Finite Group Databasebeta

← #27182611

Subgroups — #27182611

Order 12928.

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×#15692282C211yesmaximal, characteristic, complemented
2C2×D8×D101C211yesmaximal, characteristic, complemented
3#20265820C211yesmaximal, characteristic, complemented
4#20265812C211yesmaximal, characteristic, complemented
5#20265427C211yesmaximal, characteristic, complemented
6#20265403C211yesmaximal, characteristic, complemented
7#20265389C211yesmaximal, characteristic, complemented
8#20265369C211yesmaximal, characteristic, complemented
9#20265326C211yesmaximal, characteristic, complemented
10#20265321C211yesmaximal, characteristic, complemented
11#20265309C211yesmaximal, characteristic, complemented
12#20265309C211yesmaximal, characteristic, complemented
13#20265290C211yesmaximal, characteristic, complemented
14#20265250C211yesmaximal, characteristic, complemented
15#496×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic, complemented
17C2×D4×D101—21no—
18C2×D4×D101C2211yescharacteristic
19C2×D4×D101C2211yescharacteristic, complemented
20C2×D4×D101C2211yescharacteristic, complemented
21#15692282—24no—
22C2×#13108061C2211yescharacteristic, complemented
23C2×D808C2211yescharacteristic
24C2×#13108059C2211yescharacteristic
25C2×#13108059C2211yescharacteristic, complemented
26C2×#13108059C2211yescharacteristic, complemented
27D8×D101—24no—
28C2×#13108057C2211yescharacteristic, complemented
29C2×C8×D101C2211yescharacteristic, complemented
30#15692237—21no—
31#15692231C2211yescharacteristic, complemented
32#15692228—21no—
33#15692226—21no—
34#15692226C2211yescharacteristic
35#15692226C2211yescharacteristic, complemented
36#15692226C2211yescharacteristic, complemented
37#15692225—21no—
38#15692223—21no—
39#15692223—21no—
40C4×D404C2211yescharacteristic
41C4×D404C2211yescharacteristic, complemented
42C4×D404C2211yescharacteristic, complemented
43C22⋊C4×D101—21no—
44#15692187C2211yescharacteristic, complemented
45C4⋊#11717579C2211yescharacteristic, complemented
46#15692179C2211yescharacteristic, complemented
47#15692177C2211yescharacteristic, complemented
48#15692177C2211yescharacteristic, complemented
49#15692176C2211yescharacteristic, complemented
50#15692176C2211yescharacteristic, complemented
51#15692176C2211yescharacteristic, complemented
52#15692167C2211yescharacteristic, complemented
53#15692167C2211yescharacteristic, complemented
54C2×SD16×C101C2211yescharacteristic, complemented
55C2×SD16×C101C2211yescharacteristic, complemented
56C2×D8×C101C2211yescharacteristic, complemented
57C2×D8×C101C2211yescharacteristic, complemented
58#127×C101C2211yescharacteristic, complemented
59#124×C101C2211yescharacteristic, complemented
60#97×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C23×D101—21no—
63C23×D101—21no—
64#13108075—22no—
65#13108075—41no—
66C2×D404—21no—
67C2×D404D411yescharacteristic
68C2×D404C2311yescharacteristic
69C2×D404C2311yescharacteristic
70C2×C22⋊D101—21no—
71C2×C22⋊D101—21no—
72C2×C22⋊D101—21no—
73C2×C22⋊D101—21no—
74C2×C22⋊D101—21no—
75C2×C22⋊D101—21no—
