Finite Group Databasebeta

← #27180394

Subgroups — #27180394

Order 12928.

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

normal subgroups only

This group has 754 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×#13108074C211yesmaximal, characteristic
2C2×#15692288C212yesmaximal, complemented
3C2×#15692234C211yesmaximal, characteristic, complemented
4#20265341C214yesmaximal, complemented
5#20265335C212yesmaximal, complemented
6#20265330C212yesmaximal, complemented
7#20265179C212yesmaximal, complemented
8#407×C101C211yesmaximal, characteristic, complemented
9C2×#13108074—28no—
10C2×#13108074—21no—
11C2×#13108074—21no—
12C2×#13108074C2212yes—
13C2×#13108074C2212yes—
14C22×Dic202C2211yescharacteristic
15C22×Dic202C2211yescharacteristic, complemented
16C22×D404C2211yescharacteristic
17C22×C22⋊D101—21no—
18C22×C4×D101—21no—
19#15692288—28no—
20C2×Dic404C2214yescomplemented
21C2×#13108063C2214yescomplemented
22#15692234—21no—
23#15692234—21no—
24#15692234C2212yes—
25#15692234C2212yescomplemented
26C2×C4⋊Dic101C2211yescharacteristic, complemented
27C2×#13108053—21no—
28C2×C22⋊Dic101—21no—
29#15692180C2214yescomplemented
30#15692177C2214yescomplemented
31C23×C4×C101C2211yescharacteristic
32C2×M16×C101C2212yescomplemented
33C22⋊C8×C101C2212yes—
34C22⋊C8×C101C2212yescomplemented
35C23×D101—21no—
36#13108074—28no—
37#13108074—28no—
38#13108074—24no—
39#13108074—24no—
40#13108074—24no—
41#13108074D418yes—
42C2×Dic202—21no—
43C2×Dic202—24no—
44C2×Dic202—24no—
45C2×Dic202—21no—
46C2×Dic202C2312yes—
47C2×Dic202C2312yes—
48C2×D404—21no—
49C2×D404—24no—
50C2×D404C2312yes—
51C2×C22⋊D101—28no—
52C2×C22⋊D101—22no—
53C2×C22⋊D101—22no—
54C22×Dic101—21no—
55C22×Dic101—21no—
56C22×Dic101—21no—
57C2×C4×D101—28no—
58C2×C4×D101—22no—
59C2×C4×D101—22no—
60Dic404—28no—
61#13108063—28no—
62C4⋊Dic101—21no—
63C4⋊Dic101C2312yes—
64#13108053—22no—
65#13108053—22no—
66C22⋊Dic101—22no—
67C22⋊Dic101—22no—
68C24×C101D411yescharacteristic
69C22×C4×C101—24no—
70C22×C4×C101D411yescharacteristic
71C22×C4×C101D411yescharacteristic
72C22×C4×C101D411yescharacteristic
73C22×C4×C101C2311yescharacteristic
74C22×C4×C101C2311yescharacteristic
75C22×C4×C101C2311yescharacteristic
76M16×C101—24no—
77C2×C8×C101C2314yes—
78C23×C101—22no—
79C23×C101—22no—
80C23×C101—41no—
81C23×C101C2×D411yescharacteristic
82C23×C101C2×D411yescharacteristic
83C23×C101C2×D411yescharacteristic
84C22×D101—24no—
85C22×D101—22no—
86C22×D101—42no—
87C22×D101—42no—
88C2×C4×C101—22no—
89C2×C4×C101—22no—
90C2×C4×C101—22no—
91C2×C4×C101—22no—
92C2×C4×C101—22no—
93C2×C4×C101C2×D414yes—
94C2×C4×C101C2×D411yescharacteristic
95C2×C4×C101C2×D411yescharacteristic
96C2×C4×C101C2×D411yescharacteristic
97C2×C4×C101C2411yescharacteristic
98C22⋊D101—28no—
99C22⋊D101—28no—
100C2×Dic101—24no—
101C2×Dic101—22no—
102C2×Dic101—22no—
103C2×Dic101—22no—
104C2×Dic101—42no—
105C2×Dic101—42no—
106D404—24no—
107D404—22no—
108D404C2×D414yes—
109C4×D101—28no—
110C4×D101—28no—
111Dic202—24no—
112Dic202—22no—
113Dic202—24no—
114Dic202—42no—
115Dic202C2×D414yes—
116C8×C101—24no—
117C22×C101—21no—
118C22×C101—21no—
119C22×C101—21no—
120C22×C101—21no—
121C22×C101—21no—
122C22×C101—21no—
123C22×C101—21no—
124C22×C101—44no—
125C22×C101#12012yes—
126C22×C101#13712yescomplemented
127C22×C101C22×D411yescharacteristic
128C2×D101—24no—
129C2×D101—48no—
130C2×D101—42no—
131C2×D101—42no—
132C4×C101—21no—
133C4×C101—21no—
134C4×C101#12012yes—
135C4×C101C22×D412yes—
136Dic101—24no—
137Dic101—42no—
138Dic101—42no—
139C2×C101—22no—
140C2×C101—22no—
141C2×C101—41no—
142C2×C101C2×#12011yescharacteristic
143C2×C101C2×#13711yescharacteristic
144C2×C101#57511yescharacteristic
145D101—44no—
146#3007—1011nomaximal, Sylow
147C101#300711yescharacteristic, complemented, Sylow
