Finite Group Databasebeta

← #318364

Subgroups — #318364

Order 512.

606 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 170 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#28284C212yesmaximal
2#22248C211yesmaximal, characteristic
3#22238C211yesmaximal, characteristic
4C2×#1676C212yesmaximal
5#10483C214yesmaximal
6#8810C211yesmaximal, characteristic
7#8649C212yesmaximal
8#8590C212yesmaximal
9C2×#500C2211yescharacteristic
10C2×#477C2214yes—
11C2×C8×Q8C2212yes—
12C4×C4⋊Q8C2212yes—
13C4×C4⋊Q8C2211yescharacteristic
14#2068—24no—
15#1990C2212yes—
16#1984C2212yes—
17#1981—22no—
18#1947—22no—
19#1921—24no—
20#1916—24no—
21#1842C2212yes—
22#1834C2211yescharacteristic
23#1831C2211yescharacteristic
24#1803C2214yes—
25#1744C2211yescharacteristic
26#1676—28no—
27C2×#335C2212yes—
28C2×#335C2212yes—
29C2×#333C2214yes—
30#1296C2214yes—
31C2×C4⋊Q8D411yescharacteristic
32C2×C4⋊Q8C2312yes—
33C2×C4×Q8—22no—
34C2×C4×Q8C2312yes—
35#500—22no—
36#500D414yes—
37#477—28no—
38#477—44no—
39C8×Q8—24no—
40C8×Q8—24no—
41C2×#107—22no—
42C2×#107—24no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8C2314yes—
47C2×C4⋊C8C2312yes—
48C2×#103—24no—
49C2×#103—24no—
50C2×C4×C8—22no—
51C2×C4×C8C2311yescharacteristic
52C2×C4×C8C2311yescharacteristic
53#391—22no—
54#391D412yes—
55#389—22no—
56#389—22no—
57#389—22no—
58#384D411yescharacteristic
59#384C2312yes—
60C4×C4⋊C4—22no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4—22no—
63C43C2311yescharacteristic
64#340—22no—
65#340—24no—
66#340—24no—
67#336—24no—
68#336—22no—
69#336D414yes—
70#335—24no—
71#335—42no—
72#333—28no—
73C22×Q8—22no—
74C22×Q8C2×D412yes—
75C22×C8—22no—
76C22×C8—24no—
77C22×C8C2×D412yes—
78C22×C8C2×D412yes—
79C4⋊Q8—22no—
80C4⋊Q8—22no—
81C4⋊Q8—22no—
82C4⋊Q8—22no—
83C4⋊Q8Q1614yes—
84C4⋊Q8C2×D414yes—
85C4×Q8—22no—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×Q8—84no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4C2×D412yes—
100C2×C42—22no—
101C2×C42—42no—
102C2×C42C2×D412yes—
103C2×C42C2×D411yescharacteristic
104C2×C42C2×D411yescharacteristic
105C2×C42#4112yes—
106C2×C42C2411yescharacteristic
107#107—28no—
108#107—48no—
109C4⋊C8—22no—
110C4⋊C8—24no—
111C4⋊C8—22no—
112C4⋊C8—24no—
113C4⋊C8—44no—
114C4⋊C8—44no—
115C4⋊C8—82no—
116#103—48no—
117#103—84no—
118C4×C8—21no—
119C4×C8—21no—
120C4×C8—44no—
121C4×C8C2×D412yes—
122C4×C8#4112yes—
123#95—24no—
124#95—42no—
125#95—42no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—24no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—42no—
133C2×Q8—82no—
134C2×Q8—82no—
135C2×Q8C2×Q1614yes—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4#12112yes—
141C22×C4#12112yes—
142C22×C4C22×D411yescharacteristic
143C22×C4C22×D411yescharacteristic
144C22×C4C2×#4111yescharacteristic
145C2×C8—24no—
146C2×C8—22no—
147C2×C8—24no—
148C2×C8—42no—
149C2×C8—44no—
150C2×C8—44no—
151C2×C8—44no—
152C2×C8—44no—
153C2×C8#12114yes—
154C4⋊C4—28no—
155C4⋊C4—24no—
156C4⋊C4—24no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—44no—
165C4⋊C4—42no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C4⋊C4—84no—
169C42—21no—
170C42—24no—
171C42—21no—
172C42—42no—
173C42—42no—
174C42—42no—
175C42—84no—
176C42C2×Q1612yes—
177C42#13512yes—
178C42C22×D411yescharacteristic
179C42C2×#4111yescharacteristic
180C42H(5,𝔽2)11yescharacteristic
181C422−1+411yescharacteristic
182C23C2×#12111yescharacteristic
183Q8—24no—
184Q8—24no—
185Q8—42no—
186Q8—42no—
187Q8—84no—
188Q8—84no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4#46714yes—
207C2×C4C2×#12111yescharacteristic
208C2×C4#53711yescharacteristic
209C2×C4D4211yescharacteristic
210C2×C4#54711yescharacteristic
211C2×C4#54711yescharacteristic
212C2×C4#55511yescharacteristic
213C2×C4C22×Q1611yescharacteristic
214C2×C4C2×#13511yescharacteristic
215C2×C4#57611yescharacteristic
216C2×C4#57811yescharacteristic
217C2×C4#57814yes—
218C8—24no—
219C8—42no—
220C8—84no—
221C8—82no—
222C22#253711yescharacteristic
223C22#304412yes—
224C22C2×#46711yescharacteristic
225C22#306111yescharacteristic
226C22#306812yes—
227C4—21no—
228C4—21no—
229C4—24no—
230C4—44no—
231C4—42no—
232C4—82no—
233C4—84no—
234C4—162no—
235C4#304412yes—
236C4#326112yes—
237C4D4×Q1611yescharacteristic
238C4#328711yescharacteristic
239C4#333811yescharacteristic
240C4#334211yescharacteristic
241C2#1940512yes—
242C2#1953512yes—
243C2#2822811yescharacteristic
244C2#2824311yescharacteristic
245C2#2826111yescharacteristic
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
#282842
#222482
#222382
C2×#16762
#104832
#88102
#86492
#85902