Finite Group Databasebeta

← #27180372

Subgroups — #27180372

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×#13108074C211yesmaximal, characteristic, complemented
2C2×#15692285C211yesmaximal, characteristic, complemented
3C2×#15692280C211yesmaximal, characteristic, complemented
4#20265337C214yesmaximal, complemented
5#20265320C214yesmaximal, complemented
6#20265185C211yesmaximal, characteristic, complemented
7#20265181C211yesmaximal, characteristic, complemented
8#20265141C211yesmaximal, characteristic, complemented
9#525×C101C211yesmaximal, characteristic, complemented
10C2×#13108074—24no—
11C2×#13108074—24no—
12C2×#13108074—22no—
13C2×#13108074C2214yes—
14C22×Dic202C2211yescharacteristic
15C22×D404C2211yescharacteristic
16C22×C22⋊D101—21no—
17C22×C4×D101—21no—
18#15692285—24no—
19#15692280—24no—
20C2×#13108064C2212yescomplemented
21C2×#13108062C2212yescomplemented
22C2×#13108061C2212yescomplemented
23C2×#13108059C2212yescomplemented
24C2×#13108058C2211yescharacteristic
25C2×#13108058C2211yescharacteristic
26#15692178C2214yescomplemented
27#15692175C2214yescomplemented
28C22⋊#11717579C2214yescomplemented
29C2×#41×C101C2211yescharacteristic
30C23×C4×C101C2211yescharacteristic
31#123×C101—21no—
32C22⋊Q8×C101C2212yescomplemented
33#121×C101C2212yescomplemented
34#120×C101—21no—
35C4×D4×C101—22no—
36#117×C101C2211yescharacteristic
37C23×D101—21no—
38#13108074—216no—
39#13108074—28no—
40#13108074—28no—
41C2×Dic202—22no—
42C2×Dic202—21no—
43C2×Dic202D414yescomplemented
44C2×Dic202C2312yes—
45C2×D404—22no—
46C2×D404—21no—
47C2×D404D414yescomplemented
48C2×D404C2312yes—
49C2×C22⋊D101—28no—
50C2×C22⋊D101—24no—
51C22×Dic101—21no—
52C2×C4×D101—28no—
53C2×C4×D101—24no—
54#13108064—24no—
55#13108062—24no—
56#13108061—24no—
57#13108059—24no—
58#13108058—22no—
59#13108058—22no—
60C2×#11717579C2312yes—
61C2×#11717579C2312yes—
62C24×C101D411yescharacteristic
63#41×C101—22no—
64#41×C101—22no—
65C2×Q8×C101C2311yescharacteristic
66C2×D4×C101—22no—
67C2×D4×C101—21no—
68C2×D4×C101C2311yescharacteristic
69C22×C4×C101—22no—
70C22×C4×C101—22no—
71C22×C4×C101—21no—
72C22×C4×C101D411yescharacteristic
73C22×C4×C101D412yes—
74C22×C4×C101C2312yes—
75C22×C4×C101C2311yescharacteristic
76C4⋊C4×C101—22no—
77C4⋊C4×C101C2312yes—
78C22⋊C4×C101—21no—
79C22⋊C4×C101—22no—
80C22⋊C4×C101—22no—
81C42×C101—21no—
82C23×C101—22no—
83C23×C101—21no—
84C23×C101—42no—
85C23×C101C2×D412yes—
86C23×C101C2×D411yescharacteristic
87C22×D101—24no—
88C22×D101—22no—
89C22×D101—44no—
90Q8×C101—21no—
91Q8×C101—21no—
92D4×C101—21no—
93D4×C101—21no—
94D4×C101—44no—
95D4×C101—42no—
96C2×C4×C101—22no—
97C2×C4×C101—24no—
98C2×C4×C101—22no—
99C2×C4×C101—22no—
100C2×C4×C101—22no—
101C2×C4×C101—21no—
102C2×C4×C101—42no—
103C2×C4×C101C2×D412yes—
104C2×C4×C101C2×D412yes—
105C2×C4×C101C2×D412yes—
106C2×C4×C101C2×D411yescharacteristic
107C2×C4×C101C2411yescharacteristic
108C22⋊D101—216no—
109C2×Dic101—24no—
110C2×Dic101—22no—
111D404—24no—
112D404—22no—
113D404C2×D414yes—
114C4×D101—216no—
115Dic202—24no—
116Dic202—22no—
117Dic202C2×D414yes—
118#11717579—22no—
119#11717579—22no—
120C22×C101—22no—
121C22×C101—22no—
122C22×C101—22no—
123C22×C101—22no—
124C22×C101—44no—
125C22×C101—41no—
126C22×C101—42no—
127C22×C101#12012yes—
128C22×C101C22×D411yescharacteristic
129C2×D101—24no—
130C2×D101—48no—
131C2×D101—44no—
132C4×C101—22no—
133C4×C101—41no—
134C4×C101—42no—
135C4×C101#12012yes—
136C4×C101C22×D411yescharacteristic
137C4×C101C22×D411yescharacteristic
138Dic101—24no—
139C2×C101—22no—
140C2×C101—42no—
141C2×C101—81no—
142C2×C101C2×#12011yescharacteristic
143C2×C101#57512yes—
144D101—44no—
145#3005—1011nomaximal, Sylow
