Finite Group Databasebeta

← #27182971

Subgroups — #27182971

Order 12928.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265869C211yesmaximal, characteristic, complemented
2#20265821C211yesmaximal, characteristic
3#20265530C211yesmaximal, characteristic, complemented
4#20265508C211yesmaximal, characteristic, complemented
5#20265500C211yesmaximal, characteristic, complemented
6#20265479C211yesmaximal, characteristic, complemented
7#20265416C211yesmaximal, characteristic, complemented
8#20265403C211yesmaximal, characteristic, complemented
9#20265397C211yesmaximal, characteristic, complemented
10#20265336C211yesmaximal, characteristic, complemented
11SD16×Dic101C211yesmaximal, characteristic, complemented
12#20265240C211yesmaximal, characteristic, complemented
13#20265170C211yesmaximal, characteristic, complemented
14C2×#15692176C211yesmaximal, characteristic, complemented
15#450×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C2×#13108064C2211yescharacteristic, complemented
18C22×#11717579C2211yescharacteristic
19C2×C8×D101C2211yescharacteristic, complemented
20#15692250—21no—
21#15692246—21no—
22#15692246C2211yescharacteristic, complemented
23#15692242C2211yescharacteristic
24#15692239—21no—
25#15692236—21no—
26#15692234C2211yescharacteristic
27#15692231C2211yescharacteristic, complemented
28#15692227—21no—
29#15692226C2211yescharacteristic
30#15692224—21no—
31Q8×Dic101C2211yescharacteristic, complemented
32D4×Dic101C2211yescharacteristic
33C4×C22⋊D101C2211yescharacteristic
34#15692213C2211yescharacteristic, complemented
35#15692211—21no—
36C2×C4⋊Dic101C2211yescharacteristic
37C2×C22⋊Dic101—21no—
38#15692188C2211yescharacteristic, complemented
39#15692187C2211yescharacteristic, complemented
40#15692187C2211yescharacteristic, complemented
41#15692184C2211yescharacteristic, complemented
42#15692180C2211yescharacteristic, complemented
43#15692180C2211yescharacteristic, complemented
44#15692179C2211yescharacteristic, complemented
45#15692178C2211yescharacteristic, complemented
46#15692176C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692176C2211yescharacteristic
49#15692176C2211yescharacteristic
50#15692167C2211yescharacteristic, complemented
51C8×Dic101C2211yescharacteristic, complemented
52C22×D4×C101C2211yescharacteristic
53C2×SD16×C101C2211yescharacteristic
54C2×SD16×C101C2211yescharacteristic
55C22⋊Q8×C101C2211yescharacteristic
56#102×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58C22⋊C8×C101C2211yescharacteristic
59#13108073—22no—
60#13108073—41no—
61C2×Dic202C2311yescharacteristic
62C2×C22⋊D101—21no—
63C2×C22⋊D101—21no—
64C2×C22⋊D101D411yescharacteristic
65C22×Dic101—21no—
66C22×Dic101—21no—
67C22×Dic101—21no—
68C2×C4×D101C2311yescharacteristic
69#13108064—41no—
70C2×#11717579—41no—
71C2×#11717579C2311yescharacteristic
72C2×#11717579C2311yescharacteristic
73C8×D101—41no—
74C4⋊Dic101—21no—
75C4⋊Dic101—21no—
76C4⋊Dic101—21no—
77C4⋊Dic101C2311yescharacteristic
78C4⋊Dic101C2311yescharacteristic
79C4⋊Dic101C2311yescharacteristic
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053D411yescharacteristic
84C22⋊Dic101—21no—
85C22⋊Dic101—21no—
86C22⋊Dic101—21no—
87C22⋊Dic101—21no—
88C22⋊Dic101—21no—
89C22⋊Dic101D411yescharacteristic
90#13108051—21no—
91#13108051—21no—
92#13108051D411yescharacteristic
93C4×Dic101—21no—
94C4×Dic101—21no—
