Finite Group Databasebeta

← #44560304

Subgroups — #44560304

Order 32128.

462 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×D251C211yesmaximal, characteristic
2#117×D251C211yesmaximal, characteristic, complemented
3#30632012C211yesmaximal, characteristic, complemented
4#30632011C211yesmaximal, characteristic, complemented
5#30632008C211yesmaximal, characteristic, complemented
6#30632007C211yesmaximal, characteristic, complemented
7#30632004C211yesmaximal, characteristic, complemented
8#30632002C211yesmaximal, characteristic, complemented
9#30631999C211yesmaximal, characteristic, complemented
10#30631728C211yesmaximal, characteristic, complemented
11#30631726C211yesmaximal, characteristic, complemented
12#30631690C211yesmaximal, characteristic, complemented
13#30631690C211yesmaximal, characteristic, complemented
14#30631649C211yesmaximal, characteristic, complemented
15#482×C251C211yesmaximal, characteristic, complemented
16C2×D4×D251—21no—
17C2×D4×D251C2211yescharacteristic
18C22×C4×D251C2211yescharacteristic
19C2×#16860793C2211yescharacteristic
20C2×#16860793C2211yescharacteristic
21#22548690C2211yescharacteristic
22#22548689C2211yescharacteristic
23#22548688—21no—
24#22548687—21no—
25#22548686C2211yescharacteristic
26#22548674C2211yescharacteristic, complemented
27#22548673C2211yescharacteristic
28#22548673C2211yescharacteristic
29#22548672—21no—
30#22548671—21no—
31C4⋊C4×D251C2211yescharacteristic
32C4⋊C4×D251C2211yescharacteristic, complemented
33C4⋊C4×D251C2211yescharacteristic, complemented
34C22⋊C4×D251—21no—
35C22⋊C4×D251—21no—
36C42×D251—21no—
37#22548649C2211yescharacteristic, complemented
38#22548648C2211yescharacteristic, complemented
39#22548648C2211yescharacteristic, complemented
40#22548647C2211yescharacteristic, complemented
41#22548646C2211yescharacteristic, complemented
42#22548646C2211yescharacteristic, complemented
43#22548637C2211yescharacteristic, complemented
44#22548637C2211yescharacteristic, complemented
45#22548636C2211yescharacteristic, complemented
46#22548636C2211yescharacteristic, complemented
47#22548635C2211yescharacteristic, complemented
48#22548635C2211yescharacteristic, complemented
49C22⋊(C251⋊C8)C2211yescharacteristic, complemented
50(C2×D251)⋊C8C2211yescharacteristic, complemented
51(C2×D251)⋊C8C2211yescharacteristic, complemented
52#121×C251C2211yescharacteristic
53#117×C251C2211yescharacteristic, complemented
54#107×C251C2211yescharacteristic, complemented
55#106×C251C2211yescharacteristic, complemented
56#102×C251C2211yescharacteristic, complemented
57#102×C251C2211yescharacteristic, complemented
58C22⋊C8×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60C23×D251D411yescharacteristic
61C2×D1004—21no—
62C2×D1004C2311yescharacteristic
63C2×C22⋊D251—21no—
64C2×C22⋊D251—21no—
65C2×C22⋊D251—21no—
66D4×D251—22no—
67D4×D251—22no—
68D4×D251—44no—
69C22×Dic251D411yescharacteristic
70C2×C4×D251—22no—
71C2×C4×D251—22no—
72C2×C4×D251—21no—
73C2×C4×D251—21no—
74C2×C4×D251—21no—
75C2×C4×D251D411yescharacteristic
76C2×C4×D251D411yescharacteristic
77C2×C4×D251C2311yescharacteristic
78C2×C4×D251C2311yescharacteristic
79#16860793—22no—
80#16860793—22no—
81C2×C251⋊C8C2311yescharacteristic
82C2×C251⋊C8C2311yescharacteristic
83C4⋊Dic251C2311yescharacteristic
84C4⋊Dic251C2311yescharacteristic
85C4⋊Dic251C2311yescharacteristic
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88Dic251⋊C4—21no—
89C22⋊Dic251—21no—
90C22⋊Dic251—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95C4×Dic251—21no—
96C4×Dic251—21no—
97C4×Dic251—21no—
98C2×D4×C251—21no—
99C2×D4×C251C2311yescharacteristic
