Finite Group Databasebeta

← #6392789

Subgroups — #6392789

Order 512.

1656 conjugacy classes of subgroups.

The subgroup lattice has 259 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2C2×#2982C211yesmaximal, characteristic
3C2×Q8×M16C211yesmaximal, characteristic
4#22746C218yesmaximal, complemented
5Q8×#108C218yesmaximal, complemented
6#21993C216yesmaximal
7#21370C216yesmaximal
8C22×C4×Q8C2211yescharacteristic
9#2984—24no—
10#2983—23no—
11#2982—21no—
12#2982C418yes—
13#2982C416yes—
14#2982C412yes—
15Q8×M16—24no—
16Q8×M16C2218yes—
17Q8×M16C2214yes—
18Q8×M16C2214yes—
19C2×#446—23no—
20C2×C8×Q8—21no—
21C2×C8×Q8C2218yes—
22#2920C2216yes—
23C2×#431C2216yes—
24C2×#423C2213yes—
25C2×C4×M16C2213yes—
26#1950C22112yes—
27#1950C22112yes—
28#1950C22124yes—
29#1855C2214yes—
30#1855C22112yes—
31#1848C22112yes—
32#1837C22112yes—
33C4×#108C22124yes—
34C23×Q8Q811yescharacteristic
35C22×M16C2311yescharacteristic
36C22×M16C2313yes—
37C2×C4×Q8C2×C414yes—
38C2×C4×Q8C2×C413yes—
39C2×C4×Q8C2×C411yescharacteristic
40C2×C4×Q8D411yescharacteristic
41C2×C4×Q8D413yes—
42C2×C4×Q8D414yes—
43C2×C4×Q8Q814yes—
44C2×C4×Q8C2314yes—
45C22×C4⋊C4Q813yes—
46C22×C42C2313yes—
47#446—224no—
48#446—212no—
49#446—212no—
50C8×Q8—28no—
51C8×Q8—24no—
52C8×Q8—24no—
53C8×Q8C2×C4148yes—
54C8×Q8C2×C4116yes—
55#432—26no—
56#432—26no—
57#431—26no—
58#431—26no—
59#431C2×C4112yes—
60#431C2×C4112yes—
61#431C2×C4112yes—
62#431C2×C4112yes—
63C2×#108C2318yes—
64C2×#108C23124yes—
65#423—26no—
66#423C2316yes—
67#423C23112yes—
68#423C2316yes—
69C2×C4⋊C8—23no—
70C2×C4⋊C8—23no—
71C2×C4⋊C8C23112yes—
72#407C2314yes—
73C2×C22⋊C8C2314yes—
74C4×M16—26no—
75C4×M16C23112yes—
76C4×M16C23112yes—
77C2×#97—23no—
78C2×C4×C8—23no—
79C2×C4×C8C23112yes—
80#341C23124yes—
81#341C2318yes—
82C22×Q8—28no—
83C22×Q8C4⋊C414yes—
84C22×Q8C4⋊C413yes—
85C22×Q8C4⋊C411yescharacteristic
86C22×Q8C2×Q814yes—
87C23×C4C2×Q813yes—
88C23×C4C2×Q813yes—
89C23×C4C2411yescharacteristic
90C2×M16—21no—
91C2×M16—24no—
92C2×M16—23no—
93C2×M16—46no—
94C2×M16C2416yes—
95C2×M16C2414yes—
96C2×M16C2414yes—
97C2×M16C2412yes—
98C22×C8—21no—
99C22×C8—23no—
100C22×C8C2418yes—
101C4×Q8C4⋊C4112yes—
102C4×Q8C4⋊C4112yes—
103C4×Q8C4⋊C414yes—
104C4×Q8C4⋊C414yes—
105C4×Q8C22×C4112yes—
106C4×Q8C22×C414yes—
107C4×Q8C2×D414yes—
108C4×Q8C2×D4112yes—
109C2×C4⋊C4C4⋊C416yes—
110C2×C4⋊C4C4⋊C4112yes—
111C2×C4⋊C4C4⋊C413yes—
112C2×C4⋊C4C4⋊C413yes—
113C2×C4⋊C4C2×Q8112yes—
114C2×C42C22×C416yes—
115C2×C42C22×C413yes—
116C2×C42C22×C413yes—
117C2×C42C2×D413yes—
118C2×C42C2×D413yes—
119C2×C42C2×D416yes—
120C2×C42C2×Q816yes—
121C2×C42C2416yes—
122#108—216no—
123#108—424no—
124C4⋊C8—212no—
125C4⋊C8—212no—
126C4⋊C8—26no—
127C4⋊C8—26no—
128C4⋊C8—26no—
129C4⋊C8—26no—
130C4⋊C8C22×C4124yes—
131C4⋊C8C22×C4124yes—
132C22⋊C8—24no—
