Finite Group Databasebeta

← #27182607

Subgroups — #27182607

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×#15692279C211yesmaximal, characteristic, complemented
2C2×D8×D101C211yesmaximal, characteristic, complemented
3#20265822C211yesmaximal, characteristic, complemented
4#20265812C211yesmaximal, characteristic, complemented
5#20265564C211yesmaximal, characteristic, complemented
6#20265429C211yesmaximal, characteristic, complemented
7#20265405C211yesmaximal, characteristic, complemented
8#20265393C211yesmaximal, characteristic, complemented
9#20265329C211yesmaximal, characteristic, complemented
10#20265319C211yesmaximal, characteristic, complemented
11#20265312C211yesmaximal, characteristic, complemented
12#20265309C211yesmaximal, characteristic, complemented
13#20265287C211yesmaximal, characteristic, complemented
14#20265247C211yesmaximal, characteristic, complemented
15#459×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic, complemented
17C2×D4×D101—21no—
18C2×D4×D101C2211yescharacteristic
19C2×D4×D101C2211yescharacteristic, complemented
20C2×D4×D101C2211yescharacteristic, complemented
21#15692279—24no—
22C2×#13108063C2211yescharacteristic, complemented
23C2×#13108062C2211yescharacteristic, complemented
24C2×#13108062C2211yescharacteristic, complemented
25C2×D808C2211yescharacteristic
26C2×#13108059C2211yescharacteristic
27C2×#13108059C2211yescharacteristic, complemented
28C2×#13108059C2211yescharacteristic, complemented
29D8×D101—24no—
30C2×#13108057C2211yescharacteristic, complemented
31C2×C8×D101C2211yescharacteristic, complemented
32#15692243C2211yescharacteristic, complemented
33#15692239—21no—
34#15692235C2211yescharacteristic, complemented
35#15692230C2211yescharacteristic, complemented
36#15692229—21no—
37#15692226—21no—
38#15692226C2211yescharacteristic
39#15692226C2211yescharacteristic, complemented
40#15692225—21no—
41#15692223—21no—
42#15692223—21no—
43C4×D404C2211yescharacteristic
44C4×D404C2211yescharacteristic, complemented
45C2×#13108051—21no—
46#15692186C2211yescharacteristic, complemented
47#15692180C2211yescharacteristic, complemented
48#15692177C2211yescharacteristic, complemented
49#15692176C2211yescharacteristic, complemented
50#15692175C2211yescharacteristic, complemented
51#15692167C2211yescharacteristic, complemented
52#15692167C2211yescharacteristic, complemented
53#15692166C2211yescharacteristic, complemented
54C2×D8×C101C2211yescharacteristic, complemented
55C2×D8×C101C2211yescharacteristic, complemented
56#127×C101C2211yescharacteristic, complemented
57C4×D4×C101C2211yescharacteristic, complemented
58C4⋊C8×C101C2211yescharacteristic, complemented
59#102×C101C2211yescharacteristic, complemented
60#102×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C23×D101—21no—
63C23×D101—21no—
64#13108073—22no—
65#13108073—41no—
66C2×Dic202C2311yescharacteristic
67C2×D404—21no—
68C2×D404D411yescharacteristic
69C2×D404C2311yescharacteristic
70C2×D404C2311yescharacteristic
71C2×C22⋊D101—21no—
72C2×C22⋊D101—21no—
73C2×C22⋊D101—21no—
74C2×C22⋊D101—21no—
75C2×C22⋊D101—21no—
76C2×C22⋊D101—21no—
77D4×D101—22no—
78D4×D101—22no—
79D4×D101—22no—
80D4×D101—41no—
81D4×D101—41no—
82D4×D101—41no—
83D4×D101—44no—
84C22×Dic101—21no—
85C2×C4×D101—21no—
86C2×C4×D101C2311yescharacteristic
87C2×C4×D101C2311yescharacteristic
88#13108063—22no—
89#13108062—22no—
90#13108062—41no—
91D808—22no—
92#13108059—22no—
93#13108059—41no—
94#13108059D414yescomplemented
95#13108057—22no—
96C2×#11717579C2311yescharacteristic
97C2×#11717579C2311yescharacteristic
98C8×D101—22no—
99C4⋊Dic101—21no—
100C4⋊Dic101D411yescharacteristic
101C4⋊Dic101C2311yescharacteristic
102C22⋊Dic101—21no—
103C22⋊Dic101—21no—
104#13108051—21no—
105#13108051—21no—
106#13108051—21no—
107#13108051—21no—
108#13108051—21no—
109#13108051D411yescharacteristic
110#13108051D411yescharacteristic
111C2×D4×C101—21no—
112C2×D4×C101—21no—
113C2×D4×C101C2311yescharacteristic
114C2×D4×C101C2311yescharacteristic
115C2×D4×C101C2311yescharacteristic
116C22×C4×C101—21no—
117D8×C101—22no—
118D8×C101—22no—
119C2×C8×C101C2311yescharacteristic
120C2×C8×C101C2311yescharacteristic
121C4⋊C4×C101C2311yescharacteristic
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
140D4×C101—21no—
141D4×C101—21no—
142D4×C101—21no—
143D4×C101—21no—
144D4×C101—21no—
145D4×C101—42no—
146D4×C101—81no—
147D4×C101C2×D412yescomplemented
148C2×C4×C101—21no—
149C2×C4×C101—81no—
150C2×C4×C101C2×D411yescharacteristic
151C2×C4×C101C2×D411yescharacteristic
152C2×C4×C101C2411yescharacteristic
153C22⋊D101—42no—
154C22⋊D101—42no—
155C22⋊D101—42no—
156C22⋊D101—42no—
157C22⋊D101—42no—
158C22⋊D101—42no—
159C2×Dic101—21no—
