Finite Group Databasebeta

← #27181319

Subgroups — #27181319

Order 12928.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265800C211yesmaximal, characteristic, complemented
2#20265651C211yesmaximal, characteristic, complemented
3#20265501C211yesmaximal, characteristic, complemented
4#20265466C211yesmaximal, characteristic, complemented
5#20265456C211yesmaximal, characteristic, complemented
6#20265395C211yesmaximal, characteristic, complemented
7#20265395C211yesmaximal, characteristic, complemented
8#20265327C211yesmaximal, characteristic, complemented
9#20265313C211yesmaximal, characteristic
10#20265288C211yesmaximal, characteristic, complemented
11#20265280C211yesmaximal, characteristic, complemented
12#20265262C211yesmaximal, characteristic, complemented
13#20265259C211yesmaximal, characteristic, complemented
14#20265235C211yesmaximal, characteristic, complemented
15#485×C101C211yesmaximal, characteristic, complemented
16C22×D404C2211yescharacteristic
17C2×#13108061C2211yescharacteristic
18C2×#13108061C2211yescharacteristic
19C2×D808C2211yescharacteristic
20C2×D808C2211yescharacteristic
21#15692254C2211yescharacteristic, complemented
22#15692250—21no—
23#15692244C2211yescharacteristic, complemented
24#15692241—21no—
25#15692237—21no—
26#15692235—21no—
27#15692229C2211yescharacteristic
28Q8×Dic101C2211yescharacteristic, complemented
29C4×D404—21no—
30C4×D404—21no—
31#15692217C2211yescharacteristic
32#15692217C2211yescharacteristic
33#15692217C2211yescharacteristic
34#15692217C2211yescharacteristic
35#15692217C2211yescharacteristic, complemented
36#15692214C2211yescharacteristic
37C22⋊C4×D101—21no—
38C22⋊C4×D101—21no—
39#15692188C2211yescharacteristic, complemented
40#15692186C2211yescharacteristic, complemented
41#15692184C2211yescharacteristic, complemented
42#15692184C2211yescharacteristic, complemented
43#15692177C2211yescharacteristic
44#15692177C2211yescharacteristic
45#15692175C2211yescharacteristic
46#15692175C2211yescharacteristic
47#15692175C2211yescharacteristic, complemented
48#15692175C2211yescharacteristic, complemented
49C22⋊#11717579C2211yescharacteristic
50#15692165C2211yescharacteristic, complemented
51C8×Dic101C2211yescharacteristic, complemented
52C22⋊Q8×C101C2211yescharacteristic
53#117×C101C2211yescharacteristic
54#107×C101C2211yescharacteristic, complemented
55#106×C101C2211yescharacteristic, complemented
56#103×C101C2211yescharacteristic, complemented
57#103×C101C2211yescharacteristic, complemented
58C22⋊C8×C101C2211yescharacteristic
59C23×D101—21no—
60C2×Dic202—21no—
61C2×D404—21no—
62C2×D404—22no—
63C2×D404—22no—
64C2×D404C2311yescharacteristic
65C2×D404C2311yescharacteristic
66C2×D404C2311yescharacteristic
67C2×C22⋊D101—21no—
68C2×C4×D101—21no—
69C2×C4×D101—21no—
70C2×C4×D101—21no—
71C2×C4×D101—21no—
72C2×C4×D101—21no—
73#13108061—22no—
74#13108061—22no—
75D808—22no—
76D808—22no—
77C2×#11717579C2311yescharacteristic
78C2×#11717579C2311yescharacteristic
79C4⋊Dic101—21no—
80C4⋊Dic101D411yescharacteristic
81C4⋊Dic101D411yescharacteristic
82C4⋊Dic101C2311yescharacteristic
83#13108053—21no—
84C22⋊Dic101D411yescharacteristic
85C22⋊Dic101D411yescharacteristic
86#13108051—21no—
87#13108051—21no—
88#13108051—21no—
89#13108051—21no—
90#13108051—21no—
91#13108051—21no—
92#13108051—21no—
