Finite Group Databasebeta

← #27180610

Subgroups — #27180610

Order 12928.

584 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#15692280C211yesmaximal, characteristic, complemented
2C2×#15692251C211yesmaximal, characteristic, complemented
3#41×Dic101C211yesmaximal, characteristic, complemented
4#20265375C214yesmaximal, complemented
5#20265370C214yesmaximal, complemented
6#20265200C211yesmaximal, characteristic, complemented
7#20265183C211yesmaximal, characteristic, complemented
8#20265182C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C101C211yesmaximal, characteristic, complemented
10C22×D404C2211yescharacteristic
11C22×C22⋊D101—22no—
12#15692280—24no—
13C2×#13108061C2212yescomplemented
14C2×#13108059C2212yescomplemented
15C2×#13108058C2211yescharacteristic
16#15692251—21no—
17#15692251—21no—
18#15692251C2214yes—
19Q8×Dic101C2212yescomplemented
20D4×Dic101—22no—
21D4×Dic101C2212yescomplemented
22#15692214—21no—
23#15692214C2211yescharacteristic
24C2×C4×Dic101—21no—
25C2×C4×Dic101C2211yescharacteristic
26#15692184C2214yescomplemented
27#15692177C2214yescomplemented
28#15692176C2214yescomplemented
29C2×#41×C101C2211yescharacteristic
30C22×D4×C101C2211yescharacteristic
31AGL(1,ℤ/8)×C101—24no—
32C2×SD16×C101C2212yescomplemented
33C2×D8×C101C2212yescomplemented
34C2×M16×C101C2211yescharacteristic
35C23×D101—21no—
36C2×D404—22no—
37C2×D404—22no—
38C2×D404—21no—
39C2×D404C2312yes—
40C2×C22⋊D101—24no—
41C2×C22⋊D101—24no—
42C2×C22⋊D101—48no—
43C22×Dic101—22no—
44C22×Dic101D412yes—
45#13108061—24no—
46#13108059—24no—
47#13108058—22no—
48C2×#11717579C2312yes—
49C4⋊Dic101—22no—
50C4⋊Dic101C2312yes—
51C22⋊Dic101—21no—
52C22⋊Dic101—22no—
53C4×Dic101—21no—
54C4×Dic101—22no—
55C4×Dic101D411yescharacteristic
56C4×Dic101D411yescharacteristic
57C4×Dic101C2312yes—
58C24×C101—21no—
59#41×C101—22no—
60#41×C101D414yescomplemented
61C2×Q8×C101C2311yescharacteristic
62C2×D4×C101—21no—
63C2×D4×C101—22no—
64C2×D4×C101—21no—
65C2×D4×C101—22no—
66C2×D4×C101C2312yes—
67C2×D4×C101C2311yescharacteristic
68C22×C4×C101—21no—
69C22×C4×C101C2311yescharacteristic
70SD16×C101—24no—
71D8×C101—24no—
72M16×C101—22no—
73C2×C8×C101C2312yes—
74C23×C101—21no—
75C23×C101—22no—
76C23×C101—44no—
77C23×C101—42no—
78C23×C101C2×D411yescharacteristic
79C22×D101—22no—
80C22×D101—44no—
81C22×D101—42no—
82Q8×C101—21no—
83Q8×C101C2×D412yescomplemented
84D4×C101—24no—
85D4×C101—22no—
86D4×C101—21no—
87D4×C101—22no—
88D4×C101—42no—
89D4×C101C2×D412yescomplemented
90C2×C4×C101—21no—
91C2×C4×C101—24no—
92C2×C4×C101C2×D411yescharacteristic
93C2×C4×C101C2×D412yes—
94C2×C4×C101#4112yes—
95C2×C4×C101C2411yescharacteristic
96C22⋊D101—48no—
97C22⋊D101—48no—
98C2×Dic101—24no—
99C2×Dic101—22no—
100C2×Dic101—44no—
101C2×Dic101C2×D412yes—
102C2×Dic101C2×D412yes—
103D404—22no—
104D404—22no—
105D404—42no—
106C8×C101—22no—
107#11717579—22no—
108C22×C101—22no—
109C22×C101—22no—
110C22×C101—42no—
111C22×C101—42no—
112C22×C101—44no—
113C22×C101—84no—
114C22×C101#12112yes—
115C22×C101C22×D411yescharacteristic
116C2×D101—42no—
117C2×D101—44no—
118C2×D101—84no—
119C4×C101—22no—
120C4×C101#12112yes—
121C4×C101C22×D411yescharacteristic
122C4×C101C2×#4111yescharacteristic
123Dic101—22no—
124Dic101—42no—
125Dic101AGL(1,ℤ/8)14yescomplemented
126C2×C101—22no—
127C2×C101—42no—
128C2×C101—82no—
129C2×C101C2×#12111yescharacteristic
130C2×C101C2×AGL(1,ℤ/8)12yes—
131D101—82no—
132#3038—1011nomaximal, Sylow
133C101#303811yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)—1011no—
