Finite Group Databasebeta

← #27182616

Subgroups — #27182616

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
2C22×#13108059C211yesmaximal, characteristic, complemented
3#20265818C211yesmaximal, characteristic, complemented
4D4×C22⋊D101C211yesmaximal, characteristic, complemented
5#20265563C211yesmaximal, characteristic, complemented
6#20265429C211yesmaximal, characteristic, complemented
7#20265408C211yesmaximal, characteristic, complemented
8#20265394C211yesmaximal, characteristic, complemented
9#20265371C211yesmaximal, characteristic, complemented
10#20265321C211yesmaximal, characteristic, complemented
11#20265313C211yesmaximal, characteristic, complemented
12#20265312C211yesmaximal, characteristic, complemented
13#20265286C211yesmaximal, characteristic, complemented
14#20265245C211yesmaximal, characteristic, complemented
15#450×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic, complemented
17C22×D404C2211yescharacteristic, complemented
18C22×C22⋊D101—21no—
19C2×D4×D101C2211yescharacteristic, complemented
20#15692282—24no—
21C2×#13108061C2211yescharacteristic, complemented
22C2×D808C2211yescharacteristic, complemented
23C2×D808C2211yescharacteristic, complemented
24C2×#13108059—24no—
25C2×#13108059C2211yescharacteristic
26C2×#13108059C2211yescharacteristic
27C2×#13108059C2211yescharacteristic, complemented
28C2×#13108059C2211yescharacteristic, complemented
29C2×#13108057C2211yescharacteristic, complemented
30C22×#11717579C2211yescharacteristic, complemented
31#15692251C2211yescharacteristic, complemented
32#15692244C2211yescharacteristic, complemented
33#15692240—21no—
34#15692230—21no—
35#15692230C2211yescharacteristic, complemented
36#15692229C2211yescharacteristic
37#15692227—21no—
38#15692226C2211yescharacteristic, complemented
39#15692225—21no—
40#15692224—21no—
41#15692223—21no—
42D4×Dic101C2211yescharacteristic, complemented
43#15692217C2211yescharacteristic, complemented
44C4×C22⋊D101C2211yescharacteristic
45C22⋊C4×D101—21no—
46#15692186C2211yescharacteristic, complemented
47#15692184C2211yescharacteristic, complemented
48#15692179C2211yescharacteristic, complemented
49#15692177C2211yescharacteristic, complemented
50#15692177C2211yescharacteristic, complemented
51#15692175C2211yescharacteristic, complemented
52#15692167C2211yescharacteristic, complemented
53#15692165C2211yescharacteristic, complemented
54C22×D4×C101C2211yescharacteristic
55C2×SD16×C101C2211yescharacteristic, complemented
56C2×SD16×C101C2211yescharacteristic, complemented
57C22⋊Q8×C101C2211yescharacteristic, complemented
58#102×C101C2211yescharacteristic, complemented
59#102×C101C2211yescharacteristic, complemented
60C22⋊C8×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C23×D101—21no—
63#13108075—22no—
64#13108075—41no—
65C2×D404—22no—
66C2×D404—21no—
67C2×D404—21no—
68C2×D404—21no—
69C2×D404—41no—
70C2×D404C2311yescharacteristic
71C2×D404C2311yescharacteristic
72C2×D404C2311yescharacteristic
73C2×C22⋊D101—21no—
74C2×C22⋊D101—21no—
75C2×C22⋊D101—21no—
76C2×C22⋊D101—21no—
77C2×C22⋊D101—21no—
78C2×C22⋊D101—44no—
79C2×C22⋊D101D411yescharacteristic
80D4×D101—22no—
81D4×D101—41no—
82C22×Dic101—21no—
83C2×C4×D101—21no—
84C2×C4×D101—21no—
85C2×C4×D101C2311yescharacteristic
86#13108061—22no—
87D808—22no—
88D808—41no—
89#13108059—22no—
90#13108059—22no—
91#13108059—22no—
92#13108059D414yescomplemented
93#13108057—22no—
94C2×#11717579—22no—
95C2×#11717579C2311yescharacteristic
96C2×#11717579C2311yescharacteristic
97C4⋊Dic101C2311yescharacteristic
98#13108053D411yescharacteristic
99C22⋊Dic101—21no—
100C22⋊Dic101—21no—
101C22⋊Dic101D411yescharacteristic
102#13108051—21no—
103#13108051—21no—
104#13108051—21no—
105#13108051—21no—
106#13108051—21no—
107#13108051D411yescharacteristic
108C4×Dic101C2311yescharacteristic
109C24×C101—21no—
110C2×Q8×C101C2311yescharacteristic
111C2×D4×C101—22no—
112C2×D4×C101—21no—
