Finite Group Databasebeta

← #27182634

Subgroups — #27182634

Order 12928.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692283C211yesmaximal, characteristic, complemented
2C2×SD16×D101C211yesmaximal, characteristic, complemented
3#20265837C211yesmaximal, characteristic, complemented
4#20265811C211yesmaximal, characteristic, complemented
5#20265563C211yesmaximal, characteristic, complemented
6#20265414C211yesmaximal, characteristic, complemented
7#20265395C211yesmaximal, characteristic, complemented
8#20265376C211yesmaximal, characteristic, complemented
9#20265327C211yesmaximal, characteristic, complemented
10#20265320C211yesmaximal, characteristic, complemented
11#20265317C211yesmaximal, characteristic, complemented
12#20265312C211yesmaximal, characteristic, complemented
13#20265276C211yesmaximal, characteristic, complemented
14#20265246C211yesmaximal, characteristic, complemented
15#468×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic, complemented
17C2×Q8×D101C2211yescharacteristic, complemented
18C22×D404C2211yescharacteristic
19C2×D4×D101—21no—
20C2×D4×D101C2211yescharacteristic
21#15692283—24no—
22C2×#13108063C2211yescharacteristic
23C2×#13108063C2211yescharacteristic, complemented
24C2×#13108062C2211yescharacteristic, complemented
25C2×#13108061C2211yescharacteristic
26C2×#13108061C2211yescharacteristic, complemented
27SD16×D101—24no—
28C2×D808C2211yescharacteristic
29C2×D808C2211yescharacteristic, complemented
30C2×#13108059C2211yescharacteristic, complemented
31C2×C8×D101C2211yescharacteristic, complemented
32#15692251C2211yescharacteristic
33#15692244C2211yescharacteristic, complemented
34#15692241—21no—
35#15692233—21no—
36#15692230C2211yescharacteristic
37#15692230C2211yescharacteristic, complemented
38#15692228—21no—
39#15692228—21no—
40#15692225—21no—
41#15692223—21no—
42#15692220C2211yescharacteristic, complemented
43#15692217—21no—
44#15692217C2211yescharacteristic
45C4×C22⋊D101C2211yescharacteristic
46#15692184C2211yescharacteristic, complemented
47#15692178C2211yescharacteristic, complemented
48#15692175C2211yescharacteristic
49#15692175C2211yescharacteristic, complemented
50#15692175C2211yescharacteristic, complemented
51C22⋊#11717579C2211yescharacteristic, complemented
52#15692167C2211yescharacteristic, complemented
53#15692165C2211yescharacteristic, complemented
54C2×SD16×C101C2211yescharacteristic, complemented
55C2×M16×C101C2211yescharacteristic, complemented
56C22⋊Q8×C101C2211yescharacteristic, complemented
57#121×C101C2211yescharacteristic
58#107×C101C2211yescharacteristic, complemented
59#103×C101C2211yescharacteristic, complemented
60#102×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C23×D101—21no—
63#13108074—22no—
64#13108074—41no—
65C2×Dic202—21no—
66C2×Dic202C2311yescharacteristic
67Q8×D101—22no—
68Q8×D101—41no—
69C2×D404—22no—
70C2×D404—21no—
71C2×D404—21no—
72C2×D404—21no—
73C2×D404—41no—
74C2×D404C2311yescharacteristic
