Finite Group Databasebeta

← #479781

Subgroups — #479781

Order 512.

526 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 154 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#11374C211yesmaximal, characteristic
2#11362C212yesmaximal
3#11343C211yesmaximal, characteristic
4#11331C211yesmaximal, characteristic
5#11306C212yesmaximal
6#11274C212yesmaximal
7#11270C212yesmaximal
8#11252C212yesmaximal
9#11214C212yesmaximal
10C2×#430C2212yes—
11C2×#430C2211yescharacteristic
12C2×#428C2212yes—
13C22×#107C2211yescharacteristic
14#2905C2211yescharacteristic
15C2×#344C2212yes—
16C2×#344C2212yes—
17#1742C2211yescharacteristic
18C2×#343C2212yes—
19C2×#343C2212yes—
20C2×#343C2212yes—
21#1738C2212yes—
22C2×#341C2212yes—
23C2×#341C2211yescharacteristic
24#1736C2212yes—
25#1736C2212yes—
26#1733C2212yes—
27C2×#338C2212yes—
28C2×H(3,ℤ/4)C2211yescharacteristic
29#1729C2212yes—
30#1729C2211yescharacteristic
31C22×M16C2312yes—
32C22×M16C2312yes—
33C22×M16C2312yes—
34C22×M16C2311yescharacteristic
35C22×M16C2312yes—
36C23×C8C2311yescharacteristic
37C2×#117C2312yes—
38C2×#117C2311yescharacteristic
39C22×C4⋊C4C2312yes—
40#430—24no—
41#430—24no—
42#430—22no—
43#430—41no—
44C2×#108—42no—
45C2×#108—41no—
46#428—42no—
47#428—42no—
48C2×#107—22no—
49C2×#107—41no—
50C2×#107—42no—
51C2×#106—22no—
52#424—42no—
53#423—41no—
54#344—24no—
55#344—42no—
56#343—21no—
57#343—21no—
58#343—42no—
59#343—42no—
60#343—42no—
61#341—24no—
62#341—42no—
63#341Q814yes—
64#340—42no—
65#340—42no—
66#339—42no—
67#338—42no—
68H(3,ℤ/4)—41no—
69#336—41no—
70#336—42no—
71C23×C4—22no—
72C23×C4C2411yescharacteristic
73C2×M16—22no—
74C2×M16—21no—
75C2×M16—22no—
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—22no—
83C2×M16—22no—
84C2×M16—22no—
85C2×M16—22no—
86C2×M16—22no—
87C2×M16—22no—
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—22no—
93C2×M16—22no—
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—22no—
98C2×M16C2×Q814yes—
99C2×M16#4114yes—
100C2×M16#4112yes—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—21no—
104C22×C8—22no—
105C22×C8—21no—
106C22×C8—22no—
107C22×C8—21no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8C2×Q812yes—
111#117—22no—
112#117—22no—
113#117—22no—
114#117—22no—
115#117—22no—
116#117—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4#4111yescharacteristic
125C2×C4⋊C4#4111yescharacteristic
126C2×C22⋊C4—21no—
127C2×C22⋊C4—22no—
128C2×C42—21no—
129C2×C42—22no—
130#108—48no—
131#108—44no—
132#107—44no—
133#107—44no—
134#106—44no—
135C4⋊C8—44no—
136C242−1+411yescharacteristic
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—162no—
144C22×C4C22×Q811yescharacteristic
145C22×C4C2×#4111yescharacteristic
146C22×C4C2×#4111yescharacteristic
147C22×C4C2×#4112yes—
148C22×C4H(5,𝔽2)12yes—
149C22×C4H(5,𝔽2)11yescharacteristic
150C22×C4H(5,𝔽2)12yes—
151C22×C4H(5,𝔽2)11yescharacteristic
152C22×C42−1+411yescharacteristic
153C22×C42−1+412yes—
154M16—24no—
155M16—24no—
156M16—24no—
157M16—24no—
158M16—24no—
159M16—42no—
160M16—42no—
161M16—42no—
162M16—42no—
163M16—42no—
164M16—42no—
165M16—42no—
166M16—42no—
167M16—42no—
168M16—42no—
169M16—42no—
170M16—42no—
171M16—42no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—24no—
175C2×C8—24no—
176C2×C8—24no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C42—41no—
204C42—41no—
205C42—42no—
206C42—42no—
207C23—41no—
208C23—41no—
209C23#53911yescharacteristic
210C23#54112yes—
211C23#55711yescharacteristic
212C23#56212yes—
213C23#56311yescharacteristic
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—162no—
223C2×C4—162no—
224C2×C4—162no—
225C2×C4#53811yescharacteristic
226C2×C4#53812yes—
227C2×C4#53912yes—
228C2×C4#54012yes—
229C2×C4#54212yes—
230C2×C4#54412yes—
231C2×C4#54411yescharacteristic
232C2×C4#55112yes—
233C2×C4#55212yes—
234C2×C4#55411yescharacteristic
235C2×C4#55412yes—
236C2×C4#55611yescharacteristic
237C2×C4#55911yescharacteristic
238C2×C4#56011yescharacteristic
239C2×C4#56011yescharacteristic
240C2×C4#56212yes—
241C2×C4#56311yescharacteristic
242C2×C4#56312yes—
243C8—44no—
244C8—42no—
245C8—44no—
246C8—42no—
247C8—82no—
248C8—82no—
249C22—21no—
250C22—22no—
251C22—21no—
252C22—22no—
253C22—42no—
254C22—41no—
255C22—41no—
256C22#275512yes—
257C22#279411yescharacteristic
258C22#280511yescharacteristic
259C22#282611yescharacteristic
260C22#283612yes—
261C4—81no—
262C4—81no—
263C4—162no—
264C4—162no—
265C4#274512yes—
266C4#275211yescharacteristic
267C4#276812yes—
268C4#280711yescharacteristic
269C4#283011yescharacteristic
270C4#284311yescharacteristic
271C2—22no—
272C2—22no—
273C2—21no—
274C2—21no—
275C2#1724311yescharacteristic
276C2#3287011yescharacteristic
277C2#3287211yescharacteristic
subgroupG/H
center, socleC22#2826
commutator, Frattini, cosocleC23×C4C24
prime radicalC242−1+4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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
#113742
#113622
#113432
#113312
#113062
#112742
#112702
#112522
#112142