Finite Group Databasebeta

← #27180374

Subgroups — #27180374

Order 12928.

758 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15692286C211yesmaximal, characteristic, complemented
2C2×#15692281C211yesmaximal, characteristic, complemented
3#20265688C211yesmaximal, characteristic, complemented
4#20265339C214yesmaximal, complemented
5#20265322C214yesmaximal, complemented
6#20265186C211yesmaximal, characteristic, complemented
7#20265182C211yesmaximal, characteristic, complemented
8#20265141C211yesmaximal, characteristic, complemented
9C22×#41×C101C211yesmaximal, characteristic, complemented
10C2×#13108074C2211yescharacteristic
11#15692286—24no—
12#15692281—24no—
13C2×#13108064C2212yescomplemented
14C2×#13108062C2212yescomplemented
15C2×#13108061C2212yescomplemented
16C2×#13108059C2212yescomplemented
17C2×#13108058C2211yescharacteristic
18C2×#13108058C2211yescharacteristic
19#15692238—21no—
20#15692234C2212yescomplemented
21#15692229C2212yescomplemented
22#15692224—21no—
23C4×C22⋊D101—22no—
24#15692214C2211yescharacteristic
25#15692179C2214yescomplemented
26#15692176C2214yescomplemented
27C22⋊#11717579C2214yescomplemented
28C2×#41×C101—24no—
29C2×#41×C101—24no—
30C2×#41×C101—22no—
31C2×#41×C101C2214yes—
32C22×Q8×C101C2211yescharacteristic
33C22×D4×C101—21no—
34C22×D4×C101C2211yescharacteristic
35C23×C4×C101—21no—
36C23×C4×C101C2211yescharacteristic
37#13108074—22no—
38#13108074—22no—
39C2×Dic202C2311yescharacteristic
40C2×D404C2311yescharacteristic
41C2×C22⋊D101—21no—
42C2×C22⋊D101—22no—
43C2×C4×D101—21no—
44#13108064—24no—
45#13108062—24no—
46#13108061—24no—
47#13108059—24no—
48#13108058—22no—
49#13108058—22no—
50C2×#11717579C2312yes—
51C2×#11717579C2312yes—
52C4⋊Dic101C2312yes—
53#13108053—22no—
54C22⋊Dic101—21no—
55C22⋊Dic101—22no—
56#13108051—22no—
57C4×Dic101—21no—
58C24×C101—21no—
59C24×C101D411yescharacteristic
60#41×C101—216no—
61#41×C101—28no—
62#41×C101—28no—
63C2×Q8×C101—22no—
64C2×Q8×C101—21no—
65C2×Q8×C101D414yescomplemented
66C2×Q8×C101C2312yes—
67C2×D4×C101—28no—
68C2×D4×C101—24no—
69C2×D4×C101—21no—
70C2×D4×C101—22no—
71C2×D4×C101D414yescomplemented
72C2×D4×C101C2312yes—
73C22×C4×C101—28no—
74C22×C4×C101—21no—
75C22×C4×C101—24no—
76C22×C4×C101—22no—
77C22×C4×C101—22no—
78C22×C4×C101D411yescharacteristic
79C22×C4×C101D412yes—
80C22×C4×C101C2312yes—
81C22×C4×C101C2311yescharacteristic
82C23×C101—22no—
83C23×C101—24no—
84C23×C101—22no—
85C23×C101—42no—
86C23×C101—44no—
87C23×C101C2×D412yes—
88C23×C101C2×D411yescharacteristic
89C22×D101—21no—
90Q8×C101—24no—
91Q8×C101—22no—
92Q8×C101C2×D414yes—
93D4×C101—216no—
94D4×C101—24no—
95D4×C101—22no—
96D4×C101C2×D414yes—
97C2×C4×C101—216no—
98C2×C4×C101—22no—
99C2×C4×C101—24no—
100C2×C4×C101—22no—
101C2×C4×C101—24no—
102C2×C4×C101—22no—
103C2×C4×C101—22no—
104C2×C4×C101C2×D412yes—
105C2×C4×C101C2×D411yescharacteristic
106C2×C4×C101C2×D412yes—
107C2×C4×C101C2×D412yes—
108C2×C4×C101C2411yescharacteristic
109C22⋊D101—44no—
110C22⋊D101—42no—
111C2×Dic101—22no—
112C2×Dic101—21no—
113D404—21no—
114D404—21no—
115C4×D101—42no—
116Dic202—21no—
117Dic202—21no—
118#11717579—22no—
119#11717579—22no—
120C22×C101—22no—
121C22×C101—22no—
122C22×C101—24no—
123C22×C101—22no—
124C22×C101—22no—
125C22×C101—44no—
126C22×C101—48no—
127C22×C101—44no—
128C22×C101#12012yes—
129C22×C101C22×D411yescharacteristic
130C2×D101—41no—
131C2×D101—42no—
132C4×C101—22no—
133C4×C101—24no—