95C4×Dic101C2311yescharacteristic
96C24×C101—21no—
97C2×Q8×C101C2311yescharacteristic
98C2×D4×C101—22no—
99C2×D4×C101—21no—
100C2×D4×C101—41no—
101C2×D4×C101C2311yescharacteristic
102C2×D4×C101C2311yescharacteristic
103C22×C4×C101C2311yescharacteristic
104SD16×C101—22no—
105SD16×C101—41no—
106C2×C8×C101C2311yescharacteristic
107C2×C8×C101C2311yescharacteristic
108C4⋊C4×C101—21no—
109C4⋊C4×C101C2311yescharacteristic
110C22⋊C4×C101—21no—
111C23×C101—21no—
112C23×C101—21no—
113C23×C101—42no—
114C23×C101—42no—
115C23×C101—81no—
116C23×C101C2×D411yescharacteristic
117C22×D101C2×D411yescharacteristic
118Q8×C101—21no—
119Q8×C101—21no—
120D4×C101—21no—
121D4×C101—21no—
122D4×C101—21no—
123D4×C101—42no—
124D4×C101#4112yes—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101—21no—
129C2×C4×C101C2×D411yescharacteristic
130C2×C4×C101C2411yescharacteristic
131C22⋊D101—22no—
132C22⋊D101—42no—
133C22⋊D101—81no—
134C2×Dic101—21no—
135C2×Dic101—21no—
136C2×Dic101—21no—
137C2×Dic101—21no—
138C2×Dic101—42no—
139C2×Dic101—42no—
140C2×Dic101—81no—
141C2×Dic101C2×D411yescharacteristic
142C2×Dic101C2×D411yescharacteristic
143C2×Dic101C2×D411yescharacteristic
144C4×D101—21no—
145C4×D101—21no—
146Dic202—21no—
147Dic202—21no—
148C8×C101—21no—
149C8×C101—21no—
150#11717579—21no—
151#11717579—21no—
152C22×C101—21no—
153C22×C101—21no—
154C22×C101—22no—
155C22×C101—41no—
156C22×C101—41no—
157C22×C101—41no—
158C22×C101—41no—
159C22×C101—41no—
160C22×C101—84no—
161C22×C101#13512yes—
162C22×C101C22×D411yescharacteristic
163C2×D101—21no—
164C2×D101—21no—
165C2×D101—21no—
166C4×C101—21no—
167C4×C101—41no—
168C4×C101—41no—
169C4×C101C2×#4111yescharacteristic
170C4×C101H(5,𝔽2)11yescharacteristic
171Dic101—21no—
172Dic101—21no—
173Dic101—21no—
174Dic101—41no—
175Dic101—41no—
176Dic101—41no—
177Dic101—41no—
178C2×C101—22no—
179C2×C101—42no—
180C2×C101—81no—
181C2×C101#54611yescharacteristic
182C2×C101C2×#13511yescharacteristic
183C2×C101#57711yescharacteristic
184D101—41no—
185#3296—1011nomaximal, Sylow
186C101#329611yescharacteristic, complemented, Sylow
187#552—1011no—
188#546—1011no—
189#488—1011no—
190#483—1011no—
191#482—1011no—
192#478—1011no—
193#467—1011no—
194#465—1011no—
195#464—1011no—
196#452—1011no—
197#450D10111yescharacteristic, complemented
198C4×SD16—1011no—
199C8×D4—1011no—
200#416—1011no—
201C2×#102—1011no—
202C2×#41—1011no—
203C22×D4C2×D10111yescharacteristic
204C2×Q16—1011no—
205C2×SD16C2×D10111yescharacteristic
206C2×SD16C2×D10111yescharacteristic
207C22×C8—1011no—
208C22×C8—1011no—
209#126—2021no—
210#125—1011no—
211#125—2021no—
212#124—1011no—
213#123—2021no—
214#123—2021no—
215C22⋊Q8—1011no—
216C22⋊Q8—1011no—
217C22⋊Q8C2×D10111yescharacteristic
218#121—1011no—
219#121—2021no—
220#120—2021no—
221C4×Q8—1011no—
222C4×D4—1011no—
223C4×D4—1011no—
224#117—1011no—
225#117—2021no—
226C2×C4⋊C4—1011no—
227C2×C22⋊C4—2021no—
228#107—1011no—
229#106—1011no—
230#106—1011no—
231C4⋊C8—1011no—
232#103—1011no—
233#103—1011no—
234#103—1011no—
235#103—1011no—
236#102—1011no—
237#102—1011no—
238#102—1011no—
239#102—1011no—
240#102C2×D10111yescharacteristic
241#102C2×D10111yescharacteristic
242C22⋊C8—1011no—
243C22⋊C8C2×D10111yescharacteristic
244C4×C8—1011no—
245C24—21no—
246#41—4041no—
247#41—2022no—
248C2×Q8—1011no—
249C2×Q8C22×D10111yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—1011no—
