Finite Group Databasebeta

← #511689

Subgroups — #511689

Order 512.

482 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 142 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#33258C211yesmaximal, characteristic
2#33242C211yesmaximal, characteristic
3#13530C212yesmaximal
4#13481C212yesmaximal
5#13475C214yesmaximal
6#13451C212yesmaximal
7#13409C212yesmaximal
8#12879C211yesmaximal, characteristic
9C2×#500C2211yescharacteristic
10C2×#494C2211yescharacteristic
11C2×#487—21no—
12#2862—21no—
13#2859—21no—
14C2×#366C2214yes—
15C2×C4⋊C16C2212yes—
16C2×#358C2214yes—
17C2×#358C2214yes—
18#1969—21no—
19#1969—21no—
20#1969C2212yes—
21#1969C2212yes—
22#1942—21no—
23#1942—21no—
24#1942C2212yes—
25#1942C2212yes—
26#1845C2211yescharacteristic
27#1845C2211yescharacteristic
28#1834—21no—
29C42×C8C2211yescharacteristic
30#1380C2214yes—
31#1374C2212yes—
32#1374C2212yes—
33C22×Q16—21no—
34C22×Q16C2312yes—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×#125—41no—
38C22×C16C2314yes—
39#500—42no—
40#494—42no—
41#487—44no—
42C2×#107—21no—
43C2×#107—21no—
44C2×#107—21no—
45C2×#107C2312yes—
46C2×#107C2312yes—
47C2×#107C2312yes—
48C2×#103—22no—
49C2×#103—22no—
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8C2311yescharacteristic
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2311yescharacteristic
56#391—22no—
57#391—22no—
58#391—41no—
59#391—41no—
60#389—22no—
61#389—22no—
62#389—41no—
63#389—41no—
64#384—21no—
65#384—21no—
66#382—41no—
67#382—41no—
68#382—41no—
69C43D411yescharacteristic
70#366—44no—
71C4⋊C16—42no—
72#358—44no—
73#358—44no—
74#340—22no—
75#340—22no—
76C22×Q8—22no—
77C22×Q8—41no—
78C2×Q16—22no—
79C2×Q16—24no—
80C2×Q16—24no—
81C2×Q16—22no—
82C2×Q16—42no—
83C2×Q16—42no—
84C2×Q16—42no—
85C22×C8C2×D411yescharacteristic
86C22×C8C2×D411yescharacteristic
87C22×C8C2×D411yescharacteristic
88C22×C8C2411yescharacteristic
89C4⋊Q8—42no—
90C4⋊Q8—42no—
91C4⋊Q8—42no—
92C4⋊Q8—42no—
93#125—84no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C42D811yescharacteristic
107C2×C42D811yescharacteristic
108C2×C42D811yescharacteristic
109C2×C42D811yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C16—24no—
114C2×C16—24no—
115C2×C16—24no—
116#107—22no—
117#107—24no—
118#107—24no—
119#107—22no—
120#107—42no—
121#107—42no—
122#107—42no—
123#103—44no—
124#103—44no—
125C4×C8—22no—
126C4×C8—22no—
127C4×C8—21no—
128C4×C8—22no—
129C4×C8—22no—
130C4×C8—21no—
131C4×C8—41no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95—42no—
137#95—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—82no—
145C2×Q8—82no—
146C2×Q8—82no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4C2×D811yescharacteristic
152C22×C4C2×D811yescharacteristic
153C22×C4C2×D811yescharacteristic
154C22×C4C2×D811yescharacteristic
155C22×C4C2×D811yescharacteristic
156C22×C4C2×D811yescharacteristic
157C22×C4C22×D411yescharacteristic
158Q16—44no—
159Q16—44no—
160Q16—84no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8AGL(1,ℤ/8)12yes—
169C2×C8#13712yes—
170C2×C8H(5,𝔽2)11yescharacteristic
171C2×C8H(5,𝔽2)12yes—
172C2×C82−1+412yes—
173C2×C82−1+411yescharacteristic
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C4⋊C4—82no—
183C4⋊C4—84no—
184C4⋊C4—82no—
185C4⋊C4—82no—
186C4⋊C4—82no—
187C4⋊C4—84no—
188C4⋊C4—82no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—22no—
196C42—22no—
197C42—41no—
198C42Q3214yes—
199C16—44no—
200C23C22×D811yescharacteristic
201Q8—82no—
202Q8—82no—
203Q8—82no—
204Q8—82no—
205Q8—164no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4C2×Q3212yes—
232C2×C4C2×Q3212yes—
233C2×C4C2×Q3212yes—
234C2×C4#50912yes—
235C2×C4#51012yes—
236C2×C4#53511yescharacteristic
237C2×C4#53611yescharacteristic
238C2×C4#53711yescharacteristic
239C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
240C2×C4C2×#13711yescharacteristic
241C2×C4#57511yescharacteristic
242C8—22no—
243C8—22no—
244C8—41no—
245C8#56212yes—
246C8#56312yes—
247C22#320511yescharacteristic
248C22#322611yescharacteristic
249C22#324511yescharacteristic
250C22C22×Q3211yescharacteristic
251C22C2×#50911yescharacteristic
252C22C2×#51011yescharacteristic
253C22#341411yescharacteristic
254C4—22no—
255C4—22no—
256C4—22no—
257C4—22no—
258C4—22no—
259C4—41no—
260C4—162no—
261C4—162no—
262C4—162no—
263C4—162no—
264C4#282011yescharacteristic
265C4#321511yescharacteristic
266C4#323411yescharacteristic
267C4#325411yescharacteristic
268C2#3281811yescharacteristic
269C2#3377511yescharacteristic
270C2#3377911yescharacteristic
271C2#3379611yescharacteristic
272C2#3380211yescharacteristic
273C2#3381511yescharacteristic
274C2#3382011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332582
#332422
#135302
#134812
#134752
#134512
#134092
#128792