Finite Group Databasebeta

← #511706

Subgroups — #511706

Order 512.

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

normal subgroups only

This group has 478 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#33312C211yesmaximal, characteristic
2#33267C211yesmaximal, characteristic
3#13538C212yesmaximal
4#13530C212yesmaximal
5#13505C212yesmaximal
6#13481C212yesmaximal
7#13451C212yesmaximal
8#13364C212yesmaximal
9#12894C211yesmaximal, characteristic
10C2×#500—21no—
11C2×#500C2212yes—
12#2868—21no—
13#2863—21no—
14C2×#366C2214yes—
15C2×#358C2214yes—
16#1969C2211yescharacteristic
17#1969C2211yescharacteristic
18#1969C2211yescharacteristic
19#1942C2211yescharacteristic
20#1942C2211yescharacteristic
21#1942C2211yescharacteristic
22#1933—21no—
23#1933—21no—
24#1933—21no—
25#1845—21no—
26#1845—21no—
27#1845C2212yes—
28#1845C2212yes—
29C42×C8C2211yescharacteristic
30#1380C2214yes—
31#1380C2214yes—
32#1374C2212yes—
33#1374C2212yes—
34#1374C2212yes—
35C22×Q16C2311yescharacteristic
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—41no—
38C2×C4⋊Q8—41no—
39C22×C16C2314yes—
40#500—44no—
41#500—44no—
42C2×#107—22no—
43C2×#107—22no—
44C2×#107—22no—
45C2×#107C2314yes—
46C2×#107C2311yescharacteristic
47C2×#107C2311yescharacteristic
48C2×#107C2311yescharacteristic
49C2×#103—21no—
50C2×#103—21no—
51C2×#103—21no—
52C2×C4×C8D411yescharacteristic
53C2×C4×C8D411yescharacteristic
54C2×C4×C8D411yescharacteristic
55C2×C4×C8C2311yescharacteristic
56C2×C4×C8C2311yescharacteristic
57C2×C4×C8C2311yescharacteristic
58#391—21no—
59#391—21no—
60#391—21no—
61#389—21no—
62#389—21no—
63#389—21no—
64#387—41no—
65#387—41no—
66#387—41no—
67#384—22no—
68#384—22no—
69#384—41no—
70#384—41no—
71#383—41no—
72C43D411yescharacteristic
73#366—44no—
74#358—44no—
75#340—21no—
76#340—21no—
77#340—21no—
78C22×Q8—21no—
79C22×Q8—42no—
80C2×Q16—22no—
81C2×Q16—22no—
82C2×Q16—22no—
83C22×C8C2×D411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C22×C8C2411yescharacteristic
87C4⋊Q8—44no—
88C4⋊Q8—44no—
89C4⋊Q8—84no—
90C4⋊Q8—84no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C42D811yescharacteristic
104C2×C42D811yescharacteristic
105C2×C42D811yescharacteristic
106C2×C42D811yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C16—24no—
111C2×C16—24no—
112C2×C16—24no—
113#107—24no—
114#107—22no—
115#107—24no—
116#107—22no—
117#107—22no—
118#107—44no—
119#107—44no—
120#107—44no—
121#103—42no—
122#103—42no—
123#103—42no—
124C4×C8—22no—
125C4×C8—22no—
126C4×C8—22no—
127C4×C8—22no—
128C4×C8—22no—
129C4×C8—22no—
130#95—41no—
131#95—41no—
132#95—41no—
133#95—41no—
134#95—41no—
135#95—41no—
136#95—41no—
137#95—41no—
138#95—41no—
139C2×Q8—41no—
140C2×Q8—41no—
141C2×Q8—41no—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—41no—
145C2×Q8—84no—
146C2×Q8—84no—
147C2×Q8—84no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—44no—
153C22×C4C2×D811yescharacteristic
154C22×C4C2×D811yescharacteristic
155C22×C4C2×D811yescharacteristic
156C22×C4C2×D811yescharacteristic
157C22×C4C2×D811yescharacteristic
158C22×C4C2×D811yescharacteristic
159C22×C4C22×D411yescharacteristic
160Q16—44no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8AGL(1,ℤ/8)12yes—
168C2×C8#13712yes—
169C2×C8#13712yes—
170C2×C8H(5,𝔽2)12yes—
171C2×C82−1+412yes—
172C2×C82−1+412yes—
173C4⋊C4—44no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—44no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—84no—
182C4⋊C4—84no—
183C4⋊C4—84no—
184C4⋊C4—84no—
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—22no—
198C42Q3214yes—
199C16—44no—
200C23C22×D811yescharacteristic
201Q8—81no—
202Q8—81no—
203Q8—81no—
204Q8—81no—
205Q8—168no—
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—84no—
219C2×C4—84no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—84no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4C2×Q3212yes—
234C2×C4C2×Q3212yes—
235C2×C4C2×Q3212yes—
236C2×C4#50912yes—
237C2×C4#50912yes—
238C2×C4#51012yes—
239C2×C4#53511yescharacteristic
240C2×C4#53611yescharacteristic
241C2×C4#53611yescharacteristic
242C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
243C2×C4C2×#13711yescharacteristic
244C2×C4C2×#13711yescharacteristic
245C8—22no—
246C8—22no—
247C8—22no—
248C8#56314yes—
249C22#320511yescharacteristic
250C22#322611yescharacteristic
251C22#322611yescharacteristic
252C22C22×Q3211yescharacteristic
253C22C2×#50911yescharacteristic
254C22C2×#50911yescharacteristic
255C22C2×#51011yescharacteristic
256C4—22no—
257C4—22no—
258C4—22no—
259C4—22no—
260C4—22no—
261C4—22no—
262C4—161no—
263C4—161no—
264C4—161no—
265C4—161no—
266C4—164no—
267C4#282111yescharacteristic
268C4#321811yescharacteristic
269C4#323811yescharacteristic
270C4#323911yescharacteristic
271C2#3283411yescharacteristic
272C2#3377511yescharacteristic
273C2#3378311yescharacteristic
274C2#3379611yescharacteristic
275C2#3379611yescharacteristic
276C2#3380411yescharacteristic
277C2#3380511yescharacteristic
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
#333122
#332672
#135382
#135302
#135052
#134812
#134512
#133642
#128942