Finite Group Databasebeta

← #345705

Subgroups — #345705

Order 512.

505 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 144 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#26531C212yesmaximal
2#26513C212yesmaximal
3#10600C212yesmaximal
4#10594C212yesmaximal
5#10578C212yesmaximal
6#9831C211yesmaximal, characteristic
7#9598C211yesmaximal, characteristic
8#9596C211yesmaximal, characteristic
9#9058C212yesmaximal
10C2×#500C2211yescharacteristic
11#2571C2212yes—
12#2365C2212yes—
13#2101—22no—
14#2097—22no—
15#2087—22no—
16#2068—22no—
17#1990C2212yes—
18#1988C2212yes—
19#1981C2212yes—
20#1956C2212yes—
21#1949—22no—
22#1948C2212yes—
23#1947C2212yes—
24#1939C2212yes—
25#1939C2212yes—
26#1919—22no—
27#1865—22no—
28#1834C2211yescharacteristic
29#1805—22no—
30C4×#107C2211yescharacteristic
31C4×#106C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#335C2211yescharacteristic
34C2×#332C2212yes—
35C2×#322C2211yescharacteristic
36#1296C2212yes—
37#1279C2212yes—
38#1279C2212yes—
39C2×C4⋊Q8C2312yes—
40C2×C4×Q8—22no—
41#500—41no—
42#500Q814yes—
43C2×#107—22no—
44C2×#107—41no—
45C2×#107D412yes—
46C2×#106—22no—
47C2×#106D411yescharacteristic
48C2×#106D411yescharacteristic
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×#103—22no—
54C2×#103—22no—
55C2×#103—22no—
56C2×#103—22no—
57C2×C4×C8C2311yescharacteristic
58C2×C4×C8C2311yescharacteristic
59C2×C4×C8C2311yescharacteristic
60#400—42no—
61#391—22no—
62#389—22no—
63#389—22no—
64#389—22no—
65#387—42no—
66#384C2312yes—
67#382—22no—
68#382—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4C2312yes—
71C4×C4⋊C4C2312yes—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—41no—
79#340D412yes—
80#340D411yescharacteristic
81#340D411yescharacteristic
82#336—22no—
83#336—22no—
84#335—41no—
85#335—41no—
86#332—42no—
87#322—41no—
88C22×Q8—22no—
89C22×C8—21no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8C2×D411yescharacteristic
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8C2×Q814yes—
100C4×Q8—44no—
101C4×Q8—82no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4C2×D412yes—
110C2×C4⋊C4C2×D412yes—
111C2×C4⋊C4C2×D412yes—
112C2×C4⋊C4C2×D412yes—
113C2×C42—22no—
114C2×C42—22no—
115C2×C42—42no—
116C2×C42C2411yescharacteristic
117#107—44no—
118#107—42no—
119#107—82no—
120#106—22no—
121#106—41no—
122#106—44no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—44no—
128C4⋊C8—82no—
129#103—44no—
130#103—82no—
131#103—82no—
132#103—82no—
133C4×C8—21no—
134C4×C8—21no—
135C4×C8—21no—
136C4×C8—21no—
137C4×C8—21no—
138C4×C8C2×Q812yes—
139#95—22no—
140#95—22no—
141#95—42no—
142#95—42no—
143#95—42no—
144#95—42no—
145#95—42no—
146C2×Q8—24no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—42no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4C22×D411yescharacteristic
159C22×C4C22×D411yescharacteristic
160C22×C4H(5,𝔽2)11yescharacteristic
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—22no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8#13512yes—
179C4⋊C4—24no—
180C4⋊C4—24no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—82no—
196C4⋊C4—84no—
197C4⋊C4—162no—
198C42—42no—
199C42—42no—
200C42—42no—
201C42—42no—
202C42—84no—
203C42C22×Q811yescharacteristic
204C42H(5,𝔽2)11yescharacteristic
205C422−1+411yescharacteristic
206C422−1+411yescharacteristic
207C23D4211yescharacteristic
208Q8—44no—
209Q8—44no—
210Q8—82no—
211Q8—82no—
212C2×C4—22no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4#53711yescharacteristic
232C2×C4#53711yescharacteristic
233C2×C4#54311yescharacteristic
234C2×C4D4×Q811yescharacteristic
235C2×C4D4×Q811yescharacteristic
236C2×C4#56311yescharacteristic
237C2×C4C2×#13511yescharacteristic
238C2×C4#57511yescharacteristic
239C2×C4#57512yes—
240C2×C4#57611yescharacteristic
241C2×C4#57611yescharacteristic
242C2×C4#57812yes—
243C8—22no—
244C8—42no—
245C8—41no—
246C8—41no—
247C8—41no—
248C8—82no—
249C8—82no—
250C22#268311yescharacteristic
251C22#326912yes—
252C22#327311yescharacteristic
253C22#328711yescharacteristic
254C22#328912yes—
255C4—21no—
256C4—21no—
257C4—84no—
258C4—82no—
259C4—82no—
260C4—82no—
261C4—82no—
262C4—162no—
263C4#325011yescharacteristic
264C4#325611yescharacteristic
265C4#325911yescharacteristic
266C4#326111yescharacteristic
267C4#338011yescharacteristic
268C4#338911yescharacteristic
269C4#339012yes—
270C2#1987212yes—
271C2#2022312yes—
272C2#3123811yescharacteristic
273C2#3124911yescharacteristic
274C2#3125911yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#265312
#265132
#106002
#105942
#105782
#98312
#95982
#95962
#90582