Finite Group Databasebeta

← #27182610

Subgroups — #27182610

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×#15692281C211yesmaximal, characteristic, complemented
2C2×D8×D101C211yesmaximal, characteristic, complemented
3#20265824C211yesmaximal, characteristic, complemented
4D4×C22⋊D101C211yesmaximal, characteristic, complemented
5#20265565C211yesmaximal, characteristic, complemented
6#20265428C211yesmaximal, characteristic, complemented
7#20265405C211yesmaximal, characteristic, complemented
8#20265390C211yesmaximal, characteristic, complemented
9#20265371C211yesmaximal, characteristic, complemented
10#20265329C211yesmaximal, characteristic, complemented
11#20265324C211yesmaximal, characteristic, complemented
12#20265309C211yesmaximal, characteristic, complemented
13#20265286C211yesmaximal, characteristic, complemented
14#20265251C211yesmaximal, characteristic, complemented
15#447×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic, complemented
17C22×C22⋊D101—21no—
18C2×D4×D101C2211yescharacteristic
19C2×D4×D101C2211yescharacteristic, complemented
20#15692281—24no—
21C2×#13108063C2211yescharacteristic, complemented
22C2×#13108062C2211yescharacteristic, complemented
23C2×#13108062C2211yescharacteristic, complemented
24C2×D808C2211yescharacteristic, complemented
25C2×#13108059C2211yescharacteristic
26C2×#13108059C2211yescharacteristic, complemented
27C2×#13108059C2211yescharacteristic, complemented
28D8×D101—24no—
29C2×#13108058C2211yescharacteristic, complemented
30C2×C8×D101C2211yescharacteristic, complemented
31#15692251C2211yescharacteristic, complemented
32#15692242C2211yescharacteristic, complemented
33#15692237—21no—
34#15692235C2211yescharacteristic, complemented
35#15692229C2211yescharacteristic, complemented
36#15692227—21no—
37#15692226—21no—
38#15692226C2211yescharacteristic
39#15692224—21no—
40#15692223—21no—
41#15692223—21no—
42D4×Dic101C2211yescharacteristic, complemented
43#15692217C2211yescharacteristic, complemented
44C4×C22⋊D101C2211yescharacteristic
45C22⋊C4×D101—21no—
46#15692186C2211yescharacteristic, complemented
47#15692184C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692177C2211yescharacteristic, complemented
50#15692176C2211yescharacteristic, complemented
51#15692175C2211yescharacteristic, complemented
52#15692167C2211yescharacteristic, complemented
53#15692165C2211yescharacteristic, complemented
54C22×D4×C101C2211yescharacteristic, complemented
55C2×D8×C101C2211yescharacteristic
56C2×D8×C101C2211yescharacteristic, complemented
57#121×C101C2211yescharacteristic, complemented
58#102×C101C2211yescharacteristic, complemented
59#102×C101C2211yescharacteristic, complemented
60C22⋊C8×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C23×D101—21no—
63#13108074—22no—
64#13108074—41no—
65C2×Dic202C2311yescharacteristic
66C2×D404C2311yescharacteristic
67C2×D404C2311yescharacteristic
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70C2×C22⋊D101—21no—
71C2×C22⋊D101—21no—
72C2×C22⋊D101—21no—
73C2×C22⋊D101—21no—
74C2×C22⋊D101—21no—
75C2×C22⋊D101—44no—
76C2×C22⋊D101D411yescharacteristic
77D4×D101—22no—
78D4×D101—22no—
79D4×D101—41no—
80D4×D101—41no—
81C22×Dic101—21no—
82C2×C4×D101—21no—
83C2×C4×D101—21no—
84C2×C4×D101C2311yescharacteristic
85#13108063—41no—
86#13108062—22no—
87#13108062—22no—
88D808—22no—
89#13108059—22no—
90#13108059—22no—
91#13108059D414yescomplemented
92#13108058—22no—
93C2×#11717579C2311yescharacteristic