76D4×D101—22no—
77D4×D101—22no—
78D4×D101—22no—
79D4×D101—41no—
80D4×D101—41no—
81D4×D101—41no—
82D4×D101—44no—
83C2×C4×D101—21no—
84C2×C4×D101—21no—
85C2×C4×D101—21no—
86C2×C4×D101C2311yescharacteristic
87C2×C4×D101C2311yescharacteristic
88#13108061—22no—
89D808D414yescomplemented
90#13108059—22no—
91#13108059—22no—
92#13108059—22no—
93#13108057—22no—
94C2×#11717579C2311yescharacteristic
95C2×#11717579C2311yescharacteristic
96C8×D101—22no—
97C4⋊Dic101D411yescharacteristic
98C4⋊Dic101C2311yescharacteristic
99C4⋊Dic101C2311yescharacteristic
100#13108053—21no—
101C22⋊Dic101—21no—
102C22⋊Dic101—21no—
103C22⋊Dic101—21no—
104#13108051—21no—
105#13108051—21no—
106#13108051—21no—
107#13108051D411yescharacteristic
108#13108051D411yescharacteristic
109C2×Q8×C101C2311yescharacteristic
110C2×D4×C101—21no—
111C2×D4×C101—21no—
112C2×D4×C101C2311yescharacteristic
113C2×D4×C101C2311yescharacteristic
114C2×D4×C101C2311yescharacteristic
115SD16×C101—22no—
116SD16×C101—41no—
117D8×C101—22no—
118D8×C101—41no—
119C2×C8×C101C2311yescharacteristic
120C2×C8×C101C2311yescharacteristic
121C22⋊C4×C101—21no—
122C22⋊C4×C101—21no—
123C42×C101C2311yescharacteristic
124C23×C101—21no—
125C23×C101—21no—
126C23×C101—21no—
127C22×D101—21no—
128C22×D101—21no—
129C22×D101—42no—
130C22×D101—42no—
131C22×D101—42no—
132C22×D101—42no—
133C22×D101—42no—
134C22×D101—42no—
135C22×D101—81no—
136C22×D101—81no—
137C22×D101—81no—
138C22×D101C2×D411yescharacteristic
139C22×D101C2×D411yescharacteristic
140Q8×C101—21no—
141Q8×C101—21no—
142D4×C101—21no—
143D4×C101—21no—
144D4×C101—21no—
145D4×C101—21no—
146D4×C101—21no—
147D4×C101—21no—
148D4×C101—42no—
149D4×C101—81no—
150C2×C4×C101—21no—
151C2×C4×C101C2×D411yescharacteristic
152C2×C4×C101C2×D411yescharacteristic
153C2×C4×C101C2411yescharacteristic
154C22⋊D101—42no—
155C22⋊D101—42no—
156C22⋊D101—42no—
157C22⋊D101—42no—
158C22⋊D101—42no—
159C22⋊D101—81no—
160C2×Dic101—21no—
161C2×Dic101—21no—
162C2×Dic101C2×D411yescharacteristic
163C2×Dic101C2×D411yescharacteristic
164D404—22no—
165D404—21no—
166D404—21no—
167D404—42no—
168D404C2×D412yes—
169D404C2×D412yescomplemented
170C4×D101—21no—
171C4×D101—21no—
172C4×D101—21no—
173C4×D101—21no—
174C4×D101—42no—
175C4×D101—42no—
176C4×D101—81no—
177C8×C101—21no—
178C8×C101C2×D412yescomplemented
179#11717579—21no—
180#11717579—21no—
181C22×C101—41no—
182C22×C101—41no—
183C22×C101—41no—
184C22×C101—41no—
185C22×C101—41no—
186C22×C101—41no—
187C22×C101—41no—
188C22×C101—41no—
189C22×C101—41no—
190C22×C101C22×D411yescharacteristic
191C2×D101—21no—
192C2×D101—21no—
193C2×D101—21no—
194C2×D101—22no—
195C2×D101—22no—
196C2×D101—21no—
197C2×D101—21no—
198C2×D101—41no—
199C2×D101—41no—
200C2×D101—41no—
201C2×D101—41no—
202C2×D101—84no—
203C2×D101—84no—
204C2×D101—84no—
205C2×D101AGL(1,ℤ/8)12yescomplemented
206C4×C101—21no—
207C4×C101—21no—
208C4×C101—41no—
209C4×C101C22×D411yescharacteristic
210C4×C101H(5,𝔽2)11yescharacteristic
211Dic101—21no—
212Dic101—21no—
213Dic101—41no—
214Dic101—41no—
215C2×C101—81no—
216C2×C101—81no—
217C2×C101—81no—
218C2×C101D4211yescharacteristic
