Finite Group Databasebeta

← #27182619

Subgroups — #27182619

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
2C2×D8×D101C211yesmaximal, characteristic, complemented
3#20265824C211yesmaximal, characteristic, complemented
4#20265811C211yesmaximal, characteristic, complemented
5#20265565C211yesmaximal, characteristic, complemented
6#20265407C211yesmaximal, characteristic, complemented
7#20265391C211yesmaximal, characteristic, complemented
8#20265371C211yesmaximal, characteristic, complemented
9#20265329C211yesmaximal, characteristic, complemented
10#20265320C211yesmaximal, characteristic, complemented
11#20265312C211yesmaximal, characteristic, complemented
12#20265310C211yesmaximal, characteristic, complemented
13#20265288C211yesmaximal, characteristic, complemented
14#20265246C211yesmaximal, characteristic, complemented
15#469×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic, complemented
17C22×D404C2211yescharacteristic, complemented
18C2×D4×D101—21no—
19C2×D4×D101C2211yescharacteristic
20C2×D4×D101C2211yescharacteristic, complemented
21#15692283—24no—
22C2×#13108063C2211yescharacteristic, complemented
23C2×#13108063C2211yescharacteristic, complemented
24C2×#13108062C2211yescharacteristic, complemented
25C2×D808C2211yescharacteristic
26C2×D808C2211yescharacteristic, complemented
27C2×#13108059C2211yescharacteristic
28C2×#13108059C2211yescharacteristic, complemented
29C2×#13108059C2211yescharacteristic, complemented
30D8×D101—24no—
31C2×C8×D101C2211yescharacteristic, complemented
32#15692251C2211yescharacteristic, complemented
33#15692242C2211yescharacteristic, complemented
34#15692237—21no—
35#15692235C2211yescharacteristic, complemented
36#15692230C2211yescharacteristic
37#15692228—21no—
38#15692226—21no—
39#15692225—21no—
40#15692223—21no—
41#15692223—21no—
42#15692217C2211yescharacteristic, complemented
43#15692217C2211yescharacteristic, complemented
44C4×C22⋊D101C2211yescharacteristic
45C22⋊C4×D101—21no—
46#15692184C2211yescharacteristic, complemented
47#15692178C2211yescharacteristic, complemented
48#15692175C2211yescharacteristic, complemented
49#15692175C2211yescharacteristic, complemented
50#15692175C2211yescharacteristic, complemented
51C22⋊#11717579C2211yescharacteristic, complemented
52#15692167C2211yescharacteristic, complemented
53#15692165C2211yescharacteristic, complemented
54C2×D8×C101C2211yescharacteristic, complemented
55C2×M16×C101C2211yescharacteristic, complemented
56#121×C101C2211yescharacteristic, complemented
57#121×C101C2211yescharacteristic, complemented
58#106×C101C2211yescharacteristic, complemented
59#102×C101C2211yescharacteristic, complemented
60#102×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C23×D101—21no—
63C23×D101—21no—
64#13108074—22no—
65#13108074—41no—
66C2×Dic202C2311yescharacteristic
67C2×D404—22no—
68C2×D404—21no—
69C2×D404—21no—
70C2×D404—41no—
71C2×D404C2311yescharacteristic
72C2×D404C2311yescharacteristic
73C2×D404C2311yescharacteristic
74C2×C22⋊D101—21no—
75C2×C22⋊D101—21no—
76C2×C22⋊D101—21no—
77C2×C22⋊D101—21no—
78C2×C22⋊D101—21no—
79C2×C22⋊D101D411yescharacteristic
80D4×D101—22no—
81D4×D101—22no—
82D4×D101—41no—
83D4×D101—41no—
84D4×D101—44no—
85C2×C4×D101—21no—
86C2×C4×D101—21no—
87C2×C4×D101—21no—
88C2×C4×D101C2311yescharacteristic
89#13108063—22no—
90#13108063—22no—
91#13108062—41no—
92D808—22no—
93D808D414yescomplemented
94#13108059—22no—
95#13108059—22no—
96#13108059—41no—
97C2×#11717579C2311yescharacteristic
98C2×#11717579C2311yescharacteristic
99C8×D101—22no—
