Finite Group Databasebeta

← #27180596

Subgroups — #27180596

Order 12928.

548 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 153 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×#15692281C212yesmaximal, complemented
2#20265727C211yesmaximal, characteristic, complemented
3#20265392C214yesmaximal, complemented
4#20265382C212yesmaximal, complemented
5#20265372C212yesmaximal, complemented
6#20265201C211yesmaximal, characteristic, complemented
7#20265176C212yesmaximal, complemented
8C2×C4×D4×C101C211yesmaximal, characteristic, complemented
9C2×#13108074C2212yes—
10#15692281—28no—
11C2×#13108062C2214yes—
12C2×#13108059C2214yes—
13C2×#13108058C2212yes—
14#15692255C2211yescharacteristic, complemented
15#15692252C2211yescharacteristic, complemented
16#15692243C2212yescomplemented
17#15692229—21no—
18#15692229—21no—
19C4×D404—21no—
20C4×D404—21no—
21C4⋊#11717579C2212yescomplemented
22C4⋊#11717579C2212yescomplemented
23#15692178C2214yescomplemented
24#15692175C2214yescomplemented
25C22×D4×C101C2211yescharacteristic
26C23×C4×C101—21no—
27C4×D4×C101—21no—
28C4×D4×C101—21no—
29C4×D4×C101C2212yes—
30C4×D4×C101C2212yes—
31C2×C4⋊C4×C101C2211yescharacteristic
32C2×C22⋊C4×C101—21no—
33C2×C42×C101C2211yescharacteristic
34#13108074—24no—
35#13108074—24no—
36C2×Dic202C2312yes—
37C2×D404—21no—
38C2×D404—21no—
39C2×D404C2312yes—
40C2×C22⋊D101—22no—
41C2×C4×D101—22no—
42#13108062—28no—
43#13108059—28no—
44#13108058—24no—
45C2×#11717579C2314yes—
46C4⋊Dic101—21no—
47C4⋊Dic101—21no—
48#13108051—22no—
49#13108051—22no—
50C24×C101—21no—
51C2×D4×C101—21no—
52C2×D4×C101—22no—
53C2×D4×C101D414yescomplemented
54C2×D4×C101C2312yes—
55C22×C4×C101—21no—
56C22×C4×C101—22no—
57C22×C4×C101—22no—
58C22×C4×C101—44no—
59C22×C4×C101D411yescharacteristic
60C22×C4×C101D411yescharacteristic
61C22×C4×C101C2311yescharacteristic
62C4⋊C4×C101—21no—
63C4⋊C4×C101C2312yes—
64C22⋊C4×C101—22no—
65C22⋊C4×C101—22no—
66C42×C101D411yescharacteristic
67C42×C101D411yescharacteristic
68C42×C101C2311yescharacteristic
69C42×C101C2311yescharacteristic
70C23×C101—24no—
71C23×C101—22no—
72C23×C101—42no—
73C23×C101—42no—
74C23×C101C2×D411yescharacteristic
75C22×D101—22no—
76D4×C101—24no—
77D4×C101—22no—
78D4×C101C2×D414yes—
79C2×C4×C101—22no—
80C2×C4×C101—22no—
81C2×C4×C101—41no—
82C2×C4×C101—44no—
83C2×C4×C101—42no—
84C2×C4×C101—44no—
85C2×C4×C101C2×D411yescharacteristic
86C2×C4×C101C2×D411yescharacteristic
87C2×C4×C101C2×D411yescharacteristic
88C2×C4×C101C2×D411yescharacteristic
89C2×C4×C101C2×D411yescharacteristic
90C2×C4×C101C2×D412yes—
91C2×C4×C101#4112yes—
92C2×C4×C101C2411yescharacteristic
93C22⋊D101—44no—
94C2×Dic101—22no—
95D404—22no—
96D404—22no—
97D404—42no—
98D404—42no—
99C4×D101—44no—
100Dic202—22no—
101Dic202—22no—
102#11717579—24no—
103C22×C101—21no—
104C22×C101—21no—
105C22×C101—24no—
106C22×C101—48no—
107C22×C101—42no—
108C22×C101—42no—
109C22×C101#12112yes—
110C22×C101C22×D411yescharacteristic
111C2×D101—42no—
112C2×D101—42no—
113C2×D101—42no—
114C4×C101—21no—
115C4×C101—21no—
116C4×C101—21no—
117C4×C101—42no—
118C4×C101#12112yes—
119C4×C101AGL(1,ℤ/8)12yescomplemented
120C4×C101C22×D411yescharacteristic
121C4×C101C2×#4111yescharacteristic
122Dic101—42no—
123C2×C101—22no—
124C2×C101—44no—
125C2×C101C2×#12111yescharacteristic
126C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
127C2×C101#57511yescharacteristic
128D101—82no—
129#3036—1011nomaximal, Sylow
130C101#303611yescharacteristic, complemented, Sylow
131C2×AGL(1,ℤ/8)—1012no—