113C2×D4×C101—41no—
114C2×D4×C101C2311yescharacteristic
115C2×D4×C101C2311yescharacteristic
116C22×C4×C101C2311yescharacteristic
117SD16×C101—22no—
118SD16×C101—41no—
119C2×C8×C101C2311yescharacteristic
120C2×C8×C101C2311yescharacteristic
121C4⋊C4×C101—21no—
122C4⋊C4×C101C2311yescharacteristic
123C22⋊C4×C101—21no—
124C23×C101—21no—
125C23×C101—21no—
126C23×C101—42no—
127C23×C101—42no—
128C23×C101—81no—
129C23×C101C2×D411yescharacteristic
130C22×D101—21no—
131C22×D101—21no—
132C22×D101—21no—
133C22×D101—42no—
134C22×D101—42no—
135C22×D101—42no—
136C22×D101—42no—
137C22×D101—81no—
138C22×D101—81no—
139C22×D101C2×D411yescharacteristic
140Q8×C101—21no—
141Q8×C101—21no—
142D4×C101—21no—
143D4×C101—21no—
144D4×C101—21no—
145D4×C101—42no—
146D4×C101C2×D412yes—
147C2×C4×C101—21no—
148C2×C4×C101—21no—
149C2×C4×C101—21no—
150C2×C4×C101—21no—
151C2×C4×C101C2×D411yescharacteristic
152C2×C4×C101C2411yescharacteristic
153C22⋊D101—22no—
154C22⋊D101—42no—
155C22⋊D101—42no—
156C22⋊D101—42no—
157C22⋊D101—42no—
158C22⋊D101—81no—
159C2×Dic101—21no—
160C2×Dic101—42no—
161C2×Dic101C2×D411yescharacteristic
162C2×Dic101C2×D411yescharacteristic
163C2×Dic101C2×D411yescharacteristic
164D404—21no—
165D404—21no—
166D404—21no—
167D404—21no—
168D404—21no—
169D404—42no—
170D404—42no—
171D404—81no—
172D404C2×D412yescomplemented
173C4×D101—21no—
174C4×D101—21no—
175C4×D101—42no—
176C4×D101—81no—
177C8×C101—21no—
178C8×C101—21no—
179#11717579—21no—
180#11717579C2×D412yescomplemented
181C22×C101—21no—
182C22×C101—21no—
183C22×C101—22no—
184C22×C101—41no—
185C22×C101—41no—
186C22×C101—41no—
187C22×C101—41no—
188C22×C101—41no—
189C22×C101—84no—
190C22×C101AGL(1,ℤ/8)12yescomplemented
191C22×C101C22×D411yescharacteristic
192C2×D101—21no—
193C2×D101—22no—
194C2×D101—21no—
195C2×D101—21no—
196C2×D101—41no—
197C2×D101—41no—
198C2×D101—41no—
199C2×D101—41no—
200C2×D101—41no—
201C2×D101—41no—
202C2×D101—41no—
203C2×D101—41no—
204C2×D101—84no—
205C2×D101—84no—
206C4×C101—21no—
207C4×C101—41no—
208C4×C101—41no—
209C4×C101C22×D411yescharacteristic
210C4×C101H(5,𝔽2)11yescharacteristic
211Dic101—21no—
212Dic101—21no—
213Dic101—21no—
214Dic101—41no—
215C2×C101—22no—
216C2×C101—42no—
217C2×C101—81no—
218C2×C101D4211yescharacteristic
219C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C101#57611yescharacteristic
221D101—41no—
222D101—42no—
223D101—81no—
224D101—81no—
225#3264—1011nomaximal, Sylow
226C101#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1011no—
228C22×D8—1011no—
229#546—1011no—
230D42—1011no—
231#496—1011no—
232#469—1011no—
233#465—1011no—
234#463—1011no—
235#459—1011no—
236#450D10111yescharacteristic, complemented
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—2021no—
246C22×D4C2×D10111yescharacteristic
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—1011no—
256C2×D8—1011no—
257C2×D8—2024no—
258C2×M16—1011no—
259C22×C8—1011no—
260#127—1011no—
261#124—1011no—
262#123—2021no—
263C22⋊Q8C2×D10111yescharacteristic, complemented
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#102C2×D10111yescharacteristic, complemented
283#102C2×D10111yescharacteristic, complemented
284C22⋊C8—1011no—
285C22⋊C8C2×D10111yescharacteristic, complemented
286#97—1011no—
287C24—21no—
288C24—2021no—
289C24—2021no—
290#41—4041no—
291#41—2022no—
292C2×Q8C22×D10111yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—1011no—
297C2×D4—1011no—
298C2×D4—1011no—
299C2×D4—1011no—
300C2×D4—4041no—
301C2×D4—4044no—
302C2×D4—4041no—
303C2×D4—2021no—
304C2×D4—2022no—
305C2×D4—2021no—
306C2×D4—2021no—
307C2×D4—2021no—
308C2×D4—2021no—
309C2×D4—2021no—
310C2×D4—2021no—
311C2×D4—2022no—
312C2×D4—2021no—
313C2×D4C22×D10111yescharacteristic
314C2×D4C22×D10111yescharacteristic
315C22×C4—1011no—
316C22×C4—2021no—
317C22×C4—2021no—
318C22×C4—2021no—
319C22×C4C22×D10111yescharacteristic
320SD16—22no—
321SD16—41no—
322SD16—2022no—
323D8—1014no—
324D8—4041no—
