Finite Group Databasebeta

← #27181529

Subgroups — #27181529

Order 12928.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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#20265726C211yesmaximal, characteristic
2#125×D101C211yesmaximal, characteristic, complemented
3#20265544C212yesmaximal, complemented
4#20265540C212yesmaximal, complemented
5#20265534C211yesmaximal, characteristic, complemented
6#20265532C211yesmaximal, characteristic, complemented
7#20265531C211yesmaximal, characteristic, complemented
8#20265177C212yesmaximal, complemented
9#20265175C212yesmaximal, complemented
10C4×#13108057C211yesmaximal, characteristic, complemented
11#488×C101C211yesmaximal, characteristic, complemented
12C2×#13108075C2211yescharacteristic
13C2×#13108073C2211yescharacteristic
14C2×#13108057C2212yes—
15#15692253C2211yescharacteristic
16#15692251C2211yescharacteristic
17#15692247C2211yescharacteristic
18#15692246C2212yes—
19#15692245—22no—
20#15692243C2211yescharacteristic
21#15692228—22no—
22#15692215—22no—
23C4⋊C4×D101—22no—
24C4⋊C4×D101C2212yes—
25C42×D101C2211yescharacteristic
26#15692180C2212yes—
27#15692179C2212yes—
28#15692178C2212yes—
29#15692177C2212yes—
30#15692176C2212yes—
31#15692175C2212yes—
32#15692165C2212yes—
33C4×#11717579C2211yescharacteristic
34#125×C101C2211yescharacteristic
35#124×C101C2211yescharacteristic
36#103×C101C2212yes—
37#102×C101C2212yes—
38C4×C8×C101C2211yescharacteristic
39#13108075—24no—
40#13108073—24no—
41C2×Dic202C2311yescharacteristic
42C2×D404—21no—
43C2×D404C2311yescharacteristic
44C2×C22⋊D101—21no—
45C2×C22⋊D101—22no—
46C22×Dic101—21no—
47C2×C4×D101—22no—
48C2×C4×D101—21no—
49C2×C4×D101D412yes—
50C2×C4×D101C2311yescharacteristic
51#13108057—24no—
52C2×#11717579C2312yes—
53C4⋊Dic101—22no—
54C4⋊Dic101C2312yes—
55#13108053—22no—
56#13108053—22no—
57#13108053—22no—
58#13108053—22no—
59#13108051—22no—
60C4×Dic101D412yes—
61C4×Dic101C2311yescharacteristic
62C2×Q8×C101C2311yescharacteristic
63C2×D4×C101C2311yescharacteristic
64C2×C8×C101C2312yes—
65C4⋊C4×C101—22no—
66C4⋊C4×C101C2312yes—
67C22⋊C4×C101—22no—
68C42×C101C2311yescharacteristic
69C23×C101—21no—
70C22×D101—21no—
71C22×D101C2×D411yescharacteristic
72Q8×C101—22no—
73D4×C101—22no—
74C2×C4×C101—21no—
75C2×C4×C101—22no—
76C2×C4×C101C2×D412yes—
77C2×C4×C101C2411yescharacteristic
78C22⋊D101—44no—
79C22⋊D101—42no—
80C2×Dic101—21no—
81C2×Dic101—22no—
82C2×Dic101—42no—
83C2×Dic101C2×D412yes—
84C2×Dic101C2×D411yescharacteristic
85D404—22no—
86D404—42no—
87C4×D101—42no—
88C4×D101—44no—
89C4×D101—42no—
90C4×D101#4114yes—
91Dic202—22no—
92C8×C101—22no—
93#11717579—22no—
94C22×C101—41no—
95C22×C101—42no—
96C22×C101C22×D411yescharacteristic
97C2×D101—22no—
98C2×D101—41no—
99C2×D101—42no—
100C2×D101#12412yes—
101C4×C101—22no—
102C4×C101—41no—
103C4×C101—42no—
104C4×C101C2×#4112yes—
105Dic101—22no—
106Dic101—41no—
107Dic101—42no—
108Dic101#12412yes—
109C2×C101—81no—
110C2×C101C2×#12411yescharacteristic
