Finite Group Databasebeta

← #475498

Subgroups — #475498

Order 512.

541 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 541 conjugacy classes of subgroups — too many to draw. Only its 156 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
1#11361C212yesmaximal
2#11285C212yesmaximal
3#11277C212yesmaximal
4#11245C212yesmaximal
5#11234C212yesmaximal
6#11218C212yesmaximal
7#10817C211yesmaximal, characteristic
8#10814C211yesmaximal, characteristic
9#10799C211yesmaximal, characteristic
10#2911C2212yes—
11#2911C2212yes—
12#2905C2212yes—
13C2×#405C2211yescharacteristic
14C2×#344C2212yes—
15C2×#344C2212yes—
16#1740C2212yes—
17#1740C2212yes—
18#1740C2211yescharacteristic
19#1740C2211yescharacteristic
20C2×#341C2211yescharacteristic
21#1735C2211yescharacteristic
22#1735C2212yes—
23#1733C2212yes—
24C2×#338C2211yescharacteristic
25C2×H(3,ℤ/4)C2212yes—
26C2×H(3,ℤ/4)C2212yes—
27C2×H(3,ℤ/4)C2212yes—
28C2×H(3,ℤ/4)C2211yescharacteristic
29C2×H(3,ℤ/4)C2212yes—
30#1729C2212yes—
31C22×M16C2312yes—
32C22×M16C2311yescharacteristic
33C22×M16C2312yes—
34C23×C8C2311yescharacteristic
35C2×#117C2312yes—
36C2×#117C2312yes—
37C2×#117C2312yes—
38C2×#117C2312yes—
39C2×#117C2311yescharacteristic
40#428—24no—
41#428—42no—
42#428—42no—
43#428—42no—
44#427—24no—
45#427—42no—
46#424—24no—
47#424—42no—
48#423—24no—
49#405—42no—
50C2×#97Q812yes—
51C2×C4×C8Q812yes—
52#344—42no—
53#344—42no—
54#342—41no—
55#342—42no—
56#342—41no—
57#342—42no—
58#341—41no—
59#340—42no—
60#340—41no—
61#339—24no—
62#338—41no—
63H(3,ℤ/4)—24no—
64H(3,ℤ/4)—24no—
65H(3,ℤ/4)—24no—
66H(3,ℤ/4)—42no—
67H(3,ℤ/4)—41no—
68#336—24no—
69C23×C4C2411yescharacteristic
70C2×M16—22no—
71C2×M16—22no—
72C2×M16—22no—
73C2×M16—22no—
74C2×M16—21no—
75C2×M16—22no—
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—22no—
79C2×M16—21no—
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—22no—
83C2×M16—22no—
84C2×M16—22no—
85C2×M16#4114yes—
86C22×C8—22no—
87C22×C8—21no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—21no—
91C22×C8C2×Q811yescharacteristic
92C22×C8C2×Q811yescharacteristic
93C22×C8C2×Q812yes—
94C22×C8#4114yes—
95#117—22no—
96#117—22no—
97#117—22no—
98#117—22no—
99#117—22no—
100#117—22no—
101#117—22no—
102#117—22no—
103#117—22no—
104#117—22no—
105#117—22no—
106#117—22no—
107#117—22no—
108#117—22no—
109#117—22no—
110#117—22no—
111#117—22no—
112#117#4114yes—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42C2×Q812yes—
128#107—44no—
129#107—44no—
130#106—44no—
131#106—44no—
132C4⋊C8—48no—
133#97—42no—
134C4×C8—42no—
135C24H(5,𝔽2)11yescharacteristic
136C22×C4—22no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4C22×Q811yescharacteristic
142C22×C4C2×#4112yes—
143C22×C4C2×#4112yes—
144C22×C4C2×#4112yes—
145C22×C4H(5,𝔽2)12yes—
146C22×C4H(5,𝔽2)11yescharacteristic
147C22×C4H(5,𝔽2)11yescharacteristic
148C22×C42−1+411yescharacteristic
149C22×C42−1+411yescharacteristic
150C22×C42−1+411yescharacteristic
151M16—24no—
152M16—24no—
153M16—42no—
154M16—42no—
155M16—42no—
156M16—42no—
157M16—42no—
158M16—42no—
159M16—42no—
160M16—42no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—22no—
164C2×C8—24no—
165C2×C8—21no—
166C2×C8—22no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—22no—
171C2×C8—24no—
172C2×C8—24no—
173C2×C8—24no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—42no—
178C4⋊C4—24no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C22⋊C4—24no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C42—22no—
197C42—22no—
198C42—24no—
199C42—42no—
200C42—42no—
201C42—42no—
202C42—42no—
203C42—42no—
204C42—42no—
205C42—42no—
206C23—81no—
207C23#54012yes—
208C23#54411yescharacteristic
209C23#55312yes—
210C23#56211yescharacteristic
211C23#56211yescharacteristic
212C2×C4—42no—
213C2×C4—44no—
214C2×C4—82no—
215C2×C4—84no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—162no—
220C2×C4—162no—
221C2×C4—162no—
222C2×C4#53311yescharacteristic
223C2×C4#53311yescharacteristic
224C2×C4#53311yescharacteristic
225C2×C4#53812yes—
226C2×C4#53812yes—
227C2×C4#53912yes—
228C2×C4#54212yes—
229C2×C4#54212yes—
230C2×C4#54311yescharacteristic
231C2×C4#54411yescharacteristic
232C2×C4#54411yescharacteristic
233C2×C4#55112yes—
234C2×C4#55212yes—
235C2×C4#55212yes—
236C2×C4#55312yes—
237C2×C4#55612yes—
238C2×C4#56211yescharacteristic
239C2×C4#56411yescharacteristic
240C8—41no—
241C8—41no—
242C8—42no—
243C8—44no—
244C8—44no—
245C22—41no—
246C22—42no—
247C22—42no—
248C22—41no—
249C22—41no—
250C22#260611yescharacteristic
251C22#266711yescharacteristic
252C22#274112yes—
253C22#280012yes—
254C22#284511yescharacteristic
255C4—82no—
256C4—84no—
257C4—162no—
258C4—162no—
259C4—162no—
260C4#260511yescharacteristic
261C4#260611yescharacteristic
262C4#266511yescharacteristic
263C4#266511yescharacteristic
264C4#275012yes—
265C4#276712yes—
266C2—22no—
267C2—21no—
268C2—22no—
269C2—21no—
270C2—21no—
271C2#1705611yescharacteristic
subgroupG/H
centerC4#2605
commutator, Frattini, cosocleC23×C4C24
socleC2#17056
prime radicalC24H(5,𝔽2)
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#113612
#112852
#112772
#112452
#112342
#112182
#108172
#108142
#107992