135C2×AGL(1,ℤ/8)D10111yescharacteristic, complemented
136C2×#127—1011no—
137C4×#41—1011no—
138#460—1014no—
139#459—1014no—
140#423—1011no—
141#418—1011no—
142#418—1011no—
143C2×#41C2×D10111yescharacteristic
144C22×D4—1011no—
145C22×D4—2022no—
146C22×D4C2×D10111yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—2024no—
149C2×SD16—1012no—
150C2×SD16C2×D10112yescomplemented
151C2×D8—1012no—
152C2×D8C2×D10112yescomplemented
153C2×M16—1011no—
154C2×M16C2×D10111yescharacteristic
155#127—1014no—
156#127—2021no—
157#127—2021no—
158C4×Q8—1012no—
159C4×D4—1012no—
160C4×D4—2022no—
161#117—1011no—
162#117—2021no—
163C2×C42—1011no—
164C2×C42—2021no—
165C4⋊C8—1014no—
166#102—1014no—
167#102—1014no—
168C24—21no—
169C24—2021no—
170#41—22no—
171#41C22⋊D10114yescomplemented
172C2×Q8C22×D10111yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—1012no—
178C2×D4—4048no—
179C2×D4—2024no—
180C2×D4—2024no—
181C2×D4—2021no—
182C2×D4—2022no—
183C2×D4—2022no—
184C2×D4C22×D10112yes—
185C2×D4C22×D10111yescharacteristic
186C22×C4—21no—
187C22×C4—1012no—
188C22×C4—2022no—
189C22×C4C22×D10111yescharacteristic
190SD16—24no—
191SD16—2024no—
192D8—24no—
193D8—2024no—
194M16—22no—
195M16—2022no—
196C2×C8—1012no—
197C2×C8C22×D10112yes—
198C4⋊C4—1012no—
199C4⋊C4—2022no—
200C22⋊C4—2022no—
201C22⋊C4—2021no—
202C42—1011no—
203C42—1011no—
204C42—1012no—
205C42—2022no—
206C42—2021no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—4044no—
212C23—4042no—
213C23—2022no—
214C23D4×D10111yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D10112yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—4048no—
223D4—4048no—
224D4—4042no—
225D4—2022no—
226D4—2022no—
227D4C2×C22⋊D10112yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—1012no—
231C2×C4—1012no—
232C2×C4—4044no—
233C2×C4—2024no—
234C2×C4—2022no—
235C2×C4D4×D10111yescharacteristic
236C2×C4C2×C22⋊D10112yes—
237C2×C4#1310807312yes—
238C2×C4C23×D10111yescharacteristic
239C8—22no—
240C8—2022no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22—44no—
246C22—84no—
247C22—8084no—
248C22—4042no—
249C22—4044no—
250C22#1569222712yes—
251C22C2×D4×D10111yescharacteristic
252C4—22no—
253C4—1014no—
254C4—4042no—
255C4—2022no—
256C4#1569222712yes—
257C4C22×C22⋊D10111yescharacteristic
258C4C2×#1310807311yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—8082no—
263C2C2×#1569222711yescharacteristic
264C2AGL(1,ℤ/8)×D10112yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×AGL(1,ℤ/8)×C101C2
socleC22×C101C22×D4
hypercenterC2×AGL(1,ℤ/8)D101
hypocenterC101#3038
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#15692280C2semidirect
2. C2×#15692251C2semidirect
3. #41×Dic101C2semidirect
4. #20265375C2semidirect
5. #20265370C2semidirect
6. #20265200C2semidirect
7. #20265183C2semidirect
8. #20265182C2semidirect
9. C2×AGL(1,ℤ/8)×C101C2semidirect
13. C2×#13108061C22semidirect
14. C2×#13108059C22semidirect
19. Q8×Dic101C22semidirect
21. D4×Dic101C22semidirect
26. #15692184C22semidirect
27. #15692177C22semidirect
28. #15692176C22semidirect
32. C2×SD16×C101C22semidirect
33. C2×D8×C101C22semidirect
60. #41×C101D4semidirect
83. Q8×C101C2×D4semidirect
89. D4×C101C2×D4semidirect
125. Dic101AGL(1,ℤ/8)semidirect
133. C101#3038semidirect
135. C2×AGL(1,ℤ/8)D101semidirect
150. C2×SD16C2×D101semidirect
152. C2×D8C2×D101semidirect
171. #41C22⋊D101semidirect
216. Q8C2×C22⋊D101semidirect
227. D4C2×C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922802
C2×#156922512
#41×Dic1012
#202653752
#202653702
#202652002
#202651832
#202651822
C2×AGL(1,ℤ/8)×C1012
#3038101