Finite Group Databasebeta

← #27182617

Subgroups — #27182617

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×#15692283C211yesmaximal, characteristic, complemented
2C22×D808C211yesmaximal, characteristic, complemented
3#20265823C211yesmaximal, characteristic, complemented
4D4×D404C211yesmaximal, characteristic, complemented
5#20265566C211yesmaximal, characteristic, complemented
6#20265430C211yesmaximal, characteristic, complemented
7#20265406C211yesmaximal, characteristic, complemented
8#20265393C211yesmaximal, characteristic, complemented
9#20265373C211yesmaximal, characteristic, complemented
10#20265330C211yesmaximal, characteristic, complemented
11#20265324C211yesmaximal, characteristic, complemented
12#20265313C211yesmaximal, characteristic, complemented
13#20265313C211yesmaximal, characteristic, complemented
14#20265290C211yesmaximal, characteristic, complemented
15#435×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic, complemented
17C22×D404C2211yescharacteristic
18C22×D404C2211yescharacteristic, complemented
19C22×D404C2211yescharacteristic, complemented
20C2×D4×D101—21no—
21#15692283—24no—
22C2×#13108063C2211yescharacteristic, complemented
23C2×#13108063C2211yescharacteristic, complemented
24C2×#13108063C2211yescharacteristic, complemented
25C2×D808—24no—
26C2×D808C2211yescharacteristic
27C2×D808C2211yescharacteristic
28C2×D808C2211yescharacteristic, complemented
29C2×D808C2211yescharacteristic, complemented
30C2×D808C2211yescharacteristic, complemented
31C2×D808C2211yescharacteristic, complemented
32#15692252C2211yescharacteristic, complemented
33#15692243C2211yescharacteristic, complemented
34#15692240—21no—
35#15692234C2211yescharacteristic, complemented
36#15692230—21no—
37#15692229C2211yescharacteristic
38#15692229C2211yescharacteristic, complemented
39#15692229C2211yescharacteristic, complemented
40#15692228—21no—
41#15692225—21no—
42#15692225—21no—
43#15692223—21no—
44C4×D404C2211yescharacteristic, complemented
45C4×D404C2211yescharacteristic, complemented
46C22⋊C4×D101—21no—
47#15692187C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692177C2211yescharacteristic, complemented
50#15692177C2211yescharacteristic, complemented
51#15692177C2211yescharacteristic, complemented
52#15692177C2211yescharacteristic, complemented
53#15692177C2211yescharacteristic, complemented
54C2×M16×C101C2211yescharacteristic, complemented
55C22×C8×C101C2211yescharacteristic, complemented
56C4×D4×C101C2211yescharacteristic
57C4⋊C8×C101C2211yescharacteristic, complemented
58C22⋊C8×C101C2211yescharacteristic, complemented
59C22⋊C8×C101C2211yescharacteristic, complemented
60#97×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C23×D101—21no—
63C23×D101—21no—
64#13108074—22no—
65#13108074—41no—
66C2×Dic202C2311yescharacteristic
67C2×D404—22no—
68C2×D404—22no—
69C2×D404—22no—
70C2×D404—21no—
71C2×D404—21no—
72C2×D404—21no—
73C2×D404—21no—
74C2×D404—21no—
75C2×D404—41no—
76C2×D404—41no—
77C2×D404—41no—
78C2×D404C2311yescharacteristic
