Finite Group Databasebeta

← #27180373

Subgroups — #27180373

Order 12928.

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

normal subgroups only

This group has 758 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
1C22×#13108075C211yesmaximal, characteristic, complemented
2C2×#15692280C211yesmaximal, characteristic, complemented
3#20265684C211yesmaximal, characteristic, complemented
4#20265338C214yesmaximal, complemented
5#20265321C214yesmaximal, complemented
6#20265186C211yesmaximal, characteristic, complemented
7#20265183C211yesmaximal, characteristic, complemented
8#20265140C211yesmaximal, characteristic, complemented
9C2×#137×C101C211yesmaximal, characteristic, complemented
10C2×#13108075—24no—
11C2×#13108075—22no—
12C2×#13108075—24no—
13C2×#13108075C2214yes—
14C22×D404—21no—
15C22×D404C2211yescharacteristic
16C22×C4×D101—21no—
17C22×C4×D101C2211yescharacteristic
18#15692280—24no—
19C2×#13108061C2212yescomplemented
20C2×#13108059C2212yescomplemented
21C2×#13108058C2211yescharacteristic
22#15692236—21no—
23#15692231C2212yescomplemented
24#15692226C2212yescomplemented
25#15692223—21no—
26C4×C22⋊D101—22no—
27#15692214C2211yescharacteristic
28#15692179C2214yescomplemented
29#15692177C2214yescomplemented
30#15692167C2214yescomplemented
31C2×#41×C101C2211yescharacteristic
32C22×Q8×C101C2211yescharacteristic
33#137×C101—24no—
34C2×Q16×C101C2212yescomplemented
35C2×SD16×C101C2212yescomplemented
36C2×M16×C101C2211yescharacteristic
37C23×D101—21no—
38C23×D101D411yescharacteristic
39#13108075—216no—
40#13108075—28no—
41#13108075—28no—
42C2×D404—28no—
43C2×D404—24no—
44C2×D404—21no—
45C2×D404—22no—
46C2×D404D414yescomplemented
47C2×D404C2312yes—
48C2×C22⋊D101—22no—
49C22×Dic101D411yescharacteristic
50C2×C4×D101—28no—
51C2×C4×D101—22no—
52C2×C4×D101—22no—
53C2×C4×D101—24no—
54C2×C4×D101D412yes—
55C2×C4×D101C2312yes—
56#13108061—24no—
57#13108059—24no—
58#13108058—22no—
59C2×#11717579C2312yes—
60C4⋊Dic101C2312yes—
61#13108053—22no—
62C22⋊Dic101—21no—
63C22⋊Dic101—22no—
64#13108051—22no—
65C4×Dic101—21no—
66#41×C101—22no—
67#41×C101—22no—
68C2×Q8×C101—22no—
69C2×Q8×C101—21no—
70C2×Q8×C101D414yescomplemented
71C2×Q8×C101C2311yescharacteristic
72C2×Q8×C101C2312yes—
73C2×D4×C101—21no—
74C2×D4×C101C2311yescharacteristic
75C22×C4×C101—21no—
76C22×C4×C101—21no—
77C22×C4×C101C2311yescharacteristic
78Q16×C101—24no—
79SD16×C101—24no—
80M16×C101—22no—
81C2×C8×C101C2312yes—
82C23×C101—21no—
83C23×C101C2×D411yescharacteristic
84C22×D101—22no—
85C22×D101—24no—
86C22×D101—22no—
87C22×D101—42no—
88C22×D101—44no—
89C22×D101C2×D412yes—
90Q8×C101—24no—
91Q8×C101—22no—
92Q8×C101—21no—
93Q8×C101—21no—
94Q8×C101C2×D414yes—
95D4×C101—21no—
96D4×C101—21no—
97D4×C101—42no—
98C2×C4×C101—24no—
99C2×C4×C101—22no—
100C2×C4×C101—21no—
101C2×C4×C101—42no—
102C2×C4×C101C2×D411yescharacteristic
103C2×C4×C101C2×D412yes—
104C2×C4×C101C2×D412yes—
105C2×C4×C101C2411yescharacteristic
106C22⋊D101—44no—
107C2×Dic101—22no—
108C2×Dic101—22no—
109C2×Dic101C2×D412yes—
110D404—216no—
111D404—24no—
112D404—22no—
113D404C2×D414yes—
114C4×D101—216no—
115C4×D101—24no—
116C4×D101—22no—
117C4×D101—22no—
118C8×C101—22no—
119#11717579—22no—
120C22×C101—22no—
121C22×C101—41no—
122C22×C101—42no—
123C22×C101#12012yes—
124C22×C101C22×D411yescharacteristic
125C2×D101—22no—
126C2×D101—24no—
127C2×D101—22no—
128C2×D101—22no—
129C2×D101—44no—
130C2×D101—48no—
131C2×D101—44no—
132C4×C101—24no—
133C4×C101—41no—
134C4×C101#12012yes—
135C4×C101C22×D411yescharacteristic
136C4×C101C22×D411yescharacteristic
137Dic101—22no—
138Dic101—42no—
139C2×C101—22no—
140C2×C101—81no—
141C2×C101C2×#12011yescharacteristic
142C2×C101#57512yes—
143D101—42no—
144D101—44no—
145#3005—1011nomaximal, Sylow
