Finite Group Databasebeta

← #27182732

Subgroups — #27182732

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×#15692280C212yesmaximal, complemented
2D4×D404C211yesmaximal, characteristic, complemented
3#20265551C211yesmaximal, characteristic, complemented
4#20265425C212yesmaximal, complemented
5#20265379C211yesmaximal, characteristic, complemented
6#20265372C211yesmaximal, characteristic, complemented
7#20265327C212yesmaximal, complemented
8#20265322C212yesmaximal, complemented
9C4×#13108059C211yesmaximal, characteristic, complemented
10#20265255C211yesmaximal, characteristic, complemented
11#547×C101C211yesmaximal, characteristic, complemented
12C22×D404C2212yes—
13C2×D4×D101—21no—
14#15692280—28no—
15C2×#13108061C2212yes—
16C2×#13108061C2212yescomplemented
17C2×#13108061C2212yescomplemented
18C2×#13108059C2211yescharacteristic
19C2×#13108059C2212yescomplemented
20C2×#13108058C2212yescomplemented
21#15692252C2211yescharacteristic
22#15692230—21no—
23#15692229C2212yes—
24#15692225—22no—
25C4×D404C2211yescharacteristic
26#15692188C2211yescharacteristic, complemented
27C4⋊#11717579C2211yescharacteristic, complemented
28#15692179C2212yescomplemented
29#15692176C2211yescharacteristic, complemented
30#15692175C2211yescharacteristic
31#15692175C2212yescomplemented
32C22⋊#11717579C2212yescomplemented
33C4×#11717579C2211yescharacteristic, complemented
34C2×#41×C101C2212yescomplemented
35C4⋊Q8×C101C2211yescharacteristic, complemented
36#123×C101—22no—
37C22⋊Q8×C101C2212yescomplemented
38#121×C101—21no—
39C4×D4×C101C2211yescharacteristic
40C4×D4×C101C2211yescharacteristic, complemented
41C2×C4⋊C4×C101—21no—
42C23×D101—22no—
43C2×D404—24no—
44C2×D404—21no—
45C2×D404—22no—
46C2×D404—21no—
47C2×D404—42no—
48C2×D404C2312yes—
49C2×D404C2311yescharacteristic
50C2×C22⋊D101—22no—
51D4×D101—44no—
52C2×C4×D101—21no—
53#13108061—24no—
54#13108061—24no—
55#13108061—42no—
56#13108059—24no—
57#13108059D414yescomplemented
58#13108058—24no—
59C2×#11717579C2311yescharacteristic
60C2×#11717579C2312yes—
61C2×#11717579C2311yescharacteristic
62C4⋊Dic101C2311yescharacteristic
63#13108051—21no—
64#13108051—22no—
65#41×C101—24no—
66#41×C101—42no—
67C2×Q8×C101C2312yes—
68C2×D4×C101—22no—
69C2×D4×C101D411yescharacteristic
70C2×D4×C101C2311yescharacteristic
71C22×C4×C101—21no—
72C22×C4×C101—22no—
73C22×C4×C101C2312yes—
74C4⋊C4×C101—21no—
75C4⋊C4×C101—22no—
76C4⋊C4×C101—21no—
77C4⋊C4×C101D411yescharacteristic
78C4⋊C4×C101C2311yescharacteristic
79C22⋊C4×C101—22no—
80C22⋊C4×C101—21no—
81C22⋊C4×C101D412yes—
82C42×C101C2311yescharacteristic
83C23×C101—21no—
84C23×C101C2×D412yes—
85C22×D101—21no—
86C22×D101—22no—
87C22×D101—44no—
88C22×D101—44no—
89C22×D101—82no—
90Q8×C101—22no—
91Q8×C101—22no—
92D4×C101—22no—
93D4×C101—21no—
94D4×C101—44no—
95D4×C101C2×D412yescomplemented
96C2×C4×C101—22no—
97C2×C4×C101—22no—
98C2×C4×C101—21no—
99C2×C4×C101—22no—
100C2×C4×C101—44no—
101C2×C4×C101—42no—
102C2×C4×C101C2×D411yescharacteristic
103C2×C4×C101C2×D412yes—
104C2×C4×C101C2×D411yescharacteristic
105C2×C4×C101C2411yescharacteristic
106C22⋊D101—44no—
107C2×Dic101—21no—
108D404—22no—
109D404—21no—
110D404—22no—
111D404—44no—
112D404—42no—
113D404—81no—
114D404C2×D412yes—
115C4×D101—42no—
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×C101C22×D411yescharacteristic
126C2×D101—22no—
127C2×D101—41no—
128C2×D101—41no—
129C2×D101—42no—
130C2×D101—42no—
131C2×D101—42no—
132C2×D101—88no—
133C4×C101—22no—
134C4×C101—21no—
135C4×C101—42no—
136C4×C101—41no—
137C4×C101AGL(1,ℤ/8)12yescomplemented
138C4×C101C22×D411yescharacteristic
139C4×C101H(5,𝔽2)11yescharacteristic
140Dic101—41no—
141C2×C101—42no—
142C2×C101—42no—
143C2×C101D4211yescharacteristic
144C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C101#57711yescharacteristic
146D101—42no—
147D101—82no—
148#3272—1011nomaximal, Sylow
