Finite Group Databasebeta

← #27182728

Subgroups — #27182728

Order 12928.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15692281C212yesmaximal, complemented
2#20265832C211yesmaximal, characteristic, complemented
3#20265549C211yesmaximal, characteristic, complemented
4#20265421C212yesmaximal, complemented
5#20265382C211yesmaximal, characteristic, complemented
6#20265369C211yesmaximal, characteristic, complemented
7#20265328C212yesmaximal, complemented
8#20265320C212yesmaximal, complemented
9C4×#13108059C211yesmaximal, characteristic, complemented
10#20265255C211yesmaximal, characteristic, complemented
11D42×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2212yescomplemented
13#15692281—28no—
14C2×#13108062C2212yes—
15C2×#13108062C2212yescomplemented
16C2×#13108062C2212yescomplemented
17C2×#13108059C2211yescharacteristic
18C2×#13108059C2212yescomplemented
19C2×#13108058C2212yescomplemented
20#15692255C2211yescharacteristic, complemented
21#15692237—22no—
22#15692234C2212yescomplemented
23#15692226—21no—
24C4×D404C2211yescharacteristic, complemented
25C4⋊C4×D101—21no—
26#15692188C2211yescharacteristic, complemented
27C4⋊#11717579C2211yescharacteristic, complemented
28#15692178C2212yescomplemented
29#15692176C2211yescharacteristic
30#15692176C2212yescomplemented
31#15692175C2211yescharacteristic, complemented
32C22⋊#11717579C2212yescomplemented
33C4×#11717579C2211yescharacteristic, complemented
34C22×D4×C101—21no—
35C22×D4×C101C2212yes—
36#127×C101C2211yescharacteristic
37#121×C101—21no—
38#121×C101C2212yes—
39#120×C101—22no—
40C4×D4×C101C2211yescharacteristic
41C4×D4×C101C2211yescharacteristic
42#13108074—24no—
43#13108074—42no—
44C2×Dic202C2312yes—
45C2×D404C2311yescharacteristic
46C2×C22⋊D101—22no—
47C2×C4×D101—22no—
48C2×C4×D101—21no—
49#13108062—24no—
50#13108062—24no—
51#13108062—42no—
52#13108059—24no—
53#13108059D414yescomplemented
54#13108058—24no—
55C2×#11717579C2311yescharacteristic
56C2×#11717579C2312yes—
57C2×#11717579C2311yescharacteristic
58C4⋊Dic101—21no—
59C4⋊Dic101—21no—
60C4⋊Dic101C2311yescharacteristic
61#13108053—22no—
62C22⋊Dic101—22no—
63#13108051—21no—
64C24×C101—22no—
65C2×D4×C101—24no—
66C2×D4×C101—22no—
67C2×D4×C101—22no—
68C2×D4×C101—21no—
69C2×D4×C101—21no—
70C2×D4×C101—42no—
71C2×D4×C101—44no—
72C2×D4×C101D411yescharacteristic
73C2×D4×C101C2312yes—
74C2×D4×C101C2311yescharacteristic
75C22×C4×C101—21no—
76C22×C4×C101C2312yes—
77C4⋊C4×C101D411yescharacteristic
78C4⋊C4×C101C2311yescharacteristic
79C22⋊C4×C101—21no—
80C22⋊C4×C101—22no—
81C22⋊C4×C101D412yes—
82C42×C101C2311yescharacteristic
83C23×C101—21no—
84C23×C101—22no—
85C23×C101—44no—
86C23×C101—44no—
87C23×C101—82no—
88C23×C101C2×D412yes—
89C22×D101—21no—
90D4×C101—22no—
91D4×C101—22no—
92D4×C101—21no—
93D4×C101—22no—
94D4×C101—44no—
95D4×C101—44no—
96D4×C101—42no—
97D4×C101—81no—
98D4×C101C2×D412yes—
99C2×C4×C101—22no—
100C2×C4×C101—22no—
101C2×C4×C101—21no—
102C2×C4×C101—42no—
103C2×C4×C101C2×D411yescharacteristic
104C2×C4×C101C2×D412yes—
105C2×C4×C101C2×D411yescharacteristic
106C2×C4×C101C2411yescharacteristic
107C22⋊D101—44no—
108C2×Dic101—21no—
109C2×Dic101—22no—
110D404—21no—
111D404C2×D412yescomplemented
112C4×D101—44no—
113C4×D101—42no—
114Dic202—22no—
115Dic202—22no—
116#11717579—22no—
117#11717579—21no—
118#11717579C2×D412yescomplemented
119C22×C101—22no—
120C22×C101—22no—
121C22×C101—22no—
122C22×C101—22no—
123C22×C101—41no—
124C22×C101—41no—
125C22×C101—42no—
126C22×C101—42no—
127C22×C101—42no—
128C22×C101—88no—
129C22×C101C22×D411yescharacteristic
130C2×D101—22no—
131C2×D101—41no—
132C2×D101—41no—
133C4×C101—22no—
134C4×C101—21no—
135C4×C101—41no—
136C4×C101AGL(1,ℤ/8)12yescomplemented
137C4×C101C22×D411yescharacteristic
138C4×C101H(5,𝔽2)11yescharacteristic
139Dic101—42no—
140Dic101—41no—
141C2×C101—42no—
142C2×C101—42no—
143C2×C101—82no—
144C2×C101D4211yescharacteristic
145C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C101#57711yescharacteristic
