Finite Group Databasebeta

← #325694

Subgroups — #325694

Order 512.

562 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 158 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#33225C211yesmaximal, characteristic
2#29226C211yesmaximal, characteristic
3#29225C211yesmaximal, characteristic
4#21673C211yesmaximal, characteristic
5#10541C214yesmaximal
6#9745C214yesmaximal
7#9703C211yesmaximal, characteristic
8#9702C211yesmaximal, characteristic
9#8273C211yesmaximal, characteristic
10C2×#487C2212yes—
11C2×#487C2212yes—
12#2868C2211yescharacteristic
13#2868C2211yescharacteristic
14C4×#125C2211yescharacteristic
15#2100—22no—
16#2099—22no—
17#1992—21no—
18#1992—21no—
19#1988C2214yes—
20#1987C2214yes—
21#1956—22no—
22#1956—22no—
23#1939C2212yes—
24#1916—22no—
25#1836—21no—
26#1836—21no—
27#1831C2211yescharacteristic
28#1771—22no—
29#1770C2212yes—
30#1767C2211yescharacteristic
31#1767C2211yescharacteristic
32C42×C8C2211yescharacteristic
33C2×#334C2212yes—
34C2×#333C2214yes—
35C2×#332C2214yes—
36#1277C2212yes—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8C2312yes—
40C2×C4⋊Q8C2312yes—
41C2×#125D411yescharacteristic
42#487—28no—
43#487—44no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8C2312yes—
46C2×C4⋊C8C2312yes—
47C2×C4⋊C8C2312yes—
48C2×#103—24no—
49C2×#103—22no—
50C2×#103—22no—
51C2×#97—22no—
52C2×#97—21no—
53C2×#97—21no—
54C2×C4×C8D412yes—
55C2×C4×C8D412yes—
56C2×C4×C8C2311yescharacteristic
57C2×C4×C8C2311yescharacteristic
58#389D412yes—
59#387—22no—
60#384—21no—
61#384—21no—
62#384—21no—
63#384—21no—
64#384—22no—
65#384—22no—
66#382D411yescharacteristic
67C4×C4⋊C4—22no—
68C4×C4⋊C4C2312yes—
69C43C2311yescharacteristic
70#340—24no—
71#340—24no—
72#340—24no—
73#340—24no—
74#336—22no—
75#336—22no—
76#336—22no—
77#336—21no—
78#336—21no—
79#334—24no—
80#333—44no—
81#332—44no—
82C22×Q8—22no—
83C22×Q8—22no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8C2×D412yes—
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D411yescharacteristic
90C4⋊Q8—24no—
91C4⋊Q8—24no—
92C4⋊Q8—24no—
93C4⋊Q8—24no—
94C4⋊Q8—44no—
95C4⋊Q8—44no—
96C4⋊Q8—84no—
97#125—22no—
98#125—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4C2×D412yes—
110C2×C4⋊C4C2×D412yes—
111C2×C42—22no—
112C2×C42C2×D412yes—
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42#4112yes—
116C2×C42C2411yescharacteristic
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120C4⋊C8—24no—
121C4⋊C8—22no—
122C4⋊C8—44no—
123#103—48no—
124#103—44no—
125#103—44no—
126#97—44no—
127#97—42no—
128#97—42no—
129C4×C8—22no—
130C4×C8—42no—
131C4×C8—42no—
132C4×C8#4114yes—
133#95—22no—
134#95—22no—
135#95—22no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—44no—
139C2×Q8—44no—
140C2×Q8—44no—
141C2×Q8—84no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4#12112yes—
147C22×C4#12112yes—
148C22×C4C22×D411yescharacteristic
149C22×C4C22×D411yescharacteristic
150C22×C4C2×#4111yescharacteristic
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8#12412yes—
166C2×C8#12412yes—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—24no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—84no—
178C4⋊C4—84no—
179C4⋊C4—84no—
180C4⋊C4—84no—
181C42—22no—
182C42—22no—
183C42—22no—
184C42—24no—
185C42—42no—
186C42—42no—
187C42#13712yes—
188C42#13712yes—
189C42C2×#4112yes—
190C42H(5,𝔽2)12yes—
191C23C2×#12111yescharacteristic
192Q8—82no—
193Q8—82no—
194Q8—84no—
195Q8—164no—
196C2×C4—22no—
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—82no—
209C2×C4—82no—
210C2×C4—84no—
211C2×C4#47014yes—
212C2×C4#47114yes—
213C2×C4C2×#12411yescharacteristic
214C2×C4#53511yescharacteristic
215C2×C4#55012yes—
216C2×C4#55912yes—
217C2×C4C2×#13711yescharacteristic
218C2×C4C2×#13711yescharacteristic
219C2×C4#57512yes—
220C2×C4#57512yes—
221C2×C4#57812yes—
222C8—22no—
223C8—22no—
224C8—42no—
225C8—82no—
226C8—82no—
227C8—82no—
228C22#254811yescharacteristic
229C22#304012yes—
230C22C2×#47011yescharacteristic
231C22#305911yescharacteristic
232C22C2×#47112yes—
233C4—22no—
234C4—24no—
235C4—24no—
236C4—82no—
237C4—82no—
238C4—162no—
239C4—162no—
240C4#313511yescharacteristic
241C4#314211yescharacteristic
242C4#321912yes—
243C4#339812yes—
244C4#339812yes—
245C2#1930212yes—
246C2#1936512yes—
247C2#2874411yescharacteristic
248C2#2875711yescharacteristic
249C2#2876411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ 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
#332252
#292262
#292252
#216732
#105412
#97452
#97032
#97022
#82732