Finite Group Databasebeta

← #305305

Subgroups — #305305

Order 512.

686 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 158 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#26020C212yesmaximal, complemented
2#22489C212yesmaximal
3#10686C211yesmaximal, characteristic, complemented
4#10593C212yesmaximal, complemented
5#10349C211yesmaximal, characteristic, complemented
6#8889C212yesmaximal, complemented
7#8843C212yesmaximal, complemented
8#8648C212yesmaximal, complemented
9#8452C211yesmaximal, characteristic
10C2×#555C2211yescharacteristic
11C2×C8×D4C2212yes—
12#2551C2211yescharacteristic
13#2354C2212yes—
14#2067—22no—
15#2023—22no—
16#1990C2212yes—
17#1988—22no—
18#1987C2212yes—
19#1956C2211yescharacteristic
20#1947C2212yes—
21#1939C2212yes—
22#1918—22no—
23#1893—22no—
24#1875—22no—
25#1844C2211yescharacteristic
26#1834C2212yes—
27#1825—22no—
28#1802C2212yes—
29#1802C2212yes—
30C4×#102C2212yes—
31#1735—22no—
32C2×#335C2212yes—
33C2×#333C2211yescharacteristic
34C2×D4⋊C8C2212yes—
35#1296C2212yes—
36#1296C2211yescharacteristic
37#1279C2212yes—
38#1273C2211yescharacteristic
39C23×C8—22no—
40#555—24no—
41#555—41no—
42#555Q814yescomplemented
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8C2311yescharacteristic
45C2×#124—41no—
46C2×#121—22no—
47C2×C4×D4C2312yes—
48C2×#117—22no—
49C2×#117—22no—
50C8×D4—24no—
51C8×D4—42no—
52C2×#107—22no—
53C2×#106—41no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8C2312yes—
56C2×C4⋊C8C2312yes—
57C2×#103—41no—
58C2×#102—22no—
59C2×#102—22no—
60C2×#102D412yes—
61C2×#102D412yes—
62C2×C22⋊C8—22no—
63C2×C4×C8C2311yescharacteristic
64C2×C4×C8C2312yes—
65#395—41no—
66#390—22no—
67#389—22no—
68#388—42no—
69#384C2311yescharacteristic
70#384C2312yes—
71#380—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4C2312yes—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—22no—
83#340D412yes—
84#340D412yes—
85#336—22no—
86#335—42no—
87#333—41no—
88D4⋊C8—24no—
89C22×Q8—21no—
90C22×Q8—41no—
91C22×D4—41no—
92C22×D4C2×D412yes—
93C23×C4—22no—
94C22×C8—21no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—44no—
98C22×C8—44no—
99C22×C8—82no—
100C22×C8C2×D412yes—
101C22×C8C2×D412yes—
102C4⋊Q8—22no—
103C4⋊Q8—21no—
104C4⋊Q8—44no—
105C4⋊Q8—82no—
106C4⋊Q8C2×Q812yes—
107#124—44no—
108#124—44no—
109#121—28no—
110#121—44no—
111C4×D4—22no—
112C4×D4—22no—
113C4×D4C2×Q814yes—
114C4×D4#4114yes—
115#117—28no—
116#117—44no—
117#117—44no—
118#117—82no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4C2×D412yes—
127C2×C4⋊C4C2×D412yes—
128C2×C4⋊C4C2×D412yes—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—42no—
133C2×C42—22no—
134C2×C42—41no—
135C2×C42—42no—
136C2×C42C2411yescharacteristic
137#107—44no—
138#106—82no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—82no—
144#103—82no—
145#102—24no—
146#102—42no—
147#102—82no—
148#102—82no—
149C22⋊C8—44no—
150C4×C8—22no—
151C4×C8—22no—
152C4×C8—22no—
153#95—22no—
154#95—42no—
155#95—41no—
156C24—22no—
157C2×Q8—22no—
158C2×Q8—41no—
159C2×Q8—84no—
160C2×Q8—81no—
161C2×Q8—81no—
162C2×Q8—161no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—44no—
169C2×D4—161no—
170C2×D4—161no—
171C2×D4C22⋊Q814yes—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4C22×D412yes—
185C22×C4H(5,𝔽2)11yescharacteristic
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—88no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—24no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—84no—
215C4⋊C4—81no—
216C4⋊C4—82no—
217C4⋊C4C22⋊Q814yes—
218C4⋊C4#13514yes—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—48no—
222C22⋊C4—84no—
223C22⋊C4—162no—
224C42—44no—
225C42—42no—
226C42—42no—
227C42—84no—
228C42C22×Q811yescharacteristic
229C42C2×#4112yes—
230C42H(5,𝔽2)11yescharacteristic
231C23—24no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23D4211yescharacteristic
239Q8—82no—
240Q8—161no—
241Q8—164no—
242D4—24no—
243D4—24no—
244D4—42no—
245D4—42no—
246D4—164no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4C2×C22⋊Q812yes—
271C2×C4#54311yescharacteristic
272C2×C4#54612yes—
273C2×C4#55111yescharacteristic
274C2×C4C2×#13512yes—
275C2×C4#57512yes—
276C2×C4#57612yes—
277C2×C4#57812yes—
278C8—42no—
279C8—42no—
280C8—42no—
281C8—82no—
282C8—82no—
283C22—44no—
284C22—44no—
285C22—44no—
286C22—82no—
287C22—82no—
288C22—82no—
289C22—82no—
290C22#243711yescharacteristic
291C22#326912yes—
292C22#328012yes—
293C22#328811yescharacteristic
294C22#329111yescharacteristic
295C4—21no—
296C4—21no—
297C4—44no—
298C4—44no—
299C4—82no—
300C4—82no—
301C4—161no—
302C4—162no—
303C4#307612yes—
304C4#308312yes—
305C4#330212yes—
306C4#331912yes—
307C2—84no—
308C2#1978412yes—
309C2#1998711yescharacteristic
310C2#2012511yescharacteristic
311C2#2595412yes—
312C2#2598311yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#555C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #26020C2semidirect
3. #10686C2semidirect
4. #10593C2semidirect
5. #10349C2semidirect
6. #8889C2semidirect
7. #8843C2semidirect
8. #8648C2semidirect
42. #555Q8semidirect

Maximal subgroups

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

subgroupindex
#260202
#224892
#106862
#105932
#103492
#88892
#88432
#86482
#84522