Finite Group Databasebeta

← #315759

Subgroups — #315759

Order 512.

687 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 687 conjugacy classes of subgroups — too many to draw. Only its 186 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#27949C212yesmaximal
3#27875C212yesmaximal
4#20824C211yesmaximal, characteristic
5#10681C212yesmaximal, complemented
6#10680C212yesmaximal, complemented
7#9839C211yesmaximal, characteristic, complemented
8#9825C211yesmaximal, characteristic, complemented
9#8820C211yesmaximal, characteristic, complemented
10#8813C211yesmaximal, characteristic, complemented
11#8146C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×#470—22no—
14C2×C4×M16C2211yescharacteristic
15#2857C2211yescharacteristic
16#2857C2211yescharacteristic
17#2285C2212yes—
18#2085—24no—
19#2077—24no—
20#1990C2212yes—
21#1990C2212yes—
22#1990C2212yes—
23#1990C2212yes—
24#1971—22no—
25#1956C2211yescharacteristic
26#1956C2211yescharacteristic
27#1920—24no—
28#1854—24no—
29#1845C2212yes—
30#1845C2211yescharacteristic
31#1844C2212yes—
32#1844C2211yescharacteristic
33#1844C2211yescharacteristic
34#1844C2212yes—
35#1834C2212yes—
36#1834C2212yes—
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—24no—
58C2×#106—22no—
59C2×#106—24no—
60C2×#103—22no—
61C2×#103—22no—
62C2×C22⋊C8D414yes—
63C4×M16—22no—
64C4×M16—22no—
65C2×#97D412yes—
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2312yes—
68C2×C4×C8C2312yes—
69C2×C4×C8C2311yescharacteristic
70#391—21no—
71#391—21no—
72#389—22no—
73#389—22no—
74#384C2311yescharacteristic
75#384C2311yescharacteristic
76#384C2312yes—
77#384C2312yes—
78#382—22no—
79#380—22no—
80#380—22no—
81#380—22no—
82#380—22no—
83C4×C4⋊C4—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—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—84no—
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—44no—
142#107—84no—
143#106—44no—
144#106—44no—
145#106—48no—
146#106—84no—
147#103—44no—
148#103—44no—
149C22⋊C8—28no—
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—
162#95—42no—
163C24#12711yescharacteristic
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4#12712yes—
187C22×C4#12711yescharacteristic
188C22×C4C22×D411yescharacteristic
189C22×C4C22×D412yes—
190Q16—88no—
191M16—48no—
192C2×C8—24no—
193C2×C8—24no—
194C2×C8—22no—
195C2×C8—24no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—24no—
199C2×C8—24no—
200C2×C8#12314yes—
201C2×C8#13514yes—
202C2×C8AGL(1,ℤ/8)14yes—
203C2×C8#13718yes—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C22⋊C4—84no—
221C22⋊C4—82no—
222C22⋊C4—82no—
223C42—21no—
224C42—21no—
225C42—22no—
226C42—22no—
227C42—84no—
228C42C2×#4111yescharacteristic
229C42C2×#4111yescharacteristic
230C422−1+412yes—
231C23—21no—
232C23—21no—
233C23—22no—
234C23—22no—
235C23—21no—
236C23C2×#12711yescharacteristic
237Q8—82no—
238Q8—82no—
239Q8—82no—
240Q8—82no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—82no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4C2×#12311yescharacteristic
268C2×C4#53611yescharacteristic
269C2×C4#54712yes—
270C2×C4#54712yes—
271C2×C4C2×#13512yes—
272C2×C4C2×AGL(1,ℤ/8)12yes—
273C2×C4C2×#13714yes—
274C2×C4#57714yes—
275C8—28no—
276C8—42no—
277C8—44no—
278C8#48218yes—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22#250111yescharacteristic
285C22#314711yescharacteristic
286C22#315111yescharacteristic
287C22#315212yes—
288C22#315412yes—
289C4—41no—
290C4—42no—
291C4—161no—
292C4—161no—
293C4—161no—
294C4—162no—
295C4—161no—
296C4—162no—
297C4C2×#48212yes—
298C4#322812yes—
299C4#334414yes—
300C4#335614yes—
301C2—81no—
302C2#1964212yes—
303C2#1972111yescharacteristic
304C2#1974611yescharacteristic
305C2#2790711yescharacteristic
306C2#2794012yes—
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. #10681C2semidirect
6. #10680C2semidirect
7. #9839C2semidirect
8. #9825C2semidirect
9. #8820C2semidirect
10. #8813C2semidirect

Maximal subgroups

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

subgroupindex
#331762
#279492
#278752
#208242
#106812
#106802
#98392
#98252
#88202
#88132
#81462