Finite Group Databasebeta

← #27180400

Subgroups — #27180400

Order 12928.

784 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692280C211yesmaximal, characteristic, complemented
2C22×#13108059C211yesmaximal, characteristic, complemented
3#20265741C211yesmaximal, characteristic, complemented
4#20265322C214yesmaximal, complemented
5#20265311C214yesmaximal, complemented
6#20265182C211yesmaximal, characteristic, complemented
7#20265173C211yesmaximal, characteristic, complemented
8#20265144C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C101C211yesmaximal, characteristic, complemented
10C22×D404C2211yescharacteristic, complemented
11#15692280—24no—
12C2×#13108061C2212yescomplemented
13C2×#13108059—24no—
14C2×#13108059C2212yescomplemented
15C2×#13108059C2214yescomplemented
16C2×#13108058C2211yescharacteristic, complemented
17C22×#11717579C2211yescharacteristic, complemented
18#15692251—21no—
19#15692244—21no—
20#15692229C2214yescomplemented
21#15692223—22no—
22#15692214C2211yescharacteristic, complemented
23#15692179C2212yescomplemented
24#15692176C2212yescomplemented
25#15692176C2214yescomplemented
26C22⋊#11717579C2214yescomplemented
27H(5,𝔽2)×C101—24no—
28H(5,𝔽2)×C101—24no—
29C2×#41×C101—21no—
30C2×#41×C101C2211yescharacteristic
31C2×#41×C101C2211yescharacteristic
32C2×#41×C101C2212yescomplemented
33C22×D4×C101—21no—
34C22×D4×C101—22no—
35C22×D4×C101C2211yescharacteristic
36C22×D4×C101C2212yescomplemented
37C23×D101—21no—
38C2×D404—22no—
39C2×D404—22no—
40C2×D404—21no—
41C2×D404C2312yes—
42C2×C22⋊D101—24no—
43#13108061—24no—
44#13108059—28no—
45#13108059—24no—
46#13108058—22no—
47C2×#11717579—22no—
48C2×#11717579C2312yes—
49C2×#11717579C2312yes—
50C4⋊Dic101C2312yes—
51C22⋊Dic101—21no—
52#13108051—24no—
53C4×Dic101—21no—
54C24×C101—21no—
55C24×C101—22no—
56#41×C101—28no—
57#41×C101—24no—
58#41×C101—24no—
59#41×C101—22no—
60#41×C101—22no—
61#41×C101—22no—
62#41×C101D414yescomplemented
63C2×Q8×C101D411yescharacteristic
64C2×Q8×C101C2311yescharacteristic
65C2×D4×C101—216no—
66C2×D4×C101—28no—
67C2×D4×C101—21no—
68C2×D4×C101—22no—
69C2×D4×C101—24no—
70C2×D4×C101—24no—
71C2×D4×C101—21no—
72C2×D4×C101—22no—
73C2×D4×C101—22no—
74C2×D4×C101D412yes—
75C2×D4×C101D411yescharacteristic
76C2×D4×C101D414yescomplemented
77C2×D4×C101C2312yes—
78C2×D4×C101C2311yescharacteristic
79C22×C4×C101—21no—
80C22×C4×C101—22no—
81C22×C4×C101C2312yes—
82C22×C4×C101C2311yescharacteristic
83C23×C101—28no—
84C23×C101—21no—
85C23×C101—22no—
86C23×C101—24no—
87C23×C101—82no—
88C23×C101—84no—
89C23×C101C2×D412yes—
90C23×C101C2×D411yescharacteristic
91C22×D101—22no—
92C22×D101—42no—
93C22×D101—82no—
94Q8×C101—22no—
95Q8×C101—21no—
96Q8×C101C2×D412yes—
97D4×C101—28no—
98D4×C101—28no—
99D4×C101—24no—
100D4×C101—24no—
101D4×C101—22no—
102D4×C101—24no—
103D4×C101—22no—
104D4×C101—21no—
105D4×C101C2×D412yes—
106D4×C101C2×D414yes—
107C2×C4×C101—28no—
108C2×C4×C101—21no—
109C2×C4×C101—24no—
110C2×C4×C101—22no—
111C2×C4×C101—22no—
112C2×C4×C101C2×D412yes—
113C2×C4×C101C2×D411yescharacteristic
114C2×C4×C101C2×D412yes—
115C2×C4×C101C2×D412yes—
116C2×C4×C101C2411yescharacteristic
117C22⋊D101—84no—
118C2×Dic101—22no—
119D404—22no—
120D404—22no—
121D404—42no—
122#11717579—22no—
123#11717579—22no—
124C22×C101—22no—
125C22×C101—22no—
126C22×C101—24no—
127C22×C101—22no—
128C22×C101—24no—
129C22×C101—44no—
130C22×C101—42no—
131C22×C101—84no—
132C22×C101—88no—
133C22×C101#12012yes—
134C22×C101C22×D411yescharacteristic
135C2×D101—42no—
136C2×D101—44no—
137C2×D101—84no—
138C4×C101—22no—
139C4×C101—22no—
140C4×C101#12012yes—
141C4×C101C22×D411yescharacteristic
142C4×C101C22×D411yescharacteristic
143Dic101—42no—
144C2×C101—22no—
145C2×C101—42no—
146C2×C101—44no—
147C2×C101—42no—
148C2×C101C2×#12011yescharacteristic
149C2×C101#57612yes—
150D101—82no—
151#3008—1011nomaximal, Sylow
152C101#300811yescharacteristic, complemented, Sylow
153C2×H(5,𝔽2)D10111yescharacteristic, complemented
154C2×AGL(1,ℤ/8)—1011no—
