Finite Group Databasebeta

← #27183146

Subgroups — #27183146

Order 12928.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265861C211yesmaximal, characteristic, complemented
2D4×C22⋊D101C211yesmaximal, characteristic
3#20265540C211yesmaximal, characteristic, complemented
4#20265502C211yesmaximal, characteristic, complemented
5#20265489C211yesmaximal, characteristic, complemented
6#20265479C211yesmaximal, characteristic, complemented
7#20265425C211yesmaximal, characteristic, complemented
8#20265403C211yesmaximal, characteristic, complemented
9#20265375C211yesmaximal, characteristic, complemented
10#20265326C211yesmaximal, characteristic, complemented
11#20265278C211yesmaximal, characteristic, complemented
12#20265251C211yesmaximal, characteristic, complemented
13#20265182C211yesmaximal, characteristic, complemented
14#102×D101C211yesmaximal, characteristic, complemented
15#450×C101C211yesmaximal, characteristic, complemented
16C22×C22⋊D101—21no—
17C2×D4×D101C2211yescharacteristic
18C2×#13108061C2211yescharacteristic, complemented
19C2×#13108058C2211yescharacteristic, complemented
20C2×C8×D101C2211yescharacteristic, complemented
21#15692251C2211yescharacteristic
22#15692248—21no—
23#15692246C2211yescharacteristic, complemented
24#15692245—21no—
25#15692237—21no—
26#15692231C2211yescharacteristic, complemented
27#15692229C2211yescharacteristic
28#15692227—21no—
29#15692226C2211yescharacteristic
30#15692224—21no—
31#15692223—21no—
32Q8×Dic101C2211yescharacteristic, complemented
33D4×Dic101C2211yescharacteristic
34C4×C22⋊D101C2211yescharacteristic
35#15692214C2211yescharacteristic, complemented
36#15692213—21no—
37C4⋊C4×D101C2211yescharacteristic, complemented
38#15692189C2211yescharacteristic, complemented
39#15692188C2211yescharacteristic, complemented
40#15692187C2211yescharacteristic, complemented
41#15692184C2211yescharacteristic, complemented
42#15692179C2211yescharacteristic, complemented
43#15692177C2211yescharacteristic, complemented
44#15692177C2211yescharacteristic, complemented
45#15692176C2211yescharacteristic
46#15692176C2211yescharacteristic
47#15692176C2211yescharacteristic, complemented
48#15692176C2211yescharacteristic, complemented
49#15692175C2211yescharacteristic, complemented
50#15692167C2211yescharacteristic, complemented
51#15692165C2211yescharacteristic, complemented
52C22×D4×C101C2211yescharacteristic
53C2×SD16×C101C2211yescharacteristic
54C2×SD16×C101C2211yescharacteristic, complemented
55C22⋊Q8×C101C2211yescharacteristic, complemented
56#102×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic, complemented
58C22⋊C8×C101C2211yescharacteristic, complemented
59C23×D101—21no—
60C2×D404C2311yescharacteristic
61C2×C22⋊D101—21no—
62C2×C22⋊D101—21no—
63C2×C22⋊D101—21no—
64C2×C22⋊D101—21no—
65C2×C22⋊D101—21no—
66C2×C22⋊D101—44no—
67C2×C22⋊D101D411yescharacteristic
68D4×D101—22no—
69D4×D101—41no—
70C22×Dic101—21no—
71C2×C4×D101—21no—
72C2×C4×D101C2311yescharacteristic
73#13108061—41no—
74#13108058—41no—
75C2×#11717579C2311yescharacteristic
76C2×#11717579C2311yescharacteristic
77C8×D101—41no—
78C4⋊Dic101—21no—
79C4⋊Dic101—21no—
80C4⋊Dic101C2311yescharacteristic
81C4⋊Dic101C2311yescharacteristic
82C4⋊Dic101C2311yescharacteristic
83#13108053—21no—
84#13108053—21no—
85#13108053—21no—
86#13108053D411yescharacteristic
87C22⋊Dic101—21no—
88C22⋊Dic101—21no—
89C22⋊Dic101—21no—
90C22⋊Dic101D411yescharacteristic
91#13108051—21no—
92#13108051—21no—
93#13108051D411yescharacteristic
94C4×Dic101—21no—
95C4×Dic101—21no—
96C4×Dic101C2311yescharacteristic
97C24×C101—21no—
98C2×Q8×C101C2311yescharacteristic