325D8—2022no—
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—21no—
336C4⋊C4—1011no—
337C4⋊C4—1011no—
338C4⋊C4C22×D10111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—1011no—
341C22⋊C4—1011no—
342C22⋊C4—2021no—
343C22⋊C4—2021no—
344C22⋊C4—2021no—
345C22⋊C4—2021no—
346C22⋊C4—2021no—
347C22⋊C4—2021no—
348C22⋊C4—2021no—
349C42—1011no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—8081no—
356C23—8081no—
357C23—1011no—
358C23—4042no—
359C23—4042no—
360C23—4042no—
361C23—4042no—
362C23—2021no—
363C23—2021no—
364C23—2021no—
365C23D4×D10111yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—42no—
372D4—8081no—
373D4—8081no—
374D4—1012no—
375D4—4042no—
376D4—4042no—
377D4—4042no—
378D4—4042no—
379D4—4042no—
380D4—4042no—
381D4—2022no—
382D4—2021no—
383D4—2021no—
384D4—2021no—
385D4—2021no—
386D4—2021no—
387D4D4×D10112yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—8081no—
393C2×C4—1011no—
394C2×C4—1011no—
395C2×C4—1011no—
396C2×C4—4042no—
397C2×C4—4042no—
398C2×C4—2021no—
399C2×C4—2021no—
400C2×C4—2021no—
401C2×C4D4×D10111yescharacteristic
402C2×C4C23×D10111yescharacteristic
403C8—21no—
404C8—21no—
405C8—1012no—
406C8—2021no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—84no—
416C22—8084no—
417C22—8084no—
418C22—4041no—
419C22—4041no—
420C22—4041no—
421C22—4041no—
422C22—4041no—
423C22—4041no—
424C22—4041no—
425C22—4041no—
426C22—2022no—
427C22—2021no—
428C22—2021no—
429C22—2021no—
430C22#1569228212yescomplemented
431C22C2×D4×D10111yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—4041no—
436C4—2021no—
437C4—2021no—
438C4—2021no—
439C4C2×D4×D10111yescharacteristic
440C4#1569230011yescharacteristic
441C2—22no—
442C2—42no—
443C2—81no—
444C2—8081no—
445C2—8081no—
446C2—4041no—
447C2—4042no—
448C2#2026580911yescharacteristic
449C2C2×#1569228211yescharacteristic
450C2#2026598111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#450×C101C2
socleC22×C101C22×D4
hypercenter#450D101
hypocenterC101#3264
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#15692282C2semidirect
2. C22×#13108059C2semidirect
3. #20265818C2semidirect
4. D4×C22⋊D101C2semidirect
5. #20265563C2semidirect
6. #20265429C2semidirect
7. #20265408C2semidirect
8. #20265394C2semidirect
9. #20265371C2semidirect
10. #20265321C2semidirect
11. #20265313C2semidirect
12. #20265312C2semidirect
13. #20265286C2semidirect
14. #20265245C2semidirect
15. #450×C101C2semidirect
16. C2×#13108075C22semidirect
17. C22×D404C22semidirect
19. C2×D4×D101C22semidirect
21. C2×#13108061C22semidirect
22. C2×D808C22semidirect
23. C2×D808C22semidirect
27. C2×#13108059C22semidirect
28. C2×#13108059C22semidirect
29. C2×#13108057C22semidirect
30. C22×#11717579C22semidirect
31. #15692251C22semidirect
32. #15692244C22semidirect
35. #15692230C22semidirect
38. #15692226C22semidirect
42. D4×Dic101C22semidirect
43. #15692217C22semidirect
46. #15692186C22semidirect
47. #15692184C22semidirect
48. #15692179C22semidirect
49. #15692177C22semidirect
50. #15692177C22semidirect
51. #15692175C22semidirect
52. #15692167C22semidirect
53. #15692165C22semidirect
55. C2×SD16×C101C22semidirect
56. C2×SD16×C101C22semidirect
57. C22⋊Q8×C101C22semidirect
58. #102×C101C22semidirect
59. #102×C101C22semidirect
60. C22⋊C8×C101C22semidirect
92. #13108059D4semidirect
172. D404C2×D4semidirect
180. #11717579C2×D4semidirect
190. C22×C101AGL(1,ℤ/8)semidirect
226. C101#3264semidirect
236. #450D101semidirect
249. C2×SD16C2×D101semidirect
250. C2×SD16C2×D101semidirect
263. C22⋊Q8C2×D101semidirect
282. #102C2×D101semidirect
283. #102C2×D101semidirect
285. C22⋊C8C2×D101semidirect
430. C22#15692282semidirect

Maximal subgroups

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

subgroupindex
C2×#156922822
C22×#131080592
#202658182
D4×C22⋊D1012
#202655632
#202654292
#202654082
#202653942
#202653712
#202653212
#202653132
#202653122
#202652862
#202652452
#450×C1012
#3264101