146C101#300511yescharacteristic, complemented, Sylow
147C22×#41—1011no—
148C2×#137—1011no—
149C2×AGL(1,ℤ/8)—1011no—
150#525D10111yescharacteristic, complemented
151#452—1014no—
152#449—1014no—
153#419—1011no—
154#418—1011no—
155#407—1011no—
156C2×#41—1014no—
157C2×#41—2024no—
158C2×#41—2024no—
159C2×#41—2022no—
160C2×#41C2×D10111yescharacteristic
161C22×Q8—1011no—
162C22×D4—1011no—
163C22×D4—2021no—
164C23×C4—2021no—
165C23×C4C2×D10111yescharacteristic
166#137—2024no—
167AGL(1,ℤ/8)—2024no—
168C2×Q16—1012no—
169C2×SD16—1012no—
170C2×SD16—1012no—
171C2×D8—1012no—
172C2×M16—1011no—
173C2×M16—1011no—
174#123—21no—
175C22⋊Q8C2×D10112yescomplemented
176#121C2×D10112yescomplemented
177#120—21no—
178C4×D4—22no—
179#117C2×D10111yescharacteristic
180#103—1014no—
181#102—1014no—
182C22⋊C8—1014no—
183C24—2021no—
184C24C22⋊D10111yescharacteristic
185#41—22no—
186#41—22no—
187#41—2028no—
188#41—2028no—
189#41—20216no—
190C2×Q8—1014no—
191C2×Q8—1012no—
192C2×Q8—2022no—
193C2×Q8—2021no—
194C2×Q8C22×D10111yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—1014no—
198C2×D4—1012no—
199C2×D4—2028no—
200C2×D4—2024no—
201C2×D4—2022no—
202C2×D4—2021no—
203C2×D4C22×D10111yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—2028no—
208C22×C4—2021no—
209C22×C4—2024no—
210C22×C4C22⋊D10112yes—
211C22×C4C22⋊D10111yescharacteristic
212C22×C4C22×D10112yes—
213C22×C4C22×D10111yescharacteristic
214Q16—2024no—
215SD16—2024no—
216SD16—2024no—
217D8—2024no—
218M16—2022no—
219M16—2022no—
220C2×C8—1012no—
221C2×C8—1012no—
222C4⋊C4—22no—
223C4⋊C4C22×D10112yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—4044no—
232C23—2022no—
233C23—2024no—
234C23C2×C22⋊D10112yes—
235C23C2×C22⋊D10111yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—1014no—
239Q8—2022no—
240Q8—2024no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—1014no—
246D4—20216no—
247D4—2022no—
248D4—2024no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—20216no—
257C2×C4—2022no—
258C2×C4—2024no—
259C2×C4D4×D10112yes—
260C2×C4D4×D10112yes—
261C2×C4C2×C22⋊D10112yes—
262C2×C4C2×C22⋊D10111yescharacteristic
263C2×C4C23×D10111yescharacteristic
264C8—2022no—
265C8—2022no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—41no—
272C22—42no—
273C22—4048no—
274C22—4044no—
275C22—2024no—
276C22#1569222312yes—
277C22C22×C22⋊D10111yescharacteristic
278C4—22no—
279C4—41no—
280C4—42no—
281C4—2024no—
282C4#1569222312yes—
283C4C2×D4×D10111yescharacteristic
284C4C2×D4×D10111yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—4044no—
289C2C2×#1569222311yescharacteristic
290C2#2026597512yes—
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#525×C101C2
socleC22×C101C22×D4
hypercenter#525D101
hypocenterC101#3005
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#13108074C2semidirect
2. C2×#15692285C2semidirect
3. C2×#15692280C2semidirect
4. #20265337C2semidirect
5. #20265320C2semidirect
6. #20265185C2semidirect
7. #20265181C2semidirect
8. #20265141C2semidirect
9. #525×C101C2semidirect
20. C2×#13108064C22semidirect
21. C2×#13108062C22semidirect
22. C2×#13108061C22semidirect
23. C2×#13108059C22semidirect
26. #15692178C22semidirect
27. #15692175C22semidirect
28. C22⋊#11717579C22semidirect
32. C22⋊Q8×C101C22semidirect
33. #121×C101C22semidirect
43. C2×Dic202D4semidirect
47. C2×D404D4semidirect
146. C101#3005semidirect
150. #525D101semidirect
175. C22⋊Q8C2×D101semidirect
176. #121C2×D101semidirect

Maximal subgroups

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

subgroupindex
C22×#131080742
C2×#156922852
C2×#156922802
#202653372
#202653202
#202651852
#202651812
#202651412
#525×C1012
#3005101