94C2×#11717579C2311yescharacteristic
95C8×D101—22no—
96C4⋊Dic101—21no—
97C4⋊Dic101C2311yescharacteristic
98#13108053D411yescharacteristic
99C22⋊Dic101—21no—
100C22⋊Dic101—21no—
101C22⋊Dic101—21no—
102C22⋊Dic101—21no—
103C22⋊Dic101D411yescharacteristic
104#13108051—21no—
105#13108051—21no—
106#13108051—21no—
107#13108051D411yescharacteristic
108C4×Dic101C2311yescharacteristic
109C24×C101—21no—
110C2×D4×C101—22no—
111C2×D4×C101—21no—
112C2×D4×C101—21no—
113C2×D4×C101—41no—
114C2×D4×C101C2311yescharacteristic
115C2×D4×C101C2311yescharacteristic
116C2×D4×C101C2311yescharacteristic
117C22×C4×C101C2311yescharacteristic
118D8×C101—22no—
119D8×C101—41no—
120C2×C8×C101C2311yescharacteristic
121C2×C8×C101C2311yescharacteristic
122C4⋊C4×C101C2311yescharacteristic
123C22⋊C4×C101—21no—
124C23×C101—21no—
125C23×C101—21no—
126C23×C101—21no—
127C23×C101—42no—
128C23×C101—42no—
129C23×C101—81no—
130C23×C101C2×D411yescharacteristic
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
140D4×C101—21no—
141D4×C101—21no—
142D4×C101—21no—
143D4×C101—21no—
144D4×C101—21no—
145D4×C101—42no—
146D4×C101—81no—
147D4×C101C2×D412yes—
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—42no—
159C22⋊D101—42no—
160C22⋊D101—81no—
161C2×Dic101—21no—
162C2×Dic101—21no—
163C2×Dic101—42no—
164C2×Dic101C2×D411yescharacteristic
165C2×Dic101C2×D411yescharacteristic
166C2×Dic101C2×D411yescharacteristic
167D404—21no—
168D404—21no—
169D404—21no—
170D404C2×D412yescomplemented
171C4×D101—21no—
172C4×D101—21no—
173C4×D101—42no—
174C4×D101—81no—
175Dic202—21no—
176Dic202—21no—
177C8×C101—21no—
178C8×C101—21no—
179#11717579—21no—
180#11717579C2×D412yescomplemented
181C22×C101—21no—
182C22×C101—22no—
183C22×C101—21no—
184C22×C101—21no—
185C22×C101—41no—
186C22×C101—41no—
187C22×C101—41no—
188C22×C101—41no—
189C22×C101—41no—
190C22×C101—41no—
191C22×C101—41no—
192C22×C101—41no—
193C22×C101—84no—
194C22×C101C22×D411yescharacteristic
195C2×D101—21no—
196C2×D101—21no—
197C2×D101—22no—
198C2×D101—41no—
199C2×D101—41no—
200C2×D101—41no—
201C2×D101—41no—
202C2×D101—41no—
203C2×D101—84no—
204C2×D101—84no—
205C2×D101AGL(1,ℤ/8)12yescomplemented
206C4×C101—21no—
207C4×C101—41no—
208C4×C101C22×D411yescharacteristic
209C4×C101H(5,𝔽2)11yescharacteristic
210Dic101—21no—
211Dic101—21no—
212Dic101—21no—
213Dic101—41no—
214Dic101—41no—
215C2×C101—42no—
216C2×C101—41no—
217C2×C101—81no—
218C2×C101—81no—
219C2×C101D4211yescharacteristic
220C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C101#57611yescharacteristic
222D101—22no—
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#459—1011no—
236#450—1011no—
237#449—1011no—
238#447—1011no—
239#447D10111yescharacteristic, complemented
240#442—1011no—
241#435—1011no—
242C2×#41—1011no—
243C22×D4—1011no—
244C22×D4—1011no—
245C22×D4—2021no—
246C22×D4C2×D10111yescharacteristic, complemented
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
257C2×D8C2×D10111yescharacteristic, complemented
258C2×M16—1011no—
259C22×C8—1011no—
260#127—1011no—
261#124—1011no—
262#123—2021no—
263C22⋊Q8—1011no—
264#121—1011no—
265#121—1011no—
266#121—2021no—
267#121—2021no—
268#121C2×D10111yescharacteristic, complemented
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×Q8—1011no—
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—1011no—
298C2×D4—1011no—
299C2×D4—1011no—
300C2×D4—4044no—
301C2×D4—4041no—
302C2×D4—4041no—
303C2×D4—2021no—
304C2×D4—2021no—