160C2×Dic101—21no—
161C2×Dic101—42no—
162C2×Dic101C2×D411yescharacteristic
163C2×Dic101C2×D411yescharacteristic
164D404—22no—
165D404—21no—
166D404—21no—
167D404—21no—
168D404—81no—
169D404C2×D412yes—
170C4×D101—21no—
171C4×D101—21no—
172C4×D101—21no—
173C4×D101—21no—
174C4×D101—42no—
175Dic202—21no—
176Dic202—21no—
177C8×C101—21no—
178C8×C101—21no—
179#11717579—21no—
180#11717579C2×D412yescomplemented
181C22×C101—22no—
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—21no—
196C2×D101—21no—
197C2×D101—41no—
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—42no—
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—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#459D10111yescharacteristic, complemented
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×SD16—1011no—
250C2×SD16—1011no—
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#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, complemented
275C2×C22⋊C4—2021no—
276#106—1011no—
277C4⋊C8C2×D10111yescharacteristic, complemented
278#103—1011no—
279#102—1011no—
280#102—1011no—
281#102—1011no—
282#102C2×D10111yescharacteristic, complemented
283#102C2×D10111yescharacteristic, complemented
284C22⋊C8—1011no—
285C22⋊C8—1011no—
286#97—1011no—
287C24—2021no—
288C24—2021no—
289C24—2021no—
290#41—4041no—
291#41—2022no—
292C2×Q8—1011no—
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—2021no—
309C2×D4—2021no—
310C2×D4—2021no—
311C2×D4—2022no—
312C2×D4C22×D10111yescharacteristic
313C2×D4C22×D10111yescharacteristic
314C2×D4C22×D10111yescharacteristic
315C22×C4—21no—
316C22×C4—1011no—
317C22×C4—1011no—
318C22×C4—2021no—
319C22×C4—2021no—
320SD16—4041no—
321SD16—2022no—
322SD16—2022no—
323D8—22no—
324D8—22no—
325D8—1014no—
326D8—4041no—
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—2021no—
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—
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—2021no—
367Q8—2021no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—42no—
374D4—81no—
375D4—8081no—
376D4—1012no—
377D4—4042no—
378D4—4042no—
379D4—4042no—
380D4—4042no—
381D4—4042no—
382D4—4042no—
383D4—2022no—
384D4—2021no—
385D4—2021no—
386D4—2021no—
387D4D4×D10112yescomplemented
388C2×C4—21no—
389C2×C4—81no—
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—21no—
405C8—1012no—
406C8—2021no—
407C22—22no—
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—4041no—
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#1569230111yescharacteristic
441C2—42no—
442C2—81no—
443C2—81no—
444C2—8081no—
445C2—4042no—
446C2—4041no—
447C2—2022no—
448C2#2026581111yescharacteristic
449C2AGL(1,ℤ/8)×D10111yescharacteristic
450C2#2026597811yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#459×C101C2
socleC22×C101C22×D4
hypercenter#459D101
hypocenterC101#3264
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#15692279C2semidirect
2. C2×D8×D101C2semidirect
3. #20265822C2semidirect
4. #20265812C2semidirect
5. #20265564C2semidirect
6. #20265429C2semidirect
7. #20265405C2semidirect
8. #20265393C2semidirect
9. #20265329C2semidirect
10. #20265319C2semidirect
11. #20265312C2semidirect
12. #20265309C2semidirect
13. #20265287C2semidirect
14. #20265247C2semidirect
15. #459×C101C2semidirect
16. C2×#13108073C22semidirect
19. C2×D4×D101C22semidirect
20. C2×D4×D101C22semidirect
22. C2×#13108063C22semidirect
23. C2×#13108062C22semidirect
24. C2×#13108062C22semidirect
27. C2×#13108059C22semidirect
28. C2×#13108059C22semidirect
30. C2×#13108057C22semidirect
31. C2×C8×D101C22semidirect
32. #15692243C22semidirect
34. #15692235C22semidirect
35. #15692230C22semidirect
39. #15692226C22semidirect
44. C4×D404C22semidirect
46. #15692186C22semidirect
47. #15692180C22semidirect
48. #15692177C22semidirect
49. #15692176C22semidirect
50. #15692175C22semidirect
51. #15692167C22semidirect
52. #15692167C22semidirect
53. #15692166C22semidirect
54. C2×D8×C101C22semidirect
55. C2×D8×C101C22semidirect
56. #127×C101C22semidirect
57. C4×D4×C101C22semidirect
58. C4⋊C8×C101C22semidirect
59. #102×C101C22semidirect
60. #102×C101C22semidirect
94. #13108059D4semidirect
147. D4×C101C2×D4semidirect
180. #11717579C2×D4semidirect
205. C2×D101AGL(1,ℤ/8)semidirect
226. C101#3264semidirect
235. #459D101semidirect
256. C2×D8C2×D101semidirect
257. C2×D8C2×D101semidirect
260. #127C2×D101semidirect
274. C4×D4C2×D101semidirect
277. C4⋊C8C2×D101semidirect
282. #102C2×D101semidirect
283. #102C2×D101semidirect
387. D4D4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922792
C2×D8×D1012
#202658222
#202658122
#202655642
#202654292
#202654052
#202653932
#202653292
#202653192
#202653122
#202653092
#202652872
#202652472
#459×C1012
#3264101