Finite Group Databasebeta

← #44561852

Subgroups — #44561852

Order 32128.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548743C211yesmaximal, characteristic, complemented
2C2×D8×D251C211yesmaximal, characteristic, complemented
3#30632334C211yesmaximal, characteristic, complemented
4#30632321C211yesmaximal, characteristic, complemented
5#30632075C211yesmaximal, characteristic, complemented
6#30631917C211yesmaximal, characteristic, complemented
7#30631901C211yesmaximal, characteristic, complemented
8#30631881C211yesmaximal, characteristic, complemented
9#30631839C211yesmaximal, characteristic, complemented
10#30631830C211yesmaximal, characteristic, complemented
11#30631822C211yesmaximal, characteristic, complemented
12#30631820C211yesmaximal, characteristic, complemented
13#30631798C211yesmaximal, characteristic, complemented
14#30631756C211yesmaximal, characteristic, complemented
15#469×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic, complemented
17C22×D1004C2211yescharacteristic, complemented
18C2×D4×D251—21no—
19C2×D4×D251C2211yescharacteristic
20C2×D4×D251C2211yescharacteristic, complemented
21#22548743—24no—
22C2×#16860799C2211yescharacteristic, complemented
23C2×#16860799C2211yescharacteristic, complemented
24C2×#16860798C2211yescharacteristic, complemented
25C2×D2008C2211yescharacteristic
26C2×D2008C2211yescharacteristic, complemented
27C2×#16860795C2211yescharacteristic
28C2×#16860795C2211yescharacteristic, complemented
29C2×#16860795C2211yescharacteristic, complemented
30D8×D251—24no—
31C2×C8×D251C2211yescharacteristic, complemented
32#22548711C2211yescharacteristic, complemented
33#22548702C2211yescharacteristic, complemented
34#22548697—21no—
35#22548695C2211yescharacteristic, complemented
36#22548690C2211yescharacteristic
37#22548688—21no—
38#22548686—21no—
39#22548685—21no—
40#22548683—21no—
41#22548683—21no—
42#22548677C2211yescharacteristic, complemented
43#22548677C2211yescharacteristic, complemented
44C4×C22⋊D251C2211yescharacteristic
45C22⋊C4×D251—21no—
46Dic251⋊C8C2211yescharacteristic, complemented
47#22548638C2211yescharacteristic, complemented
48#22548635C2211yescharacteristic, complemented
49#22548635C2211yescharacteristic, complemented
50#22548635C2211yescharacteristic, complemented
51C22⋊(C251⋊C8)C2211yescharacteristic, complemented
52(C2×D251)⋊C8C2211yescharacteristic, complemented
53#22548625C2211yescharacteristic, complemented
54C2×D8×C251C2211yescharacteristic, complemented
55C2×M16×C251C2211yescharacteristic, complemented
56#121×C251C2211yescharacteristic, complemented
57#121×C251C2211yescharacteristic, complemented
58#106×C251C2211yescharacteristic, complemented
59#102×C251C2211yescharacteristic, complemented
60#102×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62C23×D251—21no—
63C23×D251—21no—
64#16860810—22no—
65#16860810—41no—
66C2×Dic502C2311yescharacteristic
67C2×D1004—22no—
68C2×D1004—21no—
69C2×D1004—21no—
70C2×D1004—41no—
71C2×D1004C2311yescharacteristic
72C2×D1004C2311yescharacteristic
73C2×D1004C2311yescharacteristic
74C2×C22⋊D251—21no—
75C2×C22⋊D251—21no—
76C2×C22⋊D251—21no—