147D101—42no—
148#3272—1011nomaximal, Sylow
149C101#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—1012no—
151#547—1011no—
152D42D10111yescharacteristic, complemented
153#492—1011no—
154#468—1012no—
155#461—1011no—
156#459—1011no—
157#450—1012no—
158#449—1012no—
159C4×D8—1011no—
160#436—1011no—
161C2×#41—1012no—
162C22×D4—21no—
163C22×D4C2×D10112yes—
164AGL(1,ℤ/8)—2028no—
165C2×SD16—1012no—
166C2×SD16—1012no—
167C2×SD16—1012no—
168C2×D8—1011no—
169C2×D8—1012no—
170C2×M16—1012no—
171C4⋊Q8—1011no—
172#127C2×D10111yescharacteristic
173#123—2022no—
174C22⋊Q8—1012no—
175#121—21no—
176#121—2021no—
177#121C2×D10112yes—
178#120—22no—
179C4×D4—1011no—
180C4×D4C2×D10111yescharacteristic
181C4×D4C2×D10111yescharacteristic
182C2×C4⋊C4—2021no—
183#107—1011no—
184C4⋊C8—1011no—
185#103—1012no—
186#102—1011no—
187#102—1012no—
188#102—1011no—
189C22⋊C8—1012no—
190C4×C8—1011no—
191C24—22no—
192#41—4042no—
193#41—2024no—
194C2×Q8—1012no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—1011no—
203C2×D4—2022no—
204C2×D4C22⋊D10111yescharacteristic
205C2×D4C22×D10112yes—
206C2×D4C22×D10111yescharacteristic
207C22×C4—21no—
208C22×C4—2021no—
209C22×C4—2022no—
210C22×C4C22×D10112yes—
211SD16—4042no—
212SD16—2024no—
213SD16—2024no—
214D8—1014no—
215D8—2024no—
216M16—2024no—
217C2×C8—1011no—
218C2×C8—1011no—
219C2×C8—1012no—
220C4⋊C4—1011no—
221C4⋊C4—2022no—
222C4⋊C4—2021no—
223C4⋊C4—2021no—
224C4⋊C4C22⋊D10111yescharacteristic
225C4⋊C4C22×D10111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—22no—
228C22⋊C4—2022no—
229C22⋊C4—2021no—
230C22⋊C4C22⋊D10112yes—
231C42C22×D10111yescharacteristic
232C23—21no—
233C23—22no—
234C23—44no—
235C23—44no—
236C23—82no—
237C23—2021no—
238C23C2×C22⋊D10112yes—
239Q8—2022no—
240Q8—2022no—
241D4—22no—
242D4—21no—
243D4—22no—
244D4—22no—
245D4—44no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—1012no—
250D4—4044no—
251D4—2021no—
252D4D4×D10112yes—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—4044no—
258C2×C4—4042no—
259C2×C4—2021no—
260C2×C4—2022no—
261C2×C4C2×C22⋊D10111yescharacteristic
262C2×C4C2×C22⋊D10112yes—
263C2×C4C2×C22⋊D10111yescharacteristic
264C2×C4C23×D10111yescharacteristic
265C8—1012no—
266C8—2022no—
267C8—2021no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—88no—
278C22—4041no—
279C22—4041no—
280C22—2022no—
281C22C22×C22⋊D10111yescharacteristic
282C4—21no—
283C4—22no—
284C4—41no—
285C4—4042no—
286C4—4041no—
287C4#1569228112yescomplemented
288C4C2×D4×D10111yescharacteristic
289C4#1569230011yescharacteristic
290C2—42no—
291C2—42no—
292C2—82no—
293C2—4042no—
294C2D4×C22⋊D10111yescharacteristic
295C2C2×#1569228111yescharacteristic
296C2#2026598811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingD42×C101C2
socleC22×C101C22×D4
hypercenterD42D101
hypocenterC101#3272
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#15692281C2semidirect
2. #20265832C2semidirect
3. #20265549C2semidirect
4. #20265421C2semidirect
5. #20265382C2semidirect
6. #20265369C2semidirect
7. #20265328C2semidirect
8. #20265320C2semidirect
9. C4×#13108059C2semidirect
10. #20265255C2semidirect
11. D42×C101C2semidirect
12. C2×#13108074C22semidirect
15. C2×#13108062C22semidirect
16. C2×#13108062C22semidirect
18. C2×#13108059C22semidirect
19. C2×#13108058C22semidirect
20. #15692255C22semidirect
22. #15692234C22semidirect
24. C4×D404C22semidirect
26. #15692188C22semidirect
27. C4⋊#11717579C22semidirect
28. #15692178C22semidirect
30. #15692176C22semidirect
31. #15692175C22semidirect
32. C22⋊#11717579C22semidirect
33. C4×#11717579C22semidirect
53. #13108059D4semidirect
111. D404C2×D4semidirect
118. #11717579C2×D4semidirect
136. C4×C101AGL(1,ℤ/8)semidirect
149. C101#3272semidirect
152. D42D101semidirect
287. C4#15692281semidirect

Maximal subgroups

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

subgroupindex
C2×#156922812
#202658322
#202655492
#202654212
#202653822
#202653692
#202653282
#202653202
C4×#131080592
#202652552
D42×C1012
#3272101