Finite Group Databasebeta

← #281351

Subgroups — #281351

Order 512.

636 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28273C211yesmaximal, characteristic
2#24969C212yesmaximal, complemented
3#22670C211yesmaximal, characteristic
4#10625C212yesmaximal, complemented
5#10433C212yesmaximal, complemented
6#9657C211yesmaximal, characteristic
7#9618C211yesmaximal, characteristic, complemented
8#8894C212yesmaximal, complemented
9#8815C211yesmaximal, characteristic, complemented
10#8651C211yesmaximal, characteristic, complemented
11#8207C211yesmaximal, characteristic, complemented
12C2×#446C2211yescharacteristic
13C2×#442—21no—
14C2×C4×D8C2211yescharacteristic
15#2538C2212yes—
16D4×C4⋊C4C2212yes—
17#2077—22no—
18#2065—22no—
19#2035—22no—
20#1972—21no—
21#1954C2212yes—
22#1948C2212yes—
23#1944—21no—
24#1943—21no—
25#1872—22no—
26#1864—22no—
27#1844C2211yescharacteristic
28#1844C2211yescharacteristic
29#1834C2212yes—
30#1834C2212yes—
31#1823—22no—
32#1803C2212yes—
33#1802C2212yes—
34#1802C2212yes—
35#1770C2211yescharacteristic
36C4×#103C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×Q8⋊C8C2211yescharacteristic
39C2×D4⋊C8C2212yes—
40C2×C82C2211yescharacteristic
41#1296C2212yes—
42#1296C2212yes—
43#1279C2211yescharacteristic
44#1277C2211yescharacteristic
45C22×D8D411yescharacteristic
46C2×C22⋊Q8—22no—
47C2×C4×Q8C2311yescharacteristic
48C2×C4×D4—22no—
49C2×C4×D4C2312yes—
50C22×C4⋊C4—22no—
51#446—22no—
52#446—22no—
53#442—44no—
54C4×D8—22no—
55C4×D8—22no—
56C2×#107D411yescharacteristic
57C2×#106—21no—
58C2×#106—21no—
59C2×#106—21no—
60C2×#106—21no—
61C2×#106—22no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2312yes—
66C2×#103—22no—
67C2×#103D412yes—
68C2×#102—22no—
69C2×#102—22no—
70C2×#102D412yes—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×#97—21no—
74C2×#97—21no—
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#396—42no—
79#390—22no—
80#389—22no—
81#389—22no—
82#385—42no—
83#384C2312yes—
84#384C2312yes—
85#380—22no—
86#380—22no—
87C4×C4⋊C4C2311yescharacteristic
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—42no—
95#340D412yes—
96#336—21no—
97#335—22no—
98Q8⋊C8—22no—
99D4⋊C8—24no—
100C82—22no—
101C22×Q8C2×D411yescharacteristic
102C22×D4C2×D412yes—
103C23×C4—22no—
104C23×C4—42no—
105C2×D8—22no—
106C2×D8—22no—
107C2×D8SD1614yes—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—22no—
112C22×C8C2×D412yes—
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C22⋊Q8—44no—
116C22⋊Q8—82no—
117C4×Q8—21no—
118C4×Q8—21no—
119C4×Q8—21no—
120C4×Q8—21no—
121C4×D4—22no—
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—22no—
125C4×D4—44no—
126C4×D4—82no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—82no—
138C2×C4⋊C4C2×D412yes—
139C2×C4⋊C4C2×D412yes—
140C2×C4⋊C4C2×D411yescharacteristic
141C2×C22⋊C4—22no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42C2411yescharacteristic
148#107—22no—
149#106—42no—
150#106—42no—
151#106—82no—
152#106—81no—
153#106—81no—
154C4⋊C8—22no—
155C4⋊C8—21no—
156C4⋊C8—22no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8—42no—
161#103—42no—
162#103—82no—
163#102—24no—
164#102—44no—
165#102—82no—
166C22⋊C8—82no—
167C22⋊C8—82no—
168#97—42no—
169#97—42no—
170C4×C8—21no—
171C4×C8—21no—
172C4×C8—21no—
173C4×C8#4112yes—
174C4×C8#4112yes—
175C4×C8#4112yes—
176#95—21no—
177#95—42no—
178#95—42no—
179C24—22no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4C2×SD1614yes—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—84no—
203C22×C4—84no—
204C22×C4—162no—
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D411yescharacteristic
207C22×C4H(5,𝔽2)11yescharacteristic
208D8—28no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—42no—
216C2×C8—41no—
217C2×C8—42no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8#12312yes—
226C2×C8#12312yes—
227C2×C8C2×SD1612yes—
228C2×C8#13712yes—
229C4⋊C4—22no—
230C4⋊C4—21no—
231C4⋊C4—22no—
232C4⋊C4—21no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—84no—
248C4⋊C4—84no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—84no—
252C22⋊C4—84no—
253C22⋊C4—162no—
254C42—41no—
255C42—41no—
256C42—41no—
257C42—41no—
258C42C2×#4111yescharacteristic
259C42C2×#4111yescharacteristic
260C42C2×#4111yescharacteristic
261C422−1+411yescharacteristic
262C23—24no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23D4211yescharacteristic
270Q8—42no—
271Q8—41no—
272Q8—41no—
273D4—24no—
274D4—24no—
275D4—42no—
276D4—42no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—168no—
299C2×C4C2×#12311yescharacteristic
300C2×C4#53211yescharacteristic
301C2×C4#54211yescharacteristic
302C2×C4#54711yescharacteristic
303C2×C4#54711yescharacteristic
304C2×C4#55311yescharacteristic
305C2×C4C22×SD1611yescharacteristic
306C2×C4C2×#13711yescharacteristic
307C2×C4#57512yes—
308C2×C4#57612yes—
309C2×C4#57711yescharacteristic
310C2×C4#57711yescharacteristic
311C8—22no—
312C8—42no—
313C8—42no—
314C8—81no—
315C8—81no—
316C8—82no—
317C8#48314yes—
318C22—44no—
319C22—44no—
320C22—44no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22#242211yescharacteristic
326C22#327312yes—
327C22#327811yescharacteristic
328C22D4×SD1611yescharacteristic
329C22#329012yes—
330C4—21no—
331C4—21no—
332C4—82no—
333C4—82no—
334C4—81no—
335C4—81no—
336C4—162no—
337C4—162no—
338C4C2×#48311yescharacteristic
339C4#309911yescharacteristic
340C4#317611yescharacteristic
341C4#318311yescharacteristic
342C4#333511yescharacteristic
343C4#334712yes—
344C4#335611yescharacteristic
345C2—84no—
346C2#1989112yes—
347C2#2030712yes—
348C2#2501411yescharacteristic
349C2#2504811yescharacteristic
350C2#2505011yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D8D4
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
2. #24969C2semidirect
4. #10625C2semidirect
5. #10433C2semidirect
7. #9618C2semidirect
8. #8894C2semidirect
9. #8815C2semidirect
10. #8651C2semidirect
11. #8207C2semidirect

Maximal subgroups

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

subgroupindex
#282732
#249692
#226702
#106252
#104332
#96572
#96182
#88942
#88152
#86512
#82072