134C4×C101#12012yes—
135C4×C101C22×D411yescharacteristic
136C4×C101C22×D411yescharacteristic
137Dic101—41no—
138Dic101—42no—
139C2×C101—22no—
140C2×C101—42no—
141C2×C101—44no—
142C2×C101C2×#12011yescharacteristic
143C2×C101#57512yes—
144D101—81no—
145#3005—1011nomaximal, Sylow
146C101#300511yescharacteristic, complemented, Sylow
147C22×#41D10111yescharacteristic, complemented
148C2×#137—1011no—
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—24no—
157C2×#41—24no—
158C2×#41—22no—
159C2×#41—1011no—
160C2×#41C2×D10114yes—
161C22×Q8C2×D10111yescharacteristic
162C22×D4—21no—
163C22×D4C2×D10111yescharacteristic
164C23×C4—21no—
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—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—21no—
184C24C22⋊D10111yescharacteristic
185#41—216no—
186#41—28no—
187#41—28no—
188#41—2022no—
189#41—2022no—
190C2×Q8—21no—
191C2×Q8—22no—
192C2×Q8—1011no—
193C2×Q8C22⋊D10114yescomplemented
194C2×Q8C22×D10112yes—
195C2×D4—28no—
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—1011no—
200C2×D4—2022no—
201C2×D4—2021no—
202C2×D4C22⋊D10114yescomplemented
203C2×D4C22×D10112yes—
204C22×C4—28no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—2021no—
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—1012no—
223C4⋊C4—2022no—
224C22⋊C4—2021no—
225C22⋊C4—2022no—
226C22⋊C4—2022no—
227C42—2021no—
228C23—22no—
229C23—22no—
230C23—24no—
231C23—42no—
232C23—44no—
233C23—2021no—
234C23C2×C22⋊D10112yes—
235C23C2×C22⋊D10111yescharacteristic
236Q8—22no—
237Q8—24no—
238Q8—2021no—
239Q8—2021no—
240Q8C2×C22⋊D10114yes—
241D4—216no—
242D4—22no—
243D4—24no—
244D4—4044no—
245D4—4042no—
246D4—2021no—
247D4—2021no—
248D4C2×C22⋊D10114yes—
249C2×C4—216no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—4042no—
257C2×C4—2022no—
258C2×C4—2021no—
259C2×C4C2×C22⋊D10112yes—
260C2×C4C2×C22⋊D10112yes—
261C2×C4C2×C22⋊D10112yes—
262C2×C4C2×C22⋊D10111yescharacteristic
263C2×C4C23×D10111yescharacteristic
264C8—2022no—
265C8—2022no—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—44no—
272C22—48no—
273C22—44no—
274C22—4041no—
275C22—4042no—
276C22#1569222412yes—
277C22C22×C22⋊D10111yescharacteristic
278C4—22no—
279C4—24no—
280C4—4041no—
281C4—4042no—
282C4#1569222412yes—
283C4C22×C22⋊D10111yescharacteristic
284C4C22×C22⋊D10111yescharacteristic
285C2—22no—
286C2—42no—
287C2—44no—
288C2—8081no—
289C2C2×#1569222411yescharacteristic
290C2#2026597512yes—
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC22×#41×C101C2
socleC22×C101C22×D4
hypercenterC22×#41D101
hypocenterC101#3005
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#15692286C2semidirect
2. C2×#15692281C2semidirect
3. #20265688C2semidirect
4. #20265339C2semidirect
5. #20265322C2semidirect
6. #20265186C2semidirect
7. #20265182C2semidirect
8. #20265141C2semidirect
9. C22×#41×C101C2semidirect
13. C2×#13108064C22semidirect
14. C2×#13108062C22semidirect
15. C2×#13108061C22semidirect
16. C2×#13108059C22semidirect
20. #15692234C22semidirect
21. #15692229C22semidirect
25. #15692179C22semidirect
26. #15692176C22semidirect
27. C22⋊#11717579C22semidirect
65. C2×Q8×C101D4semidirect
71. C2×D4×C101D4semidirect
146. C101#3005semidirect
147. C22×#41D101semidirect
193. C2×Q8C22⋊D101semidirect
202. C2×D4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922862
C2×#156922812
#202656882
#202653392
#202653222
#202651862
#202651822
#202651412
C22×#41×C1012
#3005101