Finite Group Databasebeta

← #314728

Subgroups — #314728

Order 512.

662 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27680C212yesmaximal
2#21597C211yesmaximal, characteristic
3C2×#1845C211yesmaximal, characteristic
4#10651C212yesmaximal, complemented
5#10617C212yesmaximal, complemented
6#9621C212yesmaximal, complemented
7#8724C212yesmaximal
8#8653C212yesmaximal, complemented
9#8595C211yesmaximal, characteristic
10C2×#446C2211yescharacteristic
11C22×#107—21no—
12C22×#107—21no—
13C22×C4×C8C2211yescharacteristic
14C4×C22⋊Q8C2211yescharacteristic
15C2×#384C2212yes—
16#2099—22no—
17#2096—22no—
18#1984C2211yescharacteristic
19#1948C2214yes—
20#1939C2211yescharacteristic
21#1916—21no—
22#1872—22no—
23#1853—22no—
24#1850—21no—
25#1845—21no—
26#1845—21no—
27#1845C2212yes—
28#1845C2212yes—
29#1803C2214yes—
30#1797C2211yescharacteristic
31#1797C2212yes—
32#1771—21no—
33#1770C2211yescharacteristic
34#1752—21no—
35C2×#334C2212yes—
36#1296C2214yes—
37#1296C2214yes—
38#1277C2212yes—
39C23×C8D411yescharacteristic
40C23×C8D411yescharacteristic
41C2×C22⋊Q8D411yescharacteristic
42C2×C4×Q8C2311yescharacteristic
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C42C2311yescharacteristic
47#446—24no—
48C2×#107—22no—
49C2×#107—22no—
50C2×#107—22no—
51C2×#107—22no—
52C2×#107—44no—
53C2×#107—44no—
54C2×#106—24no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8C2314yes—
58C2×C4⋊C8C2311yescharacteristic
59C2×C4⋊C8C2311yescharacteristic
60C2×#103—24no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—22no—
63C2×#97—21no—
64C2×#97—21no—
65C2×C4×C8—24no—
66C2×C4×C8D411yescharacteristic
67C2×C4×C8D411yescharacteristic
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70#391D411yescharacteristic
71#389D411yescharacteristic
72#387—21no—
73#385—21no—
74#384—22no—
75#384—22no—
76#384—22no—
77#384C2314yes—
78#380D411yescharacteristic
79C4×C4⋊C4—21no—
80C4×C4⋊C4C2311yescharacteristic
81C4×C22⋊C4—21no—
82#340—24no—
83#340—24no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336—22no—
89#336—21no—
90#334—24no—
91C22×Q8C2×D411yescharacteristic
92C23×C4—42no—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4#4111yescharacteristic
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—24no—
104C22×C8SD1614yes—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C22⋊Q8—24no—
110C4×Q8—22no—
111C4×Q8—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—84no—
124C2×C4⋊C4—84no—
125C2×C4⋊C4C2×D411yescharacteristic
126C2×C4⋊C4C2×D411yescharacteristic
127C2×C4⋊C4C2×D411yescharacteristic
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42#4111yescharacteristic
137C2×C42C2411yescharacteristic
138#107—44no—
139#107—44no—
140#107—44no—
141#107—44no—
142#106—84no—
143C4⋊C8—24no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—24no—
147C4⋊C8—42no—
148C4⋊C8—42no—
149#103—84no—
150C22⋊C8—42no—
151C22⋊C8—44no—
152#97—42no—
153#97—42no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8#4114yes—
158#95—21no—
159#95—21no—
160#95—21no—
161#95—21no—
162#95—21no—
163C24#12111yescharacteristic
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—44no—
181C22×C4—84no—
182C22×C4—84no—
183C22×C4—84no—
184C22×C4#12111yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4C2×SD1612yes—
188C22×C4AGL(1,ℤ/8)12yes—
189C22×C4#13714yes—
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D411yescharacteristic
192C22×C4C2×#4111yescharacteristic
193C2×C8—28no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—28no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—44no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—44no—
208C2×C8—41no—
209C2×C8—44no—
210C2×C8#12312yes—
211C2×C8#12312yes—
212C2×C8C2×SD1612yes—
213C2×C8C2×SD1612yes—
214C2×C8#13712yes—
215C2×C8#13712yes—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—84no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C4⋊C4—84no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C42—21no—
233C42—21no—
234C42—22no—
235C42—22no—
236C42—41no—
237C42—41no—
238C42—41no—
239C42—41no—
240C42C2×#4112yes—
241C422−1+412yes—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23#46012yes—
247C23#47012yes—
248C23#47312yes—
249C23C2×#12111yescharacteristic
250Q8—44no—
251C2×C4—28no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—28no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—84no—
278C2×C4—84no—
279C2×C4—84no—
280C2×C4—168no—
281C2×C4#46012yes—
282C2×C4#47012yes—
283C2×C4#47212yes—
284C2×C4C2×#12311yescharacteristic
285C2×C4#53611yescharacteristic
286C2×C4#54712yes—
287C2×C4#55212yes—
288C2×C4C22×SD1611yescharacteristic
289C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
290C2×C4C2×#13711yescharacteristic
291C2×C4C2×#13712yes—
292C2×C4C2×#13711yescharacteristic
293C2×C4#57612yes—
294C8—22no—
295C8—22no—
296C8—22no—
297C8—84no—
298C8—81no—
299C8—81no—
300C8#48314yes—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22#169014yes—
308C22#248611yescharacteristic
309C22C2×#46011yescharacteristic
310C22#303911yescharacteristic
311C22C2×#47011yescharacteristic
312C22#305811yescharacteristic
313C22C2×#47211yescharacteristic
314C22C2×#47311yescharacteristic
315C4—22no—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—164no—
324C4C2×#48311yescharacteristic
325C4#309211yescharacteristic
326C4#322911yescharacteristic
327C4#323811yescharacteristic
328C4#334614yes—
329C2—24no—
330C2C2×#169011yescharacteristic
331C2#1920311yescharacteristic
332C2#1929711yescharacteristic
333C2#1935811yescharacteristic
334C2#2764711yescharacteristic
335C2#2766611yescharacteristic
336C2#2767511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10651C2semidirect
5. #10617C2semidirect
6. #9621C2semidirect
8. #8653C2semidirect

Maximal subgroups

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

subgroupindex
#276802
#215972
C2×#18452
#106512
#106172
#96212
#87242
#86532
#85952