Finite Group Databasebeta

← #351001

Subgroups — #351001

Order 512.

586 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 174 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#33294C211yesmaximal, characteristic
2#33198C211yesmaximal, characteristic
3#33069C211yesmaximal, characteristic
4Q8×#97C211yesmaximal, characteristic
5#9902C211yesmaximal, characteristic
6#9702C211yesmaximal, characteristic
7#9576C214yesmaximal
8#9216C214yesmaximal
9#8786C211yesmaximal, characteristic
10C2×#477C2212yes—
11C2×C8×Q8C2212yes—
12#2868C2211yescharacteristic
13#2859C2211yescharacteristic
14C42×Q8C2211yescharacteristic
15#1992—22no—
16#1990C2214yes—
17#1987C2212yes—
18#1947—22no—
19#1947C2212yes—
20#1921—22no—
21#1863—22no—
22#1842C2211yescharacteristic
23#1836—21no—
24#1836—21no—
25#1831C2211yescharacteristic
26#1803—22no—
27#1803C2212yes—
28#1770—22no—
29#1770C2212yes—
30#1767—21no—
31#1767C2211yescharacteristic
32C4×#97—21no—
33C4×#97C2211yescharacteristic
34C2×#333C2214yes—
35#1296C2214yes—
36#1277C2214yes—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8C2312yes—
39C2×#125—21no—
40C2×C4×Q8D412yes—
41C2×C4×Q8C2311yescharacteristic
42#477—28no—
43C8×Q8—24no—
44C8×Q8—24no—
45C2×#107—22no—
46C2×#107—22no—
47C2×C4⋊C8—22no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8C2312yes—
50C2×C4⋊C8C2312yes—
51C2×C4⋊C8C2312yes—
52C2×#103—22no—
53C2×#103—22no—
54C2×#97—22no—
55C2×#97—21no—
56C2×#97D411yescharacteristic
57C2×#97D412yes—
58C2×#97D411yescharacteristic
59C2×C4×C8—22no—
60C2×C4×C8C2312yes—
61#389—22no—
62#389—22no—
63#384—22no—
64#384—21no—
65#384—21no—
66#384C2312yes—
67#382—21no—
68C4×C4⋊C4—21no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4D412yes—
71C4×C4⋊C4C2311yescharacteristic
72C43—21no—
73C43C2311yescharacteristic
74#340—24no—
75#340—24no—
76#340—24no—
77#340—22no—
78#340—22no—
79#336—22no—
80#336—22no—
81#336—21no—
82#333—28no—
83C22×Q8—22no—
84C22×Q8C2×D411yescharacteristic
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8C2×D412yes—
89C22×C8C2×D412yes—
90C4⋊Q8—24no—
91C4⋊Q8—24no—
92C4⋊Q8—84no—
93#125—44no—
94C4×Q8—22no—
95C4×Q8—24no—
96C4×Q8—42no—
97C4×Q8#4114yes—
98C2×C4⋊C4—21no—
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⋊C4C2×D412yes—
109C2×C4⋊C4C2×D411yescharacteristic
110C2×C42—21no—
111C2×C42—22no—
112C2×C42—21no—
113C2×C42—22no—
114C2×C42C2×D412yes—
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42#4112yes—
118C2×C42C2411yescharacteristic
119#107—44no—
120#107—44no—
121C4⋊C8—28no—
122C4⋊C8—22no—
123C4⋊C8—24no—
124C4⋊C8—24no—
125C4⋊C8—44no—
126C4⋊C8#4114yes—
127#103—44no—
128#103—44no—
129#97—22no—
130#97—42no—
131#97—44no—
132#97—42no—
133#97Q1614yes—
134C4×C8—28no—
135C4×C8—22no—
136C4×C8—22no—
137#95—22no—
138#95—21no—
139#95—42no—
140C2×Q8—21no—
141C2×Q8—22no—
142C2×Q8—44no—
143C2×Q8—84no—
144C2×Q8#12412yes—
145C2×Q8#13714yes—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4#12112yes—
152C22×C4#12112yes—
153C22×C4C22×D411yescharacteristic
154C22×C4C22×D411yescharacteristic
155C22×C4C2×#4111yescharacteristic
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—24no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—44no—
168C2×C8—42no—
169C2×C8C2×Q1614yes—
170C4⋊C4—24no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—44no—
181C4⋊C4—82no—
182C4⋊C4—84no—
183C4⋊C4—84no—
184C4⋊C4—84no—
185C4⋊C4#12412yes—
186C42—24no—
187C42—22no—
188C42—22no—
189C42—22no—
190C42—42no—
191C42—44no—
192C42—42no—
193C42—82no—
194C42C2×Q1612yes—
195C42#13712yes—
196C42C2×#4112yes—
197C42C2×#4112yes—
198C23C2×#12111yescharacteristic
199Q8—22no—
200Q8—24no—
201Q8—164no—
202Q8#49114yes—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—84no—
220C2×C4—82no—
221C2×C4#46214yes—
222C2×C4#52512yes—
223C2×C4C2×#12411yescharacteristic
224C2×C4#52912yes—
225C2×C4#53211yescharacteristic
226C2×C4C22×Q1611yescharacteristic
227C2×C4C2×#13711yescharacteristic
228C2×C4C2×#13712yes—
229C2×C4#57512yes—
230C2×C4#57512yes—
231C8—24no—
232C8—42no—
233C8—44no—
234C8—82no—
235C8—82no—
236C22#285011yescharacteristic
237C22C2×#46211yescharacteristic
238C22#304011yescharacteristic
239C22#305612yes—
240C22#306812yes—
241C4—24no—
242C4—22no—
243C4—24no—
244C4—42no—
245C4—42no—
246C4—82no—
247C4—162no—
248C4—162no—
249C4#310412yes—
250C4#311612yes—
251C4C2×#49111yescharacteristic
252C4#313511yescharacteristic
253C4#318112yes—
254C2#1917312yes—
255C2#1932612yes—
256C2#3302711yescharacteristic
257C2#3305311yescharacteristic
258C2#3306411yescharacteristic
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
#332942
#331982
#330692
Q8×#972
#99022
#97022
#95762
#92162
#87862