Finite Group Databasebeta

← #27181093

Subgroups — #27181093

Order 12928.

418 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 125 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#20265746C211yesmaximal, characteristic
2#20265657C211yesmaximal, characteristic, complemented
3#20265504C212yesmaximal, complemented
4#20265503C212yesmaximal, complemented
5#20265497C212yesmaximal, complemented
6#20265492C212yesmaximal, complemented
7#20265220C211yesmaximal, characteristic, complemented
8#20265181C211yesmaximal, characteristic, complemented
9C2×#15692177C211yesmaximal, characteristic, complemented
10#20265143C211yesmaximal, characteristic, complemented
11#427×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C22×D404C2211yescharacteristic
14C2×#13108058C2211yescharacteristic
15#15692252—21no—
16#15692243—21no—
17#15692230C2212yes—
18#15692230C2212yes—
19#15692225—22no—
20C4×Dic202—21no—
21C4×D404—21no—
22#15692215—22no—
23#15692213C2212yescomplemented
24C2×C4⋊Dic101C2211yescharacteristic
25C4⋊C4×D101C2212yescomplemented
26#15692188C2212yescomplemented
27#15692186C2212yescomplemented
28#15692177C2212yes—
29#15692177C2212yes—
30#15692175C2212yes—
31#15692175C2212yes—
32#15692167C2212yescomplemented
33#15692167C2212yescomplemented
34C22×C8×C101C2211yescharacteristic
35#117×C101C2211yescharacteristic
36#117×C101C2211yescharacteristic
37#107×C101C2211yescharacteristic, complemented
38#107×C101C2211yescharacteristic, complemented
39#106×C101C2211yescharacteristic, complemented
40#106×C101C2211yescharacteristic, complemented
41C23×D101—21no—
42#13108074—22no—
43#13108074—22no—
44C2×Dic202D411yescharacteristic
45C2×D404—22no—
46C2×D404—22no—
47C2×D404—21no—
48C2×D404—22no—
49C2×D404—22no—
50C2×D404D411yescharacteristic
51C2×D404C2312yes—
52C2×C22⋊D101D412yes—
53C22×Dic101—21no—
54C2×C4×D101—22no—
55C2×C4×D101C2312yes—
56#13108058—22no—
57C2×#11717579C2312yes—
58C4⋊Dic101—21no—
59C4⋊Dic101C2312yes—
60#13108053—22no—
61#13108051—22no—
62#13108051—22no—
63#13108051—22no—
64C4×Dic101—22no—
65C22×C4×C101C2311yescharacteristic
66C2×C8×C101—22no—
67C2×C8×C101C2311yescharacteristic
68C2×C8×C101C2311yescharacteristic
69C4⋊C4×C101C2311yescharacteristic
70C4⋊C4×C101C2311yescharacteristic
71C4⋊C4×C101C2311yescharacteristic
72C4⋊C4×C101C2311yescharacteristic
73C22⋊C4×C101—21no—
74C22⋊C4×C101—21no—
75C42×C101—21no—
76C42×C101—21no—
77C23×C101C2×D411yescharacteristic
78C22×D101—22no—
79C22×D101—42no—
80C22×D101—81no—
81C22×D101—81no—
82C22×D101C2×D412yes—
83C2×C4×C101—21no—
84C2×C4×C101—21no—
85C2×C4×C101—21no—
86C2×C4×C101—21no—
87C2×C4×C101C2×D411yescharacteristic
88C2×C4×C101#4112yes—
89C2×C4×C101#4112yes—
90C2×C4×C101C2411yescharacteristic
91C22⋊D101—24no—
92C2×Dic101—22no—
93C2×Dic101—81no—
94C2×Dic101C2×D412yes—
95D404—22no—
96D404—22no—
97D404—22no—
98D404—42no—
99D404—82no—
100D404—82no—
101C4×D101—22no—
102C4×D101—22no—
103C4×D101—82no—
104Dic202—22no—
105C8×C101—21no—
106C8×C101—21no—
107#11717579—22no—
108C22×C101—21no—
109C22×C101—21no—
110C22×C101#12312yes—
111C22×C101C22×D411yescharacteristic
112C2×D101—22no—
113C2×D101—22no—
114C2×D101—22no—
115C2×D101—42no—
116C2×D101—42no—
117C2×D101—42no—
118C2×D101—84no—
119C4×C101—41no—
120C4×C101—41no—
121C4×C101—41no—
122C4×C101—41no—
123C4×C101#12312yes—
124C4×C101C2×#4111yescharacteristic
125C4×C101C2×#4111yescharacteristic
126Dic101—22no—
127Dic101—42no—
128C2×C101—22no—
129C2×C101C2×#12311yescharacteristic
130C2×C101#57611yescharacteristic
131C2×C101#57711yescharacteristic
132D101—42no—
133D101—82no—
134#3096—1011nomaximal, Sylow
135C101#309611yescharacteristic, complemented, Sylow
136#535—1011no—
137#520—1011no—
