Finite Group Databasebeta

← #6344411

Subgroups — #6344411

Order 512.

1658 conjugacy classes of subgroups.

The subgroup lattice has 249 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60284C212yesmaximal
2C2×C8×#41C211yesmaximal, characteristic
3#17837C218yesmaximal, complemented
4#17835C218yesmaximal, complemented
5#17819C216yesmaximal, complemented
6#17793C216yesmaximal, complemented
7C2×C4×#41C2211yescharacteristic
8#2991C22116yes—
9#2982—22no—
10#2975—26no—
11#2971—28no—
12C8×#41C414yes—
13C8×#41C414yes—
14C8×#41C2214yes—
15C8×#41C2214yes—
16C2×C8×Q8C411yescharacteristic
17C2×C8×Q8C411yescharacteristic
18C2×C8×D4C413yes—
19C2×C8×D4C413yes—
20#2920C22112yes—
21C2×#431C2213yes—
22C22×C4×C8C2213yes—
23#1574C22148yescomplemented
24#1568C2214yes—
25#1568C22112yes—
26#1567C22112yescomplemented
27#1566C22112yescomplemented
28#1564C22112yescomplemented
29#1564C22112yescomplemented
30C22×#41Q811yescharacteristic
31C23×C8C2×C413yes—
32C23×C8C2×C413yes—
33C4×#41C2×C414yes—
34C4×#41D414yes—
35C4×#41Q814yes—
36C4×#41C2314yes—
37C2×C4×Q8C2×C411yescharacteristic
38C2×C4×Q8D411yescharacteristic
39C2×C4×D4C2×C413yes—
40C2×C4×D4D413yes—
41C2×#117Q813yes—
42C22×C42C2313yes—
43C8×Q8—216no—
44C8×Q8C2×C414yes—
45C8×Q8C2×C414yes—
46C8×Q8C2×C414yes—
47C8×Q8C2×C414yes—
48#436—248no—
49C8×D4C2×C4112yes—
50C8×D4C2×C4112yes—
51C8×D4C2×C4112yes—
52C8×D4C2×C4112yes—
53#432—212no—
54#432C2318yes—
55#431—212no—
56#431C2×C4112yes—
57#431C23124yes—
58#431C23112yes—
59#423C23124yes—
60C2×C4⋊C8C2×C413yes—
61C2×C4⋊C8C2×C413yes—
62#407—212no—
63C2×C22⋊C8C2×C416yes—
64C4×M16—212no—
65C2×C4×C8C2×C413yes—
66C2×C4×C8C2×C413yes—
67C2×C4×C8C2×C416yes—
68C2×C4×C8C2×C416yes—
69C2×C4×C8C2316yes—
70C2×C4×C8C2316yes—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2313yes—
73#335C2318yes—
74#335C23124yes—
75#334C2318yes—
76#334C23124yes—
77C2×#41—28no—
78C2×#41C4⋊C414yes—
79C2×#41C2×Q814yes—
80C22×Q8C4⋊C411yescharacteristic
81C22×D4C4⋊C413yes—
82C23×C4C22×C413yes—
83C23×C4C2×D413yes—
84C23×C4C2×Q813yes—
85C2×M16—224no—
86C22×C8—46no—
87C22×C8—46no—
88C22×C8C22×C411yescharacteristic
89C22×C8C22×C411yescharacteristic
90C22×C8C22×C416yes—
91C22×C8C22×C416yes—
92C22×C8C22×C413yes—
93C22×C8C22×C413yes—
94C22×C8C22×C416yes—
95C22×C8C22×C416yes—
96C4×Q8C4⋊C414yes—
97C4×Q8C4⋊C414yes—
98C4×Q8C22×C414yes—
99C4×Q8C2×D414yes—
100C4×D4C4⋊C4112yes—
101C4×D4C4⋊C4112yes—
102C4×D4C22×C4112yes—
103C4×D4C2×D4112yes—
104#117C4⋊C4112yes—
105#117C2×Q8112yes—
106C2×C4⋊C4C4⋊C413yes—
107C2×C4⋊C4C4⋊C413yes—
108C2×C22⋊C4C4⋊C416yes—
109C2×C42C22×C413yes—
110C2×C42C22×C416yes—
111C2×C42C2×D413yes—
112C2×C42C2×D416yes—
113C2×C42C2×Q813yes—
114C2×C42C2×Q816yes—
115C2×C42C2416yes—
116C2×C42C2411yescharacteristic
117C4⋊C8—224no—
118C4⋊C8C22×C4112yes—
119C4⋊C8C22×C4112yes—
120C4⋊C8C24116yes—
121C22⋊C8—28no—
122C22⋊C8—224no—
123C22⋊C8C22×C4124yes—
124#97—28no—
125C4×C8—224no—
126C4×C8C22×C411yescharacteristic
127C4×C8C22×C411yescharacteristic
128C4×C8C22×C416yes—