79C2×D404C2311yescharacteristic
80C2×D404C2311yescharacteristic
81C2×D404C2311yescharacteristic
82C2×D404C2311yescharacteristic
83C2×C22⋊D101—21no—
84C2×C22⋊D101—21no—
85C2×C22⋊D101—21no—
86C2×C22⋊D101—21no—
87D4×D101—44no—
88C2×C4×D101—21no—
89C2×C4×D101—21no—
90C2×C4×D101—21no—
91#13108063—22no—
92#13108063—22no—
93#13108063—41no—
94D808—22no—
95D808—22no—
96D808—22no—
97D808—22no—
98D808—41no—
99D808D414yescomplemented
100C4⋊Dic101C2311yescharacteristic
101C4⋊Dic101C2311yescharacteristic
102#13108053—21no—
103C22⋊Dic101—21no—
104#13108051—21no—
105#13108051—21no—
106#13108051—21no—
107#13108051—21no—
108#13108051—21no—
109#13108051—21no—
110#13108051—21no—
111C2×D4×C101D411yescharacteristic
112C22×C4×C101C2311yescharacteristic
113C22×C4×C101C2311yescharacteristic
114M16×C101—22no—
115C2×C8×C101—22no—
116C2×C8×C101C2311yescharacteristic
117C2×C8×C101C2311yescharacteristic
118C2×C8×C101C2311yescharacteristic
119C2×C8×C101C2311yescharacteristic
120C4⋊C4×C101D411yescharacteristic
121C22⋊C4×C101D411yescharacteristic
122C22⋊C4×C101D411yescharacteristic
123C42×C101C2311yescharacteristic
124C23×C101C2×D411yescharacteristic
125C23×C101C2×D411yescharacteristic
126C22×D101—21no—
127C22×D101—21no—
128C22×D101—21no—
129C22×D101—21no—
130C22×D101—21no—
131C22×D101—42no—
132C22×D101—42no—
133C22×D101—42no—
134C22×D101—42no—
135C22×D101—42no—
136C22×D101—42no—
137C22×D101—81no—
138C22×D101—81no—
139C22×D101—81no—
140D4×C101—22no—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—21no—
144C2×C4×C101—21no—
145C2×C4×C101C2×D411yescharacteristic
146C2×C4×C101C2×D411yescharacteristic
147C2×C4×C101C2×D411yescharacteristic
148C2×C4×C101C2×D411yescharacteristic
149C2×C4×C101C2411yescharacteristic
150C22⋊D101—42no—
151C22⋊D101—42no—
152C22⋊D101—42no—
153C22⋊D101—81no—
154C2×Dic101—21no—
155C2×Dic101—21no—
156C2×Dic101—21no—
157D404—21no—
158D404—21no—
159D404—21no—
160D404—21no—
161D404—21no—
162D404—21no—
163D404—21no—
164D404—21no—
165D404—42no—
166D404—42no—
167D404—42no—
168D404—42no—
169D404—81no—
170D404C2×D412yes—
171D404C2×D412yescomplemented
172C4×D101—42no—
173C4×D101—42no—
174C4×D101—81no—
175Dic202—21no—
176Dic202—21no—
177C8×C101—21no—
178C8×C101—21no—
179C8×C101—21no—
180C8×C101C2×D412yescomplemented
181C22×C101—21no—
182C22×C101—21no—
183C22×C101—21no—
184C22×C101—21no—
185C22×C101—21no—
186C22×C101AGL(1,ℤ/8)12yescomplemented
187C22×C101C22×D411yescharacteristic
188C2×D101—22no—
189C2×D101—22no—
190C2×D101—41no—
191C2×D101—41no—
192C2×D101—41no—
193C2×D101—41no—
194C2×D101—41no—
195C2×D101—41no—
196C2×D101—41no—
197C2×D101—41no—
198C2×D101—41no—
199C2×D101—41no—
200C2×D101—41no—
201C2×D101—41no—
202C2×D101—41no—
203C2×D101—84no—
204C2×D101—84no—
205C2×D101—84no—
206C4×C101—21no—
207C4×C101—21no—
208C4×C101—21no—
209C4×C101—21no—
210C4×C101C22×D411yescharacteristic
211C4×C101H(5,𝔽2)11yescharacteristic
212Dic101—41no—
213Dic101—41no—
214Dic101—41no—
215C2×C101—22no—
216C2×C101—41no—
217C2×C101D4211yescharacteristic
218C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C101#57611yescharacteristic
