Finite Group Databasebeta

← #27182625

Subgroups — #27182625

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×#15692281C211yesmaximal, characteristic, complemented
2C2×SD16×D101C211yesmaximal, characteristic, complemented
3#20265837C211yesmaximal, characteristic, complemented
4D4×C22⋊D101C211yesmaximal, characteristic, complemented
5#20265563C211yesmaximal, characteristic, complemented
6#20265425C211yesmaximal, characteristic, complemented
7#20265412C211yesmaximal, characteristic, complemented
8#20265394C211yesmaximal, characteristic, complemented
9#20265376C211yesmaximal, characteristic, complemented
10#20265324C211yesmaximal, characteristic, complemented
11#20265314C211yesmaximal, characteristic, complemented
12#20265312C211yesmaximal, characteristic, complemented
13#20265274C211yesmaximal, characteristic, complemented
14#20265251C211yesmaximal, characteristic, complemented
15#450×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic, complemented
17C2×Q8×D101C2211yescharacteristic, complemented
18C22×C22⋊D101—21no—
19C2×D4×D101C2211yescharacteristic
20#15692281—24no—
21C2×#13108063C2211yescharacteristic, complemented
22C2×#13108062C2211yescharacteristic
23C2×#13108062C2211yescharacteristic, complemented
24C2×#13108061C2211yescharacteristic, complemented
25SD16×D101—24no—
26C2×D808C2211yescharacteristic, complemented
27C2×#13108059C2211yescharacteristic
28C2×#13108059C2211yescharacteristic, complemented
29C2×#13108058C2211yescharacteristic, complemented
30C2×C8×D101C2211yescharacteristic, complemented
31#15692251C2211yescharacteristic
32#15692244C2211yescharacteristic, complemented
33#15692241—21no—
34#15692233—21no—
35#15692230C2211yescharacteristic, complemented
36#15692229C2211yescharacteristic
37#15692228—21no—
38#15692227—21no—
39#15692226C2211yescharacteristic
40#15692224—21no—
41#15692223—21no—
42#15692220C2211yescharacteristic, complemented
43D4×Dic101C2211yescharacteristic
44#15692217—21no—
45C4×C22⋊D101C2211yescharacteristic
46#15692188C2211yescharacteristic, complemented
47#15692184C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692179C2211yescharacteristic, complemented
50#15692177C2211yescharacteristic, complemented
51#15692175C2211yescharacteristic, complemented
52#15692167C2211yescharacteristic, complemented
53#15692165C2211yescharacteristic, complemented
54C22×D4×C101C2211yescharacteristic
55C2×SD16×C101C2211yescharacteristic
56C2×SD16×C101C2211yescharacteristic, complemented
57C22⋊Q8×C101C2211yescharacteristic, complemented
58#102×C101C2211yescharacteristic
59#102×C101C2211yescharacteristic, complemented
60C22⋊C8×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62#13108074—22no—
63#13108074—41no—
64C2×Dic202—21no—
65C2×Dic202C2311yescharacteristic
66Q8×D101—22no—
67Q8×D101—41no—
68C2×D404—21no—
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—
77C2×C22⋊D101—44no—
78C2×C22⋊D101D411yescharacteristic
79D4×D101—22no—
80D4×D101—41no—
81C22×Dic101—21no—
82C2×C4×D101—21no—
83C2×C4×D101—21no—
84C2×C4×D101C2311yescharacteristic
85#13108063—22no—
86#13108062—22no—
87#13108062D414yescomplemented
88#13108061—22no—
89D808—41no—
90#13108059—22no—
91#13108059—22no—
92#13108058—22no—
93C2×#11717579C2311yescharacteristic
94C2×#11717579C2311yescharacteristic
95C8×D101—22no—
96C4⋊Dic101C2311yescharacteristic
97#13108053—21no—
98#13108053D411yescharacteristic
99C22⋊Dic101—21no—
100C22⋊Dic101—21no—
101C22⋊Dic101D411yescharacteristic
102#13108051—21no—
103#13108051—21no—
104#13108051—21no—
105#13108051—21no—
106#13108051D411yescharacteristic
107C4×Dic101—21no—
108C4×Dic101C2311yescharacteristic
109C24×C101—21no—
110C2×Q8×C101C2311yescharacteristic
111C2×D4×C101—22no—