75C2×D404C2311yescharacteristic
76C2×D404C2311yescharacteristic
77C2×C22⋊D101—21no—
78C2×C22⋊D101—21no—
79C2×C22⋊D101—21no—
80C2×C22⋊D101—21no—
81C2×C22⋊D101D411yescharacteristic
82D4×D101—22no—
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#13108063D414yescomplemented
91#13108062—22no—
92#13108061—22no—
93#13108061—41no—
94D808—22no—
95D808—22no—
96#13108059—41no—
97C2×#11717579C2311yescharacteristic
98C2×#11717579C2311yescharacteristic
99C8×D101—22no—
100#13108053—21no—
101#13108053D411yescharacteristic
102C22⋊Dic101D411yescharacteristic
103#13108051—21no—
104#13108051—21no—
105#13108051—21no—
106#13108051—21no—
107#13108051—21no—
108#13108051D411yescharacteristic
109C4×Dic101—21no—
110C4×Dic101C2311yescharacteristic
111C2×Q8×C101C2311yescharacteristic
112C2×D4×C101—21no—
113C2×D4×C101C2311yescharacteristic
114C22×C4×C101C2311yescharacteristic
115SD16×C101—22no—
116M16×C101—22no—
117C2×C8×C101C2311yescharacteristic
118C2×C8×C101C2311yescharacteristic
119C4⋊C4×C101—21no—
120C4⋊C4×C101C2311yescharacteristic
121C4⋊C4×C101C2311yescharacteristic
122C22⋊C4×C101—21no—
123C22⋊C4×C101—21no—
124C23×C101—21no—
125C23×C101C2×D411yescharacteristic
126C22×D101—21no—
127C22×D101—21no—
128C22×D101—21no—
129C22×D101—42no—
130C22×D101—42no—
131C22×D101—42no—
132C22×D101—42no—
133C22×D101—81no—
134C22×D101—81no—
135C22×D101C2×D411yescharacteristic
136Q8×C101—21no—
137Q8×C101—21no—
138D4×C101—21no—
139D4×C101—21no—
140D4×C101—42no—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—21no—
144C2×C4×C101—21no—
145C2×C4×C101—21no—
146C2×C4×C101C2×D411yescharacteristic
147C2×C4×C101C2411yescharacteristic
148C22⋊D101—22no—
149C22⋊D101—42no—
150C22⋊D101—42no—
151C22⋊D101—42no—
152C22⋊D101—81no—
153C2×Dic101—21no—
154C2×Dic101C2×D411yescharacteristic
155C2×Dic101C2×D411yescharacteristic
156C2×Dic101C2×D411yescharacteristic
157D404—21no—
158D404—21no—
159D404—21no—
160D404—21no—
161D404—21no—
162D404—42no—
163D404—42no—
164D404—42no—
165D404C2×D412yes—
166C4×D101—21no—
167C4×D101—21no—
168C4×D101—42no—
169C4×D101—42no—
170C4×D101—42no—
171Dic202—21no—
172Dic202—42no—
173Dic202C2×D412yescomplemented
174C8×C101—21no—
175C8×C101C2×D412yescomplemented
176#11717579—21no—
177#11717579—21no—
178C22×C101—21no—
179C22×C101—21no—
180C22×C101—21no—
181C22×C101—41no—
182C22×C101—41no—
183C22×C101—41no—
184C22×C101C22×D411yescharacteristic
185C2×D101—21no—
186C2×D101—21no—
187C2×D101—22no—
188C2×D101—41no—
189C2×D101—41no—
190C2×D101—41no—
191C2×D101—41no—
192C2×D101—41no—
193C2×D101—41no—
194C2×D101—41no—
195C2×D101—41no—
196C2×D101—84no—
197C2×D101—84no—
198C2×D101AGL(1,ℤ/8)12yescomplemented
199C4×C101—21no—
200C4×C101—41no—
201C4×C101—41no—
202C4×C101—41no—
203C4×C101C22×D411yescharacteristic
204C4×C101H(5,𝔽2)11yescharacteristic
205Dic101—22no—
206Dic101—21no—
207Dic101—21no—
208Dic101—21no—
209C2×C101—41no—
210C2×C101—81no—
211C2×C101D4211yescharacteristic
