Finite Group Databasebeta

← #27181045

Subgroups — #27181045

Order 12928.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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
1#20265688C211yesmaximal, characteristic
2#20265518C212yesmaximal, complemented
3#20265512C211yesmaximal, characteristic, complemented
4#20265507C211yesmaximal, characteristic, complemented
5#20265501C212yesmaximal, complemented
6#20265495C211yesmaximal, characteristic, complemented
7#20265490C211yesmaximal, characteristic, complemented
8#20265213C212yesmaximal, complemented
9#20265171C212yesmaximal, complemented
10C2×C22⋊#11717579C211yesmaximal, characteristic, complemented
11C2×#117×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C22×#11717579C2212yes—
14#15692238—21no—
15#15692234C2211yescharacteristic
16#15692234C2211yescharacteristic
17#15692229C2211yescharacteristic
18#15692229C2211yescharacteristic
19#15692224—21no—
20C4×C22⋊D101—21no—
21C4×C22⋊D101—21no—
22#15692214C2211yescharacteristic
23#15692188C2212yescomplemented
24#15692188C2212yescomplemented
25#15692186C2212yescomplemented
26#15692186C2212yescomplemented
27#15692178C2212yescomplemented
28#15692178C2212yescomplemented
29#15692175C2212yescomplemented
30#15692175C2212yescomplemented
31C22⋊#11717579C2211yescharacteristic
32C22⋊#11717579C2211yescharacteristic
33C22⋊#11717579C2212yes—
34C23×C4×C101C2211yescharacteristic
35#117×C101—21no—
36#117×C101—21no—
37#117×C101C2212yes—
38#117×C101C2212yes—
39C2×C4⋊C4×C101C2211yescharacteristic
40C2×C4⋊C4×C101C2211yescharacteristic
41C2×C22⋊C4×C101—21no—
42C2×C42×C101—21no—
43#13108074—24no—
44C2×Dic202C2311yescharacteristic
45C2×D404C2311yescharacteristic
46C2×C22⋊D101—21no—
47C2×C22⋊D101—21no—
48C2×C22⋊D101—21no—
49C2×C4×D101—21no—
50C2×#11717579—24no—
51C2×#11717579C2312yes—
52C2×#11717579C2312yes—
53C4⋊Dic101C2311yescharacteristic
54C4⋊Dic101C2311yescharacteristic
55#13108053—21no—
56#13108053—21no—
57C22⋊Dic101—21no—
58C22⋊Dic101—21no—
59C22⋊Dic101—21no—
60#13108051—21no—
61#13108051—21no—
62C4×Dic101—21no—
63C24×C101D411yescharacteristic
64C22×C4×C101—24no—
65C22×C4×C101—21no—
66C22×C4×C101—21no—
67C22×C4×C101D411yescharacteristic
68C22×C4×C101D411yescharacteristic
69C22×C4×C101D411yescharacteristic
70C22×C4×C101C2311yescharacteristic
71C22×C4×C101C2311yescharacteristic
72C22×C4×C101C2311yescharacteristic
73C4⋊C4×C101—21no—
74C4⋊C4×C101—21no—
75C4⋊C4×C101C2312yes—
76C4⋊C4×C101C2312yes—
77C22⋊C4×C101—22no—
78C22⋊C4×C101—22no—
79C42×C101—22no—
80C42×C101—22no—
81C23×C101—22no—
82C23×C101—22no—
83C23×C101—41no—
84C23×C101C2×D411yescharacteristic
85C23×C101C2×D411yescharacteristic
86C23×C101C2×D411yescharacteristic
87C22×D101—21no—
88C2×C4×C101—22no—
89C2×C4×C101—22no—
90C2×C4×C101—22no—
91C2×C4×C101—22no—
92C2×C4×C101—22no—
93C2×C4×C101—22no—
94C2×C4×C101—22no—
95C2×C4×C101—42no—
96C2×C4×C101—42no—
97C2×C4×C101C2×D411yescharacteristic
98C2×C4×C101C2×D411yescharacteristic
99C2×C4×C101C2×D411yescharacteristic
100C2×C4×C101#4114yes—
101C2×C4×C101C2411yescharacteristic
102C22⋊D101—42no—
103C22⋊D101—42no—
104C22⋊D101—42no—
105C2×Dic101—21no—
106C2×Dic101—21no—
107C2×Dic101—21no—
108D404—22no—
109C4×D101—42no—
110Dic202—22no—
111#11717579—22no—
112#11717579—22no—
113C22×C101—21no—
114C22×C101—21no—
115C22×C101—21no—
116C22×C101—21no—
117C22×C101—21no—
118C22×C101—21no—
119C22×C101—21no—
120C22×C101—44no—
121C22×C101#12312yes—
122C22×C101#13512yes—
123C22×C101C22×D411yescharacteristic
124C2×D101—41no—
125C2×D101—41no—
126C2×D101—41no—
127C4×C101—21no—
128C4×C101—21no—
129C4×C101—42no—
130C4×C101—42no—
131C4×C101#12312yes—
132C4×C101C2×#4112yes—
133Dic101—41no—
134Dic101—41no—
135Dic101—41no—
136C2×C101—22no—
137C2×C101—22no—
138C2×C101—41no—
139C2×C101C2×#12311yescharacteristic
