Finite Group Databasebeta

← #27180398

Subgroups — #27180398

Order 12928.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15692301C211yesmaximal, characteristic, complemented
2C2×#15692281C211yesmaximal, characteristic, complemented
3C22×#13108059C211yesmaximal, characteristic, complemented
4#20265320C214yesmaximal, complemented
5#20265310C214yesmaximal, complemented
6#20265181C211yesmaximal, characteristic, complemented
7#20265171C211yesmaximal, characteristic, complemented
8#20265144C211yesmaximal, characteristic, complemented
9#535×C101C211yesmaximal, characteristic, complemented
10#15692301—24no—
11#15692301—24no—
12C2×#13108075—21no—
13C2×#13108074C2211yescharacteristic
14C2×#13108074C2212yescomplemented
15C22×D404C2211yescharacteristic
16C22×D404C2212yescomplemented
17C2×D4×D101—22no—
18C2×D4×D101—21no—
19#15692281—24no—
20C2×#13108062C2212yescomplemented
21C2×#13108059—24no—
22C2×#13108059C2212yescomplemented
23C2×#13108059C2214yescomplemented
24C2×#13108058C2211yescharacteristic, complemented
25C22×#11717579C2211yescharacteristic, complemented
26#15692178C2212yescomplemented
27#15692175C2212yescomplemented
28#15692175C2214yescomplemented
29C22⋊#11717579C2214yescomplemented
30C2×#41×C101C2211yescharacteristic
31C22×D4×C101C2211yescharacteristic, complemented
32#127×C101—21no—
33#124×C101—21no—
34#121×C101C2214yescomplemented
35#120×C101—22no—
36#117×C101C2211yescharacteristic, complemented
37C23×D101—21no—
38C23×D101—22no—
39#13108075—28no—
40#13108074—22no—
41#13108074—24no—
42#13108074—24no—
43#13108074D414yescomplemented
44C2×Dic202C2311yescharacteristic
45C2×D404—21no—
46C2×D404—22no—
47C2×D404—22no—
48C2×D404—24no—
49C2×D404—24no—
50C2×D404D414yescomplemented
51C2×D404C2312yes—
52C2×D404C2311yescharacteristic
53C2×C22⋊D101—21no—
54C2×C22⋊D101—22no—
55D4×D101—216no—
56D4×D101—28no—
57C2×C4×D101—21no—
58C2×C4×D101—22no—
59#13108062—24no—
60#13108059—28no—
61#13108059—24no—
62#13108058—22no—
63C2×#11717579—22no—
64C2×#11717579C2312yes—
65C2×#11717579C2312yes—
66C24×C101—21no—
67#41×C101—22no—
68#41×C101—22no—
69C2×Q8×C101D411yescharacteristic
70C2×D4×C101—24no—
71C2×D4×C101—22no—
72C2×D4×C101—22no—
73C2×D4×C101—21no—
74C2×D4×C101D412yes—
75C2×D4×C101D411yescharacteristic
76C2×D4×C101C2312yes—
77C22×C4×C101C2312yes—
78C22×C4×C101C2311yescharacteristic
79C4⋊C4×C101C2312yes—
80C22⋊C4×C101—21no—
81C22⋊C4×C101—24no—
82C42×C101—21no—
83C23×C101—22no—
84C23×C101—42no—
85C23×C101—82no—
86C23×C101C2×D412yes—
87C23×C101C2×D411yescharacteristic
88C22×D101—21no—
89C22×D101—22no—
90C22×D101—28no—
91C22×D101—24no—
92C22×D101—82no—
93C22×D101—84no—
94Q8×C101—22no—
95D4×C101—24no—
96D4×C101—22no—
97D4×C101—22no—
98D4×C101—22no—
99D4×C101—42no—
100D4×C101—84no—
101C2×C4×C101—22no—
102C2×C4×C101—22no—
103C2×C4×C101—22no—
104C2×C4×C101C2×D412yes—
105C2×C4×C101C2×D412yes—
106C2×C4×C101C2×D412yes—
107C2×C4×C101C2×D411yescharacteristic
108C2×C4×C101C2411yescharacteristic
109C22⋊D101—28no—
110C22⋊D101—24no—
111C2×Dic101—21no—
112D404—28no—
113D404—24no—
114D404—21no—
115D404—22no—
116D404C2×D412yes—
117D404C2×D414yes—
118C4×D101—28no—
119C4×D101—24no—
120Dic202—21no—
121Dic202C2×D412yes—
122#11717579—22no—
123#11717579—22no—
124C22×C101—22no—
125C22×C101—22no—
126C22×C101—24no—
127C22×C101—42no—
128C22×C101—44no—
129C22×C101—84no—
130C22×C101#12012yes—
131C22×C101C22×D411yescharacteristic
132C2×D101—24no—
133C2×D101—22no—
134C2×D101—44no—
135C2×D101—42no—
136C2×D101—84no—
137C2×D101—88no—
138C4×C101—22no—
139C4×C101—42no—
140C4×C101#12012yes—
141C4×C101C22×D411yescharacteristic
142C4×C101C22×D411yescharacteristic
143Dic101—22no—
144C2×C101—22no—
145C2×C101—42no—
146C2×C101—82no—
147C2×C101C2×#12011yescharacteristic
148C2×C101#57612yes—
149D101—44no—
150D101—42no—
151#3008—1011nomaximal, Sylow