129C4×C8C22×C416yes—
130C4×C8C22×C416yes—
131C4×C8C22×C416yes—
132C4×C8C22×C416yes—
133C4×C8C2411yescharacteristic
134C4×C8C2416yes—
135C4×C8C2411yescharacteristic
136C24C2×C4⋊C413yes—
137#41—216no—
138#41—216no—
139C2×Q8#10818yes—
140C2×Q8C2×C4⋊C414yes—
141C2×D4—212no—
142C2×D4C2×C4⋊C4112yes—
143C22×C4—212no—
144C22×C4—43no—
145C22×C4—43no—
146C22×C4—46no—
147C22×C4C2×C4⋊C416yes—
148C22×C4C2×C4⋊C413yes—
149C22×C4C2×C4⋊C416yes—
150C22×C4C2×C4⋊C413yes—
151C22×C4C2×C4⋊C416yes—
152C22×C4C2×C4⋊C413yes—
153C22×C4C23×C416yes—
154C22×C4C23×C411yescharacteristic
155C22×C4C22×D411yescharacteristic
156C22×C4C22×D416yes—
157C22×C4C22×Q811yescharacteristic
158C22×C4C22×Q816yes—
159M16—824no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—212no—
163C2×C8—212no—
164C2×C8—216no—
165C2×C8—46no—
166C2×C8—46no—
167C2×C8—43no—
168C2×C8—43no—
169C2×C8C23×C411yescharacteristic
170C2×C8C23×C411yescharacteristic
171C2×C8C23×C416yes—
172C2×C8C23×C416yes—
173C2×C8C23×C411yescharacteristic
174C2×C8C23×C411yescharacteristic
175C4⋊C4C2×C4⋊C4112yes—
176C4⋊C4C2×C4⋊C4112yes—
177C22⋊C4C2×C4⋊C4124yes—
178C42C2×C4⋊C416yes—
179C42C2×C4⋊C416yes—
180C42C2×C4⋊C416yes—
181C42C23×C411yescharacteristic
182C42C23×C416yes—
183C42C22×D416yes—
184C42C22×D411yescharacteristic
185C42C22×Q811yescharacteristic
186C42C22×Q816yes—
187C42C2511yescharacteristic
188C23—26no—
189C23—43no—
190C23—43no—
191C23C22×C4⋊C416yes—
192C23C22×C4⋊C411yescharacteristic
193Q8C2×#108116yes—
194D4—224no—
195C2×C4—22no—
196C2×C4—212no—
197C2×C4—22no—
198C2×C4—212no—
199C2×C4—212no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—26no—
203C2×C4—26no—
204C2×C4—46no—
205C2×C4—43no—
206C2×C4—46no—
207C2×C4—43no—
208C2×C4C2×#108112yes—
209C2×C4C22×C4⋊C411yescharacteristic
210C2×C4C22×C4⋊C411yescharacteristic
211C2×C4C22×C4⋊C411yescharacteristic
212C2×C4C22×C4⋊C416yes—
213C2×C4C22×C4⋊C416yes—
214C2×C4C22×C4⋊C416yes—
215C2×C4C24×C411yescharacteristic
216C2×C4C23×D411yescharacteristic
217C2×C4C23×Q811yescharacteristic
218C8—21no—
219C8—21no—
220C8—26no—
221C8—26no—
222C8—416no—
223C8#56812yes—
224C8#56812yes—
225C22—21no—
226C22—21no—
227C22—26no—
228C22—26no—
229C22—26no—
230C22—412no—
231C22C22×#10812yes—
232C22C23×C4⋊C411yescharacteristic
233C4—21no—
234C4—26no—
235C4—21no—
236C4—26no—
237C4—21no—
238C4—21no—
239C4C22×#108112yes—
240C4C2×#56811yescharacteristic
241C4C2×#56811yescharacteristic
242C4#358412yes—
243C2—22no—
244C2—46no—
245C2#6009911yescharacteristic
246C2#6010811yescharacteristic
247C2C23×#10811yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C23×C4⋊C4
prime radicalC22×#41Q8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17837C2semidirect
4. #17835C2semidirect
5. #17819C2semidirect
6. #17793C2semidirect
23. #1574C22semidirect
26. #1567C22semidirect
27. #1566C22semidirect
28. #1564C22semidirect
29. #1564C22semidirect

Maximal subgroups

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

subgroupindex
#602842
C2×C8×#412
#178372
#178352
#178192
#177932