140C2×C101C2×#13511yescharacteristic
141C2×C101#57511yescharacteristic
142D101—81no—
143#3091—1011nomaximal, Sylow
144C101#309111yescharacteristic, complemented, Sylow
145#525—1011no—
146C2×#117D10111yescharacteristic, complemented
147#485—1012no—
148#484—1011no—
149#483—1011no—
150#482—1012no—
151#481—1011no—
152#480—1011no—
153#427—1012no—
154#416—1012no—
155C2×C22⋊C8—1011no—
156C2×#41—1011no—
157C23×C4C2×D10111yescharacteristic
158C22×C8—1012no—
159#123—2021no—
160C22⋊Q8—1011no—
161C22⋊Q8—1011no—
162#121—1011no—
163#121—1011no—
164#120—2021no—
165C4×D4—2021no—
166C4×D4—2021no—
167#117—21no—
168#117—21no—
169#117—1011no—
170#117C2×D10112yes—
171#117C2×D10112yes—
172C2×C4⋊C4C2×D10111yescharacteristic
173C2×C4⋊C4C2×D10111yescharacteristic
174C2×C22⋊C4—21no—
175C2×C42—21no—
176#107—1012no—
177#107—1012no—
178#106—1012no—
179#106—1012no—
180#103—1012no—
181#103—1012no—
182#102—1012no—
183#102—1012no—
184C22⋊C8—1011no—
185C22⋊C8—1011no—
186C22⋊C8—1012no—
187C24C22⋊D10111yescharacteristic
188#41—2024no—
189C2×Q8—1011no—
190C2×D4—1011no—
191C2×D4—2021no—
192C2×D4—2021no—
193C2×D4—2021no—
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—2021no—
198C22×C4C22⋊D10111yescharacteristic
199C22×C4C22⋊D10111yescharacteristic
200C22×C4C22⋊D10111yescharacteristic
201C22×C4C22×D10111yescharacteristic
202C22×C4C22×D10111yescharacteristic
203C22×C4C22×D10111yescharacteristic
204C2×C8—1012no—
205C2×C8—1012no—
206C2×C8—2024no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—1011no—
210C4⋊C4—1011no—
211C4⋊C4—2021no—
212C4⋊C4—2021no—
213C4⋊C4C22×D10112yes—
214C4⋊C4C22×D10112yes—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—2021no—
218C22⋊C4—2021no—
219C22⋊C4—2021no—
220C22⋊C4—2021no—
221C22⋊C4—2021no—
222C42—22no—
223C42—22no—
224C42—2021no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—2021no—
229C23C2×C22⋊D10111yescharacteristic
230C23C2×C22⋊D10111yescharacteristic
231C23C2×C22⋊D10111yescharacteristic
232Q8—2022no—
233D4—4042no—
234D4—4042no—
235D4—4042no—
236D4—2022no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—4042no—
247C2×C4—2021no—
248C2×C4—2021no—
249C2×C4—2021no—
250C2×C4C2×C22⋊D10111yescharacteristic
251C2×C4C2×C22⋊D10111yescharacteristic
252C2×C4C2×C22⋊D10111yescharacteristic
253C2×C4#1310807414yes—
254C2×C4C23×D10111yescharacteristic
255C8—2022no—
256C8—2022no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—4041no—
266C22—4041no—
267C22—4041no—
268C22#1569223812yes—
269C22#1569227512yes—
270C22C22×C22⋊D10111yescharacteristic
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—4041no—
276C4—4041no—
277C4—4041no—
278C4#1569223812yes—
279C4C2×#1310807412yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—8081no—
284C2C2×#1569223811yescharacteristic
285C2C2×#1569227511yescharacteristic
286C2#2026597511yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#117×C101C2
socleC22×C101C22×D4
prime radical#20265688C2
hypercenterC2×#117D101
hypocenterC101#3091
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20265518C2semidirect
3. #20265512C2semidirect
4. #20265507C2semidirect
5. #20265501C2semidirect
6. #20265495C2semidirect
7. #20265490C2semidirect
8. #20265213C2semidirect
9. #20265171C2semidirect
10. C2×C22⋊#11717579C2semidirect
11. C2×#117×C101C2semidirect
23. #15692188C22semidirect
24. #15692188C22semidirect
25. #15692186C22semidirect
26. #15692186C22semidirect
27. #15692178C22semidirect
28. #15692178C22semidirect
29. #15692175C22semidirect
30. #15692175C22semidirect
144. C101#3091semidirect
146. C2×#117D101semidirect

Maximal subgroups

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

subgroupindex
#202656882
#202655182
#202655122
#202655072
#202655012
#202654952
#202654902
#202652132
#202651712
C2×C22⋊#117175792
C2×#117×C1012
#3091101