112C2×D4×C101—21no—
113C2×D4×C101—41no—
114C2×D4×C101C2311yescharacteristic
115C2×D4×C101C2311yescharacteristic
116C22×C4×C101C2311yescharacteristic
117SD16×C101—22no—
118SD16×C101—41no—
119C2×C8×C101C2311yescharacteristic
120C2×C8×C101C2311yescharacteristic
121C4⋊C4×C101—21no—
122C4⋊C4×C101C2311yescharacteristic
123C22⋊C4×C101—21no—
124C23×C101—21no—
125C23×C101—21no—
126C23×C101—42no—
127C23×C101—42no—
128C23×C101—81no—
129C23×C101C2×D411yescharacteristic
130C22×D101—21no—
131C22×D101—21no—
132C22×D101—42no—
133C22×D101—42no—
134C22×D101—81no—
135C22×D101C2×D411yescharacteristic
136Q8×C101—21no—
137Q8×C101—21no—
138D4×C101—21no—
139D4×C101—21no—
140D4×C101—21no—
141D4×C101—42no—
142D4×C101C2×D412yes—
143C2×C4×C101—21no—
144C2×C4×C101—21no—
145C2×C4×C101—21no—
146C2×C4×C101—21no—
147C2×C4×C101C2×D411yescharacteristic
148C2×C4×C101C2411yescharacteristic
149C22⋊D101—22no—
150C22⋊D101—42no—
151C22⋊D101—42no—
152C22⋊D101—42no—
153C22⋊D101—42no—
154C22⋊D101—42no—
155C22⋊D101—81no—
156C2×Dic101—21no—
157C2×Dic101—21no—
158C2×Dic101—42no—
159C2×Dic101C2×D411yescharacteristic
160C2×Dic101C2×D411yescharacteristic
161C2×Dic101C2×D411yescharacteristic
162D404—21no—
163D404—21no—
164D404—21no—
165D404—21no—
166D404—42no—
167C4×D101—21no—
168C4×D101—21no—
169C4×D101—42no—
170C4×D101—42no—
171Dic202—21no—
172Dic202—42no—
173Dic202C2×D412yescomplemented
174C8×C101—21no—
175C8×C101—21no—
176#11717579—21no—
177#11717579C2×D412yescomplemented
178C22×C101—21no—
179C22×C101—22no—
180C22×C101—21no—
181C22×C101—21no—
182C22×C101—41no—
183C22×C101—41no—
184C22×C101—41no—
185C22×C101—41no—
186C22×C101—41no—
187C22×C101—84no—
188C22×C101C22×D411yescharacteristic
189C2×D101—21no—
190C2×D101—21no—
191C2×D101—41no—
192C2×D101—41no—
193C2×D101—41no—
194C2×D101—41no—
195C2×D101—41no—
196C2×D101—41no—
197C2×D101—84no—
198C2×D101AGL(1,ℤ/8)12yescomplemented
199C4×C101—21no—
200C4×C101—41no—
201C4×C101—41no—
202C4×C101C22×D411yescharacteristic
203C4×C101H(5,𝔽2)11yescharacteristic
204Dic101—22no—
205Dic101—21no—
206Dic101—21no—
207Dic101—21no—
208Dic101—41no—
209C2×C101—42no—
210C2×C101—41no—
211C2×C101—81no—
212C2×C101D4211yescharacteristic
213C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C101#57711yescharacteristic
215D101—22no—
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#468—1011no—
226#466—1011no—
227#463—1011no—
228#460—1011no—
229#450D10111yescharacteristic, complemented
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—2021no—
239C22×D4C2×D10111yescharacteristic
240AGL(1,ℤ/8)—2024no—
241C2×SD16—1011no—
242C2×SD16—1011no—
243C2×SD16—1011no—
244C2×SD16—1011no—
245C2×SD16—2024no—
246C2×SD16C2×D10111yescharacteristic
247C2×SD16C2×D10111yescharacteristic, complemented
248C2×D8—1011no—
249C2×D8—1011no—
250C2×D8—1011no—
251C2×M16—1011no—
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—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#102C2×D10111yescharacteristic
276#102C2×D10111yescharacteristic, complemented
277C22⋊C8—1011no—
278C22⋊C8C2×D10111yescharacteristic, complemented
279#97—1011no—
280C24—21no—
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—22no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—1011no—
293C2×D4—1011no—
294C2×D4—1011no—
295C2×D4—4044no—
296C2×D4—4041no—
297C2×D4—2021no—
298C2×D4—2021no—
299C2×D4—2021no—
300C2×D4—2021no—
301C2×D4—2021no—
302C2×D4—2021no—
303C2×D4—2021no—
304C2×D4—2022no—
305C2×D4C22×D10111yescharacteristic
306C2×D4C22×D10111yescharacteristic
307C22×C4—1011no—
308C22×C4—2021no—
309C22×C4—2021no—
310C22×C4—2021no—
311C22×C4C22×D10111yescharacteristic
312SD16—22no—
313SD16—41no—
314SD16—1014no—
315SD16—2022no—
316SD16—2022no—
317SD16—2022no—
