Finite Group Databasebeta

← #27181090

Subgroups — #27181090

Order 12928.

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

normal subgroups only

This group has 418 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
1#20265741C211yesmaximal, characteristic
2#20265501C212yesmaximal, complemented
3#20265501C212yesmaximal, complemented
4#20265495C212yesmaximal, complemented
5#20265490C212yesmaximal, complemented
6#20265217C211yesmaximal, characteristic, complemented
7#20265213C211yesmaximal, characteristic, complemented
8#20265181C211yesmaximal, characteristic, complemented
9C2×#15692175C211yesmaximal, characteristic, complemented
10#20265144C211yesmaximal, characteristic, complemented
11#520×C101C211yesmaximal, characteristic, complemented
12C22×D404C2211yescharacteristic
13C2×#13108058C2211yescharacteristic
14C22×#11717579C2211yescharacteristic
15#15692251—21no—
16#15692244—21no—
17#15692229C2212yes—
18#15692229C2212yes—
19#15692223—22no—
20#15692214C2211yescharacteristic
21#15692188C2212yescomplemented
22#15692188C2211yescharacteristic, complemented
23#15692188C2211yescharacteristic, complemented
24#15692186C2212yescomplemented
25#15692186C2211yescharacteristic, complemented
26#15692186C2211yescharacteristic, complemented
27#15692175C2212yes—
28#15692175C2212yes—
29#15692175C2212yes—
30#15692175C2212yes—
31C22⋊#11717579C2212yescomplemented
32C22⋊#11717579C2212yescomplemented
33C2×#41×C101C2211yescharacteristic
34C4×Q8×C101—21no—
35C4×D4×C101—21no—
36C4×D4×C101—22no—
37#117×C101C2211yescharacteristic
38#117×C101C2212yescomplemented
39C2×C4⋊C4×C101C2211yescharacteristic
40C2×C4⋊C4×C101C2212yescomplemented
41C23×D101—21no—
42C2×D404—22no—
43C2×D404—22no—
44C2×D404—21no—
45C2×D404C2312yes—
46C2×C22⋊D101—22no—
47C2×C22⋊D101—22no—
48#13108058—22no—
49C2×#11717579—22no—
50C2×#11717579C2311yescharacteristic
51C2×#11717579C2312yes—
52C2×#11717579C2311yescharacteristic
53C4⋊Dic101C2311yescharacteristic
54C4⋊Dic101C2311yescharacteristic
55C22⋊Dic101—21no—
56#13108051—22no—
57#13108051—22no—
58C4×Dic101—21no—
59#41×C101—22no—
60#41×C101—22no—
61C2×Q8×C101D411yescharacteristic
62C2×D4×C101D412yes—
63C2×D4×C101D411yescharacteristic
64C22×C4×C101—21no—
65C22×C4×C101—22no—
66C22×C4×C101C2312yes—
67C22×C4×C101C2311yescharacteristic
68C4⋊C4×C101—21no—
69C4⋊C4×C101—22no—
70C4⋊C4×C101C2312yes—
71C4⋊C4×C101C2311yescharacteristic
72C4⋊C4×C101C2311yescharacteristic
73C22⋊C4×C101—21no—
74C22⋊C4×C101—22no—
75C42×C101—21no—
76C42×C101—22no—
77C23×C101C2×D412yes—
78C23×C101C2×D411yescharacteristic
79C22×D101—22no—
80C22×D101—42no—
81C22×D101—81no—
82C22×D101—81no—
83Q8×C101—22no—
84D4×C101—24no—
85D4×C101—22no—
86C2×C4×C101—22no—
87C2×C4×C101—22no—
88C2×C4×C101—21no—
89C2×C4×C101—21no—
90C2×C4×C101—22no—
91C2×C4×C101—81no—
92C2×C4×C101—82no—
93C2×C4×C101C2×D412yes—
94C2×C4×C101C2×D411yescharacteristic
95C2×C4×C101#4112yes—
96C2×C4×C101#4112yes—
97C2×C4×C101C2411yescharacteristic
98C22⋊D101—82no—
99C22⋊D101—82no—
100C2×Dic101—21no—
101C2×Dic101—21no—
102D404—22no—
103D404—22no—
104D404—42no—
105#11717579—21no—
106#11717579—21no—
107#11717579—22no—
108C22×C101—21no—
109C22×C101—21no—
110C22×C101—22no—
111C22×C101—22no—
112C22×C101—22no—
113C22×C101#12312yes—
114C22×C101C22×D411yescharacteristic
115C2×D101—42no—
116C2×D101—42no—
117C2×D101—42no—
118C2×D101—84no—
119C4×C101—22no—
120C4×C101—41no—
121C4×C101—42no—
122C4×C101—41no—
123C4×C101#12312yes—
124C4×C101C2×#4111yescharacteristic
125C4×C101C2×#4111yescharacteristic
126Dic101—41no—
127Dic101—41no—
128C2×C101—22no—
129C2×C101—42no—
130C2×C101C2×#12311yescharacteristic
131C2×C101#57611yescharacteristic
132C2×C101#57711yescharacteristic