305C2×D4—2021no—
306C2×D4—2021no—
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—1011no—
316C22×C4—2021no—
317C22×C4—2021no—
318C22×C4—2021no—
319C22×C4C22×D10111yescharacteristic
320SD16—4041no—
321SD16—2022no—
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—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—
349C42—1011no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—81no—
356C23—8081no—
357C23—8081no—
358C23—1011no—
359C23—4042no—
360C23—4042no—
361C23—4042no—
362C23—4042no—
363C23—2021no—
364C23—2021no—
365C23D4×D10111yescharacteristic
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×D10112yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—8081no—
392C2×C4—1011no—
393C2×C4—1011no—
394C2×C4—1011no—
395C2×C4—4042no—
396C2×C4—4042no—
397C2×C4—2021no—
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—22no—
409C22—21no—
410C22—21no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—84no—
420C22—8084no—
421C22—8084no—
422C22—1012no—
423C22—4041no—
424C22—4041no—
425C22—4041no—
426C22—4041no—
427C22—4041no—
428C22—2022no—
429C22—2021no—
430C22—2021no—
431C22C2×D4×D10111yescharacteristic
432C4—21no—
433C4—41no—
434C4—4041no—
435C4—4041no—
436C4—2021no—
437C4—2021no—
438C4—2021no—
439C4C2×D4×D10111yescharacteristic
440C4#1569230011yescharacteristic
441C2—41no—
442C2—42no—
443C2—81no—
444C2—81no—
445C2—8081no—
446C2—4042no—
447C2—2022no—
448C2#2026580911yescharacteristic
449C2AGL(1,ℤ/8)×D10111yescharacteristic
450C2#2026597811yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#447×C101C2
socleC22×C101C22×D4
hypercenter#447D101
hypocenterC101#3264
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#15692281C2semidirect
2. C2×D8×D101C2semidirect
3. #20265824C2semidirect
4. D4×C22⋊D101C2semidirect
5. #20265565C2semidirect
6. #20265428C2semidirect
7. #20265405C2semidirect
8. #20265390C2semidirect
9. #20265371C2semidirect
10. #20265329C2semidirect
11. #20265324C2semidirect
12. #20265309C2semidirect
13. #20265286C2semidirect
14. #20265251C2semidirect
15. #447×C101C2semidirect
16. C2×#13108074C22semidirect
19. C2×D4×D101C22semidirect
21. C2×#13108063C22semidirect
22. C2×#13108062C22semidirect
23. C2×#13108062C22semidirect
24. C2×D808C22semidirect
26. C2×#13108059C22semidirect
27. C2×#13108059C22semidirect
29. C2×#13108058C22semidirect
30. C2×C8×D101C22semidirect
31. #15692251C22semidirect
32. #15692242C22semidirect
34. #15692235C22semidirect
35. #15692229C22semidirect
42. D4×Dic101C22semidirect
43. #15692217C22semidirect
46. #15692186C22semidirect
47. #15692184C22semidirect
48. #15692180C22semidirect
49. #15692177C22semidirect
50. #15692176C22semidirect
51. #15692175C22semidirect
52. #15692167C22semidirect
53. #15692165C22semidirect
54. C22×D4×C101C22semidirect
56. C2×D8×C101C22semidirect
57. #121×C101C22semidirect
58. #102×C101C22semidirect
59. #102×C101C22semidirect
60. C22⋊C8×C101C22semidirect
91. #13108059D4semidirect
170. D404C2×D4semidirect
180. #11717579C2×D4semidirect
205. C2×D101AGL(1,ℤ/8)semidirect
226. C101#3264semidirect
239. #447D101semidirect
246. C22×D4C2×D101semidirect
257. C2×D8C2×D101semidirect
268. #121C2×D101semidirect
282. #102C2×D101semidirect
283. #102C2×D101semidirect
285. C22⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922812
C2×D8×D1012
#202658242
D4×C22⋊D1012
#202655652
#202654282
#202654052
#202653902
#202653712
#202653292
#202653242
#202653092
#202652862
#202652512
#447×C1012
#3264101