Finite Group Databasebeta

← #6344401

Subgroups — #6344401

Order 512.

2008 conjugacy classes of subgroups.

The subgroup lattice has 221 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C8×#41C211yesmaximal, characteristic
2#60099C212yesmaximal
3#17828C2116yesmaximal, complemented
4#17814C216yesmaximal, complemented
5C2×#1567C216yesmaximal, complemented
6C2×#568—22no—
7C2×C4×#41C2211yescharacteristic
8C8×#41C414yes—
9C8×#41C414yes—
10C8×#41C2218yes—
11C2×C8×Q8C412yes—
12C2×C8×D4C416yes—
13C2×#431C2213yes—
14#2906—28no—
15#2906C22116yes—
16#2905—26no—
17C2×#423C22112yes—
18C22×C4×C8C2213yes—
19#1571C22148yescomplemented
20#1567C2214yes—
21#1567C22112yes—
22#1567C22124yescomplemented
23C2×#335C22124yescomplemented
24C22×#41D411yescharacteristic
25#568—416no—
26C2×#131—216no—
27C22×M16—212no—
28C23×C8C2×C416yes—
29C4×#41C2×C414yes—
30C4×#41D414yes—
31C4×#41Q814yes—
32C4×#41C2314yes—
33C2×C4×Q8C2×C411yescharacteristic
34C2×C4×Q8Q811yescharacteristic
35C2×C4×D4C2×C413yes—
36C2×C4×D4Q813yes—
37C2×#117D413yes—
38C22×C42C2313yes—
39C8×Q8C2×C418yes—
40C8×Q8C2×C418yes—
41C8×D4C2×C4124yes—
42C8×D4C2×C4124yes—
43#431C2×C4112yes—
44#431C23112yes—
45#424—248no—
46#423—212no—
47#423—212no—
48#423C2318yes—
49#423C23124yes—
50C2×C4⋊C8—212no—
51C2×C4⋊C8C2×C416yes—
52C2×C4⋊C8C23124yes—
53C2×C22⋊C8C2×C416yes—
54C2×C4×C8C2×C416yes—
55C2×C4×C8C2×C416yes—
56C2×C4×C8C2×C416yes—
57C2×C4×C8C23112yes—
58C2×C4×C8C2311yescharacteristic
59C2×C4×C8C2313yes—
60#335C23116yes—
61#335C23148yes—
62C2×#41C4⋊C414yes—
63C2×#41D818yes—
64C2×#41Q1618yes—
65C2×#41C2×D414yes—
66C22×Q8C4⋊C411yescharacteristic
67C22×D4C4⋊C413yes—
68C23×C4C22×C413yes—
69C23×C4C2×D413yes—
70C23×C4C2×Q813yes—
71#131—464no—
72C2×M16—28no—
73C2×M16—224no—
74C2×M16—448no—
75C22×C8—224no—
76C22×C8—412no—
77C22×C8C22×C412yes—
78C22×C8C22×C4112yes—
79C22×C8C22×C416yes—
80C22×C8C22×C4112yes—
81C4×Q8C4⋊C414yes—
82C4×Q8C4⋊C414yes—
83C4×Q8C22×C414yes—
84C4×Q8C2×Q814yes—
85C4×D4C4⋊C4112yes—
86C4×D4C4⋊C4112yes—
87C4×D4C22×C4112yes—
88C4×D4C2×Q8112yes—
89#117C4⋊C4112yes—
90#117C2×D4112yes—
91C2×C4⋊C4C4⋊C413yes—
92C2×C4⋊C4C4⋊C413yes—
93C2×C22⋊C4C4⋊C416yes—
94C2×C42C22×C413yes—
95C2×C42C22×C416yes—
96C2×C42C2×D413yes—
97C2×C42C2×D416yes—
98C2×C42C2×Q813yes—
99C2×C42C2×Q816yes—
100C2×C42C2416yes—
101C2×C42C2411yescharacteristic
102C4⋊C8—28no—
103C4⋊C8—224no—
104C4⋊C8—224no—
105C4⋊C8C22×C4124yes—
106C4⋊C8C24116yes—
107C22⋊C8C22×C4124yes—
108C4×C8C22×C411yescharacteristic
109C4×C8C22×C411yescharacteristic
110C4×C8C22×C416yes—
111C4×C8C22×C4112yes—
112C4×C8C22×C4112yes—
113C4×C8C2412yes—
114C4×C8C2416yes—
115C24C2×C4⋊C413yes—
116#41#107132yes—
117#41C2×D8116yes—
