Finite Group Databasebeta

← #315777

Subgroups — #315777

Order 512.

708 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33210C211yesmaximal, characteristic
2#27961C212yesmaximal
3#27892C212yesmaximal
4#20800C211yesmaximal, characteristic
5#10680C212yesmaximal, complemented
6#10679C212yesmaximal, complemented
7#9841C211yesmaximal, characteristic, complemented
8#9819C211yesmaximal, characteristic, complemented
9#8820C211yesmaximal, characteristic, complemented
10#8816C211yesmaximal, characteristic, complemented
11#8146C211yesmaximal, characteristic
12C2×#498C2211yescharacteristic
13C2×#428—22no—
14C2×C4×M16C2211yescharacteristic
15#2857C2212yes—
16#2285C2211yescharacteristic
17C2×#384C2211yescharacteristic
18#2100—24no—
19#2036—24no—
20#1990C2212yes—
21#1990C2212yes—
22#1990C2212yes—
23#1956C2212yes—
24#1956C2212yes—
25#1943—22no—
26#1928—24no—
27#1847—22no—
28#1845C2211yescharacteristic
29#1845C2211yescharacteristic
30#1845C2211yescharacteristic
31#1845C2212yes—
32#1844C2211yescharacteristic
33#1844C2211yescharacteristic
34#1834C2212yes—
35#1834C2212yes—
36#1834C2212yes—
37#1824—24no—
38#1809—22no—
39C4×C22⋊C8C2212yes—
40C2×#322C2211yescharacteristic
41C2×C82C2211yescharacteristic
42#1273C2212yes—
43C22×M16D412yes—
44C2×C4⋊Q8C2312yes—
45C2×C22⋊Q8—22no—
46C2×#117—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—21no—
49C22×C42C2311yescharacteristic
50#498—22no—
51#498—22no—
52#428—48no—
53C2×#107—22no—
54C2×#107—22no—
55C2×#107—22no—
56C2×#107—24no—
57C2×#106—22no—
58C2×#106—22no—
59C2×#106—22no—
60C2×#103—24no—
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#336D414yes—
92#322—22no—
93C82—22no—
94C22×Q8—22no—
95C23×C4—41no—
96C23×C4C2×D412yes—
97C23×C4C2×D411yescharacteristic
98C2×M16—24no—
99C2×M16—24no—
100C2×M16Q1618yes—
101C22×C8C2×D414yes—
102C22×C8C2×D414yes—
103C22×C8C2×D412yes—
104C22×C8C2×D412yes—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C22⋊Q8—84no—
110#117—44no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—48no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—42no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—42no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—22no—
134C2×C42—41no—
135C2×C42C2×D412yes—
136C2×C42C2×D411yescharacteristic
137C2×C42C2411yescharacteristic
138#107—44no—
139#107—44no—
140#107—44no—
141#107—84no—
142#106—44no—
143#106—44no—
144#106—44no—
145#103—48no—
146#103—84no—
147C22⋊C8—44no—
148#97—24no—
149C4×C8—22no—
150C4×C8—22no—
151C4×C8—21no—
152C4×C8—21no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8#4112yes—
156C4×C8#4112yes—
157#95—42no—
158#95—42no—
159#95—42no—
160C24#12711yescharacteristic
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—44no—
164C2×Q8—44no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—82no—
179C22×C4—84no—
180C22×C4#12712yes—
181C22×C4#12711yescharacteristic
182C22×C4C2×Q1614yes—
183C22×C4C22×D411yescharacteristic
184C22×C4C22×D412yes—
185M16—216no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—24no—
189C2×C8—24no—
190C2×C8—24no—
191C2×C8—24no—
192C2×C8—24no—
193C2×C8—24no—
194C2×C8#12314yes—
195C2×C8C2×Q1614yes—
196C2×C8C2×Q1614yes—
197C2×C8AGL(1,ℤ/8)14yes—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—82no—
214C4⋊C4—84no—
215C4⋊C4—84no—
216C22⋊C4—82no—
217C22⋊C4—84no—
218C42—21no—
219C42—21no—
220C42—22no—
221C42—22no—
222C42—82no—
223C42C2×#4111yescharacteristic
224C42C2×#4111yescharacteristic
225C422−1+412yes—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—21no—
230C23#49412yes—
231C23C2×#12711yescharacteristic
232Q8—84no—
233Q8—84no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—164no—
258C2×C4#49412yes—
259C2×C4C2×#12311yescharacteristic
260C2×C4#53611yescharacteristic
261C2×C4#54712yes—
262C2×C4#54712yes—
263C2×C4C22×Q1612yes—
264C2×C4C2×AGL(1,ℤ/8)12yes—
265C2×C4#57514yes—
266C2×C4#57714yes—
267C8—24no—
268C8—44no—
269C8—44no—
270C8#48418yes—
271C22—22no—
272C22—24no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22#250111yescharacteristic
277C22C2×#49411yescharacteristic
278C22#315111yescharacteristic
279C22#315312yes—
280C22#315412yes—
281C4—22no—
282C4—42no—
283C4—161no—
284C4—162no—
285C4—161no—
286C4—161no—
287C4—161no—
288C4—162no—
289C4C2×#48412yes—
290C4#323012yes—
291C4#334814yes—
292C4#335114yes—
293C2—42no—
294C2#1963412yes—
295C2#1972111yescharacteristic
296C2#1974911yescharacteristic
297C2#2791011yescharacteristic
298C2#2794612yes—
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. #10680C2semidirect
6. #10679C2semidirect
7. #9841C2semidirect
8. #9819C2semidirect
9. #8820C2semidirect
10. #8816C2semidirect

Maximal subgroups

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

subgroupindex
#332102
#279612
#278922
#208002
#106802
#106792
#98412
#98192
#88202
#88162
#81462