138#482—1012no—
139#482—1012no—
140#481—1012no—
141#480—1012no—
142#428—1011no—
143#427D10111yescharacteristic, complemented
144#418—1011no—
145C2×#102—1011no—
146#408—1011no—
147C2×#41—1011no—
148C22×D4—1011no—
149C2×M16—1011no—
150C22×C8C2×D10111yescharacteristic
151#127—2021no—
152#124—2021no—
153#121—1012no—
154#121—1012no—
155#120—2022no—
156C4×Q8—2021no—
157C4×D4—2021no—
158C4×D4—2022no—
159#117—1012no—
160#117C2×D10111yescharacteristic
161#117C2×D10111yescharacteristic
162C2×C4⋊C4—1011no—
163C2×C4⋊C4—1012no—
164#107—1012no—
165#107C2×D10111yescharacteristic, complemented
166#107C2×D10111yescharacteristic, complemented
167#106—1012no—
168#106C2×D10111yescharacteristic, complemented
169#106C2×D10111yescharacteristic, complemented
170#102—1012no—
171#102—1012no—
172#102—1012no—
173#102—1012no—
174C22⋊C8—1012no—
175C22⋊C8—1012no—
176C24—2021no—
177#41—2022no—
178#41—2022no—
179C2×Q8—1011no—
180C2×D4—1012no—
181C2×D4—1011no—
182C2×D4—1012no—
183C2×D4—2022no—
184C2×D4—2022no—
185C2×D4—2021no—
186C2×D4—2022no—
187C2×D4—2022no—
188C22×C4—1012no—
189C22×C4—2022no—
190C22×C4—2021no—
191C22×C4C22×D10111yescharacteristic
192M16—2022no—
193C2×C8—22no—
194C2×C8—1012no—
195C2×C8C22×D10111yescharacteristic
196C2×C8C22×D10111yescharacteristic
197C4⋊C4—1012no—
198C4⋊C4—2022no—
199C4⋊C4—2021no—
200C4⋊C4C22×D10111yescharacteristic
201C4⋊C4C22×D10111yescharacteristic
202C4⋊C4C22×D10111yescharacteristic
203C4⋊C4C22×D10111yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—2022no—
207C22⋊C4—2022no—
208C22⋊C4—2022no—
209C42—21no—
210C42—21no—
211C42—2022no—
212C23—8081no—
213C23—8081no—
214C23—1012no—
215C23—4042no—
216C23—2022no—
217C23D4×D10111yescharacteristic
218Q8—2022no—
219D4—8082no—
220D4—8082no—
221D4—4042no—
222D4—2024no—
223D4—2022no—
224D4—2022no—
225D4—2022no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—8082no—
231C2×C4—8081no—
232C2×C4—1012no—
233C2×C4—2022no—
234C2×C4—2022no—
235C2×C4—2022no—
236C2×C4D4×D10111yescharacteristic
237C2×C4#1310807412yes—
238C2×C4#1310807512yes—
239C2×C4C23×D10111yescharacteristic
240C8—21no—
241C8—21no—
242C8—2022no—
243C22—21no—
244C22—21no—
245C22—8084no—
246C22—4042no—
247C22—4042no—
248C22—4042no—
249C22—2022no—
250C22—2022no—
251C22—2022no—
252C22#1569224012yes—
253C22C2×D4×D10111yescharacteristic
254C4—41no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—4042no—
259C4—2022no—
260C4#1569224012yes—
261C4C2×#1310807411yescharacteristic
262C4C2×#1310807511yescharacteristic
263C2—22no—
264C2—8082no—
265C2—4042no—
266C2C2×#1569224011yescharacteristic
267C2#2026597911yescharacteristic
268C2#2026598411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#427×C101C2
socleC22×C101C22×D4
prime radical#20265746C2
hypercenter#427D101
hypocenterC101#3096
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
2. #20265657C2semidirect
3. #20265504C2semidirect
4. #20265503C2semidirect
5. #20265497C2semidirect
6. #20265492C2semidirect
7. #20265220C2semidirect
8. #20265181C2semidirect
9. C2×#15692177C2semidirect
10. #20265143C2semidirect
11. #427×C101C2semidirect
23. #15692213C22semidirect
25. C4⋊C4×D101C22semidirect
26. #15692188C22semidirect
27. #15692186C22semidirect
32. #15692167C22semidirect
33. #15692167C22semidirect
37. #107×C101C22semidirect
38. #107×C101C22semidirect
39. #106×C101C22semidirect
40. #106×C101C22semidirect
135. C101#3096semidirect
143. #427D101semidirect
165. #107C2×D101semidirect
166. #107C2×D101semidirect
168. #106C2×D101semidirect
169. #106C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657462
#202656572
#202655042
#202655032
#202654972
#202654922
#202652202
#202651812
C2×#156921772
#202651432
#427×C1012
#3096101