152C101#300811yescharacteristic, complemented, Sylow
153C2×H(5,𝔽2)—1011no—
154C2×AGL(1,ℤ/8)—1011no—
155C22×D8—1011no—
156#535D10111yescharacteristic, complemented
157#449—1014no—
158#447—1014no—
159#418—1011no—
160#416—1011no—
161#408—1011no—
162H(5,𝔽2)—2024no—
163H(5,𝔽2)—2024no—
164C2×#41—1011no—
165C2×#41—1012no—
166C2×#41—2021no—
167C2×#41C2×D10111yescharacteristic
168C22×D4—1011no—
169C22×D4—1012no—
170C22×D4—2022no—
171C22×D4—2021no—
172C22×D4C2×D10111yescharacteristic, complemented
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—21no—
181#124—21no—
182#121C2×D10114yescomplemented
183#120—22no—
184#117C2×D10111yescharacteristic, complemented
185#103—1012no—
186#102—1012no—
187#102—1014no—
188C22⋊C8—1014no—
189C24—21no—
190C24—2022no—
191C24—2021no—
192#41—22no—
193#41—22no—
194#41—1014no—
195#41—2024no—
196#41—2024no—
197#41—2028no—
198#41—2022no—
199C2×Q8—1011no—
200C2×Q8C22⋊D10111yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—1014no—
206C2×D4—1012no—
207C2×D4—1011no—
208C2×D4—2024no—
209C2×D4—2024no—
210C2×D4—20216no—
211C2×D4—2028no—
212C2×D4—2022no—
213C2×D4—2021no—
214C2×D4—2022no—
215C2×D4—2022no—
216C2×D4—2021no—
217C2×D4C22⋊D10112yes—
218C2×D4C22⋊D10111yescharacteristic
219C2×D4C22×D10112yes—
220C22×C4—2022no—
221C22×C4—2021no—
222C22×C4C22×D10112yes—
223C22×C4C22×D10111yescharacteristic
224SD16—2024no—
225D8—2028no—
226D8—2024no—
227M16—2022no—
228C2×C8—1012no—
229C2×C8—1012no—
230C2×C8—2022no—
231C4⋊C4C22×D10112yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—82no—
238C23—8082no—
239C23—8084no—
240C23—2028no—
241C23—2024no—
242C23—2022no—
243C23—2021no—
244C23C2×C22⋊D10112yes—
245C23C2×C22⋊D10111yescharacteristic
246Q8—22no—
247Q8—1012no—
248Q8—2021no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—84no—
255D4—1014no—
256D4—1012no—
257D4—2028no—
258D4—2028no—
259D4—2024no—
260D4—2021no—
261D4—2022no—
262D4—2024no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—2028no—
267C2×C4—2021no—
268C2×C4—2024no—
269C2×C4D4×D10112yes—
270C2×C4D4×D10112yes—
271C2×C4C2×C22⋊D10112yes—
272C2×C4C2×C22⋊D10111yescharacteristic
273C2×C4C23×D10111yescharacteristic
274C8—2022no—
275C8—2022no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—8084no—
283C22—8088no—
284C22—4044no—
285C22—4042no—
286C22—2024no—
287C22—2022no—
288C22#1569222312yes—
289C22C22×C22⋊D10111yescharacteristic
290C4—22no—
291C4—42no—
292C4—2022no—
293C4#1569222312yes—
294C4C2×D4×D10111yescharacteristic
295C4C2×D4×D10111yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—4044no—
300C2—4042no—
301C2C2×#1569222311yescharacteristic
302C2#2026598012yes—
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#535×C101C2
socleC22×C101C22×D4
hypercenter#535D101
hypocenterC101#3008
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#15692301C2semidirect
2. C2×#15692281C2semidirect
3. C22×#13108059C2semidirect
4. #20265320C2semidirect
5. #20265310C2semidirect
6. #20265181C2semidirect
7. #20265171C2semidirect
8. #20265144C2semidirect
9. #535×C101C2semidirect
14. C2×#13108074C22semidirect
16. C22×D404C22semidirect
20. C2×#13108062C22semidirect
22. C2×#13108059C22semidirect
23. C2×#13108059C22semidirect
24. C2×#13108058C22semidirect
25. C22×#11717579C22semidirect
26. #15692178C22semidirect
27. #15692175C22semidirect
28. #15692175C22semidirect
29. C22⋊#11717579C22semidirect
31. C22×D4×C101C22semidirect
34. #121×C101C22semidirect
36. #117×C101C22semidirect
43. #13108074D4semidirect
50. C2×D404D4semidirect
152. C101#3008semidirect
156. #535D101semidirect
172. C22×D4C2×D101semidirect
182. #121C2×D101semidirect
184. #117C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156923012
C2×#156922812
C22×#131080592
#202653202
#202653102
#202651812
#202651712
#202651442
#535×C1012
#3008101