Finite Group Databasebeta

← #315892

Subgroups — #315892

Order 512.

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

normal subgroups only

This group has 658 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#33210C211yesmaximal, characteristic
2#27972C212yesmaximal
3#27875C212yesmaximal
4#20764C211yesmaximal, characteristic
5#10686C212yesmaximal, complemented
6#10682C212yesmaximal, complemented
7#9828C212yesmaximal, complemented
8#8821C212yesmaximal, complemented
9#8147C211yesmaximal, characteristic
10C2×#498C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12#2857C2212yes—
13#2285C2211yescharacteristic
14C2×#384C2211yescharacteristic
15#2077—24no—
16#2072—24no—
17#1990C2212yes—
18#1990C2212yes—
19#1990C2212yes—
20#1956C2212yes—
21#1956C2212yes—
22#1943—22no—
23#1920—24no—
24#1872—24no—
25#1845C2211yescharacteristic
26#1844C2212yes—
27#1836—22no—
28#1834C2212yes—
29#1834C2212yes—
30#1834C2212yes—
31#1834C2212yes—
32#1834C2212yes—
33#1809—22no—
34C4×C22⋊C8C2212yes—
35#1736—22no—
36C2×#322C2212yes—
37#1273C2211yescharacteristic
38#1273C2211yescharacteristic
39C22×M16D412yes—
40C2×C4⋊Q8C2312yes—
41C2×C22⋊Q8—22no—
42C2×#117—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—22no—
45C22×C42C2311yescharacteristic
46#498—24no—
47C2×#107—22no—
48C2×#106—22no—
49C2×#106—24no—
50C2×#103—24no—
51C2×#103—24no—
52C2×C22⋊C8D414yes—
53C4×M16—24no—
54C2×#97D412yes—
55C2×C4×C8C2311yescharacteristic
56C2×C4×C8C2312yes—
57C2×C4×C8C2312yes—
58C2×C4×C8C2311yescharacteristic
59#391—22no—
60#389—22no—
61#389—22no—
62#384—22no—
63#384—21no—
64#384C2312yes—
65#384C2312yes—
66#384C2312yes—
67#382—21no—
68#380—22no—
69#380—22no—
70#380—22no—
71C4×C4⋊C4—22no—
72#340—24no—
73#340—24no—
74#340—24no—
75#340—24no—
76#340—24no—
77#340—24no—
78#340—24no—
79#340—24no—
80#336D414yes—
81#322—24no—
82C22×Q8—22no—
83C23×C4—41no—
84C23×C4C2×D412yes—
85C23×C4C2×D411yescharacteristic
86C2×M16—24no—
87C2×M16—24no—
88C2×M16—42no—
89C22×C8C2×D412yes—
90C22×C8C2×D414yes—
91C22×C8C2×D414yes—
92C22×C8C2×D412yes—
93C4⋊Q8—22no—
94C4⋊Q8—22no—
95C4⋊Q8—22no—
96C4⋊Q8—22no—
97C22⋊Q8—48no—
98#117—44no—
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—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—44no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—84no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—42no—
117C2×C42—22no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—41no—
121C2×C42C2×D412yes—
122C2×C42C2×D411yescharacteristic
123C2×C42C2411yescharacteristic
124#107—44no—
125#106—48no—
126#106—44no—
127#103—84no—
128#103—84no—
129C22⋊C8—44no—
130#97—24no—
131C4×C8—22no—
132C4×C8—22no—
133C4×C8—22no—
134C4×C8—22no—
135C4×C8—22no—
136C4×C8#4114yes—
137#95—42no—
138#95—42no—
139#95—42no—
140C24#12711yescharacteristic
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—44no—
144C2×Q8—44no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—82no—
157C22×C4—84no—
158C22×C4#12712yes—
159C22×C4#12711yescharacteristic
160C22×C4#13714yes—
161C22×C4C22×D411yescharacteristic
162C22×C4C22×D412yes—
163M16—48no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—24no—
168C2×C8—24no—
169C2×C8—22no—
170C2×C8—24no—
171C2×C8—24no—
172C2×C8—22no—
173C2×C8—24no—
174C2×C8—24no—
175C2×C8#12314yes—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—82no—
190C4⋊C4—84no—
191C4⋊C4—84no—
192C22⋊C4—84no—
193C22⋊C4—82no—
194C42—21no—
195C42—21no—
196C42—22no—
197C42—22no—
198C42—82no—
199C42C2×#4112yes—
200C422−1+411yescharacteristic
201C422−1+411yescharacteristic
202C23—22no—
203C23—21no—
204C23—22no—
205C23—21no—
206C23#49712yes—
207C23C2×#12711yescharacteristic
208Q8—84no—
209Q8—84no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—164no—
232C2×C4#49712yes—
233C2×C4C2×#12311yescharacteristic
234C2×C4#53611yescharacteristic
235C2×C4#54712yes—
236C2×C4#54712yes—
237C2×C4C2×#13712yes—
238C2×C4#57514yes—
239C2×C4#57512yes—
240C2×C4#57714yes—
241C8—24no—
242C8—44no—
243C8—42no—
244C8—44no—
245C22—22no—
246C22—24no—
247C22—41no—
248C22—41no—
249C22—42no—
250C22#250111yescharacteristic
251C22C2×#49711yescharacteristic
252C22#315311yescharacteristic
253C22#315312yes—
254C22#315412yes—
255C4—22no—
256C4—42no—
257C4—162no—
258C4—162no—
259C4—162no—
260C4—162no—
261C4#309512yes—
262C4#323612yes—
263C4#335014yes—
264C4#335114yes—
265C2—42no—
266C2#1968612yes—
267C2#1971711yescharacteristic
268C2#1975411yescharacteristic
269C2#2791911yescharacteristic
270C2#2794712yes—
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. #10686C2semidirect
6. #10682C2semidirect
7. #9828C2semidirect
8. #8821C2semidirect

Maximal subgroups

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

subgroupindex
#332102
#279722
#278752
#207642
#106862
#106822
#98282
#88212
#81472