Finite Group Databasebeta

← #27182626

Subgroups — #27182626

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×#15692279C211yesmaximal, characteristic, complemented
2C2×SD16×D101C211yesmaximal, characteristic, complemented
3#20265835C211yesmaximal, characteristic, complemented
4#20265812C211yesmaximal, characteristic, complemented
5#20265420C211yesmaximal, characteristic, complemented
6#20265410C211yesmaximal, characteristic, complemented
7#20265389C211yesmaximal, characteristic, complemented
8#20265374C211yesmaximal, characteristic, complemented
9#20265326C211yesmaximal, characteristic, complemented
10#20265319C211yesmaximal, characteristic, complemented
11#20265314C211yesmaximal, characteristic, complemented
12#20265309C211yesmaximal, characteristic, complemented
13#20265279C211yesmaximal, characteristic, complemented
14#20265250C211yesmaximal, characteristic, complemented
15#496×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic, complemented
17C2×Q8×D101C2211yescharacteristic, complemented
18C2×D4×D101—21no—
19C2×D4×D101C2211yescharacteristic
20C2×D4×D101C2211yescharacteristic
21#15692279—24no—
22C2×#13108063C2211yescharacteristic
23C2×#13108062C2211yescharacteristic, complemented
24C2×#13108062C2211yescharacteristic, complemented
25C2×#13108061C2211yescharacteristic
26SD16×D101—24no—
27C2×#13108059C2211yescharacteristic
28C2×#13108057C2211yescharacteristic, complemented
29C2×C8×D101C2211yescharacteristic, complemented
30#15692241—21no—
31#15692232—21no—
32#15692231C2211yescharacteristic, complemented
33#15692228—21no—
34#15692226—21no—
35#15692226C2211yescharacteristic
36#15692226C2211yescharacteristic
37#15692226C2211yescharacteristic, complemented
38#15692223—21no—
39#15692223—21no—
40C4×Dic202C2211yescharacteristic, complemented
41C4×D404C2211yescharacteristic
42C4×D404C2211yescharacteristic
43#15692215—21no—
44#15692189C2211yescharacteristic, complemented
45C4⋊#11717579C2211yescharacteristic, complemented
46#15692180C2211yescharacteristic, complemented
47#15692179C2211yescharacteristic, complemented
48#15692177C2211yescharacteristic
49#15692176C2211yescharacteristic, complemented
50#15692176C2211yescharacteristic, complemented
51#15692176C2211yescharacteristic, complemented
52#15692167C2211yescharacteristic, complemented
53#15692167C2211yescharacteristic, complemented
54C2×SD16×C101C2211yescharacteristic, complemented
55C2×SD16×C101C2211yescharacteristic, complemented
56C2×D8×C101C2211yescharacteristic, complemented
57C2×D8×C101C2211yescharacteristic, complemented
58#127×C101C2211yescharacteristic
59#124×C101C2211yescharacteristic, complemented
60#97×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C23×D101—21no—
63#13108073—22no—
64#13108073—41no—
65C2×Dic202—21no—
66C2×Dic202C2311yescharacteristic
67Q8×D101—22no—
68Q8×D101—41no—
69C2×D404D411yescharacteristic
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—41no—
80D4×D101—41no—
81D4×D101—44no—
82C22×Dic101—21no—
83C2×C4×D101—21no—
84C2×C4×D101—21no—
85C2×C4×D101C2311yescharacteristic
86C2×C4×D101C2311yescharacteristic
87#13108063D414yescomplemented
88#13108062—22no—
89#13108062—22no—
90#13108061—22no—
91#13108059—22no—
92#13108057—22no—
93C2×#11717579C2311yescharacteristic
94C2×#11717579C2311yescharacteristic
95C8×D101—22no—
96C4⋊Dic101D411yescharacteristic
97C4⋊Dic101C2311yescharacteristic
98C4⋊Dic101C2311yescharacteristic
99#13108053—21no—
100#13108053—21no—
101C22⋊Dic101—21no—
102C22⋊Dic101—21no—
103C22⋊Dic101—21no—
104#13108051—21no—
105#13108051—21no—
106#13108051D411yescharacteristic
107#13108051D411yescharacteristic
108C4×Dic101—21no—
109C2×Q8×C101C2311yescharacteristic
110C2×D4×C101—21no—