93#13108051—21no—
94C4×Dic101—21no—
95C4×Dic101C2311yescharacteristic
96C4×Dic101C2311yescharacteristic
97C2×Q8×C101C2311yescharacteristic
98C22×C4×C101C2311yescharacteristic
99C2×C8×C101C2311yescharacteristic
100C2×C8×C101C2311yescharacteristic
101C4⋊C4×C101—21no—
102C4⋊C4×C101C2311yescharacteristic
103C4⋊C4×C101C2311yescharacteristic
104C4⋊C4×C101C2311yescharacteristic
105C22⋊C4×C101—21no—
106C22⋊C4×C101—21no—
107C42×C101—21no—
108C23×C101C2×D411yescharacteristic
109C22×D101—21no—
110C22×D101—21no—
111C22×D101—24no—
112C22×D101—21no—
113C22×D101—81no—
114C22×D101—81no—
115Q8×C101—21no—
116Q8×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101C2×D411yescharacteristic
124C2×C4×C101C2411yescharacteristic
125C22⋊D101—81no—
126C2×Dic101—21no—
127C2×Dic101C2×D411yescharacteristic
128C2×Dic101C2×D411yescharacteristic
129C2×Dic101C2×D411yescharacteristic
130C2×Dic101C2×D411yescharacteristic
131D404—24no—
132D404—21no—
133D404—21no—
134D404—21no—
135D404—21no—
136D404#4112yes—
137D404#4112yes—
138C4×D101—81no—
139C4×D101—81no—
140C4×D101—81no—
141C4×D101—81no—
142C4×D101—81no—
143Dic202—81no—
144C8×C101—21no—
145C8×C101—21no—
146#11717579—21no—
147#11717579—21no—
148C22×C101—21no—
149C22×C101—21no—
150C22×C101—21no—
151C22×C101C22×D411yescharacteristic
152C2×D101—22no—
153C2×D101—22no—
154C2×D101—41no—
155C2×D101—41no—
156C2×D101—41no—
157C2×D101—41no—
158C2×D101—41no—
159C2×D101—41no—
160C2×D101—41no—
161C2×D101—84no—
162C4×C101—21no—
163C4×C101—41no—
164C4×C101—41no—
165C4×C101—41no—
166C4×C101—41no—
167C4×C101C2×#4111yescharacteristic
168C4×C101C2×#4111yescharacteristic
169Dic101—21no—
170Dic101—21no—
171Dic101—21no—
172Dic101—21no—
173Dic101—41no—
174C2×C101—41no—
175C2×C101#52511yescharacteristic
176C2×C101#57611yescharacteristic
177C2×C101#57711yescharacteristic
178D101—42no—
179D101—42no—
180D101—81no—
181#3118—1011nomaximal, Sylow
182C101#311811yescharacteristic, complemented, Sylow
183#542—1011no—
184#518—1011no—
185#485D10111yescharacteristic, complemented
186#482—1011no—
187#476—1011no—
188#474—1011no—
189#463—1011no—
190#463—1011no—
191#450—1011no—
192#447—1011no—
193#442—1011no—
194#440—1011no—
195C4×SD16—1011no—
196C4×D8—1011no—
197#433—1011no—
198C22×D4—1011no—
199C2×SD16—1011no—
200C2×SD16—1011no—
201C2×D8—1011no—
202C2×D8—1011no—
203C4⋊Q8—1011no—
204#126—2021no—
205#124—1011no—
206#124—2021no—
207#123—2021no—
208C22⋊Q8—2021no—
209C22⋊Q8C2×D10111yescharacteristic
210#121—1011no—
211C4×Q8—1011no—
212C4×D4—1011no—
213C4×D4—1011no—
214C4×D4—1011no—
215C4×D4—1011no—
216C4×D4—1011no—
217C4×D4—2021no—
218C4×D4—2021no—
219#117—1011no—
220#117C2×D10111yescharacteristic
221C2×C22⋊C4—2021no—
222C2×C22⋊C4—2021no—
223#107—1011no—
224#107C2×D10111yescharacteristic, complemented
225#106—1011no—
226#106C2×D10111yescharacteristic, complemented
227C4⋊C8—1011no—
228C4⋊C8—1011no—
229#103C2×D10111yescharacteristic, complemented
230#103C2×D10111yescharacteristic, complemented
231#102—1011no—
232#102—1011no—
233#102—1011no—
234#102—1011no—
235#102—1011no—
236#102—1011no—
237C22⋊C8—1011no—
238C22⋊C8C2×D10111yescharacteristic
239#97—1011no—
240C4×C8—1011no—
241C24—2021no—
242C2×Q8—2021no—
243C2×Q8C22×D10111yescharacteristic
244C2×D4—1011no—
245C2×D4—1011no—
246C2×D4—1011no—
247C2×D4—2022no—
248C2×D4—2022no—
249C2×D4—2021no—