77C2×C22⋊D251—21no—
78C2×C22⋊D251—21no—
79C2×C22⋊D251D411yescharacteristic
80D4×D251—22no—
81D4×D251—22no—
82D4×D251—41no—
83D4×D251—41no—
84D4×D251—44no—
85C2×C4×D251—21no—
86C2×C4×D251—21no—
87C2×C4×D251—21no—
88C2×C4×D251C2311yescharacteristic
89#16860799—22no—
90#16860799—22no—
91#16860798—41no—
92D2008—22no—
93D2008D414yescomplemented
94#16860795—22no—
95#16860795—22no—
96#16860795—41no—
97C2×C251⋊C8C2311yescharacteristic
98C2×C251⋊C8C2311yescharacteristic
99C8×D251—22no—
100C4⋊Dic251—21no—
101Dic251⋊C4D411yescharacteristic
102C22⋊Dic251—21no—
103C22⋊Dic251—21no—
104C22⋊Dic251D411yescharacteristic
105(C2×D251)⋊C4—21no—
106(C2×D251)⋊C4—21no—
107(C2×D251)⋊C4—21no—
108(C2×D251)⋊C4—21no—
109(C2×D251)⋊C4D411yescharacteristic
110C4×Dic251C2311yescharacteristic
111C2×D4×C251—21no—
112C2×D4×C251—21no—
113C2×D4×C251C2311yescharacteristic
114C2×D4×C251C2311yescharacteristic
115C22×C4×C251C2311yescharacteristic
116D8×C251—22no—
117M16×C251—22no—
118C2×C8×C251C2311yescharacteristic
119C2×C8×C251C2311yescharacteristic
120C4⋊C4×C251C2311yescharacteristic
121C4⋊C4×C251C2311yescharacteristic
122C22⋊C4×C251—21no—
123C22⋊C4×C251—21no—
124C22×D251—21no—
125C22×D251—21no—
126C22×D251—21no—
127C22×D251—42no—
128C22×D251—42no—
129C22×D251—42no—
130C22×D251—42no—
131C22×D251—42no—
132C22×D251—42no—
133C22×D251—81no—
134C22×D251—81no—
135C22×D251—81no—
136C22×D251C2×D411yescharacteristic
137Dic502—21no—
138Dic502—21no—
139D1004—21no—
140D1004—21no—
141D1004—21no—
142D1004—21no—
143D1004—42no—
144D1004—42no—
145D1004C2×D412yes—
146D1004C2×D412yescomplemented
147C22⋊D251—22no—
148C22⋊D251—42no—
149C22⋊D251—42no—
150C22⋊D251—42no—
151C22⋊D251—42no—
152C22⋊D251—81no—
153C2×Dic251—21no—
154C2×Dic251C2×D411yescharacteristic
155C2×Dic251C2×D411yescharacteristic
156C2×Dic251C2×D411yescharacteristic
157C4×D251—21no—
158C4×D251—21no—
159C4×D251—42no—
160C4×D251—42no—
161C4×D251—81no—
162C251⋊C8—21no—
163C251⋊C8—21no—
164C23×C251—21no—
165C23×C251—21no—
166C23×C251C2×D411yescharacteristic
167D4×C251—21no—
168D4×C251—21no—
169D4×C251—21no—
170D4×C251—21no—
171D4×C251—42no—
172D4×C251—81no—
173C2×C4×C251—21no—
174C2×C4×C251—21no—
175C2×C4×C251—21no—
176C2×C4×C251—21no—
177C2×C4×C251C2×D411yescharacteristic
178C2×C4×C251C2411yescharacteristic
179C8×C251—21no—
180C8×C251C2×D412yescomplemented
181C22×C251—21no—
182C22×C251—21no—
183C22×C251—21no—
184C22×C251—41no—
185C22×C251—41no—
186C22×C251—41no—
187C22×C251—41no—
188C22×C251—41no—
189C22×C251—41no—
190C22×C251C22×D411yescharacteristic
191C2×D251—21no—
192C2×D251—21no—
193C2×D251—22no—
194C2×D251—22no—
195C2×D251—41no—
196C2×D251—41no—
197C2×D251—41no—
198C2×D251—41no—
199C2×D251—41no—
200C2×D251—41no—
201C2×D251—41no—
202C2×D251—84no—
203C2×D251—84no—
204C2×D251—84no—
205C2×D251AGL(1,ℤ/8)12yescomplemented
206C4×C251—21no—
207C4×C251—41no—
208C4×C251—41no—
209C4×C251C22×D411yescharacteristic
210C4×C251H(5,𝔽2)11yescharacteristic
211Dic251—21no—
212Dic251—21no—
213Dic251—21no—
214Dic251—41no—
215C2×C251—41no—
216C2×C251—81no—
217C2×C251—81no—
218C2×C251D4211yescharacteristic
