Finite Group Databasebeta

← #27181070

Subgroups — #27181070

Order 12928.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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×#15692226C211yesmaximal, characteristic
2#117×D101C211yesmaximal, characteristic, complemented
3#20265500C212yesmaximal, complemented
4#20265499C212yesmaximal, complemented
5#20265494C212yesmaximal, complemented
6#20265489C212yesmaximal, complemented
7#20265220C211yesmaximal, characteristic, complemented
8#20265219C211yesmaximal, characteristic, complemented
9#20265182C211yesmaximal, characteristic, complemented
10#20265140C211yesmaximal, characteristic, complemented
11#418×C101C211yesmaximal, characteristic, complemented
12C22×C22⋊D101—21no—
13C22×C4×D101C2211yescharacteristic
14C2×#13108058C2211yescharacteristic
15#15692226—21no—
16#15692226—21no—
17#15692226C2212yes—
18#15692226C2212yes—
19#15692214C2211yescharacteristic
20#15692213C2211yescharacteristic, complemented
21#15692213C2211yescharacteristic, complemented
22#15692212—21no—
23#15692211—21no—
24C2×C4⋊Dic101C2211yescharacteristic
25C4⋊C4×D101C2211yescharacteristic, complemented
26C4⋊C4×D101C2211yescharacteristic, complemented
27C2×C22⋊Dic101—21no—
28C22⋊C4×D101—21no—
29C42×D101—21no—
30#15692189C2212yescomplemented
31#15692188C2212yescomplemented
32#15692187C2212yescomplemented
33#15692186C2212yescomplemented
34#15692176C2212yescomplemented
35#15692176C2212yescomplemented
36#15692167C2212yescomplemented
37#15692167C2212yescomplemented
38C22×D4×C101C2211yescharacteristic
39C2×M16×C101C2211yescharacteristic
40#117×C101C2211yescharacteristic
41#102×C101C2212yescomplemented
42#102×C101C2212yescomplemented
43C23×D101D411yescharacteristic
44C2×C22⋊D101—22no—
45C2×C22⋊D101—22no—
46C2×C22⋊D101—44no—
47C22×Dic101—21no—
48C22×Dic101D411yescharacteristic
49C2×C4×D101—22no—
50C2×C4×D101—22no—
51C2×C4×D101—21no—
52C2×C4×D101—21no—
53C2×C4×D101D411yescharacteristic
54C2×C4×D101D411yescharacteristic
55C2×C4×D101C2311yescharacteristic
56C2×C4×D101C2311yescharacteristic
57#13108058—22no—
58C2×#11717579C2312yes—
59C4⋊Dic101—21no—
60C4⋊Dic101C2311yescharacteristic
61C4⋊Dic101C2311yescharacteristic
62C4⋊Dic101C2312yes—
63#13108053—21no—
64#13108053—21no—
65C22⋊Dic101—21no—
66C22⋊Dic101—22no—
67C22⋊Dic101—22no—
68#13108051—21no—
69#13108051—21no—
70C4×Dic101—21no—
71C4×Dic101—21no—
72C4×Dic101—21no—
73C24×C101—21no—
74C2×D4×C101—21no—
75C2×D4×C101—22no—
76C2×D4×C101—22no—
77C2×D4×C101C2312yes—
78C22×C4×C101C2311yescharacteristic
79M16×C101—22no—
80C2×C8×C101C2312yes—
81C4⋊C4×C101C2311yescharacteristic
82C4⋊C4×C101C2311yescharacteristic
83C22⋊C4×C101—21no—
84C42×C101—21no—
85C23×C101—22no—
86C23×C101—42no—
87C23×C101—42no—
88C23×C101—42no—
89C23×C101C2×D411yescharacteristic
90C22×D101—22no—
91C22×D101—22no—
92C22×D101—41no—
93C22×D101C2×D411yescharacteristic
94C22×D101C2×D411yescharacteristic
95D4×C101—22no—
96D4×C101—22no—
97D4×C101—42no—
98C2×C4×C101—21no—
99C2×C4×C101—21no—
100C2×C4×C101C2×D411yescharacteristic
101C2×C4×C101#4112yes—
102C2×C4×C101#4112yes—
103C2×C4×C101C2411yescharacteristic
104C22⋊D101—44no—
105C22⋊D101—44no—
106C2×Dic101—22no—
107C2×Dic101—21no—
108C2×Dic101—21no—
109C2×Dic101—22no—
110C2×Dic101—42no—
111C2×Dic101C2×D411yescharacteristic
112C2×Dic101C2×D411yescharacteristic
113C4×D101—22no—
114C4×D101—22no—
115C4×D101—21no—
116C4×D101—21no—
117C4×D101—21no—
118C4×D101—21no—
119C4×D101—42no—
120C4×D101—42no—
121C8×C101—22no—
122#11717579—22no—
123C22×C101—21no—
124C22×C101—21no—
125C22×C101—42no—
126C22×C101—42no—
127C22×C101—42no—
128C22×C101—84no—
129C22×C101#12312yes—
130C22×C101C22×D411yescharacteristic
131C2×D101—21no—
132C2×D101—21no—
133C2×D101—21no—
134C2×D101—21no—
135C2×D101—21no—
136C2×D101—44no—
137C2×D101AGL(1,ℤ/8)12yes—
138C4×C101—41no—
139C4×C101—41no—
140C4×C101#12312yes—
141C4×C101C2×#4111yescharacteristic
142C4×C101C2×#4111yescharacteristic
143Dic101—21no—
144Dic101—21no—
145Dic101—41no—
146Dic101—42no—
147Dic101—41no—
148C2×C101—22no—
149C2×C101—82no—
150C2×C101C2×#12311yescharacteristic
151C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C101#57511yescharacteristic
