Finite Group Databasebeta

← #27180382

Subgroups — #27180382

Order 12928.

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

normal subgroups only

This group has 806 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, complemented
2C2×#15692281C212yesmaximal, complemented
3#20265320C214yesmaximal, complemented
4#20265315C212yesmaximal, complemented
5#20265310C212yesmaximal, complemented
6#20265176C212yesmaximal, complemented
7#20265141C211yesmaximal, characteristic, complemented
8C2×#121×C101C211yesmaximal, characteristic, complemented
9C2×#13108074—28no—
10C2×#13108074—21no—
11C2×#13108074—21no—
12C2×#13108074C2212yes—
13C2×#13108074C2212yescomplemented
14C22×Dic202C2211yescharacteristic, complemented
15C22×D404C2211yescharacteristic, complemented
16C22×C22⋊D101—21no—
17C22×C4×D101—21no—
18#15692281—28no—
19C2×#13108062C2214yescomplemented
20C2×#13108059C2214yescomplemented
21C2×#13108058C2212yescomplemented
22#15692178C2214yescomplemented
23#15692175C2214yescomplemented
24C22⋊#11717579C2212yescomplemented
25C22⋊#11717579C2212yescomplemented
26C22×D4×C101—21no—
27C22×D4×C101C2211yescharacteristic, complemented
28C23×C4×C101C2211yescharacteristic
29#121×C101—21no—
30#121×C101—21no—
31#121×C101C2212yes—
32#121×C101C2212yescomplemented
33C2×C4⋊C4×C101C2211yescharacteristic, complemented
34C2×C22⋊C4×C101—21no—
35C23×D101—21no—
36#13108074—28no—
37#13108074—28no—
38#13108074—24no—
39#13108074—24no—
40#13108074—24no—
41#13108074D418yescomplemented
42C2×Dic202—21no—
43C2×Dic202—24no—
44C2×Dic202C2312yes—
45C2×D404—21no—
46C2×D404—24no—
47C2×D404C2312yes—
48C2×C22⋊D101—28no—
49C2×C22⋊D101—22no—
50C2×C22⋊D101—22no—
51C22×Dic101—21no—
52C2×C4×D101—28no—
53C2×C4×D101—22no—
54C2×C4×D101—22no—
55#13108062—28no—
56#13108059—28no—
57#13108058—24no—
58C2×#11717579C2314yes—
59C24×C101—21no—
60C24×C101D411yescharacteristic
61C2×D4×C101—22no—
62C2×D4×C101—22no—
63C2×D4×C101—24no—
64C2×D4×C101—21no—
65C2×D4×C101—44no—
66C2×D4×C101C2312yes—
67C22×C4×C101—24no—
68C22×C4×C101—21no—
69C22×C4×C101D411yescharacteristic
70C22×C4×C101D411yescharacteristic
71C22×C4×C101D411yescharacteristic
72C22×C4×C101C2311yescharacteristic
73C22×C4×C101C2311yescharacteristic
74C22×C4×C101C2311yescharacteristic
75C4⋊C4×C101—21no—
76C4⋊C4×C101C2312yes—
77C22⋊C4×C101—22no—
78C22⋊C4×C101—22no—
79C23×C101—22no—
80C23×C101—22no—
81C23×C101—22no—
82C23×C101—41no—
83C23×C101—42no—
84C23×C101—42no—
85C23×C101—42no—
86C23×C101C2×D411yescharacteristic
87C23×C101C2×D411yescharacteristic
88C23×C101C2×D411yescharacteristic
89C22×D101—24no—
90C22×D101—22no—
91C22×D101—42no—
92C22×D101—42no—
93D4×C101—24no—
94D4×C101—44no—
95D4×C101—44no—
96D4×C101—42no—
97C2×C4×C101—22no—
98C2×C4×C101—22no—
99C2×C4×C101—22no—
100C2×C4×C101—22no—
101C2×C4×C101—22no—
102C2×C4×C101—22no—
103C2×C4×C101—42no—
104C2×C4×C101C2×D414yes—
105C2×C4×C101C2×D411yescharacteristic
106C2×C4×C101C2×D411yescharacteristic
107C2×C4×C101C2×D411yescharacteristic
108C2×C4×C101C2411yescharacteristic
109C22⋊D101—28no—
110C22⋊D101—28no—
111C2×Dic101—24no—
112C2×Dic101—22no—
113D404—24no—
114D404—22no—
115D404C2×D414yescomplemented
116C4×D101—28no—
117C4×D101—28no—
118Dic202—24no—
119Dic202—22no—
120Dic202C2×D414yescomplemented
121#11717579—24no—
122C22×C101—21no—
123C22×C101—21no—
124C22×C101—21no—
125C22×C101—21no—
126C22×C101—21no—
127C22×C101—21no—
128C22×C101—21no—
129C22×C101—44no—
130C22×C101—42no—
131C22×C101—42no—
132C22×C101—42no—
133C22×C101—84no—
134C22×C101#12012yes—
135C22×C101AGL(1,ℤ/8)12yescomplemented
136C22×C101C22×D411yescharacteristic
137C2×D101—24no—
138C2×D101—48no—
139C2×D101—42no—
140C2×D101—42no—
141C4×C101—21no—
142C4×C101—21no—
143C4×C101—42no—
144C4×C101#12012yes—
145C4×C101C22×D412yes—
146Dic101—24no—
147C2×C101—22no—
148C2×C101—22no—
149C2×C101—41no—
150C2×C101—82no—
151C2×C101C2×#12011yescharacteristic
152C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C101#57511yescharacteristic
154D101—44no—
155#3006—1011nomaximal, Sylow
156C101#300611yescharacteristic, complemented, Sylow
157C22×#41—1011no—
158C2×AGL(1,ℤ/8)—1012no—
