Finite Group Databasebeta

← #315756

Subgroups — #315756

Order 512.

736 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33176C211yesmaximal, characteristic
2#27961C212yesmaximal
3#27876C212yesmaximal
4#20800C211yesmaximal, characteristic
5#10686C212yesmaximal, complemented
6#10684C212yesmaximal, complemented
7#9839C211yesmaximal, characteristic, complemented
8#9825C211yesmaximal, characteristic, complemented
9#8819C211yesmaximal, characteristic, complemented
10#8815C211yesmaximal, characteristic, complemented
11#8146C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×#470—22no—
14C2×#428—22no—
15C2×C4×M16C2211yescharacteristic
16#2857C2212yes—
17#2285C2211yescharacteristic
18C2×#384C2211yescharacteristic
19#2100—24no—
20#2077—24no—
21#1990C2212yes—
22#1990C2212yes—
23#1990C2212yes—
24#1990C2212yes—
25#1971—22no—
26#1956C2212yes—
27#1919—24no—
28#1847—22no—
29#1845C2211yescharacteristic
30#1845C2211yescharacteristic
31#1844C2211yescharacteristic
32#1844C2211yescharacteristic
33#1844C2211yescharacteristic
34#1844C2212yes—
35#1834C2212yes—
36#1834C2212yes—
37#1834C2212yes—
38#1824—24no—
39C4×C22⋊C8C2212yes—
40C2×#322C2211yescharacteristic
41C2×C82C2211yescharacteristic
42#1273C2212yes—
43C22×Q16—22no—
44C22×M16D412yes—
45C2×C4⋊Q8C2312yes—
46C2×C22⋊Q8—22no—
47C2×#117—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C42C2311yescharacteristic
51#494—22no—
52#494—22no—
53#470—48no—
54#428—48no—
55C2×#107—22no—
56C2×#107—22no—
57C2×#106—24no—
58C2×#106—22no—
59C2×#106—22no—
60C2×#106—22no—
61C2×#103—24no—
62C2×C22⋊C8D414yes—
63C4×M16—22no—
64C4×M16—22no—
65C2×#97D412yes—
66C2×C4×C8C2312yes—
67C2×C4×C8C2312yes—
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70#391—22no—
71#389—22no—
72#389—22no—
73#384—22no—
74#384—21no—
75#384C2312yes—
76#384C2311yescharacteristic
77#384C2311yescharacteristic
78#384C2311yescharacteristic
79#384C2311yescharacteristic
80#382—21no—
81#380—22no—
82#380—22no—
83#380—22no—
84C4×C4⋊C4—22no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#336D414yes—
93#322—22no—
94C82—22no—
95C22×Q8—22no—
96C23×C4—41no—
97C23×C4C2×D412yes—
98C23×C4C2×D411yescharacteristic
99C2×Q16—44no—
100C2×Q16—44no—
101C2×Q16—44no—
102C2×M16—24no—
103C2×M16—24no—
104C2×M16SD1618yes—
105C22×C8C2×D414yes—
106C22×C8C2×D414yes—
107C22×C8C2×D412yes—
108C22×C8C2×D412yes—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C22⋊Q8—48no—
114#117—44no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—48no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—42no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—42no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—22no—
138C2×C42—41no—
139C2×C42C2×D412yes—
140C2×C42C2×D411yescharacteristic
141C2×C42C2411yescharacteristic
142#107—44no—
143#107—44no—
144#106—48no—
145#106—44no—
146#106—44no—
147#106—44no—
148#103—48no—
149C22⋊C8—44no—
150#97—24no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8—21no—
154C4×C8—21no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8#4112yes—
158C4×C8#4112yes—
159#95—42no—
160#95—42no—
161#95—42no—
162C24#12711yescharacteristic
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—44no—
166C2×Q8—44no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—82no—
181C22×C4—84no—
182C22×C4#12712yes—
183C22×C4#12711yescharacteristic
184C22×C4C2×SD1614yes—
185C22×C4C22×D411yescharacteristic
186C22×C4C22×D412yes—
187Q16—88no—
188M16—216no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—24no—
192C2×C8—24no—
193C2×C8—24no—
194C2×C8—24no—
195C2×C8—24no—
196C2×C8—24no—
197C2×C8#12314yes—
198C2×C8C2×SD1614yes—
199C2×C8C2×SD1614yes—
200C2×C8AGL(1,ℤ/8)14yes—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—44no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—82no—
217C4⋊C4—84no—
218C4⋊C4—84no—
219C22⋊C4—82no—
220C22⋊C4—84no—
221C42—21no—
222C42—21no—
223C42—22no—
224C42—22no—
225C42—82no—
226C42C2×#4111yescharacteristic
227C42C2×#4111yescharacteristic
228C422−1+412yes—
229C23—21no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23#49212yes—
234C23C2×#12711yescharacteristic
235Q8—84no—
236Q8—84no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—164no—
261C2×C4#49212yes—
262C2×C4C2×#12311yescharacteristic
263C2×C4#53611yescharacteristic
264C2×C4#54712yes—
265C2×C4#54712yes—
266C2×C4C22×SD1612yes—
267C2×C4C2×AGL(1,ℤ/8)12yes—
268C2×C4#57514yes—
269C2×C4#57814yes—
270C8—24no—
271C8—44no—
272C8—44no—
273C8#48118yes—
274C22—22no—
275C22—24no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22#250111yescharacteristic
280C22C2×#49211yescharacteristic
281C22#315111yescharacteristic
282C22#315312yes—
283C22#315612yes—
284C4—22no—
285C4—42no—
286C4—162no—
287C4—161no—
288C4—161no—
289C4—161no—
290C4—161no—
291C4—162no—
292C4C2×#48112yes—
293C4#322712yes—
294C4#334914yes—
295C4#335214yes—
296C2—42no—
297C2#1962412yes—
298C2#1973111yescharacteristic
299C2#1974911yescharacteristic
300C2#2790611yescharacteristic
301C2#2794312yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
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
5. #10686C2semidirect
6. #10684C2semidirect
7. #9839C2semidirect
8. #9825C2semidirect
9. #8819C2semidirect
10. #8815C2semidirect

Maximal subgroups

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

subgroupindex
#331762
#279612
#278762
#208002
#106862
#106842
#98392
#98252
#88192
#88152
#81462