Finite Group Databasebeta

← #303584

Subgroups — #303584

Order 512.

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

normal subgroups only

This group has 950 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#36559C211yesmaximal, characteristic
2#33067C211yesmaximal, characteristic, complemented
3#30220C211yesmaximal, characteristic, complemented
4#28724C211yesmaximal, characteristic
5#10412C214yesmaximal, complemented
6#9702C211yesmaximal, characteristic, complemented
7#9642C211yesmaximal, characteristic
8#8450C214yesmaximal, complemented
9#8175C211yesmaximal, characteristic, complemented
10C2×#554C2212yes—
11C2×#490—21no—
12#2868C2211yescharacteristic
13#2859—21no—
14#2859C2211yescharacteristic
15#2566—22no—
16#2563C2212yes—
17#2538—22no—
18#2452—24no—
19D4×C4⋊C4—22no—
20C4×#127—21no—
21C4×#127C2211yescharacteristic
22C42×Q8C2211yescharacteristic
23#2073—22no—
24#2043—24no—
25#1992—21no—
26#1956C2214yes—
27#1948C2212yes—
28#1919—22no—
29#1865—22no—
30#1831C2211yescharacteristic
31#1803C2212yes—
32#1802C2214yes—
33#1767C2211yescharacteristic
34#1744C2211yescharacteristic
35C2×#333C2214yes—
36C2×Q8⋊C8C2212yes—
37C2×D4⋊C8C2214yes—
38#1279C2212yes—
39#554—28no—
40#554—44no—
41C2×C4⋊Q8D414yes—
42C2×C4⋊Q8C2312yes—
43C2×#127D411yescharacteristic
44C2×#125—22no—
45C2×#125—21no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8—22no—
48C2×C4×Q8D412yes—
49C2×C4×Q8C2311yescharacteristic
50C2×C4×D4—22no—
51C2×C4×D4—24no—
52C2×C4×D4—22no—
53C2×C4×D4C2312yes—
54C22×C4⋊C4—22no—
55#490—44no—
56C2×#106—22no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×#103—22no—
60C2×#103—28no—
61C2×#102—24no—
62C2×#97—21no—
63C2×C4×C8C2312yes—
64#396—28no—
65#393—24no—
66#390—24no—
67#390—22no—
68#390—22no—
69#390D412yes—
70#389—24no—
71#389—24no—
72#389—22no—
73#389—22no—
74#385—28no—
75#384—21no—
76#384—21no—
77#384—22no—
78#384D411yescharacteristic
79#384C2312yes—
80#382—22no—
81#382—21no—
82#380—24no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4D412yes—
86C4×C4⋊C4C2311yescharacteristic
87C43—21no—
88C43C2311yescharacteristic
89#340—24no—
90#340—22no—
91#340—22no—
92#336—21no—
93#336—22no—
94#336—22no—
95#333—44no—
96Q8⋊C8—24no—
97D4⋊C8—44no—
98C22×Q8C2×D411yescharacteristic
99C22×Q8C2×D414yes—
100C22×D4—22no—
101C22×D4C2×D412yes—
102C23×C4—24no—
103C23×C4—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C4⋊Q8—24no—
108C4⋊Q8—24no—
109C4⋊Q8—84no—
110#127—22no—
111#127SD1614yes—
112#125—48no—
113#125—42no—
114#125—42no—
115C22⋊Q8—416no—
116C22⋊Q8—44no—
117C22⋊Q8—44no—
118C4×Q8—22no—
119C4×Q8—42no—
120C4×Q8—42no—
121C4×Q8#4114yes—
122C4×D4—24no—
123C4×D4—24no—
124C4×D4—48no—
125C4×D4—44no—
126C4×D4—44no—
127C4×D4—88no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4C2×D412yes—
144C2×C4⋊C4C2×D412yes—
145C2×C4⋊C4C2×D411yescharacteristic
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—28no—
150C2×C42—21no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—21no—
154C2×C42C2×D412yes—
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D412yes—
157C2×C42C2×D411yescharacteristic
158C2×C42C2411yescharacteristic
159#106—44no—
160C4⋊C8—22no—
161C4⋊C8—24no—
162C4⋊C8—22no—
163#103—44no—
164#103—88no—
165#102—48no—
166#97—42no—
167C4×C8—22no—
168C4×C8—22no—
169#95—28no—
170#95—22no—
171#95—21no—
172#95—42no—
173C24—22no—
174C2×Q8—21no—
175C2×Q8—28no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—48no—
179C2×Q8#12412yes—
180C2×D4—24no—
181C2×D4—22no—
182C2×D4—24no—
183C2×D4—28no—
184C2×D4—44no—
185C2×D4—44no—
186C2×D4C2×SD1614yes—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—48no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—88no—
199C22×C4#12012yes—
200C22×C4#12012yes—
201C22×C4C22×D411yescharacteristic
202C22×C4C22×D411yescharacteristic
203C22×C4C22×D411yescharacteristic
204C2×C8—44no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C4⋊C4—21no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—48no—
218C4⋊C4—44no—
219C4⋊C4—48no—
220C4⋊C4—44no—
221C4⋊C4—48no—
222C4⋊C4—44no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—84no—
231C4⋊C4—84no—
232C4⋊C4#12412yes—
233C22⋊C4—48no—
234C22⋊C4—48no—
235C22⋊C4—44no—
236C22⋊C4—88no—
237C42—24no—
238C42—22no—
239C42—22no—
240C42—22no—
241C42—42no—
242C42—44no—
243C42—44no—
244C42—42no—
245C42C2×SD1612yes—
246C42#13712yes—
247C42C2×#4112yes—
248C42H(5,𝔽2)12yes—
249C23—24no—
250C23—42no—
251C23—44no—
252C23—42no—
253C23—44no—
254C23C2×#12011yescharacteristic
255Q8—22no—
256Q8—22no—
257Q8—42no—
258Q8—88no—
259D4—28no—
260D4—416no—
261D4—44no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—48no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—84no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—168no—
286C2×C4#44814yes—
287C2×C4C2×#12411yescharacteristic
288C2×C4#53511yescharacteristic
289C2×C4#54612yes—
290C2×C4#54612yes—
291C2×C4C22×SD1611yescharacteristic
292C2×C4C2×#13711yescharacteristic
293C2×C4#57512yes—
294C2×C4#57512yes—
295C2×C4#57812yes—
296C8—82no—
297C8—82no—
298C8—82no—
299C22—44no—
300C22—48no—
301C22—82no—
302C22—82no—
303C22—84no—
304C22#243511yescharacteristic
305C22C2×#44811yescharacteristic
306C22#300512yes—
307C22#301111yescharacteristic
308C22#301912yes—
309C4—22no—
310C4—24no—
311C4—42no—
312C4—42no—
313C4—42no—
314C4—82no—
315C4—84no—
316C4—82no—
317C4—162no—
318C4#313511yescharacteristic
319C4#314211yescharacteristic
320C4#321912yes—
321C4#329812yes—
322C4#332812yes—
323C2—84no—
324C2#1874812yes—
325C2#1882112yes—
326C2#2586211yescharacteristic
327C2#2587811yescharacteristic
328C2#2588411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
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
2. #33067C2semidirect
3. #30220C2semidirect
5. #10412C2semidirect
6. #9702C2semidirect
8. #8450C2semidirect
9. #8175C2semidirect

Maximal subgroups

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

subgroupindex
#365592
#330672
#302202
#287242
#104122
#97022
#96422
#84502
#81752