Finite Group Databasebeta

← #27180357

Subgroups — #27180357

Order 12928.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×D4×D101C211yesmaximal, characteristic, complemented
2C2×#15692280C211yesmaximal, characteristic, complemented
3#20265684C211yesmaximal, characteristic, complemented
4#20265326C214yesmaximal, complemented
5#20265309C214yesmaximal, complemented
6#20265183C211yesmaximal, characteristic, complemented
7#20265182C211yesmaximal, characteristic, complemented
8#20265140C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C101C211yesmaximal, characteristic, complemented
10C24×D101—21no—
11C22×D404C2211yescharacteristic
12C22×C22⋊D101—21no—
13C2×D4×D101—24no—
14C2×D4×D101—22no—
15C2×D4×D101—24no—
16C2×D4×D101C2214yes—
17C22×C4×D101C2211yescharacteristic
18#15692280—24no—
19C2×#13108061C2212yescomplemented
20C2×#13108059C2212yescomplemented
21C2×#13108058C2211yescharacteristic, complemented
22#15692236—21no—
23#15692231C2212yescomplemented
24#15692226C2212yescomplemented
25#15692223—21no—
26C4×C22⋊D101—22no—
27#15692214C2211yescharacteristic, complemented
28#15692177C2214yescomplemented
29#15692176C2214yescomplemented
30#15692167C2214yescomplemented
31C2×#41×C101C2211yescharacteristic, complemented
32C22×D4×C101C2211yescharacteristic
33AGL(1,ℤ/8)×C101—24no—
34C2×SD16×C101C2212yescomplemented
35C2×D8×C101C2212yescomplemented
36C2×M16×C101C2211yescharacteristic, complemented
37C23×D101—28no—
38C23×D101—21no—
39C23×D101—24no—
40C23×D101—48no—
41C23×D101D411yescharacteristic
42C2×D404—21no—
43C2×D404—22no—
44C2×D404D414yescomplemented
45C2×D404C2312yes—
46C2×C22⋊D101—28no—
47C2×C22⋊D101—22no—
48C2×C22⋊D101—24no—
49D4×D101—216no—
50D4×D101—28no—
51D4×D101—28no—
52C22×Dic101D411yescharacteristic
53C2×C4×D101—22no—
54C2×C4×D101—22no—
55C2×C4×D101D412yes—
56C2×C4×D101C2312yes—
57#13108061—24no—
58#13108059—24no—
59#13108058—22no—
60C2×#11717579C2312yescomplemented
61C4⋊Dic101C2312yescomplemented
62#13108053—22no—
63C22⋊Dic101—21no—
64C22⋊Dic101—22no—
65#13108051—22no—
66C4×Dic101—21no—
67C24×C101—21no—
68#41×C101—22no—
69#41×C101—22no—
70C2×Q8×C101C2311yescharacteristic, complemented
71C2×D4×C101—21no—
72C2×D4×C101—21no—
73C2×D4×C101—22no—
74C2×D4×C101D414yescomplemented
75C2×D4×C101C2312yes—
76C2×D4×C101C2311yescharacteristic, complemented
77C22×C4×C101—21no—
78C22×C4×C101C2311yescharacteristic
79SD16×C101—24no—
80D8×C101—24no—
81M16×C101—22no—
82C2×C8×C101C2312yescomplemented
83C23×C101—24no—
84C23×C101—22no—
85C23×C101—21no—
86C23×C101—44no—
87C23×C101C2×D411yescharacteristic
88C22×D101—22no—
89C22×D101—24no—
90C22×D101—216no—
91C22×D101—24no—
92C22×D101—22no—
93C22×D101—432no—
94C22×D101—48no—
95C22×D101—44no—
96C22×D101—48no—
97C22×D101C2×D412yes—
98Q8×C101—21no—
99Q8×C101—21no—
100D4×C101—24no—
101D4×C101—22no—