318D8—4041no—
319D8—2022no—
320D8—2022no—
321M16—2022no—
322C2×C8—1011no—
323C2×C8—1011no—
324C2×C8—2022no—
325C2×C8C22×D10111yescharacteristic
326C2×C8C22×D10111yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—1011no—
329C4⋊C4—1011no—
330C4⋊C4—2021no—
331C4⋊C4C22×D10111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—1011no—
334C22⋊C4—1011no—
335C22⋊C4—2021no—
336C22⋊C4—2021no—
337C22⋊C4—2021no—
338C22⋊C4—2021no—
339C22⋊C4—2021no—
340C22⋊C4—2021no—
341C42—1011no—
342C42—2021no—
343C23—21no—
344C23—21no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—8081no—
349C23—1011no—
350C23—4042no—
351C23—4042no—
352C23—2021no—
353C23—2021no—
354C23D4×D10111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—1012no—
358Q8—4042no—
359Q8—2021no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—8081no—
365D4—4042no—
366D4—4042no—
367D4—4042no—
368D4—4042no—
369D4—4042no—
370D4—4042no—
371D4—2022no—
372D4—2021no—
373D4—2021no—
374D4—2021no—
375D4—2021no—
376D4D4×D10112yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—1011no—
382C2×C4—1011no—
383C2×C4—1011no—
384C2×C4—4042no—
385C2×C4—4042no—
386C2×C4—4042no—
387C2×C4—2021no—
388C2×C4—2021no—
389C2×C4—2021no—
390C2×C4—2021no—
391C2×C4D4×D10111yescharacteristic
392C2×C4C23×D10111yescharacteristic
393C8—21no—
394C8—21no—
395C8—1012no—
396C8—2021no—
397C22—21no—
398C22—22no—
399C22—21no—
400C22—21no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—84no—
407C22—8084no—
408C22—1012no—
409C22—4041no—
410C22—4041no—
411C22—4041no—
412C22—4041no—
413C22—4041no—
414C22—4041no—
415C22—2021no—
416C22—2021no—
417C22C2×D4×D10111yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—4041no—
422C4—2021no—
423C4—2022no—
424C4—2021no—
425C4—2021no—
426C4C2×D4×D10111yescharacteristic
427C4#1569230011yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—8081no—
432C2—8081no—
433C2—2022no—
434C2#2026580911yescharacteristic
435C2AGL(1,ℤ/8)×D10111yescharacteristic
436C2#2026598311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#450×C101C2
socleC22×C101C22×D4
hypercenter#450D101
hypocenterC101#3265
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#15692281C2semidirect
2. C2×SD16×D101C2semidirect
3. #20265837C2semidirect
4. D4×C22⋊D101C2semidirect
5. #20265563C2semidirect
6. #20265425C2semidirect
7. #20265412C2semidirect
8. #20265394C2semidirect
9. #20265376C2semidirect
10. #20265324C2semidirect
11. #20265314C2semidirect
12. #20265312C2semidirect
13. #20265274C2semidirect
14. #20265251C2semidirect
15. #450×C101C2semidirect
16. C2×#13108074C22semidirect
17. C2×Q8×D101C22semidirect
21. C2×#13108063C22semidirect
23. C2×#13108062C22semidirect
24. C2×#13108061C22semidirect
26. C2×D808C22semidirect
28. C2×#13108059C22semidirect
29. C2×#13108058C22semidirect
30. C2×C8×D101C22semidirect
32. #15692244C22semidirect
35. #15692230C22semidirect
42. #15692220C22semidirect
46. #15692188C22semidirect
47. #15692184C22semidirect
48. #15692180C22semidirect
49. #15692179C22semidirect
50. #15692177C22semidirect
51. #15692175C22semidirect
52. #15692167C22semidirect
53. #15692165C22semidirect
56. C2×SD16×C101C22semidirect
57. C22⋊Q8×C101C22semidirect
59. #102×C101C22semidirect
60. C22⋊C8×C101C22semidirect
87. #13108062D4semidirect
173. Dic202C2×D4semidirect
177. #11717579C2×D4semidirect
198. C2×D101AGL(1,ℤ/8)semidirect
219. C101#3265semidirect
229. #450D101semidirect
247. C2×SD16C2×D101semidirect
257. C22⋊Q8C2×D101semidirect
276. #102C2×D101semidirect
278. 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×SD16×D1012
#202658372
D4×C22⋊D1012
#202655632
#202654252
#202654122
#202653942
#202653762
#202653242
#202653142
#202653122
#202652742
#202652512
#450×C1012
#3265101