219C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C101#57611yescharacteristic
221D101—22no—
222D101—41no—
223D101—42no—
224D101—42no—
225#3264—1011nomaximal, Sylow
226C101#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1011no—
228C22×D8—1011no—
229#546—1011no—
230D42—1011no—
231#496D10111yescharacteristic, complemented
232#469—1011no—
233#465—1011no—
234#463—1011no—
235#459—1011no—
236#450—1011no—
237#449—1011no—
238#447—1011no—
239#447—1011no—
240#442—1011no—
241#435—1011no—
242C2×#41—1011no—
243C22×D4—1011no—
244C22×D4—1011no—
245C22×D4—1011no—
246C22×D4—2021no—
247AGL(1,ℤ/8)—2024no—
248C2×SD16—1011no—
249C2×SD16C2×D10111yescharacteristic, complemented
250C2×SD16C2×D10111yescharacteristic, complemented
251C2×D8—1011no—
252C2×D8—1011no—
253C2×D8—1011no—
254C2×D8—1011no—
255C2×D8—2024no—
256C2×D8C2×D10111yescharacteristic, complemented
257C2×D8C2×D10111yescharacteristic, complemented
258C2×M16—1011no—
259C22×C8—1011no—
260#127C2×D10111yescharacteristic, complemented
261#124C2×D10111yescharacteristic, complemented
262#123—2021no—
263C22⋊Q8—1011no—
264#121—1011no—
265#121—1011no—
266#121—1011no—
267#121—2021no—
268#121—2021no—
269#120—2021no—
270#120—2021no—
271#120—2021no—
272C4×D4—1011no—
273C4×D4—1011no—
274C4×D4—1011no—
275C2×C22⋊C4—2021no—
276#106—1011no—
277C4⋊C8—1011no—
278#103—1011no—
279#102—1011no—
280#102—1011no—
281#102—1011no—
282#102—1011no—
283#102—1011no—
284C22⋊C8—1011no—
285C22⋊C8—1011no—
286#97C2×D10111yescharacteristic, complemented
287C24—2021no—
288C24—2021no—
289C24—2021no—
290#41—4041no—
291#41—2022no—
292C2×Q8C22×D10111yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—1011no—
296C2×D4—1011no—
297C2×D4—1011no—
298C2×D4—4044no—
299C2×D4—4041no—
300C2×D4—4041no—
301C2×D4—4041no—
302C2×D4—2021no—
303C2×D4—2021no—
304C2×D4—2021no—
305C2×D4—2021no—
306C2×D4—2022no—
307C2×D4—2022no—
308C2×D4—2022no—
309C2×D4—2021no—
310C2×D4—2021no—
311C2×D4—2021no—
312C2×D4C22×D10111yescharacteristic
313C2×D4C22×D10111yescharacteristic
314C2×D4C22×D10111yescharacteristic
315C22×C4—1011no—
316C22×C4—1011no—
317C22×C4—2021no—
318C22×C4—2021no—
319C22×C4—2021no—
320SD16—22no—
321SD16—41no—
322SD16—2022no—
323D8—22no—
324D8—41no—
325D8—1014no—
326D8—2022no—
327D8—2022no—
328D8—2022no—
329M16—2022no—
330C2×C8—1011no—
331C2×C8—1011no—
332C2×C8—2022no—
333C2×C8C22×D10111yescharacteristic
334C2×C8C22×D10111yescharacteristic
335C4⋊C4—1011no—
336C4⋊C4—1011no—
337C4⋊C4—1011no—
338C4⋊C4—2021no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—1011no—
342C22⋊C4—1011no—
343C22⋊C4—2021no—
344C22⋊C4—2021no—
345C22⋊C4—2021no—
346C22⋊C4—2021no—
347C22⋊C4—2021no—
348C22⋊C4—2021no—
349C42C22×D10111yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—8081no—
354C23—8081no—
355C23—8081no—
356C23—1011no—
357C23—1011no—
358C23—4042no—
359C23—4042no—
360C23—4042no—
361C23—4042no—
362C23—4042no—
363C23—4042no—
364C23—2021no—
365C23—2021no—
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—81no—
376D4—8081no—
377D4—1012no—
378D4—1012no—
379D4—4042no—
380D4—4042no—
381D4—4042no—
382D4—4042no—
