacteristic
102M16×C251—41no—
103C2×C8×C251—41no—
104C2×C8×C251C2311yescharacteristic
105C2×C8×C251C2311yescharacteristic
106C2×C8×C251C2311yescharacteristic
107C2×C8×C251C2311yescharacteristic
108C4⋊C4×C251D411yescharacteristic
109C22⋊C4×C251D411yescharacteristic
110C22⋊C4×C251D411yescharacteristic
111C42×C251C2311yescharacteristic
112C22×D251—21no—
113C22×D251—21no—
114C22×D251—21no—
115C22×D251—42no—
116C22×D251—42no—
117C22×D251—81no—
118Dic502—21no—
119Dic502—21no—
120Dic502—42no—
121D1004—21no—
122D1004—21no—
123D1004—21no—
124D1004—21no—
125D1004—21no—
126D1004—42no—
127D1004#4112yescomplemented
128C22⋊D251—42no—
129C22⋊D251—42no—
130C22⋊D251—42no—
131C22⋊D251—81no—
132C2×Dic251—21no—
133C2×Dic251—21no—
134C2×Dic251—21no—
135C2×Dic251—21no—
136C2×Dic251—21no—
137C2×Dic251—42no—
138C2×Dic251—42no—
139C4×D251—42no—
140C4×D251—42no—
141C4×D251—81no—
142C23×C251C2×D411yescharacteristic
143C23×C251C2×D411yescharacteristic
144D4×C251—22no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251C2×D411yescharacteristic
150C2×C4×C251C2×D411yescharacteristic
151C2×C4×C251C2×D411yescharacteristic
152C2×C4×C251C2×D411yescharacteristic
153C2×C4×C251C2411yescharacteristic
154C8×C251—21no—
155C8×C251—21no—
156C8×C251—21no—
157C8×C251—21no—
158C22×C251—21no—
159C22×C251—21no—
160C22×C251—21no—
161C22×C251—21no—
162C22×C251—21no—
163C22×C251AGL(1,ℤ/8)12yes—
164C22×C251C22×D411yescharacteristic
165C2×D251—22no—
166C2×D251—41no—
167C2×D251—41no—
168C2×D251—41no—
169C2×D251—41no—
170C2×D251—41no—
171C2×D251—41no—
172C2×D251—41no—
173C2×D251—41no—
174C2×D251—84no—
175C4×C251—21no—
176C4×C251—21no—
177C4×C251—21no—
178C4×C251—21no—
179C4×C251C2×#4111yescharacteristic
180C4×C251H(5,𝔽2)11yescharacteristic
181Dic251—41no—
182Dic251—41no—
183Dic251—41no—
184Dic251—41no—
185Dic251—41no—
186C2×C251—22no—
187C2×C251—41no—
188C2×C251#54611yescharacteristic
189C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C251#57711yescharacteristic
191D251—42no—
192D251—81no—
193D251—81no—
194C251#330611yescharacteristic, complemented, Sylow
195#3306—2511nomaximal, Sylow
196#547—2511no—
197#546—2511no—
198#489—2511no—
199#483—2511no—
200#481—2511no—
201#476—2511no—
202#469—2511no—
203#465—2511no—
204#463—2511no—
205#449—2511no—
206#447—2511no—
207#442—2511no—
208#435D25111yescharacteristic, complemented
209#417—2511no—
210C2×#102—2511no—
211C2×#41—5021no—
212C2×#41—2511no—
213C22×D4—2511no—
214C2×SD16—2511no—
215C2×D8—2511no—
216C2×D8—2511no—
217C2×M16C2×D25111yescharacteristic, complemented
218C22×C8C2×D25111yescharacteristic
219C4⋊Q8—2511no—
220#124—2511no—
221#123—5021no—
222#123—5021no—
223#123—5021no—
224C22⋊Q8—5021no—
225C22⋊Q8—2511no—
226#121—5021no—
227#121—2511no—
228#121—2511no—
229#121—2511no—
230#120—5021no—
231C4×D4—2511no—
232C4×D4—2511no—
233C4×D4C2×D25111yescharacteristic
234C2×C4⋊C4—2511no—
235C2×C4⋊C4—2511no—
236C2×C22⋊C4—5021no—
237#106—2511no—
238#106—2511no—
239#106—2511no—
240C4⋊C8C2×D25111yescharacteristic, complemented
241#103—2511no—
242#103—2511no—
243#103—2511no—
244#102—2511no—
245#102—2511no—
246#102—2511no—
247#102—2511no—
248#102—2511no—
249#102—2511no—
250#102—2511no—
251C22⋊C8C2×D25111yescharacteristic
252C22⋊C8C2×D25111yescharacteristic, complemented
253#97C2×D25111yescharacteristic, complemented
254C24—5021no—
255#41—10041no—
256#41—10044no—
257#41—5022no—
258C2×Q8—5021no—
259C2×Q8—2511no—
260C2×D4—10041no—
261C2×D4—5022no—
262C2×D4—5021no—
263C2×D4—5021no—
264C2×D4—5021no—
265C2×D4—5021no—
266C2×D4—5021no—
267C2×D4—2511no—
268C2×D4—2511no—
269C2×D4—2511no—
270C2×D4D100411yescharacteristic
271C22×C4—5021no—