111C2×C101#57512yes—
112D101—22no—
113D101—81no—
114#3142—1011nomaximal, Sylow
115C101#314211yescharacteristic, complemented, Sylow
116#532—1011no—
117C2×#125—1011no—
118#491—1012no—
119#490—1012no—
120#488—1011no—
121#488—1011no—
122#488—1011no—
123#488D10111yescharacteristic, complemented
124#417—1012no—
125#417—1012no—
126C4×M16—1011no—
127C2×#41—1011no—
128C2×#41—1011no—
129C2×M16—1012no—
130C4⋊Q8—1011no—
131#127—1011no—
132#125—1012no—
133#125—1011no—
134#125—2022no—
135#125C2×D10111yescharacteristic
136#124—1011no—
137#124C2×D10111yescharacteristic
138#121—2022no—
139C4×D4—2022no—
140C2×C4⋊C4—1012no—
141C2×C4⋊C4—2022no—
142C2×C42—1011no—
143#103—1012no—
144#103—1012no—
145#103—1012no—
146#103C2×D10112yes—
147#102—1012no—
148#102—1012no—
149#102—1012no—
150#102C2×D10112yes—
151#97—1012no—
152C4×C8—1011no—
153C4×C8C2×D10111yescharacteristic
154#41—2024no—
155#41—2024no—
156C2×Q8—1011no—
157C2×Q8C22×D10111yescharacteristic
158C2×D4—1011no—
159C2×D4—2021no—
160C2×D4—2022no—
161C2×D4—2021no—
162C2×D4C22×D10111yescharacteristic
163C22×C4—1011no—
164C22×C4—1012no—
165C22×C4—2021no—
166C22×C4—2022no—
167C22×C4—2021no—
168M16—2024no—
169C2×C8—1012no—
170C2×C8C22×D10112yes—
171C4⋊C4—22no—
172C4⋊C4—1012no—
173C4⋊C4—2022no—
174C4⋊C4—2022no—
175C4⋊C4—2022no—
176C4⋊C4—2022no—
177C4⋊C4—2022no—
178C4⋊C4C22×D10112yes—
179C22⋊C4—22no—
180C22⋊C4—2022no—
181C42—1012no—
182C42—1011no—
183C42C22×D10111yescharacteristic
184C23—21no—
185C23—1011no—
186C23—2021no—
187Q8—22no—
188Q8—2022no—
189D4—22no—
190D4—4042no—
191D4—4042no—
192D4—4044no—
193D4—2022no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—1011no—
197C2×C4—1014no—
198C2×C4—1012no—
199C2×C4—4042no—
200C2×C4—4042no—
201C2×C4—4044no—
202C2×C4—4042no—
203C2×C4—2021no—
204C2×C4—2022no—
205C2×C4D4×D10112yes—
206C2×C4C23×D10111yescharacteristic
207C8—22no—
208C8—2022no—
209C22—41no—
210C22—42no—
211C22—1012no—
212C22—4041no—
213C22—4042no—
214C22—2022no—
215C22C2×D4×D10111yescharacteristic
216C4—22no—
217C4—41no—
218C4—42no—
219C4—1012no—
220C4—4041no—
221C4—4042no—
222C4—2022no—
223C4#41×D10112yes—
224C2—81no—
225C2—8081no—
226C2—2022no—
227C2#124×D10111yescharacteristic
228C2#2026597412yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#488×C101C2
socleC22×C101C22×D4
prime radical#20265726C2
hypercenter#488D101
hypocenterC101#3142
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
2. #125×D101C2semidirect
3. #20265544C2semidirect
4. #20265540C2semidirect
5. #20265534C2semidirect
6. #20265532C2semidirect
7. #20265531C2semidirect
8. #20265177C2semidirect
9. #20265175C2semidirect
10. C4×#13108057C2semidirect
11. #488×C101C2semidirect
115. C101#3142semidirect
123. #488D101semidirect

Maximal subgroups

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

subgroupindex
#202657262
#125×D1012
#202655442
#202655402
#202655342
#202655322
#202655312
#202651772
#202651752
C4×#131080572
#488×C1012
#3142101