102D4×C101—21no—
103D4×C101—21no—
104D4×C101—42no—
105D4×C101C2×D414yes—
106C2×C4×C101—21no—
107C2×C4×C101—42no—
108C2×C4×C101C2×D411yescharacteristic
109C2×C4×C101C2×D412yes—
110C2×C4×C101C2×D412yes—
111C2×C4×C101C2411yescharacteristic
112C22⋊D101—216no—
113C22⋊D101—44no—
114C2×Dic101—22no—
115C2×Dic101—22no—
116C2×Dic101C2×D412yes—
117D404—24no—
118D404—22no—
119D404C2×D414yes—
120C4×D101—24no—
121C4×D101—22no—
122C4×D101—22no—
123C8×C101—22no—
124#11717579—22no—
125C22×C101—22no—
126C22×C101—24no—
127C22×C101—48no—
128C22×C101—44no—
129C22×C101—41no—
130C22×C101—42no—
131C22×C101#12012yes—
132C22×C101C22×D411yescharacteristic
133C2×D101—28no—
134C2×D101—22no—
135C2×D101—22no—
136C2×D101—24no—
137C2×D101—432no—
138C2×D101—48no—
139C2×D101—44no—
140C2×D101AGL(1,ℤ/8)14yescomplemented
141C4×C101—41no—
142C4×C101#12012yes—
143C4×C101C22×D411yescharacteristic
144C4×C101C22×D411yescharacteristic
145Dic101—22no—
146Dic101—42no—
147C2×C101—22no—
148C2×C101—44no—
149C2×C101—81no—
150C2×C101C2×#12011yescharacteristic
151C2×C101C2×AGL(1,ℤ/8)12yes—
152D101—24no—
153D101—44no—
154#3003—1011nomaximal, Sylow
155C101#300311yescharacteristic, complemented, Sylow
156C23×D4—1011no—
157C2×AGL(1,ℤ/8)—1011no—
158C2×AGL(1,ℤ/8)D10111yescharacteristic, complemented
159#525—1011no—
160#450—1014no—
161#447—1014no—
162#418—1011no—
163#418—1011no—
164#407—1011no—
165C25—2021no—
166C2×#41C2×D10111yescharacteristic, complemented
167C22×D4—1014no—
168C22×D4—1011no—
169C22×D4—2022no—
170C22×D4—2021no—
171C22×D4—2024no—
172C22×D4—2024no—
173C22×D4C2×D10111yescharacteristic
174C23×C4—1011no—
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—2024no—
177C2×SD16—1012no—
178C2×SD16C2×D10112yescomplemented
179C2×D8—1012no—
180C2×D8C2×D10112yescomplemented
181C2×M16—1011no—
182C2×M16C2×D10111yescharacteristic, complemented
183#123—2021no—
184C22⋊Q8—1012no—
185#121—1012no—
186#120—2021no—
187C4×D4—2022no—
188#117—1011no—
189#102—1014no—
190#102—1014no—
191C22⋊C8—1014no—
192C24—21no—
193C24—1011no—
194C24—4048no—
195C24—2024no—
196C24—2028no—
197C24—2021no—
198#41—22no—
199#41—22no—
200C2×Q8C22×D10111yescharacteristic, complemented
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—1014no—
205C2×D4—1012no—
206C2×D4—2024no—
207C2×D4—2028no—
208C2×D4—20216no—
209C2×D4—2022no—
210C2×D4—2028no—
211C2×D4—2028no—
212C2×D4—2021no—
213C2×D4—2022no—
214C2×D4C22⋊D10114yescomplemented
215C2×D4C22×D10112yes—
216C2×D4C22×D10111yescharacteristic, complemented
217C22×C4—21no—
218C22×C4—1011no—
219C22×C4—1012no—
220C22×C4—1012no—
221C22×C4—2022no—
222C22×C4—2022no—
223C22×C4C22×D10111yescharacteristic
224SD16—24no—
225SD16—2024no—
226D8—24no—
227D8—2024no—
228M16—22no—
229M16—2022no—
230C2×C8—1012no—
231C2×C8C22×D10112yescomplemented
232C4⋊C4—1012no—
233C4⋊C4—2022no—
