Finite Group Databasebeta

← #315583

Subgroups — #315583

Order 512.

787 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33170C211yesmaximal, characteristic
2#27891C212yesmaximal
3#27877C212yesmaximal, complemented
4#20797C211yesmaximal, characteristic, complemented
5#10676C211yesmaximal, characteristic, complemented
6#10674C211yesmaximal, characteristic, complemented
7#10673C211yesmaximal, characteristic, complemented
8#10671C211yesmaximal, characteristic, complemented
9#9820C212yesmaximal, complemented
10#8814C212yesmaximal, complemented
11C8×C22⋊C8C211yesmaximal, characteristic, complemented
12C2×#492C2211yescharacteristic
13C2×#465—22no—
14C2×#427—22no—
15C22×C4×C8C2211yescharacteristic
16#2857C2211yescharacteristic
17#2856C2211yescharacteristic
18#2285C2212yes—
19#2075—24no—
20#2033—24no—
21#1990C2212yes—
22#1990C2212yes—
23#1968—22no—
24#1957C2212yes—
25#1957C2212yes—
26#1957C2211yescharacteristic
27#1956C2211yescharacteristic
28#1925—24no—
29#1918—24no—
30#1846—22no—
31#1845C2211yescharacteristic
32#1845C2211yescharacteristic
33#1845C2212yes—
34#1845C2212yes—
35#1844C2212yes—
36#1844C2211yescharacteristic
37#1834C2212yes—
38#1834C2212yes—
39C4×C22⋊C8C2212yes—
40C2×C82C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×SD16—22no—
44C23×C8D412yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×#127C2311yescharacteristic
47C2×C22⋊Q8—21no—
48C2×#121—21no—
49C2×#121—22no—
50C2×#117—22no—
51C22×C42C2311yescharacteristic
52#492—24no—
53#465—48no—
54#427—48no—
55C2×#107—24no—
56C2×#107—22no—
57C2×#107—22no—
58C2×#107—24no—
59C2×#106—24no—
60C2×#106—22no—
61C2×#103—22no—
62C2×#102—24no—
63C2×#102—22no—
64C2×#102—24no—
65C2×C22⋊C8D414yes—
66C2×C4×C8—24no—
67C2×C4×C8D412yes—
68C2×C4×C8C2312yes—
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2312yes—
71C2×C4×C8C2311yescharacteristic
72#391—21no—
73#390—22no—
74#390—21no—
75#389—22no—
76#384C2311yescharacteristic
77#384C2311yescharacteristic
78#384C2312yes—
79#384C2312yes—
80#382—22no—
81#380—22no—
82#380—22no—
83#380—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86#340—24no—
87#340—24no—
88#340—24no—
89#340—24no—
90#336D414yes—
91C82—24no—
92C22×Q8—21no—
93C22×D4—21no—
94C22×D4—22no—
95C22×D4—41no—
96C23×C4C2×D412yes—
97C23×C4C2×D411yescharacteristic
98C2×SD16—44no—
99C2×SD16—44no—
100C2×SD16—44no—
101C22×C8—24no—
102C22×C8—24no—
103C22×C8D818yes—
104C22×C8C2×D414yes—
105C22×C8C2×D412yes—
106C22×C8C2×D414yes—
107C22×C8C2×D412yes—
108C4⋊Q8—22no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111#127—22no—
112#127—21no—
113#127—21no—
114C22⋊Q8—44no—
115#121—44no—
116#121—84no—
117#117—48no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—41no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—41no—
134C2×C42—22no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—42no—
138C2×C42C2×D412yes—
139C2×C42C2×D411yescharacteristic
140C2×C42C2411yescharacteristic
141#107—44no—
142#107—44no—
143#107—48no—
144#107—84no—
145#106—48no—
146#106—44no—
147#103—44no—
148#102—44no—
149#102—48no—
150#102—84no—
151C22⋊C8—28no—
152C4×C8—24no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8#4114yes—
159#95—42no—
160#95—42no—
161#95—42no—
162C24—41no—
163C24#12711yescharacteristic
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—44no—
174C2×D4—42no—
175C2×D4—44no—
176C2×D4—84no—
177C2×D4—82no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4#12712yes—
191C22×C4#12711yescharacteristic
192C22×C4C2×D814yes—
193C22×C4C22×D411yescharacteristic
194C22×C4C22×D412yes—
195SD16—88no—
196C2×C8—22no—
197C2×C8—24no—
198C2×C8—22no—
199C2×C8—24no—
200C2×C8—216no—
201C2×C8—24no—
202C2×C8—24no—
203C2×C8—24no—
204C2×C8#12314yes—
205C2×C8C2×D814yes—
206C2×C8C2×D814yes—
207C2×C8#13518yes—
208C2×C8#13714yes—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—82no—
222C22⋊C4—84no—
223C22⋊C4—84no—
224C22⋊C4—82no—
225C22⋊C4—82no—
226C42—21no—
227C42—21no—
228C42—22no—
229C42—22no—
230C42—84no—
231C42C2×#4112yes—
232C422−1+411yescharacteristic
233C422−1+411yescharacteristic
234C23—22no—
235C23—21no—
236C23—22no—
237C23—21no—
238C23—81no—
239C23—82no—
240C23—81no—
241C23—82no—
242C23—81no—
243C23#49312yes—
244C23C2×#12711yescharacteristic
245Q8—82no—
246Q8—82no—
247D4—84no—
248D4—84no—
249D4—82no—
250D4—82no—
251D4—164no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—81no—
274C2×C4#49312yes—
275C2×C4C2×#12311yescharacteristic
276C2×C4#53611yescharacteristic
277C2×C4#54712yes—
278C2×C4#54712yes—
279C2×C4C22×D812yes—
280C2×C4C2×#13514yes—
281C2×C4C2×#13712yes—
282C2×C4#57714yes—
283C8—28no—
284C8—24no—
285C8—44no—
286C8#48018yes—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—42no—
291C22—42no—
292C22—161no—
293C22—162no—
294C22—161no—
295C22—162no—
296C22—161no—
297C22#250111yescharacteristic
298C22C2×#49311yescharacteristic
299C22#314712yes—
300C22#315211yescharacteristic
301C22#315412yes—
302C4—22no—
303C4—42no—
304C4—161no—
305C4—162no—
306C4—161no—
307C4—162no—
308C4—161no—
309C4C2×#48012yes—
310C4#322612yes—
311C4#333714yes—
312C4#334014yes—
313C2—42no—
314C2—321no—
315C2#1960311yescharacteristic
316C2#1964212yes—
317C2#1966011yescharacteristic
318C2#2790911yescharacteristic
319C2#2792512yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
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
3. #27877C2semidirect
4. #20797C2semidirect
5. #10676C2semidirect
6. #10674C2semidirect
7. #10673C2semidirect
8. #10671C2semidirect
9. #9820C2semidirect
10. #8814C2semidirect
11. C8×C22⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#331702
#278912
#278772
#207972
#106762
#106742
#106732
#106712
#98202
#88142
C8×C22⋊C82