250C2×D4—2021no—
251C22×C4—2021no—
252C22×C4—2021no—
253C22×C4—2021no—
254C22×C4—2021no—
255C22×C4—2021no—
256C22×C4C22×D10111yescharacteristic
257SD16—2022no—
258SD16—2022no—
259D8—2022no—
260D8—2022no—
261C2×C8—1011no—
262C2×C8—1011no—
263C2×C8C22×D10111yescharacteristic
264C2×C8C22×D10111yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—1011no—
267C4⋊C4—1011no—
268C4⋊C4—1011no—
269C4⋊C4—2021no—
270C4⋊C4—2021no—
271C4⋊C4C22×D10111yescharacteristic
272C4⋊C4C22×D10111yescharacteristic
273C4⋊C4C22×D10111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—1011no—
277C22⋊C4—1011no—
278C22⋊C4—2021no—
279C22⋊C4—2021no—
280C22⋊C4—2021no—
281C22⋊C4—2021no—
282C22⋊C4—2021no—
283C22⋊C4—2021no—
284C22⋊C4—2021no—
285C22⋊C4—2021no—
286C42—21no—
287C42—1011no—
288C42—1011no—
289C42—2021no—
290C23—8081no—
291C23—8081no—
292C23—2024no—
293C23—2021no—
294C23—2021no—
295C23—2021no—
296C23D4×D10111yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—8081no—
300D4—8081no—
301D4—1012no—
302D4—1012no—
303D4—2024no—
304D4—2021no—
305D4—2021no—
306D4—2021no—
307D4—2021no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—8081no—
315C2×C4—8081no—
316C2×C4—8081no—
317C2×C4—8081no—
318C2×C4—8081no—
319C2×C4—1011no—
320C2×C4—1011no—
321C2×C4—1011no—
322C2×C4—1011no—
323C2×C4—2021no—
324C2×C4D4×D10111yescharacteristic
325C2×C4C23×D10111yescharacteristic
326C8—21no—
327C8—21no—
328C8—2021no—
329C8—2021no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—8084no—
334C22—4041no—
335C22—4041no—
336C22—4041no—
337C22—4041no—
338C22—4041no—
339C22—4041no—
340C22—4041no—
341C22—2022no—
342C22—2022no—
343C22C2×D4×D10111yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—4041no—
350C4—2021no—
351C4—2021no—
352C4—2021no—
353C4—2021no—
354C4#41×D10111yescharacteristic
355C4#41×D10111yescharacteristic
356C2—41no—
357C2—8081no—
358C2—4042no—
359C2—4042no—
360C2#2026568511yescharacteristic
361C2#2026597911yescharacteristic
362C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#485×C101C2
socleC22×C101C22×D4
prime radical#20265313C2
hypercenter#485D101
hypocenterC101#3118
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #20265800C2semidirect
2. #20265651C2semidirect
3. #20265501C2semidirect
4. #20265466C2semidirect
5. #20265456C2semidirect
6. #20265395C2semidirect
7. #20265395C2semidirect
8. #20265327C2semidirect
10. #20265288C2semidirect
11. #20265280C2semidirect
12. #20265262C2semidirect
13. #20265259C2semidirect
14. #20265235C2semidirect
15. #485×C101C2semidirect
21. #15692254C22semidirect
23. #15692244C22semidirect
28. Q8×Dic101C22semidirect
35. #15692217C22semidirect
39. #15692188C22semidirect
40. #15692186C22semidirect
41. #15692184C22semidirect
42. #15692184C22semidirect
47. #15692175C22semidirect
48. #15692175C22semidirect
50. #15692165C22semidirect
51. C8×Dic101C22semidirect
54. #107×C101C22semidirect
55. #106×C101C22semidirect
56. #103×C101C22semidirect
57. #103×C101C22semidirect
182. C101#3118semidirect
185. #485D101semidirect
224. #107C2×D101semidirect
226. #106C2×D101semidirect
229. #103C2×D101semidirect
230. #103C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658002
#202656512
#202655012
#202654662
#202654562
#202653952
#202653272
#202653132
#202652882
#202652802
#202652622
#202652592
#202652352
#485×C1012
#3118101