Finite Group Databasebeta

← #318869

Subgroups — #318869

Order 512.

548 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 162 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#28287C212yesmaximal
2#22303C211yesmaximal, characteristic
3#22247C211yesmaximal, characteristic
4#10598C212yesmaximal
5#10590C212yesmaximal
6#9740C212yesmaximal
7#9622C212yesmaximal
8#8809C211yesmaximal, characteristic
9#8586C212yesmaximal
10C2×#500C2211yescharacteristic
11C2×#479—22no—
12C4×C4⋊Q8C2212yes—
13C4×#125C2211yescharacteristic
14#2068—22no—
15#2066—22no—
16#1990C2212yes—
17#1981C2212yes—
18#1981C2212yes—
19#1948C2212yes—
20#1947C2212yes—
21#1940—22no—
22#1939C2212yes—
23#1918—22no—
24#1918—22no—
25#1917—22no—
26#1842C2211yescharacteristic
27#1842C2212yes—
28#1834C2211yescharacteristic
29#1805—22no—
30#1770C2212yes—
31#1744C2211yescharacteristic
32C2×#334C2212yes—
33C2×#334C2211yescharacteristic
34C2×#334C2211yescharacteristic
35C2×#332C2212yes—
36C2×#332C2212yes—
37#1296C2212yes—
38#1296C2212yes—
39C2×C4⋊Q8C2312yes—
40C2×#125D411yescharacteristic
41C2×C4×Q8—22no—
42#500—22no—
43#500Q814yes—
44#479—48no—
45C2×#107—22no—
46C2×#107—22no—
47C2×#106—22no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8C2312yes—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2312yes—
55C2×#103—22no—
56C2×#103—22no—
57C2×#103—22no—
58C2×#103—22no—
59C2×#97—22no—
60C2×C4×C8C2311yescharacteristic
61C2×C4×C8C2311yescharacteristic
62#391—22no—
63#389—22no—
64#389—22no—
65#389D412yes—
66#387—22no—
67#384C2312yes—
68#382D411yescharacteristic
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4C2312yes—
73C43C2311yescharacteristic
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#336—22no—
80#336—22no—
81#336D412yes—
82#336D412yes—
83#334—24no—
84#334—41no—
85#334—41no—
86#332—42no—
87#332—42no—
88C22×Q8—22no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8C2×D411yescharacteristic
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D412yes—
95C4⋊Q8—22no—
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C4⋊Q8C2×Q814yes—
99#125—24no—
100C4×Q8—44no—
101C4×Q8—44no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4C2×D412yes—
111C2×C4⋊C4C2×D412yes—
112C2×C42—22no—
113C2×C42—42no—
114C2×C42C2×D412yes—
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42#4112yes—
118C2×C42C2411yescharacteristic
119#107—28no—
120#107—44no—
121#106—44no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—44no—
129C4⋊C8—82no—
130C4⋊C8—82no—
131C4⋊C8—82no—
132#103—44no—
133#103—44no—
134#103—82no—
135#103—82no—
136#97—82no—
137C4×C8—21no—
138C4×C8—21no—
139C4×C8C2×Q812yes—
140C4×C8#4112yes—
141#95—22no—
142#95—22no—
143#95—22no—
144#95—42no—
145#95—42no—
146C2×Q8—24no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—42no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4#12112yes—
157C22×C4#12112yes—
158C22×C4C22×D411yescharacteristic
159C22×C4C22×D411yescharacteristic
160C22×C4C2×#4111yescharacteristic
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8C22⋊Q814yes—
177C4⋊C4—24no—
178C4⋊C4—24no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—82no—
191C4⋊C4—82no—
192C4⋊C4—84no—
193C42—22no—
194C42—22no—
195C42—22no—
196C42—42no—
197C42—42no—
198C42—42no—
199C42—84no—
200C42AGL(1,ℤ/8)14yes—
201C42C22×Q811yescharacteristic
202C42C2×#4111yescharacteristic
203C42H(5,𝔽2)11yescharacteristic
204C422−1+411yescharacteristic
205C23C2×#12111yescharacteristic
206Q8—44no—
207Q8—44no—
208Q8—82no—
209Q8—82no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4#46914yes—
229C2×C4C2×C22⋊Q811yescharacteristic
230C2×C4#53711yescharacteristic
231C2×C4D4×Q811yescharacteristic
232C2×C4#55011yescharacteristic
233C2×C4#55411yescharacteristic
234C2×C4#55611yescharacteristic
235C2×C4C2×AGL(1,ℤ/8)12yes—
236C2×C4#57712yes—
237C2×C4#57712yes—
238C2×C4#57812yes—
239C8—24no—
240C8—41no—
241C8—41no—
242C8—82no—
243C8—82no—
244C8—82no—
245C22#253811yescharacteristic
246C22#304612yes—
247C22C2×#46911yescharacteristic
248C22#306011yescharacteristic
249C22#306912yes—
250C4—21no—
251C4—21no—
252C4—24no—
253C4—42no—
254C4—82no—
255C4—84no—
256C4—82no—
257C4—162no—
258C4#308212yes—
259C4#326011yescharacteristic
260C4#326111yescharacteristic
261C4#338412yes—
262C4#339712yes—
263C2#1945212yes—
264C2#1954812yes—
265C2#2827311yescharacteristic
266C2#2829411yescharacteristic
267C2#2830011yescharacteristic
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
#282872
#223032
#222472
#105982
#105902
#97402
#96222
#88092
#85862