146C101#300511yescharacteristic, complemented, Sylow
147C22×#41—1011no—
148C2×#137D10111yescharacteristic, complemented
149C2×AGL(1,ℤ/8)—1011no—
150#525—1011no—
151#452—1014no—
152#449—1014no—
153#419—1011no—
154#418—1011no—
155#407—1011no—
156C2×#41—1014no—
157C2×#41—2022no—
158C2×#41—2024no—
159C2×#41—2024no—
160C2×#41C2×D10111yescharacteristic
161C22×Q8C2×D10111yescharacteristic
162C22×D4—1011no—
163C22×D4—2021no—
164C23×C4—1011no—
165C23×C4—2021no—
166#137—24no—
167AGL(1,ℤ/8)—2024no—
168C2×Q16C2×D10112yescomplemented
169C2×SD16—1012no—
170C2×SD16C2×D10112yescomplemented
171C2×D8—1012no—
172C2×M16—1011no—
173C2×M16C2×D10111yescharacteristic
174#123—2021no—
175C22⋊Q8—1012no—
176#121—1012no—
177#120—2021no—
178C4×D4—2022no—
179#117—1011no—
180#103—1014no—
181#102—1014no—
182C22⋊C8—1014no—
183C24—1011no—
184C24—2021no—
185#41—22no—
186#41—22no—
187#41—2028no—
188#41—20216no—
189#41—2028no—
190C2×Q8—21no—
191C2×Q8—22no—
192C2×Q8C22⋊D10114yescomplemented
193C2×Q8C22×D10111yescharacteristic
194C2×Q8C22×D10112yes—
195C2×D4—21no—
196C2×D4—1014no—
197C2×D4—1012no—
198C2×D4—2024no—
199C2×D4—2028no—
200C2×D4—2022no—
201C2×D4—2021no—
202C2×D4—2022no—
203C2×D4C22×D10111yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—1011no—
207C22×C4—1012no—
208C22×C4—1012no—
209C22×C4—2024no—
210C22×C4—2028no—
211C22×C4—2022no—
212C22×C4—2022no—
213C22×C4C22×D10111yescharacteristic
214Q16—24no—
215SD16—24no—
216SD16—2024no—
217D8—2024no—
218M16—22no—
219M16—2022no—
220C2×C8—1012no—
221C2×C8C22×D10112yes—
222C4⋊C4—1012no—
223C4⋊C4—2022no—
224C22⋊C4—2022no—
225C22⋊C4—2022no—
226C22⋊C4—2021no—
227C42—2021no—
228C23—21no—
229C23—1012no—
230C23—4044no—
231C23—4042no—
232C23—2022no—
233C23—2024no—
234C23—2022no—
235C23D4×D10111yescharacteristic
236Q8—22no—
237Q8—24no—
238Q8—21no—
239Q8—21no—
240Q8C2×C22⋊D10114yes—
241D4—21no—
242D4—21no—
243D4—42no—
244D4—1014no—
245D4—4044no—
246D4—20216no—
247D4—2022no—
248D4—2024no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—1012no—
254C2×C4—20216no—
255C2×C4—2022no—
256C2×C4—2022no—
257C2×C4—2022no—
258C2×C4—2022no—
259C2×C4—2024no—
260C2×C4D4×D10112yes—
261C2×C4D4×D10111yescharacteristic
262C2×C4C2×C22⋊D10112yes—
263C2×C4C23×D10111yescharacteristic
264C8—22no—
265C8—2022no—
266C22—22no—
267C22—41no—
268C22—42no—
269C22—4048no—
270C22—4044no—
271C22—4044no—
272C22—2024no—
273C22—2022no—
274C22—2022no—
275C22—2022no—
276C22#1569222312yes—
277C22C2×D4×D10111yescharacteristic
278C4—24no—
279C4—41no—
280C4—4042no—
281C4—2022no—
282C4#1569222312yes—
283C4C2×D4×D10111yescharacteristic
284C4C22×C22⋊D10111yescharacteristic
285C2—22no—
286C2—81no—
287C2—4044no—
288C2—4042no—
289C2C2×#1569222311yescharacteristic
290C2#2026597612yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#137×C101C2
socleC22×C101C22×D4
hypercenterC2×#137D101
hypocenterC101#3005
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#13108075C2semidirect
2. C2×#15692280C2semidirect
3. #20265684C2semidirect
4. #20265338C2semidirect
5. #20265321C2semidirect
6. #20265186C2semidirect
7. #20265183C2semidirect
8. #20265140C2semidirect
9. C2×#137×C101C2semidirect
19. C2×#13108061C22semidirect
20. C2×#13108059C22semidirect
23. #15692231C22semidirect
24. #15692226C22semidirect
28. #15692179C22semidirect
29. #15692177C22semidirect
30. #15692167C22semidirect
34. C2×Q16×C101C22semidirect
35. C2×SD16×C101C22semidirect
46. C2×D404D4semidirect
70. C2×Q8×C101D4semidirect
146. C101#3005semidirect
148. C2×#137D101semidirect
168. C2×Q16C2×D101semidirect
170. C2×SD16C2×D101semidirect
192. C2×Q8C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C22×#131080752
C2×#156922802
#202656842
#202653382
#202653212
#202651862
#202651832
#202651402
C2×#137×C1012
#3005101