100C22×C4×C251C2311yescharacteristic
101C2×C8×C251C2311yescharacteristic
102C2×C8×C251C2311yescharacteristic
103C4⋊C4×C251C2311yescharacteristic
104C4⋊C4×C251C2311yescharacteristic
105C4⋊C4×C251C2311yescharacteristic
106C22⋊C4×C251—21no—
107C22⋊C4×C251—21no—
108C42×C251—21no—
109C22×D251—22no—
110C22×D251—22no—
111C22×D251—21no—
112C22×D251—41no—
113C22×D251—42no—
114C22×D251—42no—
115C22×D251—42no—
116C22×D251C2×D411yescharacteristic
117C22×D251C2×D411yescharacteristic
118D1004—21no—
119D1004—21no—
120D1004—42no—
121C22⋊D251—42no—
122C22⋊D251—42no—
123C22⋊D251—42no—
124C2×Dic251—22no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251—21no—
128C2×Dic251C2×D411yescharacteristic
129C2×Dic251C2×D411yescharacteristic
130C4×D251—22no—
131C4×D251—21no—
132C4×D251—21no—
133C4×D251—22no—
134C4×D251—42no—
135C4×D251—42no—
136C4×D251—42no—
137C4×D251#4112yes—
138C4×D251#4112yes—
139C251⋊C8—21no—
140C251⋊C8—21no—
141C23×C251—21no—
142C23×C251C2×D411yescharacteristic
143D4×C251—21no—
144D4×C251—21no—
145D4×C251—42no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251—21no—
150C2×C4×C251—21no—
151C2×C4×C251C2×D411yescharacteristic
152C2×C4×C251C2411yescharacteristic
153C8×C251—21no—
154C8×C251—21no—
155C22×C251—21no—
156C22×C251—21no—
157C22×C251—21no—
158C22×C251—41no—
159C22×C251—41no—
160C22×C251—41no—
161C22×C251C22×D411yescharacteristic
162C2×D251—21no—
163C2×D251—21no—
164C2×D251—21no—
165C2×D251—21no—
166C2×D251—44no—
167C2×D251—41no—
168C2×D251—41no—
169C2×D251—41no—
170C2×D251—84no—
171C2×D251#12312yes—
172C2×D251AGL(1,ℤ/8)12yes—
173C4×C251—21no—
174C4×C251—41no—
175C4×C251—41no—
176C4×C251—41no—
177C4×C251C2×#4111yescharacteristic
178C4×C251C2×#4111yescharacteristic
179Dic251—21no—
180Dic251—41no—
181Dic251—41no—
182Dic251—41no—
183Dic251#12312yes—
184C2×C251—41no—
185C2×C251—81no—
186C2×C251C2×#12311yescharacteristic
187C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
188C2×C251#57511yescharacteristic
189D251—22no—
190D251—22no—
191D251—81no—
192C251#309411yescharacteristic, complemented, Sylow
193#3094—2511nomaximal, Sylow
194C2×#121—2511no—
195C2×#117—2511no—
196#482—2511no—
197#482—2511no—
198#482—2511no—
199#482D25111yescharacteristic, complemented
200#481—2511no—
201#481—2511no—
202#480—2511no—
203#480—2511no—
204#428—2511no—
205#428—2511no—
206#418—2511no—
207#418—2511no—
208#407—2511no—
209C22×D4—5021no—
210C22×D4—2511no—
211C23×C4—2511no—
212C2×M16—2511no—
213C2×M16—2511no—
214#121—5021no—
215#121—5021no—
216#121—2511no—
217#121—2511no—
218#121—2511no—
219#121C2×D25111yescharacteristic
220#117—5021no—
221#117—5021no—
222#117—2511no—
223#117—2511no—
224#117—2511no—
225#117C2×D25111yescharacteristic, complemented
226C2×C4⋊C4—2511no—
227C2×C4⋊C4—2511no—
228C2×C4⋊C4—2511no—
229C2×C22⋊C4—5021no—
230C2×C22⋊C4—5021no—
231C2×C42—5021no—
232#107—2511no—
233#107—2511no—
234#107—2511no—
235#107C2×D25111yescharacteristic, complemented
236#106—2511no—
237#106—2511no—
238#106—2511no—
239#106C2×D25111yescharacteristic, complemented
240#102—2511no—
241#102—2511no—
242#102—2511no—
243#102—2511no—
244#102—2511no—
245#102—2511no—
246#102C2×D25111yescharacteristic, complemented
247#102C2×D25111yescharacteristic, complemented
248C22⋊C8—2511no—
249C22⋊C8—2511no—
250C22⋊C8—2511no—
251C22⋊C8C2×D25111yescharacteristic, complemented
252C24—5021no—
253C24—2511no—
254C2×D4—21no—
255C2×D4—10044no—
256C2×D4—5021no—
257C2×D4—5021no—
258C2×D4—5021no—
259C2×D4—5022no—
260C2×D4—5022no—
261C2×D4—5021no—
262C2×D4—2511no—
263C2×D4C22×D25111yescharacteristic
264C22×C4—5021no—
265C22×C4—5021no—
266C22×C4—5021no—
267C22×C4—5022no—
268C22×C4—5022no—
269C22×C4—2511no—
