Finite Group Databasebeta

← #315762

Subgroups — #315762

Order 512.

671 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 671 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33176C211yesmaximal, characteristic
2#27962C212yesmaximal
3#27878C212yesmaximal
4#20824C211yesmaximal, characteristic
5#10682C212yesmaximal, complemented
6#10679C212yesmaximal, complemented
7#9839C211yesmaximal, characteristic, complemented
8#9825C211yesmaximal, characteristic, complemented
9#8820C211yesmaximal, characteristic, complemented
10#8815C211yesmaximal, characteristic, complemented
11#8146C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×#470—22no—
14C2×C4×M16C2211yescharacteristic
15#2857C2211yescharacteristic
16#2857C2211yescharacteristic
17#2285C2212yes—
18#2101—24no—
19#2076—24no—
20#1990C2212yes—
21#1990C2212yes—
22#1990C2212yes—
23#1990C2212yes—
24#1971—22no—
25#1956C2211yescharacteristic
26#1956C2211yescharacteristic
27#1921—24no—
28#1845C2211yescharacteristic
29#1845C2212yes—
30#1844C2211yescharacteristic
31#1844C2211yescharacteristic
32#1834C2212yes—
33#1834C2212yes—
34#1834C2212yes—
35#1834C2212yes—
36#1825—24no—
37#1809—22no—
38#1776—22no—
39C4×C22⋊C8C2212yes—
40C2×#322C2211yescharacteristic
41C2×C82C2211yescharacteristic
42#1273C2212yes—
43C22×Q16—22no—
44C22×M16D412yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×#117—22no—
50C22×C42C2311yescharacteristic
51#494—22no—
52#494—22no—
53#470—48no—
54C2×#107—22no—
55C2×#107—24no—
56C2×#106—22no—
57C2×#106—22no—
58C2×#103—22no—
59C2×#103—22no—
60C2×C22⋊C8D414yes—
61C4×M16—22no—
62C4×M16—22no—
63C2×#97D412yes—
64C2×C4×C8C2311yescharacteristic
65C2×C4×C8C2312yes—
66C2×C4×C8C2312yes—
67C2×C4×C8C2311yescharacteristic
68#391—21no—
69#391—21no—
70#389—22no—
71#389—22no—
72#384C2311yescharacteristic
73#384C2311yescharacteristic
74#384C2312yes—
75#384C2312yes—
76#382—22no—
77#380—22no—
78#380—22no—
79#380—22no—
80#380—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83#340—24no—
84#340—24no—
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—21no—
95C22×Q8—21no—
96C23×C4C2×D412yes—
97C23×C4C2×D411yescharacteristic
98C2×Q16—44no—
99C2×Q16—44no—
100C2×Q16—44no—
101C2×M16—24no—
102C2×M16—24no—
103C2×M16—24no—
104C22×C8C2×D412yes—
105C22×C8C2×D414yes—
106C22×C8C2×D414yes—
107C22×C8C2×D412yes—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C22⋊Q8—44no—
115C22⋊Q8—44no—
116#117—48no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
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—22no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—22no—
137C2×C42—42no—
138C2×C42C2×D412yes—
139C2×C42C2×D411yescharacteristic
140C2×C42C2411yescharacteristic
141#107—48no—
142#107—44no—
143#106—44no—
144#106—44no—
145#103—44no—
146#103—44no—
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—
160#95—42no—
161C24#12711yescharacteristic
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4#12712yes—
185C22×C4#12711yescharacteristic
186C22×C4C22×D411yescharacteristic
187C22×C4C22×D412yes—
188Q16—88no—
189M16—48no—
190C2×C8—22no—
191C2×C8—24no—
192C2×C8—24no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—24no—
196C2×C8—24no—
197C2×C8—24no—
198C2×C8—24no—
199C2×C8#12314yes—
200C2×C8#13514yes—
201C2×C8AGL(1,ℤ/8)14yes—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—44no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—82no—
217C4⋊C4—82no—
218C22⋊C4—84no—
219C22⋊C4—82no—
220C22⋊C4—82no—
221C42—21no—
222C42—21no—
223C42—22no—
224C42—22no—
225C42—84no—
226C42C2×#4111yescharacteristic
227C42C2×#4111yescharacteristic
228C422−1+412yes—
229C23—21no—
230C23—21no—
231C23—22no—
232C23—22no—
233C23—21no—
234C23C2×#12711yescharacteristic
235Q8—82no—
236Q8—82no—
237Q8—82no—
238Q8—82no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4C2×#12311yescharacteristic
266C2×C4#53611yescharacteristic
267C2×C4#54712yes—
268C2×C4#54712yes—
269C2×C4C2×#13512yes—
270C2×C4C2×AGL(1,ℤ/8)12yes—
271C2×C4#57514yes—
272C2×C4#57814yes—
273C8—42no—
274C8—44no—
275C8—44no—
276C8#48218yes—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22—42no—
282C22#250111yescharacteristic
283C22#314711yescharacteristic
284C22#315111yescharacteristic
285C22#315312yes—
286C22#315612yes—
287C4—41no—
288C4—42no—
289C4—161no—
290C4—161no—
291C4—161no—
292C4—161no—
293C4—162no—
294C4—162no—
295C4C2×#48212yes—
296C4#322812yes—
297C4#334914yes—
298C4#335214yes—
299C2—81no—
300C2#1964312yes—
301C2#1973111yescharacteristic
302C2#1975011yescharacteristic
303C2#2790711yescharacteristic
304C2#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. #10682C2semidirect
6. #10679C2semidirect
7. #9839C2semidirect
8. #9825C2semidirect
9. #8820C2semidirect
10. #8815C2semidirect

Maximal subgroups

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

subgroupindex
#331762
#279622
#278782
#208242
#106822
#106792
#98392
#98252
#88202
#88152
#81462