133D101—82no—
134#3096—1011nomaximal, Sylow
135C101#309611yescharacteristic, complemented, Sylow
136#535—1011no—
137#520D10111yescharacteristic, complemented
138#482—1012no—
139#482—1012no—
140#481—1012no—
141#480—1012no—
142#428—1011no—
143#427—1011no—
144#418—1011no—
145C2×#102—1011no—
146#408—1011no—
147C2×#41C2×D10111yescharacteristic
148C22×D4—1011no—
149C2×M16—1011no—
150C22×C8—1011no—
151#127—2021no—
152#124—2021no—
153#121—1012no—
154#121—1012no—
155#120—2022no—
156C4×Q8—21no—
157C4×D4—21no—
158C4×D4—22no—
159#117—1011no—
160#117C2×D10111yescharacteristic
161#117C2×D10112yescomplemented
162C2×C4⋊C4C2×D10111yescharacteristic
163C2×C4⋊C4C2×D10112yescomplemented
164#107—1011no—
165#107—1011no—
166#107—1012no—
167#106—1011no—
168#106—1011no—
169#106—1012no—
170#102—1012no—
171#102—1012no—
172#102—1012no—
173#102—1012no—
174C22⋊C8—1012no—
175C22⋊C8—1012no—
176C24—2021no—
177#41—22no—
178#41—22no—
179C2×Q8C22⋊D10111yescharacteristic
180C2×D4—1012no—
181C2×D4—2022no—
182C2×D4—2022no—
183C2×D4—2021no—
184C2×D4—2022no—
185C2×D4—2022no—
186C2×D4C22⋊D10112yes—
187C2×D4C22⋊D10111yescharacteristic
188C22×C4—21no—
189C22×C4—22no—
190C22×C4C22×D10112yes—
191C22×C4C22×D10111yescharacteristic
192M16—2022no—
193C2×C8—1011no—
194C2×C8—1012no—
195C2×C8—1011no—
196C2×C8—2022no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—1011no—
200C4⋊C4—1011no—
201C4⋊C4C22×D10112yes—
202C4⋊C4C22×D10111yescharacteristic
203C4⋊C4C22×D10111yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—22no—
206C22⋊C4—2021no—
207C22⋊C4—2022no—
208C22⋊C4—2022no—
209C42—21no—
210C42—22no—
211C42—2021no—
212C23—8081no—
213C23—8081no—
214C23—4042no—
215C23—2022no—
216C23C2×C22⋊D10112yes—
217C23C2×C22⋊D10111yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—8082no—
222D4—8082no—
223D4—4042no—
224D4—2022no—
225D4—2022no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—2021no—
234C2×C4—2021no—
235C2×C4C2×C22⋊D10112yes—
236C2×C4C2×C22⋊D10111yescharacteristic
237C2×C4#1310807412yes—
238C2×C4#1310807412yes—
239C2×C4C23×D10111yescharacteristic
240C8—2021no—
241C8—2021no—
242C8—2022no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—8084no—
249C22—4042no—
250C22—4042no—
251C22—4042no—
252C22#1569223812yes—
253C22C22×C22⋊D10111yescharacteristic
254C4—22no—
255C4—41no—
256C4—41no—
257C4—42no—
258C4—4041no—
259C4—4041no—
260C4#1569223812yes—
261C4C2×#1310807411yescharacteristic
262C4C2×#1310807411yescharacteristic
263C2—22no—
264C2—42no—
265C2—8082no—
266C2C2×#1569223811yescharacteristic
267C2#2026598011yescharacteristic
268C2#2026598511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#520×C101C2
socleC22×C101C22×D4
prime radical#20265741C2
hypercenter#520D101
hypocenterC101#3096
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×#41 ◁ #520

Complements

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

NH ≅ G/Ntype
2. #20265501C2semidirect
3. #20265501C2semidirect
4. #20265495C2semidirect
5. #20265490C2semidirect
6. #20265217C2semidirect
7. #20265213C2semidirect
8. #20265181C2semidirect
9. C2×#15692175C2semidirect
10. #20265144C2semidirect
11. #520×C101C2semidirect
21. #15692188C22semidirect
22. #15692188C22semidirect
23. #15692188C22semidirect
24. #15692186C22semidirect
25. #15692186C22semidirect
26. #15692186C22semidirect
31. C22⋊#11717579C22semidirect
32. C22⋊#11717579C22semidirect
38. #117×C101C22semidirect
40. C2×C4⋊C4×C101C22semidirect
135. C101#3096semidirect
137. #520D101semidirect
161. #117C2×D101semidirect
163. C2×C4⋊C4C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657412
#202655012
#202654952
#202654902
#202652172
#202652132
#202651812
C2×#156921752
#202651442
#520×C1012
#3096101