Finite Group Databasebeta

← #315876

Subgroups — #315876

Order 512.

702 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 178 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#33176C211yesmaximal, characteristic
2#27948C212yesmaximal
3#27877C212yesmaximal
4#20764C211yesmaximal, characteristic
5#10681C212yesmaximal, complemented
6#10680C212yesmaximal, complemented
7#9835C212yesmaximal, complemented
8#8819C212yesmaximal, complemented
9#8147C211yesmaximal, characteristic
10C2×#494C2211yescharacteristic
11C2×#470—22no—
12C2×C4×M16C2211yescharacteristic
13#2857C2212yes—
14#2285C2211yescharacteristic
15C2×#384C2211yescharacteristic
16#2084—24no—
17#2075—24no—
18#1990C2212yes—
19#1990C2212yes—
20#1990C2212yes—
21#1990C2212yes—
22#1971—22no—
23#1956C2212yes—
24#1918—24no—
25#1853—24no—
26#1845C2212yes—
27#1844C2212yes—
28#1844C2212yes—
29#1844C2211yescharacteristic
30#1836—22no—
31#1834C2212yes—
32#1834C2212yes—
33#1834C2212yes—
34C4×C22⋊C8C2212yes—
35#1736—22no—
36C2×#322C2212yes—
37#1273C2211yescharacteristic
38#1273C2211yescharacteristic
39C22×Q16—22no—
40C22×M16D412yes—
41C2×C4⋊Q8C2312yes—
42C2×C22⋊Q8—22no—
43C2×#117—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—22no—
46C22×C42C2311yescharacteristic
47#494—24no—
48#470—48no—
49C2×#107—24no—
50C2×#106—22no—
51C2×#106—24no—
52C2×#106—24no—
53C2×#103—24no—
54C2×C22⋊C8D414yes—
55C4×M16—24no—
56C2×#97D412yes—
57C2×C4×C8C2311yescharacteristic
58C2×C4×C8C2312yes—
59C2×C4×C8C2312yes—
60C2×C4×C8C2311yescharacteristic
61#391—22no—
62#389—22no—
63#389—22no—
64#384—22no—
65#384—21no—
66#384C2312yes—
67#384C2312yes—
68#384C2312yes—
69#382—21no—
70#380—22no—
71#380—22no—
72#380—22no—
73C4×C4⋊C4—22no—
74#340—24no—
75#340—24no—
76#340—24no—
77#340—24no—
78#340—24no—
79#340—24no—
80#340—24no—
81#336D414yes—
82#322—24no—
83C22×Q8—22no—
84C23×C4—41no—
85C23×C4C2×D412yes—
86C23×C4C2×D411yescharacteristic
87C2×Q16—44no—
88C2×Q16—44no—
89C2×Q16—44no—
90C2×M16—24no—
91C2×M16—24no—
92C2×M16—42no—
93C22×C8C2×D412yes—
94C22×C8C2×D414yes—
95C22×C8C2×D412yes—
96C22×C8C2×D414yes—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C22⋊Q8—48no—
102#117—44no—
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—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—84no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—42no—
121C2×C42—22no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—41no—
125C2×C42C2×D412yes—
126C2×C42C2×D411yescharacteristic
127C2×C42C2411yescharacteristic
128#107—48no—
129#106—44no—
130#106—84no—
131#106—84no—
132#103—48no—
133C22⋊C8—28no—
134#97—24no—
135C4×C8—22no—
136C4×C8—22no—
137C4×C8—22no—
138C4×C8—22no—
139C4×C8—22no—
140C4×C8#4114yes—
141#95—42no—
142#95—42no—
143#95—42no—
144C24#12711yescharacteristic
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—44no—
148C2×Q8—44no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—82no—
161C22×C4—84no—
162C22×C4#12712yes—
163C22×C4#12711yescharacteristic
164C22×C4AGL(1,ℤ/8)14yes—
165C22×C4C22×D411yescharacteristic
166C22×C4C22×D412yes—
167Q16—88no—
168M16—48no—
169C2×C8—24no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—24no—
173C2×C8—22no—
174C2×C8—24no—
175C2×C8—22no—
176C2×C8—24no—
177C2×C8—22no—
178C2×C8—24no—
179C2×C8#12314yes—
180C2×C8#13718yes—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—82no—
195C4⋊C4—84no—
196C4⋊C4—84no—
197C22⋊C4—82no—
198C22⋊C4—84no—
199C42—21no—
200C42—21no—
201C42—22no—
202C42—22no—
203C42—82no—
204C42C2×#4112yes—
205C422−1+411yescharacteristic
206C422−1+411yescharacteristic
207C23—22no—
208C23—21no—
209C23—22no—
210C23—21no—
211C23#49612yes—
212C23C2×#12711yescharacteristic
213Q8—84no—
214Q8—84no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—164no—
237C2×C4#49612yes—
238C2×C4C2×#12311yescharacteristic
239C2×C4#53611yescharacteristic
240C2×C4#54712yes—
241C2×C4#54712yes—
242C2×C4C2×AGL(1,ℤ/8)12yes—
243C2×C4C2×#13714yes—
244C2×C4#57512yes—
245C2×C4#57714yes—
246C8—28no—
247C8—24no—
248C8—42no—
249C8—44no—
250C22—22no—
251C22—24no—
252C22—41no—
253C22—42no—
254C22—41no—
255C22#250111yescharacteristic
256C22C2×#49611yescharacteristic
257C22#315212yes—
258C22#315311yescharacteristic
259C22#315412yes—
260C4—22no—
261C4—42no—
262C4—162no—
263C4—162no—
264C4—162no—
265C4—162no—
266C4#309412yes—
267C4#323412yes—
268C4#334414yes—
269C4#335614yes—
270C2—42no—
271C2#1967212yes—
272C2#1971711yescharacteristic
273C2#1975111yescharacteristic
274C2#2791811yescharacteristic
275C2#2794012yes—
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
5. #10681C2semidirect
6. #10680C2semidirect
7. #9835C2semidirect
8. #8819C2semidirect

Maximal subgroups

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

subgroupindex
#331762
#279482
#278772
#207642
#106812
#106802
#98352
#88192
#81472