Finite Group Databasebeta

← #475651

Subgroups — #475651

Order 512.

674 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 186 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#23746C212yesmaximal
2#21020C211yesmaximal, characteristic
3C2×#1846C211yesmaximal, characteristic
4#11366C212yesmaximal
5#11366C212yesmaximal
6#11355C212yesmaximal
7#11270C212yesmaximal
8#11239C212yesmaximal
9#10822C211yesmaximal, characteristic
10C22×#107C2212yes—
11C22×#106C2212yes—
12C2×#424C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14#2296C2211yescharacteristic
15C2×#382C2212yes—
16#2101—28no—
17#1916—24no—
18#1846—24no—
19C2×#343C2214yes—
20C2×#343C2214yes—
21C2×#341C2212yes—
22C2×#341C2212yes—
23#1736C2214yes—
24#1736C2214yes—
25#1729C2211yescharacteristic
26#1729C2211yescharacteristic
27#1729C2211yescharacteristic
28#1728C2211yescharacteristic
29#1728C2212yes—
30C22×M16C2311yescharacteristic
31C22×M16C2311yescharacteristic
32C22×M16C2314yes—
33C23×C8C2311yescharacteristic
34C23×C8C2311yescharacteristic
35C2×#117C2311yescharacteristic
36C2×#117C2311yescharacteristic
37C22×C4⋊C4C2314yes—
38C22×C42C2311yescharacteristic
39C2×#107—24no—
40C2×#107—44no—
41C2×#106—24no—
42C2×#106—24no—
43C2×#106—24no—
44#424—24no—
45C2×C4⋊C8—22no—
46C2×C4×C8—24no—
47C2×C4×C8D414yes—
48#389D414yes—
49#385—22no—
50#382—24no—
51#382—24no—
52C4×C4⋊C4—22no—
53C2×#95—24no—
54#343—44no—
55#343—44no—
56#341—24no—
57#341—24no—
58#340—28no—
59#340—28no—
60#336—22no—
61#336—22no—
62#336—22no—
63#336—24no—
64#336—22no—
65C23×C4—24no—
66C23×C4#4112yes—
67C23×C4C2411yescharacteristic
68C2×M16—24no—
69C2×M16—24no—
70C2×M16—22no—
71C2×M16—22no—
72C2×M16—22no—
73C2×M16—24no—
74C2×M16—22no—
75C2×M16—24no—
76C2×M16—24no—
77C2×M16—22no—
78C2×M16—22no—
79C2×M16—24no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—24no—
87C22×C8C2×D412yes—
88C22×C8C2×D412yes—
89C22×C8#4114yes—
90C22×C8#4114yes—
91#117—22no—
92#117—22no—
93#117—22no—
94#117—22no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4C2×D412yes—
103C2×C4⋊C4C2×D412yes—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42C2×D412yes—
111C2×C42C2×D412yes—
112#107—48no—
113#106—48no—
114C4⋊C8—44no—
115C4×C8—28no—
116#95—24no—
117#95—48no—
118C24C2×#4111yescharacteristic
119C22×C4—24no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—44no—
123C22×C4—44no—
124C22×C4—44no—
125C22×C4—44no—
126C22×C4—88no—
127C22×C4#12114yes—
128C22×C4C22×D411yescharacteristic
129C22×C4C22×D411yescharacteristic
130C22×C4C2×#4112yes—
131C22×C4C2×#4112yes—
132C22×C4H(5,𝔽2)12yes—
133C22×C42−1+412yes—
134C22×C42−1+412yes—
135C22×C42−1+412yes—
136M16—44no—
137M16—44no—
138M16—44no—
139M16—44no—
140M16—44no—
141M16—44no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—28no—
147C2×C8—28no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—44no—
155C2×C8—44no—
156C2×C8—44no—
157C2×C8#12314yes—
158C2×C8#12414yes—
159C4⋊C4—24no—
160C4⋊C4—24no—
161C4⋊C4—48no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C22⋊C4—42no—
167C22⋊C4—42no—
168C42—22no—
169C42—22no—
170C42—24no—
171C42—41no—
172C42—41no—
173C42—41no—
174C42—41no—
175C23—24no—
176C23C2×#12111yescharacteristic
177C23#54112yes—
178C23#55212yes—
179C23#55912yes—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—48no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4—84no—
189C2×C4—88no—
190C2×C4—164no—
191C2×C4—164no—
192C2×C4C2×#12311yescharacteristic
193C2×C4C2×#12411yescharacteristic
194C2×C4#53511yescharacteristic
195C2×C4#53611yescharacteristic
196C2×C4#53611yescharacteristic
197C2×C4#53611yescharacteristic
198C2×C4#53612yes—
199C2×C4#53914yes—
200C2×C4#54114yes—
201C2×C4#54712yes—
202C2×C4#55012yes—
203C2×C4#55214yes—
204C2×C4#56314yes—
205C8—24no—
206C8—24no—
207C8—81no—
208C8—81no—
209C8—84no—
210C22—24no—
211C22—24no—
212C22—24no—
213C22#248611yescharacteristic
214C22#254811yescharacteristic
215C22#262111yescharacteristic
216C22#279414yes—
217C22#306714yes—
218C4—44no—
219C4—82no—
220C4—82no—
221C4—164no—
222C4#250311yescharacteristic
223C4#262311yescharacteristic
224C4#264211yescharacteristic
225C4#272114yes—
226C4#282111yescharacteristic
227C2—24no—
228C2#1706311yescharacteristic
229C2#2767412yes—
230C2#2876512yes—
231C2#2988812yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#237462
#210202
C2×#18462
#113662
#113552
#112702
#112392
#108222