99C2×D4×C101—22no—
100C2×D4×C101—21no—
101C2×D4×C101—41no—
102C2×D4×C101C2311yescharacteristic
103C2×D4×C101C2311yescharacteristic
104C22×C4×C101C2311yescharacteristic
105SD16×C101—22no—
106SD16×C101—41no—
107C2×C8×C101C2311yescharacteristic
108C2×C8×C101C2311yescharacteristic
109C4⋊C4×C101—21no—
110C4⋊C4×C101C2311yescharacteristic
111C22⋊C4×C101—21no—
112C23×C101—21no—
113C23×C101—21no—
114C23×C101—42no—
115C23×C101—42no—
116C23×C101—81no—
117C23×C101C2×D411yescharacteristic
118C22×D101—21no—
119C22×D101—42no—
120C22×D101—42no—
121C22×D101—81no—
122C22×D101C2×D411yescharacteristic
123Q8×C101—21no—
124Q8×C101—21no—
125D4×C101—21no—
126D4×C101—21no—
127D4×C101—21no—
128D4×C101—42no—
129D4×C101#4112yes—
130C2×C4×C101—21no—
131C2×C4×C101—21no—
132C2×C4×C101—21no—
133C2×C4×C101—21no—
134C2×C4×C101C2×D411yescharacteristic
135C2×C4×C101C2411yescharacteristic
136C22⋊D101—22no—
137C22⋊D101—42no—
138C22⋊D101—42no—
139C22⋊D101—42no—
140C22⋊D101—42no—
141C22⋊D101—81no—
142C2×Dic101—21no—
143C2×Dic101—21no—
144C2×Dic101—21no—
145C2×Dic101—42no—
146C2×Dic101C2×D411yescharacteristic
147C2×Dic101C2×D411yescharacteristic
148C2×Dic101C2×D411yescharacteristic
149D404—21no—
150D404—21no—
151C4×D101—21no—
152C4×D101—21no—
153C4×D101—42no—
154C8×C101—21no—
155C8×C101—21no—
156#11717579—21no—
157#11717579—21no—
158C22×C101—22no—
159C22×C101—21no—
160C22×C101—21no—
161C22×C101—21no—
162C22×C101—41no—
163C22×C101—41no—
164C22×C101—41no—
165C22×C101—41no—
166C22×C101—41no—
167C22×C101—84no—
168C22×C101C22×D411yescharacteristic
169C2×D101—21no—
170C2×D101—21no—
171C2×D101—41no—
172C2×D101—41no—
173C2×D101—41no—
174C2×D101—84no—
175C2×D101AGL(1,ℤ/8)12yes—
176C4×C101—21no—
177C4×C101—41no—
178C4×C101—41no—
179C4×C101C2×#4111yescharacteristic
180C4×C101H(5,𝔽2)11yescharacteristic
181Dic101—21no—
182Dic101—21no—
183Dic101—21no—
184Dic101—41no—
185Dic101—41no—
186Dic101—41no—
187C2×C101—42no—
188C2×C101—41no—
189C2×C101—81no—
190C2×C101#54611yescharacteristic
191C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C101#57711yescharacteristic
193D101—22no—
194D101—81no—
195#3310—1011nomaximal, Sylow
196C101#331011yescharacteristic, complemented, Sylow
197#552—1011no—
198D42—1011no—
199#490—1011no—
200#482—1011no—
201#480—1011no—
202#478—1011no—
203#468—1011no—
204#465—1011no—
205#460—1011no—
206#450—1011no—
207#450D10111yescharacteristic, complemented
208#440—1011no—
209#435—1011no—
210#418—1011no—
211C2×#102—1011no—
212C22×D4—1011no—
213C22×D4—2021no—
214C22×D4C2×D10111yescharacteristic
215C2×SD16—1011no—
216C2×SD16C2×D10111yescharacteristic
217C2×SD16C2×D10111yescharacteristic, complemented
218C2×M16—1011no—
219C22×C8—1011no—
220#127—1011no—
221#126—2021no—
222#125—1011no—
223#125—2021no—
224#123—2021no—
225C22⋊Q8—1011no—
226C22⋊Q8C2×D10111yescharacteristic, complemented
227#121—1011no—
228#121—1011no—
229#121—2021no—
230#120—2021no—
231#120—2021no—
232C4×Q8—1011no—
233C4×D4—1011no—
234C4×D4—1011no—
235#117—1011no—
236#117—2021no—
237C2×C4⋊C4—1011no—
238#107—1011no—
239#107—1011no—
240#106—1011no—
241C4⋊C8—1011no—
242#103—1011no—
243#102—1011no—
244#102—1011no—
245#102—1011no—
246#102—1011no—
247#102—1011no—
248#102—1011no—
249#102—1011no—
250#102C2×D10111yescharacteristic
251#102C2×D10111yescharacteristic, complemented
252C22⋊C8—1011no—
253C22⋊C8C2×D10111yescharacteristic, complemented
254#97—1011no—
255C24—21no—
256C24—2021no—
257C2×Q8C22×D10111yescharacteristic
258C2×D4—22no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—1011no—
262C2×D4—1011no—
263C2×D4—4044no—
264C2×D4—4041no—
265C2×D4—2021no—
266C2×D4—2021no—
267C2×D4—2021no—
268C2×D4—2021no—