219C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C251#57611yescharacteristic
221D251—22no—
222D251—42no—
223D251—42no—
224D251—81no—
225C251#326411yescharacteristic, complemented, Sylow
226#3264—2511nomaximal, Sylow
227C2×AGL(1,ℤ/8)—2511no—
228C22×D8—2511no—
229#546—2511no—
230D42—2511no—
231#496—2511no—
232#469D25111yescharacteristic, complemented
233#465—2511no—
234#463—2511no—
235#459—2511no—
236#450—2511no—
237#449—2511no—
238#447—2511no—
239#447—2511no—
240#442—2511no—
241#435—2511no—
242C2×#41—2511no—
243C22×D4—5021no—
244C22×D4—2511no—
245C22×D4—2511no—
246C22×D4—2511no—
247AGL(1,ℤ/8)—5024no—
248C2×SD16—2511no—
249C2×SD16—2511no—
250C2×SD16—2511no—
251C2×D8—5024no—
252C2×D8—2511no—
253C2×D8—2511no—
254C2×D8—2511no—
255C2×D8—2511no—
256C2×D8—2511no—
257C2×D8C2×D25111yescharacteristic, complemented
258C2×M16C2×D25111yescharacteristic, complemented
259C22×C8—2511no—
260#127—2511no—
261#124—2511no—
262#123—5021no—
263C22⋊Q8—2511no—
264#121—5021no—
265#121—5021no—
266#121—2511no—
267#121C2×D25111yescharacteristic, complemented
268#121C2×D25111yescharacteristic, complemented
269#120—5021no—
270#120—5021no—
271#120—5021no—
272C4×D4—2511no—
273C4×D4—2511no—
274C4×D4—2511no—
275C2×C22⋊C4—5021no—
276#106C2×D25111yescharacteristic, complemented
277C4⋊C8—2511no—
278#103—2511no—
279#102—2511no—
280#102—2511no—
281#102—2511no—
282#102C2×D25111yescharacteristic, complemented
283#102C2×D25111yescharacteristic, complemented
284C22⋊C8—2511no—
285C22⋊C8—2511no—
286#97—2511no—
287C24—5021no—
288C24—5021no—
289C24—5021no—
290#41—10041no—
291#41—5022no—
292C2×Q8—2511no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—10041no—
296C2×D4—10044no—
297C2×D4—10041no—
298C2×D4—10041no—
299C2×D4—5021no—
300C2×D4—5022no—
301C2×D4—5022no—
302C2×D4—5022no—
303C2×D4—5021no—
304C2×D4—5021no—
305C2×D4—5021no—
306C2×D4—5021no—
307C2×D4—5021no—
308C2×D4—5021no—
309C2×D4—2511no—
310C2×D4—2511no—
311C2×D4—2511no—
312C2×D4—2511no—
313C2×D4C22×D25111yescharacteristic
314C2×D4C22×D25111yescharacteristic
315C22×C4—5021no—
316C22×C4—5021no—
317C22×C4—5021no—
318C22×C4—2511no—
319C22×C4C22×D25111yescharacteristic
320SD16—10041no—
321SD16—5022no—
322SD16—5022no—
323D8—22no—
324D8—10041no—
325D8—5022no—
326D8—5022no—
327D8—5022no—
328D8—2514no—
329M16—22no—
330C2×C8—5022no—
331C2×C8—2511no—
332C2×C8—2511no—
333C2×C8C22×D25111yescharacteristic
334C2×C8C22×D25111yescharacteristic
335C4⋊C4—5021no—
336C4⋊C4—2511no—
337C4⋊C4C22×D25111yescharacteristic
338C4⋊C4C22×D25111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—5021no—
342C22⋊C4—5021no—
343C22⋊C4—5021no—
344C22⋊C4—5021no—
345C22⋊C4—5021no—
346C22⋊C4—5021no—
347C22⋊C4—2511no—
348C22⋊C4—2511no—
349C42—2511no—
350C23—21no—
351C23—21no—
352C23—20081no—
353C23—20081no—
354C23—20081no—
355C23—10042no—
356C23—10042no—
357C23—10042no—
358C23—10042no—
359C23—10042no—
360C23—10042no—
361C23—5021no—
362C23—5021no—
363C23—5021no—
364C23—2511no—
365C23D4×D25111yescharacteristic
366Q8—5021no—
367Q8—5021no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—81no—
374D4—20081no—
375D4—10042no—
376D4—10042no—
377D4—10042no—
378D4—10042no—