234C22⋊C4—2022no—
235C22⋊C4—2022no—
236C22⋊C4—2021no—
237C42—2021no—
238C23—22no—
239C23—24no—
240C23—21no—
241C23—44no—
242C23—1012no—
243C23—4044no—
244C23—4048no—
245C23—40432no—
246C23—4048no—
247C23—2024no—
248C23—20216no—
249C23—2022no—
250C23—2024no—
251C23—2022no—
252C23D4×D10111yescharacteristic
253Q8—21no—
254Q8—21no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—24no—
259D4—42no—
260D4—1014no—
261D4—4044no—
262D4—20216no—
263D4—2022no—
264D4—2024no—
265D4C2×C22⋊D10114yes—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—1012no—
269C2×C4—2022no—
270C2×C4—2022no—
271C2×C4—2022no—
272C2×C4—2022no—
273C2×C4—2024no—
274C2×C4D4×D10112yes—
275C2×C4D4×D10111yescharacteristic
276C2×C4C2×C22⋊D10112yes—
277C2×C4C23×D10111yescharacteristic
278C8—22no—
279C8—2022no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—41no—
285C22—42no—
286C22—1014no—
287C22—4048no—
288C22—40432no—
289C22—4044no—
290C22—2028no—
291C22—2024no—
292C22—2022no—
293C22—2022no—
294C22#1569222312yes—
295C22C2×D4×D10111yescharacteristic
296C4—41no—
297C4—4042no—
298C4—2022no—
299C4#1569222312yes—
300C4C2×D4×D10111yescharacteristic
301C4C22×C22⋊D10111yescharacteristic
302C2—22no—
303C2—44no—
304C2—81no—
305C2—4044no—
306C2—2024no—
307C2C2×#1569222311yescharacteristic
308C2AGL(1,ℤ/8)×D10112yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×AGL(1,ℤ/8)×C101C2
socleC22×C101C22×D4
hypercenterC2×AGL(1,ℤ/8)D101
hypocenterC101#3003
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×D4×D101C2semidirect
2. C2×#15692280C2semidirect
3. #20265684C2semidirect
4. #20265326C2semidirect
5. #20265309C2semidirect
6. #20265183C2semidirect
7. #20265182C2semidirect
8. #20265140C2semidirect
9. C2×AGL(1,ℤ/8)×C101C2semidirect
19. C2×#13108061C22semidirect
20. C2×#13108059C22semidirect
21. C2×#13108058C22semidirect
23. #15692231C22semidirect
24. #15692226C22semidirect
27. #15692214C22semidirect
28. #15692177C22semidirect
29. #15692176C22semidirect
30. #15692167C22semidirect
31. C2×#41×C101C22semidirect
34. C2×SD16×C101C22semidirect
35. C2×D8×C101C22semidirect
36. C2×M16×C101C22semidirect
44. C2×D404D4semidirect
60. C2×#11717579C23semidirect
61. C4⋊Dic101C23semidirect
70. C2×Q8×C101C23semidirect
74. C2×D4×C101D4semidirect
76. C2×D4×C101C23semidirect
82. C2×C8×C101C23semidirect
140. C2×D101AGL(1,ℤ/8)semidirect
155. C101#3003semidirect
158. C2×AGL(1,ℤ/8)D101semidirect
166. C2×#41C2×D101semidirect
178. C2×SD16C2×D101semidirect
180. C2×D8C2×D101semidirect
182. C2×M16C2×D101semidirect
200. C2×Q8C22×D101semidirect
214. C2×D4C22⋊D101semidirect
216. C2×D4C22×D101semidirect
231. C2×C8C22×D101semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1012
C2×#156922802
#202656842
#202653262
#202653092
#202651832
#202651822
#202651402
C2×AGL(1,ℤ/8)×C1012
#3003101