212C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C101#57711yescharacteristic
214D101—22no—
215D101—42no—
216D101—81no—
217D101—81no—
218#3265—1011nomaximal, Sylow
219C101#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1011no—
221C22×SD16—1011no—
222#548—1011no—
223D42—1011no—
224#496—1011no—
225#468D10111yescharacteristic, complemented
226#466—1011no—
227#463—1011no—
228#460—1011no—
229#450—1011no—
230#449—1011no—
231#448—1011no—
232#447—1011no—
233#440—1011no—
234#435—1011no—
235C2×#41—1011no—
236C22×Q8—1011no—
237C22×D4—1011no—
238C22×D4—1011no—
239C22×D4—2021no—
240AGL(1,ℤ/8)—2024no—
241C2×SD16—1011no—
242C2×SD16—1011no—
243C2×SD16—1011no—
244C2×SD16—1011no—
245C2×SD16—1011no—
246C2×SD16—2024no—
247C2×SD16C2×D10111yescharacteristic, complemented
248C2×D8—1011no—
249C2×D8—1011no—
250C2×D8—1011no—
251C2×M16C2×D10111yescharacteristic, complemented
252C22×C8—1011no—
253#127—1011no—
254#124—1011no—
255#124—2021no—
256C22⋊Q8—2021no—
257C22⋊Q8C2×D10111yescharacteristic, complemented
258#121—1011no—
259#121—1011no—
260#121—2021no—
261#121—2021no—
262#121C2×D10111yescharacteristic
263#120—2021no—
264#120—2021no—
265C4×Q8—1011no—
266C4×D4—1011no—
267C4×D4—1011no—
268C4×D4—2021no—
269#107C2×D10111yescharacteristic, complemented
270C4⋊C8—1011no—
271#103—1011no—
272#103C2×D10111yescharacteristic, complemented
273#102—1011no—
274#102—1011no—
275#102—1011no—
276#102C2×D10111yescharacteristic, complemented
277C22⋊C8—1011no—
278C22⋊C8—1011no—
279#97—1011no—
280C24—2021no—
281C24—2021no—
282#41—4041no—
283#41—2022no—
284C2×Q8—1011no—
285C2×Q8—4041no—
286C2×Q8—2021no—
287C2×Q8—2022no—
288C2×Q8C22×D10111yescharacteristic
289C2×D4—21no—
290C2×D4—1011no—
291C2×D4—1011no—
292C2×D4—1011no—
293C2×D4—1011no—
294C2×D4—4041no—
295C2×D4—4044no—
296C2×D4—4041no—
297C2×D4—2022no—
298C2×D4—2021no—
299C2×D4—2021no—
300C2×D4—2021no—
301C2×D4—2021no—
302C2×D4—2021no—
303C2×D4—2022no—
304C2×D4—2021no—
305C2×D4—2021no—
306C2×D4C22×D10111yescharacteristic
307C22×C4—1011no—
308C22×C4—2021no—
309C22×C4—2021no—
310C22×C4—2021no—
311C22×C4C22×D10111yescharacteristic
312SD16—22no—
313SD16—1014no—
314SD16—4041no—
315SD16—2022no—
316SD16—2022no—
317SD16—2022no—
318D8—4041no—
319D8—2022no—
320D8—2022no—
321M16—22no—
322C2×C8—1011no—
323C2×C8—1011no—
324C2×C8—2022no—
325C2×C8C22×D10111yescharacteristic
326C2×C8C22×D10111yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—1011no—
329C4⋊C4—2021no—
330C4⋊C4C22×D10111yescharacteristic
331C4⋊C4C22×D10111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—1011no—
335C22⋊C4—1011no—
336C22⋊C4—2021no—
337C22⋊C4—2021no—
338C22⋊C4—2021no—
339C22⋊C4—2021no—
340C22⋊C4—2021no—
341C42—1011no—
342C42—2021no—
343C23—21no—
344C23—8081no—
345C23—8081no—
346C23—1011no—
347C23—4042no—
348C23—4042no—
349C23—4042no—
350C23—4042no—
351C23—2021no—
352C23—2021no—
353C23—2021no—
354C23D4×D10111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—1012no—
358Q8—4042no—
359Q8—2021no—
360D4—21no—
361D4—21no—