379D4—10042no—
380D4—10042no—
381D4—5022no—
382D4—5021no—
383D4—5021no—
384D4—5021no—
385D4—5021no—
386D4—2512no—
387D4—2512no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—20081no—
393C2×C4—10042no—
394C2×C4—10042no—
395C2×C4—5021no—
396C2×C4—5021no—
397C2×C4—5021no—
398C2×C4—2511no—
399C2×C4—2511no—
400C2×C4—2511no—
401C2×C4D4×D25111yescharacteristic
402C2×C4C23×D25111yescharacteristic
403C8—21no—
404C8—5021no—
405C8—5021no—
406C8D4×D25112yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—20084no—
417C22—20084no—
418C22—20084no—
419C22—10041no—
420C22—10041no—
421C22—10041no—
422C22—10041no—
423C22—10041no—
424C22—10041no—
425C22—10041no—
426C22—5022no—
427C22—5022no—
428C22—5021no—
429C22—5021no—
430C22—2512no—
431C22C2×D4×D25111yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—10041no—
436C4—5021no—
437C4—5021no—
438C4—5021no—
439C4C2×D4×D25111yescharacteristic
440C4#2254876111yescharacteristic
441C2—41no—
442C2—81no—
443C2—81no—
444C2—20081no—
445C2—10042no—
446C2—10042no—
447C2—5022no—
448C2#3063232111yescharacteristic
449C2AGL(1,ℤ/8)×D25111yescharacteristic
450C2#3063249111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#469×C251C2
socleC22×C251C22×D4
hypercenter#469D251
hypocenterC251#3264
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#22548743C2semidirect
2. C2×D8×D251C2semidirect
3. #30632334C2semidirect
4. #30632321C2semidirect
5. #30632075C2semidirect
6. #30631917C2semidirect
7. #30631901C2semidirect
8. #30631881C2semidirect
9. #30631839C2semidirect
10. #30631830C2semidirect
11. #30631822C2semidirect
12. #30631820C2semidirect
13. #30631798C2semidirect
14. #30631756C2semidirect
15. #469×C251C2semidirect
16. C2×#16860810C22semidirect
17. C22×D1004C22semidirect
20. C2×D4×D251C22semidirect
22. C2×#16860799C22semidirect
23. C2×#16860799C22semidirect
24. C2×#16860798C22semidirect
26. C2×D2008C22semidirect
28. C2×#16860795C22semidirect
29. C2×#16860795C22semidirect
31. C2×C8×D251C22semidirect
32. #22548711C22semidirect
33. #22548702C22semidirect
35. #22548695C22semidirect
42. #22548677C22semidirect
43. #22548677C22semidirect
46. Dic251⋊C8C22semidirect
47. #22548638C22semidirect
48. #22548635C22semidirect
49. #22548635C22semidirect
50. #22548635C22semidirect
51. C22⋊(C251⋊C8)C22semidirect
52. (C2×D251)⋊C8C22semidirect
53. #22548625C22semidirect
54. C2×D8×C251C22semidirect
55. C2×M16×C251C22semidirect
56. #121×C251C22semidirect
57. #121×C251C22semidirect
58. #106×C251C22semidirect
59. #102×C251C22semidirect
60. #102×C251C22semidirect
93. D2008D4semidirect
146. D1004C2×D4semidirect
180. C8×C251C2×D4semidirect
205. C2×D251AGL(1,ℤ/8)semidirect
225. C251#3264semidirect
232. #469D251semidirect
257. C2×D8C2×D251semidirect
258. C2×M16C2×D251semidirect
267. #121C2×D251semidirect
268. #121C2×D251semidirect
276. #106C2×D251semidirect
282. #102C2×D251semidirect
283. #102C2×D251semidirect
406. C8D4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487432
C2×D8×D2512
#306323342
#306323212
#306320752
#306319172
#306319012
#306318812
#306318392
#306318302
#306318222
#306318202
#306317982
#306317562
#469×C2512
#3264251