Finite Group Databasebeta

← #475652

Subgroups — #475652

Order 512.

718 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 194 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#23745C212yesmaximal
2#21020C211yesmaximal, characteristic
3C2×#1844C211yesmaximal, characteristic
4#11366C212yesmaximal
5#11366C212yesmaximal
6#11354C212yesmaximal
7#11271C212yesmaximal
8#11238C212yesmaximal
9#10822C211yesmaximal, characteristic
10C22×#106C2212yes—
11C22×#106C2212yes—
12C2×#424C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14#2296C2211yescharacteristic
15C2×#384C2212yes—
16#2100—24no—
17#2099—24no—
18#1916—24no—
19#1844—24no—
20C2×#343C2214yes—
21C2×#343C2214yes—
22C2×#341C2212yes—
23C2×#341C2212yes—
24#1736C2214yes—
25#1736C2214yes—
26#1729C2211yescharacteristic
27#1729C2211yescharacteristic
28#1729C2211yescharacteristic
29#1728C2211yescharacteristic
30#1728C2212yes—
31C22×M16C2311yescharacteristic
32C22×M16C2311yescharacteristic
33C22×M16C2314yes—
34C23×C8C2311yescharacteristic
35C23×C8C2311yescharacteristic
36C2×#117C2311yescharacteristic
37C2×#117C2311yescharacteristic
38C22×C4⋊C4—24no—
39C22×C4⋊C4C2314yes—
40C22×C42C2311yescharacteristic
41C2×#106—24no—
42C2×#106—24no—
43C2×#106—24no—
44C2×#106—24no—
45C2×#106—44no—
46#424—24no—
47C2×C4⋊C8—22no—
48C2×C4×C8—24no—
49C2×C4×C8D414yes—
50#389D414yes—
51#385—22no—
52#384—22no—
53#384—24no—
54#384—22no—
55C4×C4⋊C4—22no—
56#343—44no—
57#343—44no—
58#341—24no—
59#341—24no—
60#340—28no—
61#340—28no—
62#336—22no—
63#336—22no—
64#336—22no—
65#336—22no—
66#336—22no—
67#336—22no—
68C23×C4—24no—
69C23×C4#4112yes—
70C23×C4C2411yescharacteristic
71C2×M16—24no—
72C2×M16—24no—
73C2×M16—22no—
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—24no—
77C2×M16—22no—
78C2×M16—24no—
79C2×M16—24no—
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—24no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—24no—
90C22×C8C2×D412yes—
91C22×C8C2×D412yes—
92C22×C8#4114yes—
93C22×C8#4114yes—
94#117—22no—
95#117—22no—
96#117—22no—
97#117—22no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4—88no—
108C2×C4⋊C4C2×D412yes—
109C2×C4⋊C4C2×D412yes—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42C2×D412yes—
117C2×C42C2×D412yes—
118#106—48no—
119#106—48no—
120C4⋊C8—44no—
121C4×C8—28no—
122#95—24no—
123C24C2×#4111yescharacteristic
124C22×C4—24no—
125C22×C4—28no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—44no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—88no—
132C22×C4#12114yes—
133C22×C4C22×D411yescharacteristic
134C22×C4C22×D411yescharacteristic
135C22×C4C2×#4112yes—
136C22×C4C2×#4112yes—
137C22×C4H(5,𝔽2)12yes—
138C22×C42−1+412yes—
139C22×C42−1+412yes—
140C22×C42−1+412yes—
141M16—44no—
142M16—44no—
143M16—44no—
144M16—44no—
145M16—44no—
146M16—44no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—28no—
152C2×C8—28no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—44no—
160C2×C8—44no—
161C2×C8—44no—
162C2×C8#12314yes—
163C2×C8#12414yes—
164C4⋊C4—24no—
165C4⋊C4—24no—
166C4⋊C4—48no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—88no—
172C4⋊C4—88no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C42—22no—
176C42—22no—
177C42—24no—
178C42—41no—
179C42—41no—
180C42—41no—
181C42—41no—
182C23—24no—
183C23C2×#12111yescharacteristic
184C23#54112yes—
185C23#55212yes—
186C23#55912yes—
187C2×C4—216no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—84no—
195C2×C4—88no—
196C2×C4—164no—
197C2×C4—164no—
198C2×C4#47118yes—
199C2×C4C2×#12311yescharacteristic
200C2×C4C2×#12411yescharacteristic
201C2×C4#53511yescharacteristic
202C2×C4#53611yescharacteristic
203C2×C4#53611yescharacteristic
204C2×C4#53611yescharacteristic
205C2×C4#53612yes—
206C2×C4#53914yes—
207C2×C4#54114yes—
208C2×C4#54712yes—
209C2×C4#55012yes—
210C2×C4#55214yes—
211C2×C4#56314yes—
212C8—24no—
213C8—24no—
214C8—81no—
215C8—81no—
216C8—84no—
217C22—24no—
218C22—24no—
219C22—24no—
220C22#248611yescharacteristic
221C22#254811yescharacteristic
222C22#262111yescharacteristic
223C22#279414yes—
224C22C2×#47114yes—
225C4—28no—
226C4—82no—
227C4—82no—
228C4—164no—
229C4#250311yescharacteristic
230C4#262311yescharacteristic
231C4#264211yescharacteristic
232C4#272114yes—
233C4#282111yescharacteristic
234C2—24no—
235C2#1706311yescharacteristic
236C2#2767212yes—
237C2#2876412yes—
238C2#2988712yes—
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
#237452
#210202
C2×#18442
#113662
#113542
#112712
#112382
#108222