220D101—42no—
221D101—42no—
222D101—81no—
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#450—1011no—
237#449—1011no—
238#447—1011no—
239#447—1011no—
240#442—1011no—
241#435D10111yescharacteristic, complemented
242C2×#41—1011no—
243C22×D4—1011no—
244C22×D4—1011no—
245C22×D4—1011no—
246C22×D4—2021no—
247AGL(1,ℤ/8)—2024no—
248C2×SD16—1011no—
249C2×SD16—1011no—
250C2×SD16—1011no—
251C2×D8—1011no—
252C2×D8—1011no—
253C2×D8—1011no—
254C2×D8—1011no—
255C2×D8—1011no—
256C2×D8—1011no—
257C2×D8—2024no—
258C2×M16C2×D10111yescharacteristic, complemented
259C22×C8C2×D10111yescharacteristic, complemented
260#127—1011no—
261#124—1011no—
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×D4C2×D10111yescharacteristic
275C2×C22⋊C4—2021no—
276#106—1011no—
277C4⋊C8C2×D10111yescharacteristic, complemented
278#103—1011no—
279#102—1011no—
280#102—1011no—
281#102—1011no—
282#102—1011no—
283#102—1011no—
284C22⋊C8C2×D10111yescharacteristic, complemented
285C22⋊C8C2×D10111yescharacteristic, complemented
286#97C2×D10111yescharacteristic, complemented
287C24—2021no—
288C24—2021no—
289C24—2021no—
290#41—4041no—
291#41—2022no—
292C2×Q8—1011no—
293C2×D4—1011no—
294C2×D4—1011no—
295C2×D4—1011no—
296C2×D4—1011no—
297C2×D4—1011no—
298C2×D4—4041no—
299C2×D4—4041no—
300C2×D4—4041no—
301C2×D4—4044no—
302C2×D4—2022no—
303C2×D4—2022no—
304C2×D4—2022no—
305C2×D4—2021no—
306C2×D4—2021no—
307C2×D4—2021no—
308C2×D4—2021no—
309C2×D4—2021no—
310C2×D4—2021no—
311C2×D4—2021no—
312C2×D4—2021no—
313C2×D4—2021no—
314C2×D4D40411yescharacteristic
315C22×C4—2021no—
316C22×C4—2021no—
317C22×C4—2021no—
318C22×C4C22×D10111yescharacteristic
319C22×C4C22×D10111yescharacteristic
320SD16—4041no—
321SD16—2022no—
322SD16—2022no—
323D8—1014no—
324D8—4041no—
325D8—2022no—
326D8—2022no—
327D8—2022no—
328D8—2022no—
329M16—22no—
330C2×C8—22no—
331C2×C8C22×D10111yescharacteristic
332C2×C8C22×D10111yescharacteristic
333C2×C8C22×D10111yescharacteristic
334C2×C8C22×D10111yescharacteristic
335C4⋊C4—1011no—
336C4⋊C4—1011no—
337C4⋊C4—2021no—
338C4⋊C4D40411yescharacteristic
339C22⋊C4—2021no—
340C22⋊C4—2021no—
341C22⋊C4—2021no—
342C22⋊C4—2021no—
343C22⋊C4—2021no—
344C22⋊C4—2021no—
345C22⋊C4—2021no—
346C22⋊C4—2021no—
347C22⋊C4D40411yescharacteristic
348C22⋊C4D40411yescharacteristic
349C42C22×D10111yescharacteristic
350C23—8081no—
351C23—8081no—
352C23—8081no—
353C23—4042no—
354C23—4042no—
355C23—4042no—
356C23—4042no—
357C23—4042no—
358C23—4042no—
359C23—2021no—
360C23—2021no—
361C23—2021no—
362C23—2021no—
363C23—2021no—
364C23C2×D40411yescharacteristic
365C23C2×D40411yescharacteristic
366Q8—2021no—
367Q8—2021no—
368D4—22no—
369D4—8081no—
370D4—8081no—
371D4—1012no—
372D4—1012no—
373D4—4042no—
374D4—4042no—
375D4—4042no—
376D4—4042no—
377D4—4042no—
378D4—4042no—
379D4—4042no—
380D4—2021no—
381D4—2021no—
382D4—2021no—
383D4—2021no—
384D4—2021no—
385D4—2021no—
386D4—2021no—
387D4—2021no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—8081no—
393C2×C4—4042no—
394C2×C4—4042no—
