Finite Group Databasebeta

← #281357

Subgroups — #281357

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28273C211yesmaximal, characteristic
2#24970C212yesmaximal, complemented
3#22813C211yesmaximal, characteristic
4#10625C212yesmaximal, complemented
5#10433C212yesmaximal, complemented
6#9657C211yesmaximal, characteristic
7#9617C211yesmaximal, characteristic, complemented
8#8897C212yesmaximal, complemented
9#8815C211yesmaximal, characteristic, complemented
10#8651C211yesmaximal, characteristic, complemented
11#8207C211yesmaximal, characteristic, complemented
12C2×#446C2211yescharacteristic
13C2×C4×D8C2211yescharacteristic
14#2538C2212yes—
15#2354C2212yes—
16#2075—22no—
17#2065—22no—
18#2036—22no—
19#1972—21no—
20#1972—21no—
21#1954C2212yes—
22#1947C2212yes—
23#1944—21no—
24#1926—22no—
25#1873—22no—
26#1844C2211yescharacteristic
27#1844C2211yescharacteristic
28#1834C2212yes—
29#1834C2212yes—
30#1825—22no—
31#1803C2212yes—
32#1802C2212yes—
33#1802C2212yes—
34#1776—21no—
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×D4C2312yes—
49C2×#117—22no—
50#446—22no—
51#446—22no—
52C4×D8—22no—
53C4×D8—22no—
54C2×#107—22no—
55C2×#107D411yescharacteristic
56C2×#106—21no—
57C2×#106—21no—
58C2×#106—21no—
59C2×#106—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×C4⋊C8C2312yes—
64C2×#103—22no—
65C2×#103D412yes—
66C2×#102—22no—
67C2×#102—22no—
68C2×#102D412yes—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×#97—21no—
72C2×#97—21no—
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#396—42no—
77#390—22no—
78#390—22no—
79#389—22no—
80#388—42no—
81#384C2312yes—
82#384C2312yes—
83#380—22no—
84#380—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4C2311yescharacteristic
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—42no—
94#340D412yes—
95#336—21no—
96#335—22no—
97Q8⋊C8—22no—
98D4⋊C8—24no—
99C82—22no—
100C22×Q8C2×D411yescharacteristic
101C22×D4C2×D412yes—
102C23×C4—22no—
103C2×D8—22no—
104C2×D8—22no—
105C2×D8—22no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—22no—
110C22×C8C2×D412yes—
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C22⋊Q8—44no—
114C22⋊Q8—82no—
115C4×Q8—21no—
116C4×Q8—21no—
117C4×Q8—21no—
118C4×Q8—21no—
119C4×D4—22no—
120C4×D4—22no—
121C4×D4—22no—
122C4×D4—22no—
123#117—44no—
124#117—82no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C2×D412yes—
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2×D411yescharacteristic
136C2×C22⋊C4—22no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—42no—
144C2×C42C2411yescharacteristic
145#107—22no—
146#107—82no—
147#106—42no—
148#106—42no—
149#106—81no—
150#106—81no—
151C4⋊C8—22no—
152C4⋊C8—21no—
153C4⋊C8—22no—
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—42no—
158#103—42no—
159#103—82no—
160#102—24no—
161#102—82no—
162#102—82no—
163C22⋊C8—82no—
164C22⋊C8—82no—
165#97—42no—
166#97—42no—
167C4×C8—21no—
168C4×C8—21no—
169C4×C8—21no—
170C4×C8#4112yes—
171C4×C8#4112yes—
172C4×C8#4112yes—
173#95—21no—
174#95—42no—
175#95—42no—
176C24—22no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D411yescharacteristic
201C22×C4H(5,𝔽2)11yescharacteristic
202D8—44no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—42no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8#12312yes—
220C2×C8#12312yes—
221C2×C8#13512yes—
222C2×C8#13712yes—
223C4⋊C4—22no—
224C4⋊C4—21no—
225C4⋊C4—22no—
226C4⋊C4—21no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C4⋊C4—84no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—84no—
245C22⋊C4—84no—
246C22⋊C4—162no—
247C22⋊C4—162no—
248C42—41no—
249C42—41no—
250C42—41no—
251C42—41no—
252C42—84no—
253C42C2×#4111yescharacteristic
254C42C2×#4111yescharacteristic
255C42C2×#4111yescharacteristic
256C422−1+411yescharacteristic
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23D4211yescharacteristic
265Q8—42no—
266Q8—41no—
267Q8—41no—
268D4—42no—
269D4—42no—
270D4—42no—
271D4—42no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4C2×#12311yescharacteristic
294C2×C4#53211yescharacteristic
295C2×C4#54211yescharacteristic
296C2×C4#54711yescharacteristic
297C2×C4#54711yescharacteristic
298C2×C4#55311yescharacteristic
299C2×C4C2×#13511yescharacteristic
300C2×C4C2×#13711yescharacteristic
301C2×C4#57512yes—
302C2×C4#57612yes—
303C2×C4#57711yescharacteristic
304C2×C4#57711yescharacteristic
305C8—22no—
306C8—42no—
307C8—42no—
308C8—81no—
309C8—81no—
310C8—82no—
311C8#48514yes—
312C22—82no—
313C22—82no—
314C22—82no—
315C22—82no—
316C22—82no—
317C22—82no—
318C22—82no—
319C22#242211yescharacteristic
320C22#327312yes—
321C22#327811yescharacteristic
322C22#328411yescharacteristic
323C22#329012yes—
324C4—21no—
325C4—21no—
326C4—82no—
327C4—81no—
328C4—81no—
329C4—82no—
330C4—162no—
331C4—162no—
332C4C2×#48511yescharacteristic
333C4#309911yescharacteristic
334C4#317611yescharacteristic
335C4#318311yescharacteristic
336C4#334011yescharacteristic
337C4#334712yes—
338C4#335611yescharacteristic
339C2—162no—
340C2#1994612yes—
341C2#2030712yes—
342C2#2501511yescharacteristic
343C2#2504811yescharacteristic
344C2#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. #24970C2semidirect
4. #10625C2semidirect
5. #10433C2semidirect
7. #9617C2semidirect
8. #8897C2semidirect
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
#249702
#228132
#106252
#104332
#96572
#96172
#88972
#88152
#86512
#82072