383D4—4042no—
384D4—4042no—
385D4—2022no—
386D4—2021no—
387D4—2021no—
388C2×C4—21no—
389C2×C4—8081no—
390C2×C4—1011no—
391C2×C4—1011no—
392C2×C4—4042no—
393C2×C4—4042no—
394C2×C4—2021no—
395C2×C4—2021no—
396C2×C4—2021no—
397C2×C4—2021no—
398C2×C4—2021no—
399C2×C4—2021no—
400C2×C4D4×D10111yescharacteristic
401C2×C4D4×D10111yescharacteristic
402C2×C4C23×D10111yescharacteristic
403C8—21no—
404C8—2021no—
405C8—2021no—
406C8D4×D10112yescomplemented
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—8084no—
417C22—8084no—
418C22—8084no—
419C22—1012no—
420C22—4041no—
421C22—4041no—
422C22—4041no—
423C22—4041no—
424C22—2022no—
425C22—2021no—
426C22—2022no—
427C22—2021no—
428C22—2021no—
429C22—2021no—
430C22—2021no—
431C22C2×D4×D10111yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—4041no—
436C4—4041no—
437C4—2021no—
438C4—2021no—
439C4C2×D4×D10111yescharacteristic
440C4#1569230011yescharacteristic
441C2—81no—
442C2—81no—
443C2—81no—
444C2—4042no—
445C2—4041no—
446C2—4042no—
447C2—2022no—
448C2#2026581211yescharacteristic
449C2AGL(1,ℤ/8)×D10111yescharacteristic
450C2#2026598111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#496×C101C2
socleC22×C101C22×D4
hypercenter#496D101
hypocenterC101#3264
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#15692282C2semidirect
2. C2×D8×D101C2semidirect
3. #20265820C2semidirect
4. #20265812C2semidirect
5. #20265427C2semidirect
6. #20265403C2semidirect
7. #20265389C2semidirect
8. #20265369C2semidirect
9. #20265326C2semidirect
10. #20265321C2semidirect
11. #20265309C2semidirect
12. #20265309C2semidirect
13. #20265290C2semidirect
14. #20265250C2semidirect
15. #496×C101C2semidirect
16. C2×#13108075C22semidirect
19. C2×D4×D101C22semidirect
20. C2×D4×D101C22semidirect
22. C2×#13108061C22semidirect
25. C2×#13108059C22semidirect
26. C2×#13108059C22semidirect
28. C2×#13108057C22semidirect
29. C2×C8×D101C22semidirect
31. #15692231C22semidirect
35. #15692226C22semidirect
36. #15692226C22semidirect
41. C4×D404C22semidirect
42. C4×D404C22semidirect
44. #15692187C22semidirect
45. C4⋊#11717579C22semidirect
46. #15692179C22semidirect
47. #15692177C22semidirect
48. #15692177C22semidirect
49. #15692176C22semidirect
50. #15692176C22semidirect
51. #15692176C22semidirect
52. #15692167C22semidirect
53. #15692167C22semidirect
54. C2×SD16×C101C22semidirect
55. C2×SD16×C101C22semidirect
56. C2×D8×C101C22semidirect
57. C2×D8×C101C22semidirect
58. #127×C101C22semidirect
59. #124×C101C22semidirect
60. #97×C101C22semidirect
89. D808D4semidirect
169. D404C2×D4semidirect
178. C8×C101C2×D4semidirect
205. C2×D101AGL(1,ℤ/8)semidirect
226. C101#3264semidirect
231. #496D101semidirect
249. C2×SD16C2×D101semidirect
250. C2×SD16C2×D101semidirect
256. C2×D8C2×D101semidirect
257. C2×D8C2×D101semidirect
260. #127C2×D101semidirect
261. #124C2×D101semidirect
286. #97C2×D101semidirect
406. C8D4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922822
C2×D8×D1012
#202658202
#202658122
#202654272
#202654032
#202653892
#202653692
#202653262
#202653212
#202653092
#202652902
#202652502
#496×C1012
#3264101