395C2×C4—2021no—
396C2×C4—2021no—
397C2×C4—2021no—
398C2×C4C2×D40411yescharacteristic
399C2×C4C2×D40411yescharacteristic
400C2×C4C2×D40411yescharacteristic
401C2×C4C2×D40411yescharacteristic
402C2×C4C23×D10111yescharacteristic
403C8—21no—
404C8—21no—
405C8—21no—
406C8D4×D10112yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—8084no—
413C22—8084no—
414C22—8084no—
415C22—4041no—
416C22—4041no—
417C22—4041no—
418C22—4041no—
419C22—4041no—
420C22—4041no—
421C22—4041no—
422C22—4041no—
423C22—4041no—
424C22—4041no—
425C22—4041no—
426C22—4041no—
427C22—4041no—
428C22—2022no—
429C22—2022no—
430C22#1569228312yescomplemented
431C22C22×D40411yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—4041no—
437C4—4041no—
438C4—4041no—
439C4C2×D4×D10111yescharacteristic
440C4#1569230111yescharacteristic
441C2—22no—
442C2—41no—
443C2—8081no—
444C2—8081no—
445C2—8081no—
446C2—4042no—
447C2—4042no—
448C2D4×D40411yescharacteristic
449C2C2×#1569228311yescharacteristic
450C2#2026598211yescharacteristic
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#435×C101C2
socleC22×C101C22×D4
hypercenter#435D101
hypocenterC101#3264
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#15692283C2semidirect
2. C22×D808C2semidirect
3. #20265823C2semidirect
4. D4×D404C2semidirect
5. #20265566C2semidirect
6. #20265430C2semidirect
7. #20265406C2semidirect
8. #20265393C2semidirect
9. #20265373C2semidirect
10. #20265330C2semidirect
11. #20265324C2semidirect
12. #20265313C2semidirect
13. #20265313C2semidirect
14. #20265290C2semidirect
15. #435×C101C2semidirect
16. C2×#13108074C22semidirect
18. C22×D404C22semidirect
19. C22×D404C22semidirect
22. C2×#13108063C22semidirect
23. C2×#13108063C22semidirect
24. C2×#13108063C22semidirect
28. C2×D808C22semidirect
29. C2×D808C22semidirect
30. C2×D808C22semidirect
31. C2×D808C22semidirect
32. #15692252C22semidirect
33. #15692243C22semidirect
35. #15692234C22semidirect
38. #15692229C22semidirect
39. #15692229C22semidirect
44. C4×D404C22semidirect
45. C4×D404C22semidirect
47. #15692187C22semidirect
48. #15692180C22semidirect
49. #15692177C22semidirect
50. #15692177C22semidirect
51. #15692177C22semidirect
52. #15692177C22semidirect
53. #15692177C22semidirect
54. C2×M16×C101C22semidirect
55. C22×C8×C101C22semidirect
57. C4⋊C8×C101C22semidirect
58. C22⋊C8×C101C22semidirect
59. C22⋊C8×C101C22semidirect
60. #97×C101C22semidirect
99. D808D4semidirect
171. D404C2×D4semidirect
180. C8×C101C2×D4semidirect
186. C22×C101AGL(1,ℤ/8)semidirect
226. C101#3264semidirect
241. #435D101semidirect
258. C2×M16C2×D101semidirect
259. C22×C8C2×D101semidirect
277. C4⋊C8C2×D101semidirect
284. C22⋊C8C2×D101semidirect
285. C22⋊C8C2×D101semidirect
286. #97C2×D101semidirect
406. C8D4×D101semidirect
430. C22#15692283semidirect

Maximal subgroups

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

subgroupindex
C2×#156922832
C22×D8082
#202658232
D4×D4042
#202655662
#202654302
#202654062
#202653932
#202653732
#202653302
#202653242
#202653132
#202652902
#435×C1012
#3264101