155C22×D8—1011no—
156#535—1011no—
157#449—1014no—
158#447—1014no—
159#418—1011no—
160#416—1011no—
161#408—1011no—
162H(5,𝔽2)—24no—
163H(5,𝔽2)—24no—
164C2×#41—21no—
165C2×#41C2×D10111yescharacteristic
166C2×#41C2×D10111yescharacteristic
167C2×#41C2×D10112yescomplemented
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—1011no—
171C22×D4C2×D10111yescharacteristic
172C22×D4C2×D10112yescomplemented
173AGL(1,ℤ/8)—2024no—
174C2×SD16—1012no—
175C2×D8—1014no—
176C2×D8—1012no—
177C2×D8—2024no—
178C2×M16—1011no—
179C22×C8—1011no—
180#127—2021no—
181#124—2021no—
182#121—1014no—
183#120—2022no—
184#117—1011no—
185#103—1012no—
186#102—1012no—
187#102—1014no—
188C22⋊C8—1014no—
189C24—21no—
190C24—22no—
191C24—2021no—
192#41—28no—
193#41—24no—
194#41—24no—
195#41—22no—
196#41—22no—
197#41—22no—
198#41C22⋊D10114yescomplemented
199C2×Q8C22⋊D10111yescharacteristic
200C2×Q8C22×D10111yescharacteristic
201C2×D4—216no—
202C2×D4—28no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—24no—
206C2×D4—24no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—1012no—
211C2×D4—2024no—
212C2×D4—2021no—
213C2×D4—2022no—
214C2×D4—2022no—
215C2×D4C22⋊D10112yes—
216C2×D4C22⋊D10111yescharacteristic
217C2×D4C22⋊D10114yescomplemented
218C2×D4C22×D10112yes—
219C2×D4C22×D10111yescharacteristic
220C22×C4—21no—
221C22×C4—22no—
222C22×C4C22×D10112yes—
223C22×C4C22×D10111yescharacteristic
224SD16—2024no—
225D8—2028no—
226D8—2024no—
227M16—2022no—
228C2×C8—1012no—
229C2×C8—1012no—
230C2×C8—2022no—
231C4⋊C4—1012no—
232C22⋊C4—2021no—
233C22⋊C4—2024no—
234C42—2021no—
235C23—28no—
236C23—21no—
237C23—22no—
238C23—24no—
239C23—82no—
240C23—84no—
241C23—8082no—
242C23—4042no—
243C23—2022no—
244C23C2×C22⋊D10112yes—
245C23C2×C22⋊D10111yescharacteristic
246Q8—21no—
247Q8—22no—
248Q8C2×C22⋊D10112yes—
249D4—28no—
250D4—28no—
251D4—24no—
252D4—24no—
253D4—22no—
254D4—21no—
255D4—22no—
256D4—24no—
257D4—8084no—
258D4—4042no—
259D4—2022no—
260D4—2022no—
261D4C2×C22⋊D10112yes—
262D4C2×C22⋊D10114yes—
263C2×C4—28no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—2022no—
269C2×C4C2×C22⋊D10112yes—
270C2×C4C2×C22⋊D10112yes—
271C2×C4C2×C22⋊D10112yes—
272C2×C4C2×C22⋊D10111yescharacteristic
273C2×C4C23×D10111yescharacteristic
274C8—2022no—
275C8—2022no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—22no—
280C22—24no—
281C22—44no—
282C22—42no—
283C22—84no—
284C22—88no—
285C22—8084no—
286C22—4042no—
287C22—4044no—
288C22#1569222412yes—
289C22C22×C22⋊D10111yescharacteristic
290C4—22no—
291C4—22no—
292C4—4042no—
293C4#1569222412yes—
294C4C22×C22⋊D10111yescharacteristic
295C4C22×C22⋊D10111yescharacteristic
296C2—22no—
297C2—42no—
298C2—44no—
299C2—42no—
300C2—8082no—
301C2C2×#1569222411yescharacteristic
302C2#2026598012yes—
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×H(5,𝔽2)×C101C2
socleC22×C101C22×D4
hypercenterC2×H(5,𝔽2)D101
hypocenterC101#3008
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#15692280C2semidirect
2. C22×#13108059C2semidirect
3. #20265741C2semidirect
4. #20265322C2semidirect
5. #20265311C2semidirect
6. #20265182C2semidirect
7. #20265173C2semidirect
8. #20265144C2semidirect
9. C2×H(5,𝔽2)×C101C2semidirect
10. C22×D404C22semidirect
12. C2×#13108061C22semidirect
14. C2×#13108059C22semidirect
15. C2×#13108059C22semidirect
16. C2×#13108058C22semidirect
17. C22×#11717579C22semidirect
20. #15692229C22semidirect
22. #15692214C22semidirect
23. #15692179C22semidirect
24. #15692176C22semidirect
25. #15692176C22semidirect
26. C22⋊#11717579C22semidirect
32. C2×#41×C101C22semidirect
36. C22×D4×C101C22semidirect
62. #41×C101D4semidirect
76. C2×D4×C101D4semidirect
152. C101#3008semidirect
153. C2×H(5,𝔽2)D101semidirect
167. C2×#41C2×D101semidirect
172. C22×D4C2×D101semidirect
198. #41C22⋊D101semidirect
217. C2×D4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922802
C22×#131080592
#202657412
#202653222
#202653112
#202651822
#202651732
#202651442
C2×H(5,𝔽2)×C1012
#3008101