148C22×#41—1011no—
149C2×#137—1012no—
150C2×C22⋊Q8—1011no—
151#452—1014no—
152#451—1012no—
153#450—1012no—
154#417—1012no—
155#407D10111yescharacteristic, complemented
156C2×#41—1012no—
157C2×#41—1012no—
158C2×#41—2028no—
159C2×#41—2021no—
160C2×#41—2021no—
161C22×Q8—1011no—
162C22×Q8—1011no—
163C22×D4—1011no—
164C22×D4—2021no—
165C23×C4—2021no—
166C23×C4C2×D10111yescharacteristic
167#137—2028no—
168C2×Q16—1014no—
169C2×SD16—1014no—
170C2×M16C2×D10112yescomplemented
171C22⋊Q8—1012no—
172C22⋊Q8—1012no—
173C22⋊Q8—2021no—
174C22⋊Q8—2021no—
175C2×C4⋊C4—1011no—
176C2×C4⋊C4—2021no—
177C2×C22⋊C4—2021no—
178#103—1014no—
179#102—1014no—
180C22⋊C8C2×D10112yes—
181C22⋊C8C2×D10112yescomplemented
182C24—2021no—
183C24D40411yescharacteristic
184#41—1018no—
185#41—2024no—
186#41—2024no—
187#41—2028no—
188#41—2028no—
189#41—2024no—
190C2×Q8—1012no—
191C2×Q8—1012no—
192C2×Q8—2024no—
193C2×Q8—2024no—
194C2×Q8—2021no—
195C2×Q8—2021no—
196C2×D4—1012no—
197C2×D4—2024no—
198C2×D4—2028no—
199C2×D4—2022no—
200C2×D4—2022no—
201C2×D4—2021no—
202C22×C4—24no—
203C22×C4—2028no—
204C22×C4—2021no—
205C22×C4—2021no—
206C22×C4—2021no—
207C22×C4—2022no—
208C22×C4—2022no—
209C22×C4D40411yescharacteristic
210C22×C4D40411yescharacteristic
211C22×C4D40411yescharacteristic
212C22×C4C22×D10111yescharacteristic
213C22×C4C22×D10111yescharacteristic
214C22×C4C22×D10111yescharacteristic
215Q16—2028no—
216SD16—2028no—
217M16—24no—
218C2×C8C22×D10114yes—
219C4⋊C4—1012no—
220C4⋊C4—2022no—
221C4⋊C4—2022no—
222C4⋊C4—2021no—
223C22⋊C4—2022no—
224C22⋊C4—2022no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—4042no—
229C23—4042no—
230C23—2024no—
231C23—2022no—
232C23C2×D40411yescharacteristic
233C23C2×D40411yescharacteristic
234C23C2×D40411yescharacteristic
235Q8—1014no—
236Q8—4042no—
237Q8—2024no—
238Q8—2024no—
239Q8—2022no—
240D4—1014no—
241D4—2024no—
242D4—2028no—
243D4—2028no—
244D4—2022no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—4042no—
251C2×C4—4042no—
252C2×C4—2028no—
253C2×C4—2022no—
254C2×C4—2022no—
255C2×C4—2024no—
256C2×C4—2022no—
257C2×C4—2028no—
258C2×C4D4×D10114yes—
259C2×C4C2×D40411yescharacteristic
260C2×C4C2×D40411yescharacteristic
261C2×C4C2×D40411yescharacteristic
262C2×C4C23×D10111yescharacteristic
263C8—24no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—4048no—
273C22—4042no—
274C22—4042no—
275C22—2024no—
276C22#1569222512yes—
277C22#1569228812yescomplemented
278C22C22×D40411yescharacteristic
279C4—21no—
280C4—21no—
281C4—4042no—
282C4—4042no—
283C4—2024no—
284C4#1569222512yes—
285C4C2×D4×D10112yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—4044no—
290C2C2×#1569222511yescharacteristic
291C2C2×#1569228811yescharacteristic
292C2#2026597711yescharacteristic
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#407×C101C2
socleC22×C101C22×D4
prime radicalC22×#13108074C2
hypercenter#407D101
hypocenterC101#3007
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#15692288C2semidirect
3. C2×#15692234C2semidirect
4. #20265341C2semidirect
5. #20265335C2semidirect
6. #20265330C2semidirect
7. #20265179C2semidirect
8. #407×C101C2semidirect
15. C22×Dic202C22semidirect
20. C2×Dic404C22semidirect
21. C2×#13108063C22semidirect
25. #15692234C22semidirect
26. C2×C4⋊Dic101C22semidirect
29. #15692180C22semidirect
30. #15692177C22semidirect
32. C2×M16×C101C22semidirect
34. C22⋊C8×C101C22semidirect
126. C22×C101#137semidirect
147. C101#3007semidirect
155. #407D101semidirect
170. C2×M16C2×D101semidirect
181. C22⋊C8C2×D101semidirect
277. C22#15692288semidirect

Maximal subgroups

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

subgroupindex
C22×#131080742
C2×#156922882
C2×#156922342
#202653412
#202653352
#202653302
#202651792
#407×C1012
#3007101