159C2×#121D10111yescharacteristic, complemented
160#449—1014no—
161#448—1012no—
162#447—1012no—
163#417—1012no—
164#407—1011no—
165C2×#41—1012no—
166C2×#41—1012no—
167C2×#41—2028no—
168C2×#41—2021no—
169C2×#41—2021no—
170C22×Q8—1011no—
171C22×D4—21no—
172C22×D4—1011no—
173C22×D4—2021no—
174C22×D4C2×D10111yescharacteristic, complemented
175C23×C4—2021no—
176C23×C4C2×D10111yescharacteristic
177AGL(1,ℤ/8)—2028no—
178C2×SD16—1014no—
179C2×D8—1014no—
180C2×M16—1012no—
181#121—21no—
182#121—21no—
183#121C2×D10112yes—
184#121C2×D10112yescomplemented
185C2×C4⋊C4C2×D10111yescharacteristic, complemented
186C2×C22⋊C4—21no—
187#103—1014no—
188#102—1014no—
189C22⋊C8—1012no—
190C22⋊C8—1012no—
191C24—21no—
192C24—2021no—
193C24C22⋊D10111yescharacteristic
194#41—1018no—
195#41—2024no—
196#41—2024no—
197#41—2028no—
198#41—2028no—
199#41—2024no—
200C2×Q8—1012no—
201C2×Q8—2024no—
202C2×Q8—2021no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—24no—
206C2×D4—21no—
207C2×D4—44no—
208C2×D4—1012no—
209C2×D4—2024no—
210C2×D4—2028no—
211C2×D4—2022no—
212C2×D4—2022no—
213C2×D4—2021no—
214C2×D4C22×D10112yes—
215C22×C4—24no—
216C22×C4—21no—
217C22×C4—2028no—
218C22×C4—2021no—
219C22×C4—2022no—
220C22×C4—2022no—
221C22×C4C22⋊D10111yescharacteristic
222C22×C4C22⋊D10111yescharacteristic
223C22×C4C22⋊D10111yescharacteristic
224C22×C4C22×D10111yescharacteristic
225C22×C4C22×D10111yescharacteristic
226C22×C4C22×D10111yescharacteristic
227SD16—2028no—
228D8—2028no—
229M16—2024no—
230C2×C8—1014no—
231C4⋊C4—21no—
232C4⋊C4C22×D10112yes—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—41no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—4042no—
243C23—4042no—
244C23—2024no—
245C23—2022no—
246C23C2×C22⋊D10111yescharacteristic
247C23C2×C22⋊D10111yescharacteristic
248C23C2×C22⋊D10111yescharacteristic
249Q8—1014no—
250Q8—2024no—
251Q8—2022no—
252D4—24no—
253D4—44no—
254D4—44no—
255D4—42no—
256D4—1014no—
257D4—2024no—
258D4—2028no—
259D4—2028no—
260D4—2022no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—42no—
268C2×C4—2028no—
269C2×C4—2024no—
270C2×C4—2022no—
271C2×C4—2028no—
272C2×C4D4×D10114yes—
273C2×C4C2×C22⋊D10111yescharacteristic
274C2×C4C2×C22⋊D10111yescharacteristic
275C2×C4C2×C22⋊D10111yescharacteristic
276C2×C4C23×D10111yescharacteristic
277C8—2024no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—44no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—84no—
290C22—4048no—
291C22—4042no—
292C22—4042no—
293C22—2024no—
294C22#1569222312yes—
295C22#1569228112yescomplemented
296C22C22×C22⋊D10111yescharacteristic
297C4—21no—
298C4—21no—
299C4—42no—
300C4—2024no—
301C4#1569222312yes—
302C4C2×D4×D10112yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—82no—
307C2—4044no—
308C2C2×#1569222311yescharacteristic
309C2C2×#1569228111yescharacteristic
310C2#2026597511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#121×C101C2
socleC22×C101C22×D4
hypercenterC2×#121D101
hypocenterC101#3006
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#13108074C2semidirect
2. C2×#15692281C2semidirect
3. #20265320C2semidirect
4. #20265315C2semidirect
5. #20265310C2semidirect
6. #20265176C2semidirect
7. #20265141C2semidirect
8. C2×#121×C101C2semidirect
13. C2×#13108074C22semidirect
14. C22×Dic202C22semidirect
15. C22×D404C22semidirect
19. C2×#13108062C22semidirect
20. C2×#13108059C22semidirect
21. C2×#13108058C22semidirect
22. #15692178C22semidirect
23. #15692175C22semidirect
24. C22⋊#11717579C22semidirect
25. C22⋊#11717579C22semidirect
27. C22×D4×C101C22semidirect
32. #121×C101C22semidirect
33. C2×C4⋊C4×C101C22semidirect
41. #13108074D4semidirect
115. D404C2×D4semidirect
120. Dic202C2×D4semidirect
135. C22×C101AGL(1,ℤ/8)semidirect
156. C101#3006semidirect
159. C2×#121D101semidirect
174. C22×D4C2×D101semidirect
184. #121C2×D101semidirect
185. C2×C4⋊C4C2×D101semidirect
295. C22#15692281semidirect

Maximal subgroups

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

subgroupindex
C22×#131080742
C2×#156922812
#202653202
#202653152
#202653102
#202651762
#202651412
C2×#121×C1012
#3006101