272C22×C4—5021no—
273C22×C4—5021no—
274C22×C4—5021no—
275C22×C4—5021no—
276C22×C4C22×D25111yescharacteristic
277C22×C4C22×D25111yescharacteristic
278SD16—10041no—
279D8—10041no—
280D8—5022no—
281M16—41no—
282C2×C8—41no—
283C2×C8C22×D25111yescharacteristic
284C2×C8C22×D25111yescharacteristic
285C2×C8C22×D25111yescharacteristic
286C2×C8C22×D25111yescharacteristic
287C4⋊C4—5021no—
288C4⋊C4—5021no—
289C4⋊C4—5021no—
290C4⋊C4—5021no—
291C4⋊C4—2511no—
292C4⋊C4—2511no—
293C4⋊C4—2511no—
294C4⋊C4—2511no—
295C4⋊C4D100411yescharacteristic
296C22⋊C4—5021no—
297C22⋊C4—5021no—
298C22⋊C4—5021no—
299C22⋊C4—5021no—
300C22⋊C4—5021no—
301C22⋊C4—5021no—
302C22⋊C4—5021no—
303C22⋊C4—5021no—
304C22⋊C4D100411yescharacteristic
305C22⋊C4D100411yescharacteristic
306C42C22×D25111yescharacteristic
307C23—20081no—
308C23—10042no—
309C23—10042no—
310C23—5021no—
311C23—5021no—
312C23—5021no—
313C23C2×D100411yescharacteristic
314C23C2×D100411yescharacteristic
315Q8—10042no—
316Q8—5021no—
317Q8—5021no—
318D4—22no—
319D4—20081no—
320D4—10042no—
321D4—10042no—
322D4—10042no—
323D4—10042no—
324D4—5021no—
325D4—5021no—
326D4—5021no—
327D4—5021no—
328D4—5021no—
329D4—2512no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—20081no—
335C2×C4—10042no—
336C2×C4—10042no—
337C2×C4—10042no—
338C2×C4—10042no—
339C2×C4—5021no—
340C2×C4—5021no—
341C2×C4—5021no—
342C2×C4—5021no—
343C2×C4—5021no—
344C2×C4C2×D100411yescharacteristic
345C2×C4C2×D100411yescharacteristic
346C2×C4C2×D100411yescharacteristic
347C2×C4C2×D100411yescharacteristic
348C2×C4C23×D25111yescharacteristic
349C8—21no—
350C8—21no—
351C8—21no—
352C8—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—20084no—
359C22—10041no—
360C22—10041no—
361C22—10041no—
362C22—10041no—
363C22—10041no—
364C22—10041no—
365C22—10041no—
366C22—10041no—
367C22—5022no—
368C22#2254874312yes—
369C22C22×D100411yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—10041no—
375C4—10041no—
376C4—10041no—
377C4—10041no—
378C4—10041no—
379C4#41×D25111yescharacteristic
380C4#2254876011yescharacteristic
381C2—22no—
382C2—41no—
383C2—20081no—
384C2—20081no—
385C2—10042no—
386C2#3063233211yescharacteristic
387C2C2×#2254874311yescharacteristic
388C2#3063249911yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#435×C251C2
socleC22×C251C22×D4
hypercenter#435D251
hypocenterC251#3306
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #30632341C2semidirect
2. #30632333C2semidirect
3. #30632047C2semidirect
4. #30632018C2semidirect
5. #30632006C2semidirect
6. #30631978C2semidirect
7. #30631940C2semidirect
8. #30631916C2semidirect
9. #30631903C2semidirect
10. #30631834C2semidirect
11. #30631823C2semidirect
12. #30631800C2semidirect
13. #30631689C2semidirect
14. C2×#22548637C2semidirect
15. #435×C251C2semidirect
19. C2×#16860799C22semidirect
22. #22548715C22semidirect
31. #22548689C22semidirect
35. C4×D1004C22semidirect
37. C2×C4⋊Dic251C22semidirect
39. #22548647C22semidirect
40. #22548647C22semidirect
41. #22548647C22semidirect
42. #22548640C22semidirect
43. #22548640C22semidirect
44. #22548640C22semidirect
51. #22548637C22semidirect
52. C2×M16×C251C22semidirect
55. C4⋊C8×C251C22semidirect
57. C22⋊C8×C251C22semidirect
58. #97×C251C22semidirect
127. D1004#41semidirect
194. C251#3306semidirect
208. #435D251semidirect
217. C2×M16C2×D251semidirect
240. C4⋊C8C2×D251semidirect
252. C22⋊C8C2×D251semidirect
253. #97C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323412
#306323332
#306320472
#306320182
#306320062
#306319782
#306319402
#306319162
#306319032
#306318342
#306318232
#306318002
#306316892
C2×#225486372
#435×C2512
#3306251