Finite Group Databasebeta

← #27181504

Subgroups — #27181504

Order 12928.

484 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 129 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#20265730C211yesmaximal, characteristic, complemented
2C2×#15692251C211yesmaximal, characteristic
3#20265540C214yesmaximal, complemented
4#20265522C214yesmaximal, complemented
5#20265183C211yesmaximal, characteristic, complemented
6#20265182C211yesmaximal, characteristic, complemented
7#20265181C211yesmaximal, characteristic, complemented
8M16×Dic101C211yesmaximal, characteristic, complemented
9#418×C101C211yesmaximal, characteristic, complemented
10C22×D404C2211yescharacteristic
11C22×C22⋊D101—22no—
12C2×#13108058C2211yescharacteristic
13#15692254C2212yescomplemented
14#15692251—22no—
15#15692251C2212yes—
16#15692251C2212yes—
17#15692246C2212yescomplemented
18#15692232—22no—
19C4×Dic202—22no—
20#15692214C2211yescharacteristic
21C2×C4×Dic101C2211yescharacteristic
22#15692177C2214yescomplemented
23#15692176C2214yescomplemented
24#15692175C2214yescomplemented
25#15692165C2212yescomplemented
26C8×Dic101C2212yescomplemented
27C22×D4×C101C2211yescharacteristic
28C2×M16×C101C2211yescharacteristic
29#117×C101C2211yescharacteristic
30#102×C101C2214yescomplemented
31C23×D101—21no—
32C2×Dic202—22no—
33C2×D404—22no—
34C2×D404—22no—
35C2×D404—21no—
36C2×D404C2312yes—
37C2×C22⋊D101—24no—
38C2×C22⋊D101—24no—
39C2×C22⋊D101—48no—
40C22×Dic101D412yes—
41#13108058—22no—
42C2×#11717579C2312yes—
43C4⋊Dic101—22no—
44C4⋊Dic101C2312yes—
45#13108053—22no—
46#13108053—22no—
47C22⋊Dic101—21no—
48C4×Dic101—21no—
49C4×Dic101D412yes—
50C4×Dic101C2311yescharacteristic
51C4×Dic101C2311yescharacteristic
52C24×C101—21no—
53C2×D4×C101—21no—
54C2×D4×C101—22no—
55C2×D4×C101—22no—
56C2×D4×C101C2312yes—
57C22×C4×C101C2311yescharacteristic
58M16×C101—22no—
59C2×C8×C101C2312yes—
60C4⋊C4×C101C2312yes—
61C22⋊C4×C101—21no—
62C42×C101—21no—
63C23×C101—22no—
64C23×C101—44no—
65C23×C101—42no—
66C23×C101C2×D411yescharacteristic
67C22×D101—22no—
68C22×D101—44no—
69C22×D101—42no—
70D4×C101—22no—
71D4×C101—22no—
72D4×C101—42no—
73C2×C4×C101—22no—
74C2×C4×C101C2×D411yescharacteristic
75C2×C4×C101#4112yes—
76C2×C4×C101#4112yes—
77C2×C4×C101C2411yescharacteristic
78C22⋊D101—48no—
79C22⋊D101—48no—
80C2×Dic101—24no—
81C2×Dic101—22no—
82C2×Dic101C2×D412yes—
83C2×Dic101C2×D412yes—
84D404—22no—
85D404—22no—
86D404—42no—
87Dic202—44no—
88C8×C101—22no—
89#11717579—22no—
90C22×C101—22no—
91C22×C101—42no—
92C22×C101—44no—
93C22×C101—84no—
94C22×C101#12412yes—
95C22×C101C22×D411yescharacteristic
96C2×D101—42no—
97C2×D101—44no—
98C2×D101—84no—
99C4×C101—42no—
100C4×C101#12412yes—
101C4×C101C2×#4111yescharacteristic
102C4×C101C2×#4111yescharacteristic
103Dic101—22no—
104Dic101—42no—
105Dic101AGL(1,ℤ/8)14yes—
106C2×C101—22no—
107C2×C101—82no—
108C2×C101C2×#12411yescharacteristic
109C2×C101C2×AGL(1,ℤ/8)12yes—
110D101—82no—
111#3138—1011nomaximal, Sylow