133C22⋊C8—24no—
134C22⋊C8C22×C418yes—
135C22⋊C8C22×C418yes—
136#97—212no—
137#97—212no—
138C4×C8—212no—
139C4×C8—212no—
140C4×C8C22×C4124yes—
141C4×C8C22×C4124yes—
142C24C22×Q811yescharacteristic
143C2×Q8—216no—
144C2×Q8—216no—
145C2×Q8—212no—
146C2×Q8#10818yescomplemented
147C2×Q8C2×C4⋊C4112yes—
148C2×Q8C2×C4⋊C414yes—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—212no—
152C22×C4—83no—
153C22×C4C2×C4⋊C413yes—
154C22×C4C2×C4⋊C416yes—
155C22×C4C2×C4⋊C416yes—
156C22×C4C2×C4⋊C413yes—
157C22×C4C2×C4⋊C413yes—
158C22×C4C2×C4⋊C413yes—
159C22×C4C23×C411yescharacteristic
160C22×C4C23×C411yescharacteristic
161C22×C4C22×D411yescharacteristic
162C22×C4C22×D411yescharacteristic
163C22×C4C22×Q811yescharacteristic
164C22×C4C22×Q816yes—
165C22×C4C22×Q816yes—
166C22×C4C2511yescharacteristic
167M16—24no—
168M16—22no—
169M16—28no—
170M16—28no—
171M16—212no—
172M16—46no—
173M162−1+414yes—
174C2×C8—24no—
175C2×C8—22no—
176C2×C8—28no—
177C2×C8—28no—
178C2×C8—24no—
179C2×C8—24no—
180C2×C8—26no—
181C2×C8—46no—
182C2×C8C23×C4116yes—
183C4⋊C4C2×C4⋊C4124yes—
184C4⋊C4C2×C4⋊C4112yes—
185C4⋊C4C2×C4⋊C4112yes—
186C42C2×C4⋊C416yes—
187C42C2×C4⋊C416yes—
188C42C2×C4⋊C416yes—
189C42C2×C4⋊C416yes—
190C42C23×C416yes—
191C42C23×C416yes—
192C42C22×D416yes—
193C42C22×D416yes—
194C23—22no—
195C23—22no—
196C23—41no—
197C23C22×C4⋊C411yescharacteristic
198C23C22×C4⋊C411yescharacteristic
199C23C23×Q811yescharacteristic
200Q8—224no—
201Q8C2×#108116yes—
202C2×C4—26no—
203C2×C4—22no—
204C2×C4—212no—
205C2×C4—212no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—43no—
213C2×C4—43no—
214C2×C4—46no—
215C2×C4—46no—
216C2×C4C2×#108112yes—
217C2×C4C22×C4⋊C411yescharacteristic
218C2×C4C22×C4⋊C411yescharacteristic
219C2×C4C22×C4⋊C416yes—
220C2×C4C22×C4⋊C416yes—
221C2×C4C22×C4⋊C416yes—
222C2×C4C22×C4⋊C416yes—
223C2×C4C24×C411yescharacteristic
224C2×C4C23×D411yescharacteristic
225C2×C4C2×2−1+412yes—
226C2×C4#58512yes—
227C8—216no—
228C8—24no—
229C8—44no—
230C8—46no—
231C8—44no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—44no—
240C22C22×#10812yes—
241C22C23×C4⋊C411yescharacteristic
242C22#350212yes—
243C4—21no—
244C4—21no—
245C4—26no—
246C4—26no—
247C4—26no—
248C4C22×#108112yes—
249C4C4×2−1+412yes—
250C4#348611yescharacteristic
251C4#348711yescharacteristic
252C2—22no—
253C2—22no—
254C2—41no—
255C2#3424911yescharacteristic
256C2C23×#10811yescharacteristic
257C2#6011311yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C23×C4⋊C4
prime radicalC24C22×Q8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #22746C2semidirect
5. Q8×#108C2semidirect
146. C2×Q8#108semidirect

Maximal subgroups

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

subgroupindex
#602482
C2×#29822
C2×Q8×M162
#227462
Q8×#1082
#219932
#213702