254C2×D4—2021no—
255C2×D4—2021no—
256C2×D4C22×D10111yescharacteristic
257C2×D4C22×D10111yescharacteristic
258C22×C4—1011no—
259C22×C4—2021no—
260C22×C4—2021no—
261C22×C4—2021no—
262C22×C4C22×D10111yescharacteristic
263Q16—4041no—
264SD16—22no—
265SD16—41no—
266C2×C8—1011no—
267C2×C8—1011no—
268C2×C8—4041no—
269C2×C8—4041no—
270C2×C8C22×D10111yescharacteristic
271C2×C8C22×D10111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—1011no—
274C4⋊C4—1011no—
275C4⋊C4—1011no—
276C4⋊C4—1011no—
277C4⋊C4—2021no—
278C4⋊C4—2021no—
279C4⋊C4—2021no—
280C4⋊C4—2021no—
281C4⋊C4—2021no—
282C4⋊C4—2021no—
283C4⋊C4C22×D10111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—1011no—
286C22⋊C4—1011no—
287C22⋊C4—2021no—
288C22⋊C4—2021no—
289C22⋊C4—2021no—
290C22⋊C4—2021no—
291C22⋊C4—2021no—
292C22⋊C4—2021no—
293C22⋊C4—2021no—
294C42—1011no—
295C42—2021no—
296C42—2021no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—1011no—
303C23D4×D10111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2021no—
307Q8—2021no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—8081no—
313D4—4042no—
314D4—2022no—
315D4#1310807312yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—8081no—
321C2×C4—1011no—
322C2×C4—1011no—
323C2×C4—1011no—
324C2×C4—4042no—
325C2×C4—4042no—
326C2×C4—2021no—
327C2×C4—2021no—
328C2×C4—2021no—
329C2×C4—2021no—
330C2×C4—2021no—
331C2×C4—2021no—
332C2×C4D4×D10111yescharacteristic
333C2×C4C23×D10111yescharacteristic
334C8—21no—
335C8—21no—
336C8—2021no—
337C8—2021no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—84no—
347C22—2021no—
348C22—2021no—
349C22—2021no—
350C22#1569227612yes—
351C22C2×D4×D10111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—4041no—
356C4—4041no—
357C4—4041no—
358C4—4041no—
359C4—2021no—
360C4—2021no—
361C4—2021no—
362C4C2×#1310807311yescharacteristic
363C4#1569230011yescharacteristic
364C2—22no—
365C2—42no—
366C2—81no—
367C2—4041no—
368C2#2026581411yescharacteristic
369C2C2×#1569227611yescharacteristic
370C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#450×C101C2
socleC22×C101C22×D4
prime radical#20265821C2
hypercenter#450D101
hypocenterC101#3296
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. #20265869C2semidirect
3. #20265530C2semidirect
4. #20265508C2semidirect
5. #20265500C2semidirect
6. #20265479C2semidirect
7. #20265416C2semidirect
8. #20265403C2semidirect
9. #20265397C2semidirect
10. #20265336C2semidirect
11. SD16×Dic101C2semidirect
12. #20265240C2semidirect
13. #20265170C2semidirect
14. C2×#15692176C2semidirect
15. #450×C101C2semidirect
17. C2×#13108064C22semidirect
19. C2×C8×D101C22semidirect
22. #15692246C22semidirect
27. #15692231C22semidirect
31. Q8×Dic101C22semidirect
34. #15692213C22semidirect
38. #15692188C22semidirect
39. #15692187C22semidirect
40. #15692187C22semidirect
41. #15692184C22semidirect
42. #15692180C22semidirect
43. #15692180C22semidirect
44. #15692179C22semidirect
45. #15692178C22semidirect
50. #15692167C22semidirect
51. C8×Dic101C22semidirect
186. C101#3296semidirect
197. #450D101semidirect

Maximal subgroups

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

subgroupindex
#202658692
#202658212
#202655302
#202655082
#202655002
#202654792
#202654162
#202654032
#202653972
#202653362
SD16×Dic1012
#202652402
#202651702
C2×#156921762
#450×C1012
#3296101