Finite Group Databasebeta

← #319332

Subgroups — #319332

Order 512.

780 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 182 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#33172C211yesmaximal, characteristic
2#27876C212yesmaximal
3#25672C212yesmaximal
4#21318C211yesmaximal, characteristic
5#10680C212yesmaximal, complemented
6#10318C212yesmaximal, complemented
7#9828C212yesmaximal, complemented
8#9826C212yesmaximal, complemented
9#8144C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#498C2212yes—
12C2×#494C2211yescharacteristic
13C2×#487C2212yes—
14C2×#467—22no—
15C22×C4×C8C2211yescharacteristic
16#2857C2212yes—
17#2285C2211yescharacteristic
18#2077—24no—
19#2035—24no—
20#1991—22no—
21#1990C2212yes—
22#1990C2212yes—
23#1990C2212yes—
24#1990C2212yes—
25#1990C2212yes—
26#1969—22no—
27#1956C2212yes—
28#1956C2212yes—
29#1919—24no—
30#1865—24no—
31#1845C2211yescharacteristic
32#1844C2212yes—
33#1834C2212yes—
34C4×C22⋊C8C2212yes—
35#1735—22no—
36C2×#322C2211yescharacteristic
37C2×#322C2211yescharacteristic
38#1273C2212yes—
39C22×Q16—22no—
40C23×C8D412yes—
41#533—24no—
42#533—41no—
43#533—41no—
44C2×C4⋊Q8C2312yes—
45C2×C4⋊Q8C2312yes—
46C2×#125—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C4×Q8—22no—
50C2×#117—21no—
51C2×#117—21no—
52C22×C42C2311yescharacteristic
53#498—24no—
54#498—42no—
55#494—41no—
56#494—41no—
57#487—24no—
58#487—42no—
59#467—48no—
60C2×#107—22no—
61C2×#106—24no—
62C2×#106—24no—
63C2×#103—24no—
64C2×#103—24no—
65C2×#103—24no—
66C2×C22⋊C8D414yes—
67C2×C4×C8—41no—
68C2×C4×C8—41no—
69C2×C4×C8D412yes—
70C2×C4×C8C2312yes—
71C2×C4×C8C2312yes—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#391—22no—
75#389—22no—
76#389—22no—
77#384C2312yes—
78#384C2312yes—
79#382—21no—
80#380—22no—
81#380—22no—
82#380—22no—
83C4×C4⋊C4—22no—
84#340—24no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—24no—
89#340—24no—
90#336D414yes—
91#322—41no—
92#322—41no—
93C22×Q8—22no—
94C22×Q8—22no—
95C23×C4C2×D412yes—
96C23×C4C2×D411yescharacteristic
97C2×Q16—44no—
98C2×Q16—44no—
99C2×Q16—44no—
100C22×C8—42no—
101C22×C8—42no—
102C22×C8SD1618yes—
103C22×C8C2×D414yes—
104C22×C8C2×D414yes—
105C22×C8C2×D412yes—
106C22×C8C2×D412yes—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8#4114yes—
115#125—24no—
116#125—42no—
117C22⋊Q8—44no—
118C22⋊Q8—44no—
119C22⋊Q8—48no—
120C4×Q8—44no—
121C4×Q8—44no—
122#117—24no—
123#117—42no—
124#117—44no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—42no—
140C2×C42—22no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—41no—
144C2×C42—41no—
145C2×C42C2×D412yes—
146C2×C42C2×D411yescharacteristic
147C2×C42C2411yescharacteristic
148#107—44no—
149#106—48no—
150#106—48no—
151#103—48no—
152#103—48no—
153#103—48no—
154C22⋊C8—44no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—21no—
158C4×C8—21no—
159C4×C8—22no—
160C4×C8—22no—
161C4×C8—21no—
162C4×C8—21no—
163C4×C8—42no—
164#95—42no—
165#95—42no—
166#95—42no—
167C24#12711yescharacteristic
168C2×Q8—24no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×Q8—44no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4#12712yes—
188C22×C4#12711yescharacteristic
189C22×C4C2×SD1614yes—
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D412yes—
192Q16—88no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—24no—
198C2×C8—24no—
199C2×C8—24no—
200C2×C8—24no—
201C2×C8—24no—
202C2×C8—24no—
203C2×C8—48no—
204C2×C8C2×SD1614yes—
205C2×C8C2×SD1614yes—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—48no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—84no—
222C22⋊C4—44no—
223C22⋊C4—82no—
224C22⋊C4—84no—
225C42—21no—
226C42—21no—
227C42—22no—
228C42—22no—
229C42—44no—
230C42—82no—
231C42C2×#4112yes—
232C42H(5,𝔽2)11yescharacteristic
233C42H(5,𝔽2)11yescharacteristic
234C23—22no—
235C23—21no—
236C23—22no—
237C23—21no—
238C23#49212yes—
239C23C2×#12711yescharacteristic
240Q8—48no—
241Q8—84no—
242Q8—84no—
243Q8—84no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4#49212yes—
266C2×C4#52511yescharacteristic
267C2×C4#53411yescharacteristic
268C2×C4#54612yes—
269C2×C4#54612yes—
270C2×C4C22×SD1612yes—
271C2×C4#57512yes—
272C2×C4#57514yes—
273C2×C4#57814yes—
274C8—24no—
275C8—24no—
276C8—44no—
277C8—44no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—42no—
282C22—42no—
283C22#253911yescharacteristic
284C22C2×#49211yescharacteristic
285C22#315012yes—
286C22#315311yescharacteristic
287C22#315612yes—
288C4—22no—
289C4—42no—
290C4—84no—
291C4—162no—
292C4—162no—
293C4—162no—
294C4#310312yes—
295C4#318712yes—
296C4#331314yes—
297C4#331514yes—
298C2—42no—
299C2#1960811yescharacteristic
300C2#1965311yescharacteristic
301C2#1969612yes—
302C2#2830511yescharacteristic
303C2#2835012yes—
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. #10318C2semidirect
7. #9828C2semidirect
8. #9826C2semidirect

Maximal subgroups

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

subgroupindex
#331722
#278762
#256722
#213182
#106802
#103182
#98282
#98262
#81442