132#532—1011no—
133C2×C4×D4D10111yescharacteristic, complemented
134#463—1014no—
135#461—1012no—
136#459—1012no—
137#423—1011no—
138#417—1012no—
139C2×#41—1012no—
140C22×D4C2×D10111yescharacteristic
141C23×C4—21no—
142AGL(1,ℤ/8)—2028no—
143C2×SD16—1014no—
144C2×D8—1014no—
145C2×M16—1012no—
146C4⋊Q8—1011no—
147#127—1011no—
148#124—1012no—
149#121—2021no—
150#121—2021no—
151C4×D4—21no—
152C4×D4—21no—
153C4×D4—2021no—
154C4×D4—2021no—
155C4×D4C2×D10112yes—
156C4×D4C2×D10112yes—
157C2×C4⋊C4C2×D10111yescharacteristic
158C2×C22⋊C4—21no—
159C2×C42C2×D10111yescharacteristic
160C4⋊C8—1012no—
161C4⋊C8—1012no—
162#103—1014no—
163#102—1014no—
164C24—21no—
165#41—2024no—
166#41—2024no—
167C2×Q8—1012no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—1012no—
171C2×D4—2022no—
172C2×D4—2021no—
173C2×D4—2021no—
174C2×D4D40414yescomplemented
175C2×D4C22×D10112yes—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—44no—
180C22×C4—2022no—
181C22×C4C22⋊D10111yescharacteristic
182C22×C4C22⋊D10111yescharacteristic
183C22×C4C22×D10111yescharacteristic
184SD16—2028no—
185D8—2028no—
186M16—2024no—
187C2×C8—1014no—
188C4⋊C4—21no—
189C4⋊C4—2021no—
190C4⋊C4—2021no—
191C4⋊C4C22×D10112yes—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—2022no—
195C22⋊C4—2022no—
196C42C22⋊D10111yescharacteristic
197C42C22⋊D10111yescharacteristic
198C42C22×D10111yescharacteristic
199C42C22×D10111yescharacteristic
200C23—22no—
201C23—24no—
202C23—42no—
203C23—42no—
204C23—2022no—
205C23C2×C22⋊D10111yescharacteristic
206Q8—2022no—
207Q8—2022no—
208D4—22no—
209D4—24no—
210D4—4042no—
211D4—4044no—
212D4—4042no—
213D4—2022no—
214D4—2022no—
215D4C2×D40414yes—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—44no—
222C2×C4—4044no—
223C2×C4—2022no—
224C2×C4C2×C22⋊D10111yescharacteristic
225C2×C4C2×C22⋊D10111yescharacteristic
226C2×C4C2×C22⋊D10111yescharacteristic
227C2×C4C2×C22⋊D10111yescharacteristic
228C2×C4C2×C22⋊D10111yescharacteristic
229C2×C4C2×D40412yes—
230C2×C4#1310807412yes—
231C2×C4C23×D10111yescharacteristic
232C8—2024no—
233C22—21no—
234C22—21no—
235C22—24no—
236C22—48no—
237C22—42no—
238C22—42no—
239C22—4042no—
240C22—4042no—
241C22—4042no—
242C22#1569222912yes—
243C22C22×C22⋊D10111yescharacteristic
244C4—21no—
245C4—21no—
246C4—21no—
247C4—42no—
248C4—4042no—
249C4#1569222912yes—
250C4#1569228112yescomplemented
251C4C22×D40411yescharacteristic
252C4C2×#1310807411yescharacteristic
253C2—22no—
254C2—44no—
255C2—8082no—
256C2C2×#1569222911yescharacteristic
257C2C2×#1569228111yescharacteristic
258C2#2026597511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×C4×D4×C101C2
socleC22×C101C22×D4
hypercenterC2×C4×D4D101
hypocenterC101#3036
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15692281C2semidirect
2. #20265727C2semidirect
3. #20265392C2semidirect
4. #20265382C2semidirect
5. #20265372C2semidirect
6. #20265201C2semidirect
7. #20265176C2semidirect
8. C2×C4×D4×C101C2semidirect
14. #15692255C22semidirect
15. #15692252C22semidirect
16. #15692243C22semidirect
21. C4⋊#11717579C22semidirect
22. C4⋊#11717579C22semidirect
23. #15692178C22semidirect
24. #15692175C22semidirect
53. C2×D4×C101D4semidirect
119. C4×C101AGL(1,ℤ/8)semidirect
130. C101#3036semidirect
133. C2×C4×D4D101semidirect
174. C2×D4D404semidirect
250. C4#15692281semidirect

Maximal subgroups

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

subgroupindex
C2×#156922812
#202657272
#202653922
#202653822
#202653722
#202652012
#202651762
C2×C4×D4×C1012
#3036101