Finite Group Databasebeta

← #266692

Subgroups — #266692

Order 512.

698 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 152 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#46007C211yesmaximal, characteristic
2#29211C212yesmaximal
3#21657C211yesmaximal, characteristic
4#10585C212yesmaximal
5#10361C212yesmaximal
6#9890C212yesmaximal
7#9004C212yesmaximal
8#8913C212yesmaximal
9#8584C211yesmaximal, characteristic
10C2×#491C2211yescharacteristic
11#2862C2212yes—
12#2845C2212yes—
13#2806—22no—
14#2755—24no—
15#2546—22no—
16#2451—22no—
17#2446—22no—
18#2382C2212yes—
19C4×C4⋊Q8C2211yescharacteristic
20C4×#125C2211yescharacteristic
21#2058—22no—
22#2057—22no—
23#1993C2214yes—
24#1992C2211yescharacteristic
25#1960C2211yescharacteristic
26#1949—24no—
27#1933—22no—
28#1847C2214yes—
29#1836C2211yescharacteristic
30#1767C2212yes—
31#1763—22no—
32C4×#97C2211yescharacteristic
33C2×#330C2212yes—
34#1295C2214yes—
35#1295C2212yes—
36#1295C2214yes—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8D412yes—
39C2×C4⋊Q8D412yes—
40C2×C4⋊Q8C2311yescharacteristic
41C2×#125C2311yescharacteristic
42C2×C4×Q8—22no—
43C2×C4×Q8—21no—
44#491—22no—
45#491Q814yescomplemented
46C2×#107—24no—
47C2×#106—24no—
48C2×C4⋊C8—22no—
49C2×#103—22no—
50C2×#103—22no—
51C2×#97—21no—
52C2×#97C2311yescharacteristic
53C2×#97C2314yes—
54C2×#97C2311yescharacteristic
55C2×C4×C8—21no—
56#400—24no—
57#398—24no—
58#398—24no—
59#398—24no—
60#398—24no—
61#395—24no—
62#391—22no—
63#391—24no—
64#391—22no—
65#391—22no—
66#391—22no—
67#391—22no—
68#391—21no—
69#389—22no—
70#389—24no—
71#389—24no—
72#389—22no—
73#389D412yes—
74#387—24no—
75#387—24no—
76#387—24no—
77#387—22no—
78#384C2311yescharacteristic
79#382D414yes—
80#382C2311yescharacteristic
81#382C2314yes—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4D412yes—
85C43C2311yescharacteristic
86#340—24no—
87#340—24no—
88#340—22no—
89#340—22no—
90#336—22no—
91#330—24no—
92C22×Q8—22no—
93C22×Q8—22no—
94C22×Q8—21no—
95C22×C8—24no—
96C22×C8—21no—
97C22×C8—21no—
98C4⋊Q8—21no—
99C4⋊Q8—22no—
100C4⋊Q8—82no—
101C4⋊Q8—82no—
102C4⋊Q8—84no—
103C4⋊Q8C2×Q812yes—
104#125—22no—
105#125C2×Q812yes—
106#125#4112yes—
107C4×Q8—44no—
108C4×Q8—84no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4C2×D412yes—
122C2×C4⋊C4C2×D414yes—
123C2×C4⋊C4C2×D414yes—
124C2×C4⋊C4C2×D412yes—
125C2×C42—22no—
126C2×C42—24no—
127C2×C42—41no—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D412yes—
130C2×C42C2×D412yes—
131C2×C42C2×D411yescharacteristic
132C2×C42C2411yescharacteristic
133#107—84no—
134#106—84no—
135C4⋊C8—28no—
136#103—28no—
137#103—44no—
138#97—21no—
139#97—24no—
140#97—22no—
141#97—24no—
142#97—42no—
143#97C2×Q812yes—
144C4×C8—42no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95—22no—
149#95—22no—
150#95—24no—
151#95—24no—
152#95—22no—
153#95—24no—
154#95—24no—
155#95—42no—
156C2×Q8—22no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—41no—
162C2×Q8—44no—
163C2×Q8—44no—
164C2×Q8—82no—
165C2×Q8—82no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4#12011yescharacteristic
173C22×C4#12011yescharacteristic
174C22×C4#12012yes—
175C22×C4C22×D411yescharacteristic
176C22×C4C22×D412yes—
177C2×C8—22no—
178C2×C8—28no—
179C2×C8—22no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—44no—
184C2×C8—41no—
185C2×C8—44no—
186C4⋊C4—22no—
187C4⋊C4—24no—
188C4⋊C4—24no—
189C4⋊C4—24no—
190C4⋊C4—42no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C4⋊C4—82no—
199C4⋊C4—84no—
200C4⋊C4—84no—
201C4⋊C4—84no—
202C4⋊C4—84no—
203C4⋊C4—82no—
204C4⋊C4C22⋊Q814yes—
205C42—22no—
206C42—24no—
207C42—41no—
208C42—42no—
209C42—44no—
210C42—82no—
211C42—84no—
212C42C22×Q811yescharacteristic
213C42C2×#4111yescharacteristic
214C422−1+412yes—
215C23C2×#12011yescharacteristic
216Q8—44no—
217Q8—88no—
218Q8—82no—
219Q8—164no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—24no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—42no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—82no—
246C2×C4#45412yes—
247C2×C4#45614yes—
248C2×C4C2×C22⋊Q811yescharacteristic
249C2×C4#53611yescharacteristic
250C2×C4#54812yes—
251C2×C4D4×Q812yes—
252C8—48no—
253C8—42no—
254C8—42no—
255C22#240911yescharacteristic
256C22C2×#45411yescharacteristic
257C22C2×#45612yes—
258C22#301811yescharacteristic
259C22#301912yes—
260C4—22no—
261C4—22no—
262C4—22no—
263C4—24no—
264C4—21no—
265C4—21no—
266C4—44no—
267C4—41no—
268C4—84no—
269C4—82no—
270C4—84no—
271C4—82no—
272C4—161no—
273C2#1900111yescharacteristic
274C2#1904912yes—
275C2#1908411yescharacteristic
276C2#2431311yescharacteristic
277C2#2432312yes—
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
45. #491Q8semidirect

Maximal subgroups

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

subgroupindex
#460072
#292112
#216572
#105852
#103612
#98902
#90042
#89132
#85842