111C2×D4×C101—21no—
112C2×D4×C101C2311yescharacteristic
113C2×D4×C101C2311yescharacteristic
114C2×D4×C101C2311yescharacteristic
115SD16×C101—22no—
116SD16×C101—41no—
117D8×C101—22no—
118D8×C101—41no—
119C2×C8×C101C2311yescharacteristic
120C2×C8×C101C2311yescharacteristic
121C22⋊C4×C101—21no—
122C22⋊C4×C101—21no—
123C42×C101C2311yescharacteristic
124C23×C101—21no—
125C23×C101—21no—
126C23×C101—21no—
127C22×D101—21no—
128C22×D101—42no—
129C22×D101—42no—
130C22×D101—42no—
131C22×D101—42no—
132C22×D101—81no—
133C22×D101—81no—
134C22×D101C2×D411yescharacteristic
135C22×D101C2×D411yescharacteristic
136Q8×C101—21no—
137Q8×C101—21no—
138D4×C101—21no—
139D4×C101—21no—
140D4×C101—21no—
141D4×C101—21no—
142D4×C101—21no—
143D4×C101—21no—
144D4×C101—42no—
145D4×C101—81no—
146C2×C4×C101—21no—
147C2×C4×C101C2×D411yescharacteristic
148C2×C4×C101C2×D411yescharacteristic
149C2×C4×C101C2411yescharacteristic
150C22⋊D101—42no—
151C22⋊D101—42no—
152C22⋊D101—42no—
153C22⋊D101—42no—
154C22⋊D101—42no—
155C22⋊D101—42no—
156C2×Dic101—21no—
157C2×Dic101—21no—
158C2×Dic101—21no—
159C2×Dic101—42no—
160C2×Dic101C2×D411yescharacteristic
161C2×Dic101C2×D411yescharacteristic
162D404—22no—
163D404—21no—
164D404C2×D412yes—
165C4×D101—21no—
166C4×D101—21no—
167C4×D101—21no—
168C4×D101—21no—
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—41no—
179C22×C101—41no—
180C22×C101—41no—
181C22×C101—41no—
182C22×C101—41no—
183C22×C101—41no—
184C22×C101—41no—
185C22×C101—41no—
186C22×C101—41no—
187C22×C101C22×D411yescharacteristic
188C2×D101—21no—
189C2×D101—21no—
190C2×D101—21no—
191C2×D101—22no—
192C2×D101—21no—
193C2×D101—21no—
194C2×D101—41no—
195C2×D101—41no—
196C2×D101—84no—
197C2×D101—84no—
198C2×D101AGL(1,ℤ/8)12yescomplemented
199C4×C101—21no—
200C4×C101—21no—
201C4×C101—41no—
202C4×C101C22×D411yescharacteristic
203C4×C101H(5,𝔽2)11yescharacteristic
204Dic101—22no—
205Dic101—21no—
206Dic101—21no—
207Dic101—41no—
208Dic101—41no—
209C2×C101—81no—
210C2×C101—81no—
211C2×C101—81no—
212C2×C101D4211yescharacteristic
213C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C101#57711yescharacteristic
215D101—22no—
216D101—41no—
217D101—42no—
218#3265—1011nomaximal, Sylow
219C101#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1011no—
221C22×SD16—1011no—
222#548—1011no—
223D42—1011no—
224#496D10111yescharacteristic, complemented
225#468—1011no—
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—2024no—
246C2×SD16C2×D10111yescharacteristic, complemented
247C2×SD16C2×D10111yescharacteristic, complemented
248C2×D8—1011no—
249C2×D8C2×D10111yescharacteristic, complemented
250C2×D8C2×D10111yescharacteristic, complemented
251C2×M16—1011no—
252C22×C8—1011no—
253#127C2×D10111yescharacteristic
254#124—2021no—
255#124C2×D10111yescharacteristic, complemented
256C22⋊Q8—1011no—
257C22⋊Q8—2021no—
258#121—1011no—
259#121—1011no—
260#121—1011no—
261#121—2021no—
262#121—2021no—
263#120—2021no—
264#120—2021no—
265C4×Q8—1011no—
266C4×D4—1011no—
267C4×D4—1011no—
268C4×D4—2021no—
269#107—1011no—
270C4⋊C8—1011no—
271#103—1011no—
272#103—1011no—
273#102—1011no—
274#102—1011no—
275#102—1011no—
276#102—1011no—
277C22⋊C8—1011no—
278C22⋊C8—1011no—
279#97C2×D10111yescharacteristic, complemented
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—21no—
291C2×D4—1011no—
292C2×D4—1011no—
293C2×D4—4044no—
294C2×D4—4041no—
295C2×D4—4041no—
296C2×D4—2021no—
297C2×D4—2021no—
298C2×D4—2022no—
299C2×D4—2022no—
300C2×D4—2021no—
301C2×D4—2021no—
302C2×D4—2021no—
303C2×D4—2021no—
304C2×D4C22×D10111yescharacteristic
305C2×D4C22×D10111yescharacteristic
306C2×D4C22×D10111yescharacteristic
307C22×C4—1011no—
308C22×C4—1011no—
309C22×C4—2021no—