112C101#313811yescharacteristic, complemented, Sylow
113#533—1011no—
114C2×#127—1011no—
115#490—1014no—
116#486—1014no—
117#418—1011no—
118#418—1011no—
119#418—1011no—
120#418D10111yescharacteristic, complemented
121C4×M16—1011no—
122C22×D4—1011no—
123C22×D4—2022no—
124C22×D4C2×D10111yescharacteristic
125C2×M16—1011no—
126C2×M16C2×D10111yescharacteristic
127C4⋊Q8—1012no—
128#127—1012no—
129#127—1012no—
130#127—2022no—
131#125—1012no—
132C22⋊Q8—2022no—
133C4×Q8—2022no—
134#117—1011no—
135#117C2×D10111yescharacteristic
136C2×C42—1011no—
137#102—1014no—
138#102—1014no—
139#102—1014no—
140#102C2×D10114yescomplemented
141#97—1012no—
142C4×C8—1012no—
143C24—21no—
144C24—2021no—
145C2×Q8—2022no—
146C2×D4—21no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—1012no—
150C2×D4—4048no—
151C2×D4—2024no—
152C2×D4—2024no—
153C2×D4—2021no—
154C2×D4—2022no—
155C2×D4—2022no—
156C2×D4C22×D10112yes—
157C22×C4—1012no—
158C22×C4C22×D10111yescharacteristic
159M16—22no—
160M16—2022no—
161C2×C8—1012no—
162C2×C8C22×D10112yes—
163C4⋊C4—1012no—
164C4⋊C4—2022no—
165C4⋊C4—2022no—
166C4⋊C4—2022no—
167C4⋊C4C22×D10112yes—
168C22⋊C4—21no—
169C22⋊C4—2021no—
170C42—21no—
171C42—1012no—
172C42—1011no—
173C42—1011no—
174C42—2021no—
175C23—22no—
176C23—44no—
177C23—42no—
178C23—4044no—
179C23—4042no—
180C23—2022no—
181C23D4×D10111yescharacteristic
182Q8—4044no—
183D4—22no—
184D4—22no—
185D4—42no—
186D4—4048no—
187D4—4048no—
188D4—4042no—
189D4—2022no—
190D4—2022no—
191C2×C4—22no—
192C2×C4—1012no—
193C2×C4—1012no—
194C2×C4—2024no—
195C2×C4—2022no—
196C2×C4D4×D10111yescharacteristic
197C2×C4#1310807312yes—
198C2×C4#1310807412yes—
199C2×C4C23×D10111yescharacteristic
200C8—22no—
201C8—2022no—
202C22—22no—
203C22—42no—
204C22—44no—
205C22—84no—
206C22—8084no—
207C22—4042no—
208C22—4044no—
209C22#1569224112yes—
210C22C2×D4×D10111yescharacteristic
211C4—42no—
212C4—1014no—
213C4—4042no—
214C4—2022no—
215C4#1569224112yes—
216C4C2×#1310807311yescharacteristic
217C4C2×#1310807411yescharacteristic
218C2—22no—
219C2—82no—
220C2—8082no—
221C2C2×#1569224111yescharacteristic
222C2AGL(1,ℤ/8)×D10112yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#418×C101C2
socleC22×C101C22×D4
prime radicalC2×#15692251C2
hypercenter#418D101
hypocenterC101#3138
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
1. #20265730C2semidirect
3. #20265540C2semidirect
4. #20265522C2semidirect
5. #20265183C2semidirect
6. #20265182C2semidirect
7. #20265181C2semidirect
8. M16×Dic101C2semidirect
9. #418×C101C2semidirect
13. #15692254C22semidirect
17. #15692246C22semidirect
22. #15692177C22semidirect
23. #15692176C22semidirect
24. #15692175C22semidirect
25. #15692165C22semidirect
26. C8×Dic101C22semidirect
30. #102×C101C22semidirect
112. C101#3138semidirect
120. #418D101semidirect
140. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657302
C2×#156922512
#202655402
#202655222
#202651832
#202651822
#202651812
M16×Dic1012
#418×C1012
#3138101