149C101#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—1012no—
151#547D10111yescharacteristic, complemented
152D42—1011no—
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×#41C2×D10112yescomplemented
162C22×D4—1012no—
163C22×D4—2021no—
164AGL(1,ℤ/8)—2028no—
165C2×SD16—1012no—
166C2×SD16—1012no—
167C2×SD16—1012no—
168C2×D8—1011no—
169C2×D8—1012no—
170C2×M16—1012no—
171C4⋊Q8C2×D10111yescharacteristic, complemented
172#127—1011no—
173#123—22no—
174C22⋊Q8C2×D10112yescomplemented
175#121—21no—
176#121—1012no—
177#121—2021no—
178#120—2022no—
179C4×D4—1011no—
180C4×D4C2×D10111yescharacteristic
181C4×D4C2×D10111yescharacteristic, complemented
182C2×C4⋊C4—21no—
183#107—1011no—
184C4⋊C8—1011no—
185#103—1012no—
186#102—1011no—
187#102—1011no—
188#102—1012no—
189C22⋊C8—1012no—
190C4×C8—1011no—
191C24—2022no—
192#41—24no—
193#41—42no—
194C2×Q8C22×D10112yes—
195C2×D4—22no—
196C2×D4—1011no—
197C2×D4—1012no—
198C2×D4—4042no—
199C2×D4—4044no—
200C2×D4—2024no—
201C2×D4—2022no—
202C2×D4—2021no—
203C2×D4—2022no—
204C2×D4—2021no—
205C2×D4C22⋊D10111yescharacteristic
206C2×D4C22×D10111yescharacteristic
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—2021no—
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—21no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—1011no—
224C4⋊C4C22⋊D10111yescharacteristic
225C4⋊C4C22×D10111yescharacteristic
226C22⋊C4—22no—
227C22⋊C4—21no—
228C22⋊C4—2021no—
229C22⋊C4—2022no—
230C22⋊C4C22⋊D10112yes—
231C42C22×D10111yescharacteristic
232C23—21no—
233C23—8082no—
234C23—4044no—
235C23—4044no—
236C23—2022no—
237C23—2021no—
238C23C2×C22⋊D10112yes—
239Q8—22no—
240Q8—22no—
241D4—21no—
242D4—22no—
243D4—44no—
244D4—8081no—
245D4—1012no—
246D4—4044no—
247D4—4044no—
248D4—4042no—
249D4—2022no—
250D4—2022no—
251D4—2021no—
252D4D4×D10112yescomplemented
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—4042no—
260C2×C4—2021no—
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—8088no—
275C22—4042no—
276C22—4042no—
277C22—4042no—
278C22—4041no—
279C22—4041no—
280C22—2022no—
281C22C22×C22⋊D10111yescharacteristic
282C4—21no—
283C4—22no—
284C4—42no—
285C4—41no—
286C4—4041no—
287C4#1569228012yescomplemented
288C4C2×D4×D10111yescharacteristic
289C4#1569230011yescharacteristic
290C2—42no—
291C2—42no—
292C2—8082no—
293C2—4042no—
294C2D4×C22⋊D10111yescharacteristic
295C2C2×#1569228011yescharacteristic
296C2#2026598511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#547×C101C2
socleC22×C101C22×D4
hypercenter#547D101
hypocenterC101#3272
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#15692280C2semidirect
2. D4×D404C2semidirect
3. #20265551C2semidirect
4. #20265425C2semidirect
5. #20265379C2semidirect
6. #20265372C2semidirect
7. #20265327C2semidirect
8. #20265322C2semidirect
9. C4×#13108059C2semidirect
10. #20265255C2semidirect
11. #547×C101C2semidirect
16. C2×#13108061C22semidirect
17. C2×#13108061C22semidirect
19. C2×#13108059C22semidirect
20. C2×#13108058C22semidirect
26. #15692188C22semidirect
27. C4⋊#11717579C22semidirect
28. #15692179C22semidirect
29. #15692176C22semidirect
31. #15692175C22semidirect
32. C22⋊#11717579C22semidirect
33. C4×#11717579C22semidirect
34. C2×#41×C101C22semidirect
35. C4⋊Q8×C101C22semidirect
37. C22⋊Q8×C101C22semidirect
40. C4×D4×C101C22semidirect
57. #13108059D4semidirect
95. D4×C101C2×D4semidirect
118. #11717579C2×D4semidirect
137. C4×C101AGL(1,ℤ/8)semidirect
149. C101#3272semidirect
151. #547D101semidirect
161. C2×#41C2×D101semidirect
171. C4⋊Q8C2×D101semidirect
174. C22⋊Q8C2×D101semidirect
181. C4×D4C2×D101semidirect
252. D4D4×D101semidirect
287. C4#15692280semidirect

Maximal subgroups

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

subgroupindex
C2×#156922802
D4×D4042
#202655512
#202654252
#202653792
#202653722
#202653272
#202653222
C4×#131080592
#202652552
#547×C1012
#3272101