Finite Group Databasebeta

← #315584

Subgroups — #315584

Order 512.

736 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 206 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#33172C211yesmaximal, characteristic
2#27877C214yesmaximal
3#20797C211yesmaximal, characteristic
4#10675C214yesmaximal, complemented
5#9825C211yesmaximal, characteristic, complemented
6#9825C211yesmaximal, characteristic, complemented
7#8818C211yesmaximal, characteristic, complemented
8#8816C211yesmaximal, characteristic, complemented
9C8×C22⋊C8C211yesmaximal, characteristic
10C2×#494C2211yescharacteristic
11C2×#467—22no—
12C2×#427—22no—
13C22×C4×C8C2211yescharacteristic
14#2857C2212yes—
15#2285C2212yes—
16#2075—28no—
17#1990C2214yes—
18#1990C2214yes—
19#1969—22no—
20#1956C2212yes—
21#1918—28no—
22#1846—22no—
23#1845C2211yescharacteristic
24#1845C2211yescharacteristic
25#1845C2211yescharacteristic
26#1845C2214yes—
27#1844C2212yes—
28#1834C2214yes—
29C4×C22⋊C8C2212yes—
30C2×C82C2211yescharacteristic
31C2×C82C2211yescharacteristic
32#1273C2212yes—
33C22×Q16—22no—
34C23×C8D412yes—
35C2×C4⋊Q8C2312yes—
36C2×C22⋊Q8—22no—
37C2×#117—22no—
38C22×C42C2311yescharacteristic
39#494—22no—
40#494—22no—
41#467—48no—
42#427—48no—
43C2×#107—28no—
44C2×#107—22no—
45C2×#107—22no—
46C2×#107—22no—
47C2×#106—24no—
48C2×#103—24no—
49C2×C22⋊C8D414yes—
50C2×C4×C8—22no—
51C2×C4×C8—22no—
52C2×C4×C8D412yes—
53C2×C4×C8C2312yes—
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2312yes—
56C2×C4×C8C2311yescharacteristic
57#391—22no—
58#389—24no—
59#384C2312yes—
60#384C2312yes—
61#384C2312yes—
62#382—22no—
63#380—24no—
64#380—24no—
65C4×C4⋊C4—24no—
66#340—28no—
67#340—28no—
68#340—28no—
69#336D414yes—
70C82—22no—
71C82—22no—
72C22×Q8—22no—
73C23×C4C2×D412yes—
74C23×C4C2×D411yescharacteristic
75C2×Q16—44no—
76C2×Q16—44no—
77C2×Q16—44no—
78C22×C8—24no—
79C22×C8—24no—
80C22×C8D818yes—
81C22×C8C2×D414yes—
82C22×C8C2×D412yes—
83C22×C8C2×D414yes—
84C22×C8C2×D412yes—
85C4⋊Q8—22no—
86C4⋊Q8—22no—
87C4⋊Q8—24no—
88C22⋊Q8—48no—
89#117—48no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—22no—
103C2×C42—42no—
104C2×C42C2×D412yes—
105C2×C42C2×D411yescharacteristic
106C2×C42C2411yescharacteristic
107#107—44no—
108#107—44no—
109#107—44no—
110#107—416no—
111#106—48no—
112#103—48no—
113C22⋊C8—28no—
114C4×C8—24no—
115C4×C8—22no—
116C4×C8—22no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—22no—
120C4×C8—22no—
121C4×C8#4112yes—
122C4×C8#4112yes—
123#95—44no—
124#95—44no—
125C24#12711yescharacteristic
126C2×Q8—42no—
127C2×Q8—42no—
128C2×Q8—44no—
129C2×Q8—44no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4#12712yes—
142C22×C4#12711yescharacteristic
143C22×C4C2×D814yes—
144C22×C4C22×D411yescharacteristic
145C22×C4C22×D412yes—
146Q16—88no—
147C2×C8—22no—
148C2×C8—24no—
149C2×C8—22no—
150C2×C8—24no—
151C2×C8—216no—
152C2×C8—24no—
153C2×C8—24no—
154C2×C8—24no—
155C2×C8#12314yes—
156C2×C8C2×D814yes—
157C2×C8C2×D814yes—
158C2×C8#13518yes—
159C2×C8#13714yes—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—44no—
165C4⋊C4—48no—
166C4⋊C4—48no—
167C4⋊C4—48no—
168C4⋊C4—84no—
169C22⋊C4—84no—
170C22⋊C4—84no—
171C42—21no—
172C42—21no—
173C42—22no—
174C42—22no—
175C42—84no—
176C42C2×#4111yescharacteristic
177C42C2×#4111yescharacteristic
178C422−1+412yes—
179C23—22no—
180C23—21no—
181C23—22no—
182C23—21no—
183C23#49312yes—
184C23C2×#12711yescharacteristic
185Q8—84no—
186Q8—84no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—84no—
201C2×C4—84no—
202C2×C4—82no—
203C2×C4—84no—
204C2×C4—82no—
205C2×C4#49312yes—
206C2×C4C2×#12311yescharacteristic
207C2×C4#53611yescharacteristic
208C2×C4#54712yes—
209C2×C4#54712yes—
210C2×C4C22×D812yes—
211C2×C4C2×#13514yes—
212C2×C4C2×#13712yes—
213C2×C4#57814yes—
214C8—28no—
215C8—24no—
216C8—44no—
217C8#48018yes—
218C22—22no—
219C22—22no—
220C22—22no—
221C22—42no—
222C22—42no—
223C22#250111yescharacteristic
224C22C2×#49311yescharacteristic
225C22#314712yes—
226C22#315211yescharacteristic
227C22#315612yes—
228C4—22no—
229C4—42no—
230C4—162no—
231C4—162no—
232C4—162no—
233C4—162no—
234C4C2×#48012yes—
235C4#322612yes—
236C4#333614yes—
237C4#334214yes—
238C2—42no—
239C2#1960311yescharacteristic
240C2#1964812yes—
241C2#1966211yescharacteristic
242C2#2790911yescharacteristic
243C2#2792712yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #10675C2semidirect
5. #9825C2semidirect
6. #9825C2semidirect
7. #8818C2semidirect
8. #8816C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#331722
#278772
#207972
#106752
#98252
#88182
#88162
C8×C22⋊C82