Finite Group Databasebeta

← #341534

Subgroups — #341534

Order 512.

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

normal subgroups only

This group has 470 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#28828C212yesmaximal
2#26507C212yesmaximal
3#10674C211yesmaximal, characteristic
4#10652C212yesmaximal
5#10576C212yesmaximal
6#10352C211yesmaximal, characteristic
7#9567C212yesmaximal
8#9533C212yesmaximal
9#8928C211yesmaximal, characteristic
10#2571C2212yes—
11#2571C2211yescharacteristic
12#2369C2211yescharacteristic
13#2101—22no—
14#2086—22no—
15#2085—22no—
16#2085—22no—
17#1990C2212yes—
18#1987C2212yes—
19#1956C2211yescharacteristic
20#1949—22no—
21#1948C2212yes—
22#1948C2212yes—
23#1948C2212yes—
24#1942—22no—
25#1939C2212yes—
26#1920—22no—
27#1846—22no—
28#1845C2212yes—
29#1845C2211yescharacteristic
30C4×#107C2212yes—
31C8×C4⋊C4C2212yes—
32C2×#333C2211yescharacteristic
33#1296C2212yes—
34#1296C2211yescharacteristic
35#1279C2212yes—
36#1279C2212yes—
37#1277C2212yes—
38#1273C2211yescharacteristic
39C2×C4⋊Q8C2311yescharacteristic
40C2×#107—22no—
41C2×#107—22no—
42C2×#107—41no—
43C2×#107D412yes—
44C2×#107D412yes—
45C2×#106—22no—
46C2×#106—22no—
47C2×#106—22no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8C2312yes—
50C2×C4⋊C8C2312yes—
51C2×#103—41no—
52C2×C4×C8—22no—
53C2×C4×C8C2311yescharacteristic
54C2×C4×C8C2312yes—
55#400—42no—
56#400—41no—
57#391—41no—
58#389—22no—
59#389—22no—
60#389—22no—
61#389—22no—
62#384C2311yescharacteristic
63#384C2312yes—
64#382—22no—
65#382—22no—
66#382—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4C2312yes—
69C4×C4⋊C4C2312yes—
70#340—22no—
71#340—22no—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340D412yes—
78#340D412yes—
79#336—22no—
80#336—22no—
81#333Q814yes—
82C22×Q8—21no—
83C22×C8—22no—
84C22×C8—21no—
85C22×C8—22no—
86C22×C8C2×D412yes—
87C22×C8C2×D412yes—
88C4⋊Q8—22no—
89C4⋊Q8—21no—
90C4⋊Q8C2×Q812yes—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4C2×D412yes—
101C2×C4⋊C4C2×D412yes—
102C2×C4⋊C4C2×D412yes—
103C2×C4⋊C4C2×D412yes—
104C2×C42—22no—
105C2×C42—22no—
106C2×C42—41no—
107C2×C42C2411yescharacteristic
108#107—24no—
109#107—42no—
110#107—44no—
111#107—82no—
112#107—82no—
113#106—44no—
114#106—82no—
115#106—82no—
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—44no—
120C4⋊C8C2×Q814yes—
121#103—82no—
122C4×C8—22no—
123C4×C8—22no—
124C4×C8—22no—
125C4×C8—44no—
126#95—22no—
127#95—22no—
128#95—42no—
129#95—41no—
130#95—42no—
131#95—42no—
132#95—42no—
133#95—42no—
134C2×Q8—22no—
135C2×Q8—41no—
136C2×Q8—81no—
137C2×Q8—81no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4—41no—
145C22×C4C22×D412yes—
146C22×C4H(5,𝔽2)11yescharacteristic
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—24no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8#13514yes—
162C4⋊C4—24no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—81no—
179C4⋊C4—82no—
180C4⋊C4—161no—
181C4⋊C4—161no—
182C4⋊C4—162no—
183C42—42no—
184C42—42no—
185C42—42no—
186C42—42no—
187C42—161no—
188C42C22×Q811yescharacteristic
189C422−1+411yescharacteristic
190C422−1+412yes—
191C23D4211yescharacteristic
192Q8—82no—
193Q8—161no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—81no—
215C2×C4—82no—
216C2×C4#53612yes—
217C2×C4#54411yescharacteristic
218C2×C4D4×Q812yes—
219C2×C4#56311yescharacteristic
220C2×C4C2×#13512yes—
221C2×C4#57512yes—
222C2×C4#57712yes—
223C2×C4#57812yes—
224C8—24no—
225C8—42no—
226C8—44no—
227C8—42no—
228C8—82no—
229C22#267711yescharacteristic
230C22#327112yes—
231C22#328312yes—
232C22#328911yescharacteristic
233C22#329211yescharacteristic
234C4—21no—
235C4—21no—
236C4—82no—
237C4—82no—
238C4—82no—
239C4—82no—
240C4—82no—
241C4—162no—
242C4—161no—
243C4#323212yes—
244C4#324212yes—
245C4#338112yes—
246C4#339012yes—
247C2#1986612yes—
248C2#2001911yescharacteristic
249C2#2015011yescharacteristic
250C2#3095212yes—
251C2#3097811yescharacteristic
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
#288282
#265072
#106742
#106522
#105762
#103522
#95672
#95332
#89282