310C22×C4—2021no—
311C22×C4—2021no—
312SD16—22no—
313SD16—41no—
314SD16—1014no—
315SD16—2022no—
316SD16—2022no—
317SD16—2022no—
318D8—22no—
319D8—41no—
320D8—2022no—
321M16—2022no—
322C2×C8—1011no—
323C2×C8—1011no—
324C2×C8—2022no—
325C2×C8C22×D10111yescharacteristic
326C2×C8C22×D10111yescharacteristic
327C4⋊C4—1011no—
328C4⋊C4—1011no—
329C4⋊C4—1011no—
330C4⋊C4—2021no—
331C4⋊C4—2021no—
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—2021no—
342C42C22×D10111yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—8081no—
347C23—8081no—
348C23—1011no—
349C23—1011no—
350C23—4042no—
351C23—4042no—
352C23—4042no—
353C23—4042no—
354C23—2021no—
355Q8—21no—
356Q8—21no—
357Q8—1012no—
358Q8—4042no—
359Q8—2021no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—81no—
368D4—1012no—
369D4—4042no—
370D4—4042no—
371D4—4042no—
372D4—4042no—
373D4—4042no—
374D4—4042no—
375D4—2022no—
376D4—2021no—
377C2×C4—21no—
378C2×C4—1011no—
379C2×C4—1011no—
380C2×C4—4042no—
381C2×C4—4042no—
382C2×C4—4042no—
383C2×C4—2021no—
384C2×C4—2021no—
385C2×C4—2021no—
386C2×C4—2021no—
387C2×C4—2021no—
388C2×C4—2021no—
389C2×C4—2021no—
390C2×C4D4×D10111yescharacteristic
391C2×C4D4×D10111yescharacteristic
392C2×C4C23×D10111yescharacteristic
393C8—21no—
394C8—2021no—
395C8—2021no—
396C8D4×D10112yescomplemented
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—8084no—
407C22—8084no—
408C22—1012no—
409C22—4041no—
410C22—4041no—
411C22—2021no—
412C22—2021no—
413C22—2022no—
414C22—2021no—
415C22—2021no—
416C22—2021no—
417C22C2×D4×D10111yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—4041no—
422C4—4041no—
423C4—2021no—
424C4—2021no—
425C4—2022no—
426C4C2×D4×D10111yescharacteristic
427C4#1569230011yescharacteristic
428C2—81no—
429C2—81no—
430C2—81no—
431C2—4042no—
432C2—4041no—
433C2—2022no—
434C2#2026581211yescharacteristic
435C2AGL(1,ℤ/8)×D10111yescharacteristic
436C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#496×C101C2
socleC22×C101C22×D4
hypercenter#496D101
hypocenterC101#3265
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#15692279C2semidirect
2. C2×SD16×D101C2semidirect
3. #20265835C2semidirect
4. #20265812C2semidirect
5. #20265420C2semidirect
6. #20265410C2semidirect
7. #20265389C2semidirect
8. #20265374C2semidirect
9. #20265326C2semidirect
10. #20265319C2semidirect
11. #20265314C2semidirect
12. #20265309C2semidirect
13. #20265279C2semidirect
14. #20265250C2semidirect
15. #496×C101C2semidirect
16. C2×#13108073C22semidirect
17. C2×Q8×D101C22semidirect
23. C2×#13108062C22semidirect
24. C2×#13108062C22semidirect
28. C2×#13108057C22semidirect
29. C2×C8×D101C22semidirect
32. #15692231C22semidirect
37. #15692226C22semidirect
40. C4×Dic202C22semidirect
44. #15692189C22semidirect
45. C4⋊#11717579C22semidirect
46. #15692180C22semidirect
47. #15692179C22semidirect
49. #15692176C22semidirect
50. #15692176C22semidirect
51. #15692176C22semidirect
52. #15692167C22semidirect
53. #15692167C22semidirect
54. C2×SD16×C101C22semidirect
55. C2×SD16×C101C22semidirect
56. C2×D8×C101C22semidirect
57. C2×D8×C101C22semidirect
59. #124×C101C22semidirect
60. #97×C101C22semidirect
87. #13108063D4semidirect
173. Dic202C2×D4semidirect
175. C8×C101C2×D4semidirect
198. C2×D101AGL(1,ℤ/8)semidirect
219. C101#3265semidirect
224. #496D101semidirect
246. C2×SD16C2×D101semidirect
247. C2×SD16C2×D101semidirect
249. C2×D8C2×D101semidirect
250. C2×D8C2×D101semidirect
255. #124C2×D101semidirect
279. #97C2×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×#156922792
C2×SD16×D1012
#202658352
#202658122
#202654202
#202654102
#202653892
#202653742
#202653262
#202653192
#202653142
#202653092
#202652792
#202652502
#496×C1012
#3265101