118#41C2×Q16116yes—
119C2×Q8#10718yes—
120C2×Q8C2×C4⋊C414yes—
121C2×D4#107124yes—
122C2×D4C2×C4⋊C4112yes—
123C22×C4—46no—
124C22×C4—46no—
125C22×C4C2×C4⋊C416yes—
126C22×C4C2×C4⋊C413yes—
127C22×C4C2×C4⋊C413yes—
128C22×C4C2×C4⋊C416yes—
129C22×C4C2×C4⋊C413yes—
130C22×C4C2×C4⋊C416yes—
131C22×C4C2×D8112yes—
132C22×C4C2×Q16112yes—
133C22×C4C23×C416yes—
134C22×C4C23×C411yescharacteristic
135C22×C4C22×D411yescharacteristic
136C22×C4C22×D416yes—
137C22×C4C22×Q811yescharacteristic
138C22×C4C22×Q816yes—
139M16—416no—
140M16—448no—
141C2×C8—24no—
142C2×C8—224no—
143C2×C8—216no—
144C2×C8—412no—
145C2×C8—46no—
146C2×C8—448no—
147C2×C8C23×C412yes—
148C2×C8C23×C4112yes—
149C2×C8C23×C412yes—
150C4⋊C4C2×C4⋊C4112yes—
151C4⋊C4C2×C4⋊C4112yes—
152C22⋊C4C2×C4⋊C4124yes—
153C42C2×C4⋊C416yes—
154C42C2×C4⋊C416yes—
155C42C2×C4⋊C416yes—
156C42C23×C411yescharacteristic
157C42C23×C416yes—
158C42C22×D411yescharacteristic
159C42C22×D416yes—
160C42C22×Q811yescharacteristic
161C42C22×Q816yes—
162C42C2511yescharacteristic
163C23—43no—
164C23—43no—
165C23C2×#107112yes—
166C23C22×C4⋊C416yes—
167C23C22×C4⋊C411yescharacteristic
168Q8C2×#107116yes—
169D4C2×#107148yes—
170C2×C4—22no—
171C2×C4—212no—
172C2×C4—22no—
173C2×C4—212no—
174C2×C4—46no—
175C2×C4—43no—
176C2×C4—43no—
177C2×C4—46no—
178C2×C4C2×#107112yes—
179C2×C4C2×#107124yes—
180C2×C4C22×C4⋊C411yescharacteristic
181C2×C4C22×C4⋊C416yes—
182C2×C4C22×C4⋊C416yes—
183C2×C4C22×C4⋊C411yescharacteristic
184C2×C4C22×C4⋊C416yes—
185C2×C4C22×C4⋊C411yescharacteristic
186C2×C4C22×D8112yes—
187C2×C4C22×D812yes—
188C2×C4C22×Q1612yes—
189C2×C4C22×Q16112yes—
190C2×C4C24×C411yescharacteristic
191C2×C4C23×D411yescharacteristic
192C2×C4C23×Q811yescharacteristic
193C8—22no—
194C8—212no—
195C8—416no—
196C8#56814yescomplemented
197C22—21no—
198C22—26no—
199C22—21no—
200C22—26no—
201C22—412no—
202C22C22×#10712yes—
203C22C22×#107112yes—
204C22C23×C4⋊C411yescharacteristic
205C4—21no—
206C4—26no—
207C4—21no—
208C4—26no—
209C4C22×#10712yes—
210C4C22×#107112yes—
211C4#291612yes—
212C4C2×#56812yes—
213C4C23×D811yescharacteristic
214C4C23×Q1611yescharacteristic
215C2—22no—
216C2—212no—
217C2#6009911yescharacteristic
218C2C23×#10711yescharacteristic
219C2C2×#291611yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C23×C4⋊C4
prime radicalC22×#41D4
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. #17828C2semidirect
4. #17814C2semidirect
5. C2×#1567C2semidirect
19. #1571C22semidirect
22. #1567C22semidirect
23. C2×#335C22semidirect
196. C8#568semidirect

Maximal subgroups

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

subgroupindex
C2×C8×#412
#600992
#178282
#178142
C2×#15672