270C22×C4—2511no—
271C22×C4—2511no—
272C22×C4—2511no—
273C22×C4—2511no—
274C22×C4C22×D25111yescharacteristic
275M16—5022no—
276M16—5022no—
277C2×C8—2511no—
278C2×C8—2511no—
279C2×C8C22×D25111yescharacteristic
280C2×C8C22×D25111yescharacteristic
281C4⋊C4—5021no—
282C4⋊C4—5021no—
283C4⋊C4—5021no—
284C4⋊C4—2511no—
285C4⋊C4—2511no—
286C4⋊C4—2511no—
287C4⋊C4C22×D25111yescharacteristic
288C4⋊C4C22×D25111yescharacteristic
289C4⋊C4C22×D25111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—5021no—
293C22⋊C4—5021no—
294C22⋊C4—5021no—
295C22⋊C4—5021no—
296C22⋊C4—5021no—
297C22⋊C4—5021no—
298C42—21no—
299C42—5021no—
300C42—5021no—
301C42—5021no—
302C23—21no—
303C23—10041no—
304C23—10042no—
305C23—10042no—
306C23—10042no—
307C23—5022no—
308C23—5022no—
309C23—5021no—
310C23—2511no—
311C23—2511no—
312C23D4×D25111yescharacteristic
313D4—21no—
314D4—21no—
315D4—42no—
316D4—10042no—
317D4—10042no—
318D4—10042no—
319D4—10042no—
320D4—5021no—
321D4—5021no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—10042no—
328C2×C4—10042no—
329C2×C4—10042no—
330C2×C4—5022no—
331C2×C4—5022no—
332C2×C4—5021no—
333C2×C4—5021no—
334C2×C4—5021no—
335C2×C4—5022no—
336C2×C4—5021no—
337C2×C4—5021no—
338C2×C4—2511no—
339C2×C4—2511no—
340C2×C4—2512no—
341C2×C4—2512no—
342C2×C4D4×D25111yescharacteristic
343C2×C4C23×D25111yescharacteristic
344C8—21no—
345C8—21no—
346C8—5021no—
347C8—5021no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—20084no—
355C22—10044no—
356C22—10041no—
357C22—10041no—
358C22—10041no—
359C22—5021no—
360C22—5021no—
361C22—5021no—
362C22—5021no—
363C22—2512no—
364C22—2512no—
365C22C2×D4×D25111yescharacteristic
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—10041no—
371C4—10041no—
372C4—10041no—
373C4—5021no—
374C4—2512no—
375C4#41×D25111yescharacteristic
376C4#41×D25111yescharacteristic
377C2—41no—
378C2—81no—
379C2—20081no—
380C2—5022no—
381C2—5022no—
382C2#123×D25111yescharacteristic
383C2AGL(1,ℤ/8)×D25111yescharacteristic
384C2#3063248411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#482×C251C2
socleC22×C251C22×D4
prime radical#121×D251C2
hypercenter#482D251
hypocenterC251#3094
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #117×D251C2semidirect
3. #30632012C2semidirect
4. #30632011C2semidirect
5. #30632008C2semidirect
6. #30632007C2semidirect
7. #30632004C2semidirect
8. #30632002C2semidirect
9. #30631999C2semidirect
10. #30631728C2semidirect
11. #30631726C2semidirect
12. #30631690C2semidirect
13. #30631690C2semidirect
14. #30631649C2semidirect
15. #482×C251C2semidirect
26. #22548674C22semidirect
32. C4⋊C4×D251C22semidirect
33. C4⋊C4×D251C22semidirect
37. #22548649C22semidirect
38. #22548648C22semidirect
39. #22548648C22semidirect
40. #22548647C22semidirect
41. #22548646C22semidirect
42. #22548646C22semidirect
43. #22548637C22semidirect
44. #22548637C22semidirect
45. #22548636C22semidirect
46. #22548636C22semidirect
47. #22548635C22semidirect
48. #22548635C22semidirect
49. C22⋊(C251⋊C8)C22semidirect
50. (C2×D251)⋊C8C22semidirect
51. (C2×D251)⋊C8C22semidirect
53. #117×C251C22semidirect
54. #107×C251C22semidirect
55. #106×C251C22semidirect
56. #102×C251C22semidirect
57. #102×C251C22semidirect
58. C22⋊C8×C251C22semidirect
192. C251#3094semidirect
199. #482D251semidirect
225. #117C2×D251semidirect
235. #107C2×D251semidirect
239. #106C2×D251semidirect
246. #102C2×D251semidirect
247. #102C2×D251semidirect
251. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#121×D2512
#117×D2512
#306320122
#306320112
#306320082
#306320072
#306320042
#306320022
#306319992
#306317282
#306317262
#306316902
#306316492
#482×C2512
#3094251