362D4—42no—
363D4—8081no—
364D4—1012no—
365D4—4042no—
366D4—4042no—
367D4—4042no—
368D4—4042no—
369D4—4042no—
370D4—4042no—
371D4—2022no—
372D4—2021no—
373D4—2021no—
374D4—2021no—
375D4—2021no—
376D4—2021no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—1011no—
383C2×C4—1011no—
384C2×C4—1011no—
385C2×C4—4042no—
386C2×C4—4042no—
387C2×C4—4042no—
388C2×C4—2021no—
389C2×C4—2021no—
390C2×C4—2021no—
391C2×C4D4×D10111yescharacteristic
392C2×C4C23×D10111yescharacteristic
393C8—21no—
394C8—2021no—
395C8—2021no—
396C8D4×D10112yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—8084no—
404C22—8084no—
405C22—1012no—
406C22—4041no—
407C22—4041no—
408C22—4041no—
409C22—4041no—
410C22—4041no—
411C22—4041no—
412C22—4041no—
413C22—4041no—
414C22—2022no—
415C22—2021no—
416C22—2021no—
417C22C2×D4×D10111yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—2021no—
423C4—2021no—
424C4—2022no—
425C4—2021no—
426C4C2×D4×D10111yescharacteristic
427C4#1569230111yescharacteristic
428C2—41no—
429C2—81no—
430C2—8081no—
431C2—8081no—
432C2—4042no—
433C2—2022no—
434C2#2026581111yescharacteristic
435C2AGL(1,ℤ/8)×D10111yescharacteristic
436C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#468×C101C2
socleC22×C101C22×D4
hypercenter#468D101
hypocenterC101#3265
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#15692283C2semidirect
2. C2×SD16×D101C2semidirect
3. #20265837C2semidirect
4. #20265811C2semidirect
5. #20265563C2semidirect
6. #20265414C2semidirect
7. #20265395C2semidirect
8. #20265376C2semidirect
9. #20265327C2semidirect
10. #20265320C2semidirect
11. #20265317C2semidirect
12. #20265312C2semidirect
13. #20265276C2semidirect
14. #20265246C2semidirect
15. #468×C101C2semidirect
16. C2×#13108074C22semidirect
17. C2×Q8×D101C22semidirect
23. C2×#13108063C22semidirect
24. C2×#13108062C22semidirect
26. C2×#13108061C22semidirect
29. C2×D808C22semidirect
30. C2×#13108059C22semidirect
31. C2×C8×D101C22semidirect
33. #15692244C22semidirect
37. #15692230C22semidirect
42. #15692220C22semidirect
46. #15692184C22semidirect
47. #15692178C22semidirect
49. #15692175C22semidirect
50. #15692175C22semidirect
51. C22⋊#11717579C22semidirect
52. #15692167C22semidirect
53. #15692165C22semidirect
54. C2×SD16×C101C22semidirect
55. C2×M16×C101C22semidirect
56. C22⋊Q8×C101C22semidirect
58. #107×C101C22semidirect
59. #103×C101C22semidirect
60. #102×C101C22semidirect
90. #13108063D4semidirect
173. Dic202C2×D4semidirect
175. C8×C101C2×D4semidirect
198. C2×D101AGL(1,ℤ/8)semidirect
219. C101#3265semidirect
225. #468D101semidirect
247. C2×SD16C2×D101semidirect
251. C2×M16C2×D101semidirect
257. C22⋊Q8C2×D101semidirect
269. #107C2×D101semidirect
272. #103C2×D101semidirect
276. #102C2×D101semidirect
396. 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×SD16×D1012
#202658372
#202658112
#202655632
#202654142
#202653952
#202653762
#202653272
#202653202
#202653172
#202653122
#202652762
#202652462
#468×C1012
#3265101