100C4⋊Dic101—21no—
101#13108053D411yescharacteristic
102C22⋊Dic101—21no—
103C22⋊Dic101—21no—
104C22⋊Dic101D411yescharacteristic
105#13108051—21no—
106#13108051—21no—
107#13108051—21no—
108#13108051—21no—
109#13108051D411yescharacteristic
110C4×Dic101C2311yescharacteristic
111C2×D4×C101—21no—
112C2×D4×C101—21no—
113C2×D4×C101C2311yescharacteristic
114C2×D4×C101C2311yescharacteristic
115C22×C4×C101C2311yescharacteristic
116D8×C101—22no—
117M16×C101—22no—
118C2×C8×C101C2311yescharacteristic
119C2×C8×C101C2311yescharacteristic
120C4⋊C4×C101C2311yescharacteristic
121C4⋊C4×C101C2311yescharacteristic
122C22⋊C4×C101—21no—
123C22⋊C4×C101—21no—
124C23×C101—21no—
125C23×C101—21no—
126C23×C101C2×D411yescharacteristic
127C22×D101—21no—
128C22×D101—21no—
129C22×D101—21no—
130C22×D101—42no—
131C22×D101—42no—
132C22×D101—42no—
133C22×D101—42no—
134C22×D101—42no—
135C22×D101—42no—
136C22×D101—81no—
137C22×D101—81no—
138C22×D101—81no—
139C22×D101C2×D411yescharacteristic
140D4×C101—21no—
141D4×C101—21no—
142D4×C101—21no—
143D4×C101—21no—
144D4×C101—42no—
145D4×C101—81no—
146C2×C4×C101—21no—
147C2×C4×C101—21no—
148C2×C4×C101—21no—
149C2×C4×C101—21no—
150C2×C4×C101C2×D411yescharacteristic
151C2×C4×C101C2411yescharacteristic
152C22⋊D101—22no—
153C22⋊D101—42no—
154C22⋊D101—42no—
155C22⋊D101—42no—
156C22⋊D101—42no—
157C22⋊D101—81no—
158C2×Dic101—21no—
159C2×Dic101C2×D411yescharacteristic
160C2×Dic101C2×D411yescharacteristic
161C2×Dic101C2×D411yescharacteristic
162D404—21no—
163D404—21no—
164D404—21no—
165D404—21no—
166D404—42no—
167D404—42no—
168D404C2×D412yes—
169D404C2×D412yescomplemented
170C4×D101—21no—
171C4×D101—21no—
172C4×D101—42no—
173C4×D101—42no—
174C4×D101—81no—
175Dic202—21no—
176Dic202—21no—
177C8×C101—21no—
178C8×C101C2×D412yescomplemented
179#11717579—21no—
180#11717579—21no—
181C22×C101—21no—
182C22×C101—21no—
183C22×C101—21no—
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—22no—
194C2×D101—22no—
195C2×D101—41no—
196C2×D101—41no—
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—41no—
208C4×C101—41no—
209C4×C101C22×D411yescharacteristic
210C4×C101H(5,𝔽2)11yescharacteristic
211Dic101—21no—
212Dic101—21no—
213Dic101—21no—
214Dic101—41no—
215C2×C101—41no—
216C2×C101—81no—
217C2×C101—81no—
218C2×C101D4211yescharacteristic
219C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C101#57611yescharacteristic
221D101—22no—
222D101—42no—
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#469D10111yescharacteristic, complemented
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×SD16—1011no—
250C2×SD16—1011no—
251C2×D8—1011no—
252C2×D8—1011no—
253C2×D8—1011no—
254C2×D8—1011no—
255C2×D8—1011no—
256C2×D8—2024no—
257C2×D8C2×D10111yescharacteristic, complemented
258C2×M16C2×D10111yescharacteristic, complemented
259C22×C8—1011no—
260#127—1011no—
261#124—1011no—
262#123—2021no—
263C22⋊Q8—1011no—
264#121—1011no—
265#121—2021no—
266#121—2021no—
267#121C2×D10111yescharacteristic, complemented
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#106C2×D10111yescharacteristic, complemented
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⋊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—1011no—
299C2×D4—4041no—
300C2×D4—4044no—
301C2×D4—4041no—
302C2×D4—4041no—
303C2×D4—2022no—
304C2×D4—2021no—
305C2×D4—2021no—
306C2×D4—2021no—
307C2×D4—2021no—
308C2×D4—2021no—
309C2×D4—2022no—
310C2×D4—2022no—
311C2×D4—2021no—
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—4041no—
321SD16—2022no—
322SD16—2022no—
323D8—22no—
324D8—1014no—
325D8—4041no—
