Finite Group Databasebeta

← #475650

Subgroups — #475650

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×#1845C211yesmaximal, characteristic
4#11365C212yesmaximal
5#11365C212yesmaximal
6#11354C212yesmaximal
7#11270C212yesmaximal
8#11239C212yesmaximal
9#10822C211yesmaximal, characteristic
10C22×#107C2212yes—
11C22×#107C2212yes—
12C2×#424C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14#2296C2211yescharacteristic
15C2×#384C2212yes—
16#2100—24no—
17#2099—24no—
18#1916—24no—
19#1845—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×#107—24no—
42C2×#107—24no—
43C2×#107—24no—
44C2×#107—24no—
45C2×#107—44no—
46#424—24no—
47C2×C4⋊C8—22no—
48C2×C4×C8—24no—
49C2×C4×C8D414yes—
50#389D412yes—
51#389D412yes—
52#385—22no—
53#384—22no—
54#384—22no—
55#384—24no—
56C4×C4⋊C4—22no—
57#343—44no—
58#343—44no—
59#341—24no—
60#341—24no—
61#340—28no—
62#340—28no—
63#336—22no—
64#336—22no—
65#336—22no—
66#336—22no—
67#336—22no—
68#336—22no—
69C23×C4—24no—
70C23×C4#4112yes—
71C23×C4C2411yescharacteristic
72C2×M16—24no—
73C2×M16—24no—
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—22no—
77C2×M16—24no—
78C2×M16—22no—
79C2×M16—24no—
80C2×M16—24no—
81C2×M16—22no—
82C2×M16—22no—
83C2×M16—24no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—24no—
91C22×C8C2×D412yes—
92C22×C8C2×D412yes—
93C22×C8#4114yes—
94C22×C8#4114yes—
95#117—22no—
96#117—22no—
97#117—22no—
98#117—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—44no—
108C2×C4⋊C4—44no—
109C2×C4⋊C4—88no—
110C2×C4⋊C4C2×D412yes—
111C2×C4⋊C4C2×D412yes—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D412yes—
121#107—48no—
122#107—48no—
123C4⋊C8—44no—
124C4×C8—28no—
125#95—22no—
126#95—22no—
127C24C2×#4111yescharacteristic
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—24no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—44no—
134C22×C4—44no—
135C22×C4—44no—
136C22×C4—88no—
137C22×C4#12112yes—
138C22×C4#12112yes—
139C22×C4C22×D411yescharacteristic
140C22×C4C22×D411yescharacteristic
141C22×C4C2×#4112yes—
142C22×C4C2×#4112yes—
143C22×C4H(5,𝔽2)12yes—
144C22×C42−1+412yes—
145C22×C42−1+412yes—
146C22×C42−1+412yes—
147M16—44no—
148M16—44no—
149M16—44no—
150M16—44no—
151M16—44no—
152M16—44no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—28no—
158C2×C8—28no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—44no—
166C2×C8—44no—
167C2×C8—44no—
168C2×C8#12314yes—
169C2×C8#12414yes—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—88no—
179C4⋊C4—88no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C42—22no—
183C42—22no—
184C42—22no—
185C42—22no—
186C42—41no—
187C42—41no—
188C42—41no—
189C42—41no—
190C23—22no—
191C23—22no—
192C23C2×#12111yescharacteristic
193C23#54112yes—
194C23#55212yes—
195C23#55912yes—
196C2×C4—216no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—84no—
208C2×C4—88no—
209C2×C4—164no—
210C2×C4—164no—
211C2×C4#47214yes—
212C2×C4#47314yes—
213C2×C4C2×#12311yescharacteristic
214C2×C4C2×#12411yescharacteristic
215C2×C4#53511yescharacteristic
216C2×C4#53611yescharacteristic
217C2×C4#53611yescharacteristic
218C2×C4#53611yescharacteristic
219C2×C4#53612yes—
220C2×C4#53914yes—
221C2×C4#54114yes—
222C2×C4#54712yes—
223C2×C4#55012yes—
224C2×C4#55214yes—
225C2×C4#56314yes—
226C8—24no—
227C8—24no—
228C8—81no—
229C8—81no—
230C8—84no—
231C22—24no—
232C22—22no—
233C22—22no—
234C22—24no—
235C22#248611yescharacteristic
236C22#254811yescharacteristic
237C22#262111yescharacteristic
238C22#279414yes—
239C22C2×#47212yes—
240C22C2×#47312yes—
241C4—24no—
242C4—24no—
243C4—82no—
244C4—82no—
245C4—164no—
246C4#250311yescharacteristic
247C4#262311yescharacteristic
248C4#264211yescharacteristic
249C4#272114yes—
250C4#282111yescharacteristic
251C2—24no—
252C2#1706311yescharacteristic
253C2#2767512yes—
254C2#2876211yescharacteristic
255C2#2876311yescharacteristic
256C2#2988612yes—
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×#18452
#113652
#113542
#112702
#112392
#108222