Finite Group Databasebeta

← #507648

Subgroups — #507648

Order 512.

438 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 166 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#33257C211yesmaximal, characteristic
2C4×#499C211yesmaximal, characteristic
3#13507C212yesmaximal
4#13453C212yesmaximal
5#13178C212yesmaximal
6#13178C212yesmaximal
7#13168C212yesmaximal
8#13167C212yesmaximal
9#12653C211yesmaximal, characteristic
10C2×#499C2211yescharacteristic
11#2864—21no—
12C4×#125—21no—
13C2×#367C2212yes—
14C2×#367C2212yes—
15C2×#366C2212yes—
16C2×#366C2212yes—
17C2×C4⋊C16C2212yes—
18C2×#346C2211yescharacteristic
19C2×#346C2211yescharacteristic
20#1991—21no—
21#1942C2212yes—
22#1942C2212yes—
23#1942C2212yes—
24#1941—21no—
25#1941—21no—
26#1941—21no—
27#1846C2211yescharacteristic
28#1846C2211yescharacteristic
29#1846C2211yescharacteristic
30#1835—21no—
31C4×#107C2212yes—
32C4×#106—21no—
33C42×C8C2211yescharacteristic
34#1380C2212yes—
35#1380C2212yes—
36#1380C2212yes—
37#1380C2212yes—
38#1374C2212yes—
39C2×#125—21no—
40C22×C16C2312yes—
41C22×C16C2312yes—
42#499—24no—
43C2×#107C2312yes—
44C2×#107C2312yes—
45C2×#107C2312yes—
46C2×#107C2312yes—
47C2×#106—21no—
48C2×#106—21no—
49C2×#106—21no—
50C2×#106—21no—
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8D411yescharacteristic
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2311yescharacteristic
56C2×C4×C8C2311yescharacteristic
57#389—22no—
58#389—22no—
59#389—22no—
60#387—42no—
61#387—42no—
62#387—42no—
63#382—21no—
64#382—21no—
65#382—21no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—42no—
68C43D411yescharacteristic
69#367—24no—
70#367—42no—
71#366—24no—
72#366—42no—
73C4⋊C16Q818yes—
74#346—41no—
75#346—41no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80C22×C8C2×D411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C22×C8C2411yescharacteristic
84#125—24no—
85#125—42no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C42—42no—
95C2×C42D812yes—
96C2×C42D812yes—
97C2×C42C2×D411yescharacteristic
98C2×C42C2×D411yescharacteristic
99C2×C42C2×D411yescharacteristic
100C2×C16—22no—
101C2×C16—22no—
102C2×C16—22no—
103C2×C16—22no—
104C2×C16C2×Q814yes—
105C2×C16C2×Q814yes—
106#107—22no—
107#107—22no—
108#107—22no—
109#107—22no—
110#107—22no—
111#107—22no—
112#107—22no—
113#107#4114yes—
114#106—24no—
115#106—42no—
116#106—42no—
117#106—42no—
118C4×C8—22no—
119C4×C8—21no—
120C4×C8—21no—
121C4×C8—41no—
122C4×C8—41no—
123C4×C8Q1614yes—
124C4×C8C2×Q814yes—
125#95—42no—
126#95—42no—
127#95—42no—
128#95—41no—
129#95—41no—
130#95—41no—
131#95—41no—
132#95—42no—
133#95—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4C2×D812yes—
139C22×C4C2×D812yes—
140C22×C4C2×D811yescharacteristic
141C22×C4C2×D811yescharacteristic
142C22×C4C22×D411yescharacteristic
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8C4⋊Q812yes—
150C2×C8C2×Q1612yes—
151C2×C8C2×Q1612yes—
152C2×C8C22×Q812yes—
153C2×C8C2×#4112yes—
154C2×C8H(5,𝔽2)11yescharacteristic
155C2×C82−1+411yescharacteristic
156C4⋊C4—24no—
157C4⋊C4—48no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—84no—
166C4⋊C4—84no—
167C4⋊C4—84no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—42no—
173C42—42no—
174C42—84no—
175C42C4⋊Q812yes—
176C42C2×Q1612yes—
177C16—24no—
178C16—24no—
179C23C22×D811yescharacteristic
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—44no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4#50012yes—
201C2×C4#50012yes—
202C2×C4#52511yescharacteristic
203C2×C4C2×C4⋊Q811yescharacteristic
204C2×C4#53711yescharacteristic
205C2×C4C22×Q1611yescharacteristic
206C2×C4#57511yescharacteristic
207C2×C4#57511yescharacteristic
208C8—22no—
209C8—22no—
210C8—22no—
211C8#55412yes—
212C8#55712yes—
213C22#310011yescharacteristic
214C22C2×#50011yescharacteristic
215C22#324511yescharacteristic
216C22#341412yes—
217C22#341412yes—
218C4—22no—
219C4—22no—
220C4—42no—
221C4—42no—
222C4—84no—
223C4—162no—
224C4—162no—
225C4—162no—
226C4#258411yescharacteristic
227C4#310411yescharacteristic
228C4#316611yescharacteristic
229C4#324911yescharacteristic
230C2#2931211yescharacteristic
231C2#3367112yes—
232C2#3372811yescharacteristic
233C2#3373211yescharacteristic
234C2#3382212yes—
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
#332572
C4×#4992
#135072
#134532
#131782
#131682
#131672
#126532