326D8—2022no—
327D8—2022no—
328D8—2022no—
329M16—22no—
330C2×C8—1011no—
331C2×C8—1011no—
332C2×C8—2022no—
333C2×C8C22×D10111yescharacteristic
334C2×C8C22×D10111yescharacteristic
335C4⋊C4—1011no—
336C4⋊C4—2021no—
337C4⋊C4C22×D10111yescharacteristic
338C4⋊C4C22×D10111yescharacteristic
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—
349C42—1011no—
350C23—21no—
351C23—21no—
352C23—8081no—
353C23—8081no—
354C23—8081no—
355C23—1011no—
356C23—4042no—
357C23—4042no—
358C23—4042no—
359C23—4042no—
360C23—4042no—
361C23—4042no—
362C23—2021no—
363C23—2021no—
364C23—2021no—
365C23D4×D10111yescharacteristic
366Q8—2021no—
367Q8—2021no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—81no—
374D4—8081no—
375D4—1012no—
376D4—1012no—
377D4—4042no—
378D4—4042no—
379D4—4042no—
380D4—4042no—
381D4—4042no—
382D4—4042no—
383D4—2022no—
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—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—2021no—
405C8—2021no—
406C8D4×D10112yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
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—4041no—
426C22—4041no—
427C22—2022no—
428C22—2022no—
429C22—2021no—
430C22—2021no—
431C22C2×D4×D10111yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—4041no—
436C4—2021no—
437C4—2021no—
438C4—2021no—
439C4C2×D4×D10111yescharacteristic
440C4#1569230111yescharacteristic
441C2—41no—
442C2—81no—
443C2—81no—
444C2—8081no—
445C2—4042no—
446C2—4042no—
447C2—2022no—
448C2#2026581111yescharacteristic
449C2AGL(1,ℤ/8)×D10111yescharacteristic
450C2#2026598111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#469×C101C2
socleC22×C101C22×D4
hypercenter#469D101
hypocenterC101#3264
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#15692283C2semidirect
2. C2×D8×D101C2semidirect
3. #20265824C2semidirect
4. #20265811C2semidirect
5. #20265565C2semidirect
6. #20265407C2semidirect
7. #20265391C2semidirect
8. #20265371C2semidirect
9. #20265329C2semidirect
10. #20265320C2semidirect
11. #20265312C2semidirect
12. #20265310C2semidirect
13. #20265288C2semidirect
14. #20265246C2semidirect
15. #469×C101C2semidirect
16. C2×#13108074C22semidirect
17. C22×D404C22semidirect
20. C2×D4×D101C22semidirect
22. C2×#13108063C22semidirect
23. C2×#13108063C22semidirect
24. C2×#13108062C22semidirect
26. C2×D808C22semidirect
28. C2×#13108059C22semidirect
29. C2×#13108059C22semidirect
31. C2×C8×D101C22semidirect
32. #15692251C22semidirect
33. #15692242C22semidirect
35. #15692235C22semidirect
42. #15692217C22semidirect
43. #15692217C22semidirect
46. #15692184C22semidirect
47. #15692178C22semidirect
48. #15692175C22semidirect
49. #15692175C22semidirect
50. #15692175C22semidirect
51. C22⋊#11717579C22semidirect
52. #15692167C22semidirect
53. #15692165C22semidirect
54. C2×D8×C101C22semidirect
55. C2×M16×C101C22semidirect
56. #121×C101C22semidirect
57. #121×C101C22semidirect
58. #106×C101C22semidirect
59. #102×C101C22semidirect
60. #102×C101C22semidirect
93. D808D4semidirect
169. D404C2×D4semidirect
178. C8×C101C2×D4semidirect
205. C2×D101AGL(1,ℤ/8)semidirect
226. C101#3264semidirect
232. #469D101semidirect
257. C2×D8C2×D101semidirect
258. C2×M16C2×D101semidirect
267. #121C2×D101semidirect
268. #121C2×D101semidirect
276. #106C2×D101semidirect
282. #102C2×D101semidirect
283. #102C2×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×#156922832
C2×D8×D1012
#202658242
#202658112
#202655652
#202654072
#202653912
#202653712
#202653292
#202653202
#202653122
#202653102
#202652882
#202652462
#469×C1012
#3264101