269C2×D4—2021no—
270C2×D4—2022no—
271C2×D4C22×D10111yescharacteristic
272C2×D4C22×D10111yescharacteristic
273C22×C4—1011no—
274C22×C4—2021no—
275C22×C4—2021no—
276C22×C4C22×D10111yescharacteristic
277SD16—22no—
278SD16—41no—
279SD16—4041no—
280M16—4041no—
281C2×C8—1011no—
282C2×C8—1011no—
283C2×C8—4041no—
284C2×C8C22×D10111yescharacteristic
285C2×C8C22×D10111yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—1011no—
288C4⋊C4—1011no—
289C4⋊C4—1011no—
290C4⋊C4—1011no—
291C4⋊C4—2021no—
292C4⋊C4—2021no—
293C4⋊C4—2021no—
294C4⋊C4—2021no—
295C4⋊C4—2021no—
296C4⋊C4C22×D10111yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—1011no—
299C22⋊C4—1011no—
300C22⋊C4—2021no—
301C22⋊C4—2021no—
302C22⋊C4—2021no—
303C22⋊C4—2021no—
304C22⋊C4—2021no—
305C42—1011no—
306C42—2021no—
307C42—2021no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—8081no—
314C23—1011no—
315C23—4042no—
316C23—4042no—
317C23—2021no—
318C23D4×D10111yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—8081no—
326D4—4042no—
327D4—4042no—
328D4—4042no—
329D4—4042no—
330D4—2022no—
331D4—2021no—
332D4—2021no—
333D4#1310807312yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—1011no—
339C2×C4—1011no—
340C2×C4—1011no—
341C2×C4—4042no—
342C2×C4—4042no—
343C2×C4—2021no—
344C2×C4—2021no—
345C2×C4—2021no—
346C2×C4—2021no—
347C2×C4—2021no—
348C2×C4D4×D10111yescharacteristic
349C2×C4C23×D10111yescharacteristic
350C8—21no—
351C8—21no—
352C8—2021no—
353C8—2021no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—84no—
364C22—8084no—
365C22—1012no—
366C22—4041no—
367C22—4041no—
368C22—4041no—
369C22—2021no—
370C22—2021no—
371C22C2×D4×D10111yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—4041no—
376C4—4041no—
377C4—4041no—
378C4—2021no—
379C4—2021no—
380C4—2021no—
381C4C2×#1310807311yescharacteristic
382C4#1569230011yescharacteristic
383C2—41no—
384C2—42no—
385C2—81no—
386C2—8081no—
387C2—2022no—
388C2#2026581411yescharacteristic
389C2AGL(1,ℤ/8)×D10111yescharacteristic
390C2#2026598311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#450×C101C2
socleC22×C101C22×D4
prime radicalD4×C22⋊D101C2
hypercenter#450D101
hypocenterC101#3310
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. #20265861C2semidirect
3. #20265540C2semidirect
4. #20265502C2semidirect
5. #20265489C2semidirect
6. #20265479C2semidirect
7. #20265425C2semidirect
8. #20265403C2semidirect
9. #20265375C2semidirect
10. #20265326C2semidirect
11. #20265278C2semidirect
12. #20265251C2semidirect
13. #20265182C2semidirect
14. #102×D101C2semidirect
15. #450×C101C2semidirect
18. C2×#13108061C22semidirect
19. C2×#13108058C22semidirect
20. C2×C8×D101C22semidirect
23. #15692246C22semidirect
26. #15692231C22semidirect
32. Q8×Dic101C22semidirect
35. #15692214C22semidirect
37. C4⋊C4×D101C22semidirect
38. #15692189C22semidirect
39. #15692188C22semidirect
40. #15692187C22semidirect
41. #15692184C22semidirect
42. #15692179C22semidirect
43. #15692177C22semidirect
44. #15692177C22semidirect
47. #15692176C22semidirect
48. #15692176C22semidirect
49. #15692175C22semidirect
50. #15692167C22semidirect
51. #15692165C22semidirect
54. C2×SD16×C101C22semidirect
55. C22⋊Q8×C101C22semidirect
57. #102×C101C22semidirect
58. C22⋊C8×C101C22semidirect
196. C101#3310semidirect
207. #450D101semidirect
217. C2×SD16C2×D101semidirect
226. C22⋊Q8C2×D101semidirect
251. #102C2×D101semidirect
253. C22⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658612
D4×C22⋊D1012
#202655402
#202655022
#202654892
#202654792
#202654252
#202654032
#202653752
#202653262
#202652782
#202652512
#202651822
#102×D1012
#450×C1012
#3310101