Finite Group Databasebeta

← #508696

Subgroups — #508696

Order 512.

520 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 146 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#33242C211yesmaximal, characteristic
2C4×#500C211yesmaximal, characteristic
3#13531C212yesmaximal
4#13452C212yesmaximal
5#13409C212yesmaximal
6#13364C212yesmaximal
7#13132C214yesmaximal
8#12894C211yesmaximal, characteristic
9C2×#500C2211yescharacteristic
10C2×#500C2211yescharacteristic
11C2×#494C2211yescharacteristic
12C2×#487—21no—
13#2862—21no—
14C4×C4⋊Q8—21no—
15C2×#367C2214yes—
16C2×#358C2214yes—
17C2×#358C2214yes—
18#1990—21no—
19#1969—21no—
20#1969—21no—
21#1969C2212yes—
22#1969C2212yes—
23#1942—21no—
24#1942C2212yes—
25#1845C2211yescharacteristic
26C4×#107—21no—
27C4×#107C2212yes—
28C42×C8C2211yescharacteristic
29#1380C2214yes—
30#1374C2212yes—
31#1374C2212yes—
32#1374C2212yes—
33C22×Q16—21no—
34C22×Q16C2312yes—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4×Q8—41no—
39C22×C16C2314yes—
40#500—24no—
41#500—42no—
42#494—42no—
43#487—44no—
44C2×#107—21no—
45C2×#107—21no—
46C2×#107—21no—
47C2×#107C2312yes—
48C2×#107C2312yes—
49C2×#107C2312yes—
50C2×#103—22no—
51C2×#103—22no—
52C2×C4×C8D411yescharacteristic
53C2×C4×C8D411yescharacteristic
54C2×C4×C8D411yescharacteristic
55C2×C4×C8C2311yescharacteristic
56C2×C4×C8C2311yescharacteristic
57C2×C4×C8C2311yescharacteristic
58#391—22no—
59#391—22no—
60#391—41no—
61#391—41no—
62#391—41no—
63#389—22no—
64#389—41no—
65#384—21no—
66#382—41no—
67#382—41no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—41no—
70C43D411yescharacteristic
71#367—44no—
72#358—28no—
73#358—44no—
74#340—22no—
75#340—22no—
76C22×Q8—22no—
77C22×Q8—41no—
78C22×Q8—41no—
79C2×Q16—24no—
80C2×Q16—24no—
81C2×Q16—24no—
82C2×Q16—42no—
83C2×Q16—42no—
84C2×Q16—42no—
85C22×C8C2×D411yescharacteristic
86C22×C8C2×D411yescharacteristic
87C22×C8C2×D411yescharacteristic
88C22×C8C2411yescharacteristic
89C4⋊Q8—24no—
90C4⋊Q8—42no—
91C4⋊Q8—42no—
92C4⋊Q8—42no—
93C4⋊Q8—42no—
94C4⋊Q8—42no—
95C4×Q8—84no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C42—42no—
107C2×C42D811yescharacteristic
108C2×C42D811yescharacteristic
109C2×C42D811yescharacteristic
110C2×C42D811yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C16—24no—
115C2×C16—24no—
116C2×C16—24no—
117#107—24no—
118#107—22no—
119#107—24no—
120#107—24no—
121#107—42no—
122#107—42no—
123#107#4114yes—
124#103—44no—
125#103—44no—
126C4×C8—22no—
127C4×C8—22no—
128C4×C8—22no—
129C4×C8—22no—
130C4×C8—22no—
131C4×C8—41no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95—42no—
137#95—42no—
138C2×Q8—44no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—82no—
146C2×Q8—82no—
147C2×Q8—82no—
148C2×Q8—82no—
149C2×Q8—82no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4C2×D811yescharacteristic
155C22×C4C2×D811yescharacteristic
156C22×C4C2×D811yescharacteristic
157C22×C4C2×D811yescharacteristic
158C22×C4C2×D811yescharacteristic
159C22×C4C2×D811yescharacteristic
160C22×C4C22×D411yescharacteristic
161Q16—48no—
162Q16—84no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8AGL(1,ℤ/8)12yes—
171C2×C8#13712yes—
172C2×C8C2×#4112yes—
173C2×C8H(5,𝔽2)12yes—
174C2×C82−1+412yes—
175C4⋊C4—24no—
176C4⋊C4—44no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—82no—
183C4⋊C4—82no—
184C4⋊C4—82no—
185C4⋊C4—84no—
186C4⋊C4—82no—
187C4⋊C4—82no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—22no—
195C42—22no—
196C42—41no—
197C42—84no—
198C42SD3214yes—
199C16—44no—
200C23C22×D811yescharacteristic
201Q8—88no—
202Q8—82no—
203Q8—82no—
204Q8—82no—
205Q8—82no—
206Q8—164no—
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—21no—
220C2×C4—44no—
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×C4—82no—
232C2×C4C2×SD3212yes—
233C2×C4C2×SD3212yes—
234C2×C4C2×SD3212yes—
235C2×C4#51012yes—
236C2×C4#51012yes—
237C2×C4#52511yescharacteristic
238C2×C4#53511yescharacteristic
239C2×C4#53611yescharacteristic
240C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
241C2×C4C2×#13711yescharacteristic
242C2×C4#57511yescharacteristic
243C8—22no—
244C8—22no—
245C8—41no—
246C8#54214yes—
247C22#310011yescharacteristic
248C22#320511yescharacteristic
249C22#322611yescharacteristic
250C22C22×SD3211yescharacteristic
251C22C2×#51011yescharacteristic
252C22C2×#51011yescharacteristic
253C22#341411yescharacteristic
254C4—22no—
255C4—22no—
256C4—22no—
257C4—22no—
258C4—22no—
259C4—41no—
260C4—84no—
261C4—162no—
262C4—162no—
263C4—162no—
264C4#263211yescharacteristic
265C4#311211yescharacteristic
266C4#321511yescharacteristic
267C4#323511yescharacteristic
268C2#3005911yescharacteristic
269C2#3366111yescharacteristic
270C2#3367311yescharacteristic
271C2#3377211yescharacteristic
272C2#3378111yescharacteristic
273C2#3379511yescharacteristic
274C2#3380311yescharacteristic
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
#332422
C4×#5002
#135312
#134522
#134092
#133642
#131322
#128942