153D101—22no—
154D101—41no—
155#3094—1011nomaximal, Sylow
156C101#309411yescharacteristic, complemented, Sylow
157C2×#121—1011no—
158C2×#117—1011no—
159#482—1012no—
160#482—1012no—
161#481—1012no—
162#480—1012no—
163#428—1011no—
164#428—1011no—
165#418—1011no—
166#418D10111yescharacteristic, complemented
167#407—1011no—
168C22×D4—2021no—
169C22×D4C2×D10111yescharacteristic
170C23×C4—1011no—
171C2×M16—1011no—
172C2×M16C2×D10111yescharacteristic
173#121—1012no—
174#121—1012no—
175#121—2021no—
176#121—2021no—
177#117—1011no—
178#117—1011no—
179#117—1011no—
180#117—2021no—
181#117—2021no—
182#117C2×D10111yescharacteristic
183C2×C4⋊C4—1011no—
184C2×C4⋊C4—1011no—
185C2×C4⋊C4—1011no—
186C2×C22⋊C4—2021no—
187C2×C22⋊C4—2021no—
188C2×C42—2021no—
189#107—1012no—
190#107—1012no—
191#106—1012no—
192#106—1012no—
193#102—1012no—
194#102—1012no—
195#102C2×D10112yescomplemented
196#102C2×D10112yescomplemented
197C22⋊C8—1012no—
198C22⋊C8—1012no—
199C24—21no—
200C24—1011no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—4044no—
205C2×D4—2022no—
206C2×D4—2022no—
207C2×D4C22×D10112yes—
208C22×C4—1011no—
209C22×C4—1011no—
210C22×C4—1011no—
211C22×C4—1011no—
212C22×C4—1011no—
213C22×C4—2021no—
214C22×C4—2021no—
215C22×C4—2021no—
216C22×C4—2022no—
217C22×C4—2022no—
218C22×C4C22×D10111yescharacteristic
219M16—22no—
220M16—2022no—
221C2×C8—1012no—
222C2×C8C22×D10112yes—
223C4⋊C4—1011no—
224C4⋊C4—1011no—
225C4⋊C4—1012no—
226C4⋊C4—2021no—
227C4⋊C4—2021no—
228C4⋊C4—2021no—
229C4⋊C4C22×D10111yescharacteristic
230C4⋊C4C22×D10111yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—2021no—
233C22⋊C4—2021no—
234C22⋊C4—2022no—
235C22⋊C4—2022no—
236C22⋊C4—2021no—
237C42—21no—
238C42—2021no—
239C42—2021no—
240C42—2021no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—1011no—
246C23—1011no—
247C23—4041no—
248C23—2022no—
249C23—2022no—
250C23D4×D10111yescharacteristic
251D4—22no—
252D4—22no—
253D4—42no—
254D4—4044no—
255D4—4044no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—1011no—
259C2×C4—1011no—
260C2×C4—4042no—
261C2×C4—4042no—
262C2×C4—4042no—
263C2×C4—2021no—
264C2×C4—2021no—
265C2×C4—2022no—
266C2×C4—2022no—
267C2×C4—2021no—
268C2×C4—2021no—
269C2×C4—2021no—
270C2×C4—2021no—
271C2×C4—2022no—
272C2×C4—2022no—
273C2×C4D4×D10111yescharacteristic
274C2×C4#1310807312yes—
275C2×C4#1310807412yes—
276C2×C4C23×D10111yescharacteristic
277C8—22no—
278C8—2022no—
279C22—21no—
280C22—21no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—84no—
285C22—1012no—
286C22—4044no—
287C22—2021no—
288C22—2021no—
289C22—2021no—
290C22—2021no—
291C22—2021no—
292C22#1569223712yes—
293C22C2×D4×D10111yescharacteristic
294C4—41no—
295C4—41no—
296C4—4042no—
297C4—4041no—
298C4—4041no—
299C4—2021no—
300C4—2021no—
301C4#1569223712yes—
302C4C2×#1310807311yescharacteristic
303C4C2×#1310807411yescharacteristic
304C2—22no—
305C2—82no—
306C2—4041no—
307C2—2022no—
308C2C2×#1569223711yescharacteristic
309C2AGL(1,ℤ/8)×D10111yescharacteristic
310C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#418×C101C2
socleC22×C101C22×D4
prime radicalC2×#15692226C2
hypercenter#418D101
hypocenterC101#3094
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #117×D101C2semidirect
3. #20265500C2semidirect
4. #20265499C2semidirect
5. #20265494C2semidirect
6. #20265489C2semidirect
7. #20265220C2semidirect
8. #20265219C2semidirect
9. #20265182C2semidirect
10. #20265140C2semidirect
11. #418×C101C2semidirect
20. #15692213C22semidirect
21. #15692213C22semidirect
25. C4⋊C4×D101C22semidirect
26. C4⋊C4×D101C22semidirect
30. #15692189C22semidirect
31. #15692188C22semidirect
32. #15692187C22semidirect
33. #15692186C22semidirect
34. #15692176C22semidirect
35. #15692176C22semidirect
36. #15692167C22semidirect
37. #15692167C22semidirect
41. #102×C101C22semidirect
42. #102×C101C22semidirect
156. C101#3094semidirect
166. #418D101semidirect
195. #102C2×D101semidirect
196. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922262
#117×D1012
#202655002
#202654992
#202654942
#202654892
#202652202
#202652192
#202651822
#202651402
#418×C1012
#3094101