Finite Group Databasebeta

← #266670

Subgroups — #266670

Order 512.

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

normal subgroups only

This group has 740 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, complemented
2#29145C212yesmaximal
3#21656C211yesmaximal, characteristic
4#10586C212yesmaximal, complemented
5#10360C212yesmaximal
6#9890C212yesmaximal, complemented
7#9002C212yesmaximal
8#8915C212yesmaximal, complemented
9#8584C211yesmaximal, characteristic, complemented
10C2×#490C2211yescharacteristic
11#2862C2212yes—
12#2845C2212yes—
13#2806—22no—
14#2755—24no—
15#2546—22no—
16#2451—22no—
17#2446—22no—
18#2382C2212yes—
19C4×#127C2211yescharacteristic
20C4×#125C2211yescharacteristic
21#2023—22no—
22#2023—22no—
23#1992C2211yescharacteristic
24#1961C2214yes—
25#1961C2211yescharacteristic
26#1934—22no—
27#1836C2211yescharacteristic
28#1836C2214yes—
29#1804—24no—
30#1767C2212yes—
31#1762—22no—
32C4×#97C2211yescharacteristic
33C2×#330C2212yes—
34#1295C2214yes—
35#1295C2212yes—
36#1295C2214yes—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8D412yes—
39C2×C4⋊Q8D412yes—
40C2×#127C2311yescharacteristic
41C2×#125C2311yescharacteristic
42C2×C4×Q8—22no—
43C2×C4×D4—22no—
44C2×C4×D4—21no—
45#490—22no—
46#490Q814yescomplemented
47C2×C4⋊C8—22no—
48C2×#102—22no—
49C2×#102—22no—
50C2×#102—24no—
51C2×#102—24no—
52C2×#97—21no—
53C2×#97C2311yescharacteristic
54C2×#97C2314yes—
55C2×#97C2311yescharacteristic
56C2×C4×C8—21no—
57#400—24no—
58#398—24no—
59#398—24no—
60#398—24no—
61#398—24no—
62#395—24no—
63#391—22no—
64#391—24no—
65#391—22no—
66#391—22no—
67#391—22no—
68#391—22no—
69#390—21no—
70#390—22no—
71#389—22no—
72#389—24no—
73#389—24no—
74#389D412yes—
75#387—24no—
76#387—24no—
77#387—24no—
78#387—22no—
79#384C2311yescharacteristic
80#382D414yes—
81#382C2311yescharacteristic
82#382C2314yes—
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×D4—21no—
95C22×D4—22no—
96C23×C4—41no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—24no—
100C4⋊Q8—82no—
101C4⋊Q8—82no—
102C4⋊Q8—84no—
103#127—21no—
104#127—22no—
105#127C2×Q812yes—
106#125—22no—
107#125C2×Q812yes—
108#125#4112yes—
109C4×Q8—84no—
110C4×D4—44no—
111C4×D4—48no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4C2×D412yes—
123C2×C4⋊C4C2×D414yes—
124C2×C4⋊C4C2×D414yes—
125C2×C4⋊C4C2×D412yes—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—41no—
128C2×C42—22no—
129C2×C42—24no—
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D412yes—
132C2×C42C2×D412yes—
133C2×C42C2×D411yescharacteristic
134C2×C42C2411yescharacteristic
135C4⋊C8—44no—
136#102—28no—
137#102—44no—
138#102—84no—
139#102—84no—
140#97—21no—
141#97—24no—
142#97—22no—
143#97—24no—
144#97—42no—
145#97C2×Q812yes—
146C4×C8—42no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95—22no—
151#95—22no—
152#95—24no—
153#95—24no—
154#95—22no—
155#95—24no—
156#95—24no—
157C24—41no—
158C2×Q8—44no—
159C2×Q8—44no—
160C2×Q8—44no—
161C2×Q8—44no—
162C2×Q8—82no—
163C2×Q8—82no—
164C2×D4—22no—
165C2×D4—24no—
166C2×D4—42no—
167C2×D4—44no—
168C2×D4—44no—
169C2×D4—42no—
170C2×D4—41no—
171C2×D4—42no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—41no—
177C22×C4—84no—
178C22×C4—82no—
179C22×C4#12011yescharacteristic
180C22×C4#12012yes—
181C22×C4#12011yescharacteristic
182C22×C4C22×D411yescharacteristic
183C22×C4C22×D412yes—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—44no—
191C2×C8—44no—
192C2×C8—44no—
193C4⋊C4—22no—
194C4⋊C4—24no—
195C4⋊C4—24no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—42no—
202C4⋊C4—44no—
203C4⋊C4—82no—
204C4⋊C4—84no—
205C4⋊C4—84no—
206C4⋊C4—84no—
207C4⋊C4—84no—
208C4⋊C4C22⋊Q814yes—
209C22⋊C4—84no—
210C22⋊C4—82no—
211C42—22no—
212C42—24no—
213C42—41no—
214C42—42no—
215C42—44no—
216C42—84no—
217C42C22×Q811yescharacteristic
218C42C2×#4111yescharacteristic
219C422−1+412yes—
220C23—42no—
221C23—82no—
222C23—81no—
223C23—81no—
224C23—82no—
225C23C2×#12011yescharacteristic
226Q8—88no—
227Q8—164no—
228D4—48no—
229D4—44no—
230D4—84no—
231D4—82no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—42no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—82no—
254C2×C4—81no—
255C2×C4—164no—
256C2×C4#45414yes—
257C2×C4#45412yes—
258C2×C4C2×C22⋊Q811yescharacteristic
259C2×C4#53611yescharacteristic
260C2×C4#54812yes—
261C2×C4D4×Q812yes—
262C8—42no—
263C8—42no—
264C8—84no—
265C22—84no—
266C22—82no—
267C22—161no—
268C22—162no—
269C22—161no—
270C22#240911yescharacteristic
271C22C2×#45412yes—
272C22C2×#45411yescharacteristic
273C22#301811yescharacteristic
274C22#301912yes—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—24no—
279C4—21no—
280C4—21no—
281C4—44no—
282C4—41no—
283C4—84no—
284C4—82no—
285C4—84no—
286C4—161no—
287C2—162no—
288C2#1900111yescharacteristic
289C2#1904812yes—
290C2#1908311yescharacteristic
291C2#2431311yescharacteristic
292C2#2432112yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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
1. #46007C2semidirect
4. #10586C2semidirect
6. #9890C2semidirect
8. #8915C2semidirect
9. #8584C2semidirect
46. #490Q8semidirect

Maximal subgroups

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

subgroupindex
#460072
#291452
#216562
#105862
#103602
#98902
#90022
#89152
#85842