Finite Group Databasebeta

← #319466

Subgroups — #319466

Order 512.

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

normal subgroups only

This group has 764 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#33210C211yesmaximal, characteristic
2#27972C212yesmaximal
3#25674C212yesmaximal
4#21322C211yesmaximal, characteristic
5#10679C212yesmaximal, complemented
6#10314C212yesmaximal, complemented
7#9834C212yesmaximal, complemented
8#9821C212yesmaximal, complemented
9#8146C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#500C2212yes—
12C2×#498C2211yescharacteristic
13C2×#487C2212yes—
14C2×C4×M16C2211yescharacteristic
15#2857C2212yes—
16C2×#384C2211yescharacteristic
17#2072—24no—
18#2036—24no—
19#1992—22no—
20#1990C2212yes—
21#1990C2212yes—
22#1990C2212yes—
23#1990C2212yes—
24#1956C2212yes—
25#1956C2212yes—
26#1956C2212yes—
27#1943—22no—
28#1872—24no—
29#1863—24no—
30#1845C2211yescharacteristic
31#1834C2212yes—
32#1834C2212yes—
33#1809—22no—
34C4×C22⋊C8C2212yes—
35#1736—22no—
36C2×#322C2211yescharacteristic
37C2×C82C2211yescharacteristic
38#1273C2212yes—
39C22×M16D412yes—
40#533—24no—
41#533—41no—
42#533—41no—
43C2×C4⋊Q8C2312yes—
44C2×C4⋊Q8C2312yes—
45C2×#125—21no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—22no—
48C2×C4×Q8—22no—
49C2×#117—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C42C2311yescharacteristic
53#500—24no—
54#500—42no—
55#498—41no—
56#498—41no—
57#487—24no—
58#487—42no—
59C2×#107—22no—
60C2×#107—24no—
61C2×#106—22no—
62C2×#103—24no—
63C2×#103—24no—
64C2×#103—24no—
65C2×#103—24no—
66C2×C22⋊C8D414yes—
67C4×M16—41no—
68C4×M16—41no—
69C2×#97D412yes—
70C2×C4×C8C2312yes—
71C2×C4×C8C2312yes—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#391—22no—
75#389—22no—
76#389—22no—
77#384—22no—
78#384—21no—
79#384C2312yes—
80#384C2312yes—
81#380—22no—
82#380—22no—
83#340—24no—
84#340—24no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—24no—
89#336D414yes—
90#322—41no—
91C82—41no—
92C22×Q8—22no—
93C22×Q8—22no—
94C23×C4—41no—
95C23×C4C2×D412yes—
96C23×C4C2×D411yescharacteristic
97C2×M16—42no—
98C2×M16—42no—
99C2×M16Q1618yes—
100C22×C8C2×D414yes—
101C22×C8C2×D414yes—
102C22×C8C2×D412yes—
103C22×C8C2×D412yes—
104C4⋊Q8—22no—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8#4114yes—
112#125—24no—
113#125—42no—
114C22⋊Q8—44no—
115C22⋊Q8—44no—
116C22⋊Q8—48no—
117C4×Q8—44no—
118C4×Q8—44no—
119#117—24no—
120#117—42no—
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—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—84no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—42no—
139C2×C42—22no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—41no—
143C2×C42C2×D412yes—
144C2×C42C2×D411yescharacteristic
145C2×C42C2411yescharacteristic
146#107—48no—
147#107—44no—
148#106—44no—
149#103—48no—
150#103—48no—
151#103—84no—
152#103—84no—
153C22⋊C8—44no—
154#97—42no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—21no—
158C4×C8—22no—
159C4×C8—21no—
160C4×C8—22no—
161C4×C8—21no—
162C4×C8—21no—
163#95—42no—
164#95—42no—
165C24#12711yescharacteristic
166C2×Q8—24no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—44no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—82no—
186C22×C4—84no—
187C22×C4#12712yes—
188C22×C4#12711yescharacteristic
189C22×C4C2×Q1614yes—
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D412yes—
192M16—48no—
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×C8C2×Q1614yes—
204C2×C8C2×Q1614yes—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—48no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—82no—
221C4⋊C4—84no—
222C4⋊C4—84no—
223C22⋊C4—44no—
224C22⋊C4—84no—
225C42—21no—
226C42—21no—
227C42—22no—
228C42—22no—
229C42—44no—
230C42C2×#4112yes—
231C42H(5,𝔽2)11yescharacteristic
232C42H(5,𝔽2)11yescharacteristic
233C23—22no—
234C23—21no—
235C23—22no—
236C23—21no—
237C23#49412yes—
238C23C2×#12711yescharacteristic
239Q8—48no—
240Q8—84no—
241Q8—84no—
242Q8—84no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—82no—
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—164no—
265C2×C4#49412yes—
266C2×C4#52511yescharacteristic
267C2×C4#53411yescharacteristic
268C2×C4#54612yes—
269C2×C4#54612yes—
270C2×C4C22×Q1612yes—
271C2×C4#57512yes—
272C2×C4#57514yes—
273C2×C4#57714yes—
274C8—24no—
275C8—24no—
276C8—44no—
277C8—44no—
278C22—22no—
279C22—24no—
280C22—41no—
281C22—41no—
282C22—42no—
283C22#253911yescharacteristic
284C22C2×#49411yescharacteristic
285C22#315311yescharacteristic
286C22#315312yes—
287C22#315412yes—
288C4—22no—
289C4—42no—
290C4—84no—
291C4—162no—
292C4—162no—
293C4—162no—
294C4#310412yes—
295C4#318912yes—
296C4#330714yes—
297C4#332614yes—
298C2—42no—
299C2#1963412yes—
300C2#1971711yescharacteristic
301C2#1975411yescharacteristic
302C2#2830411yescharacteristic
303C2#2836812yes—
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. #10679C2semidirect
6. #10314C2semidirect
7. #9834C2semidirect
8. #9821C2semidirect

Maximal subgroups

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

subgroupindex
#332102
#279722
#256742
#213222
#106792
#103142
#98342
#98212
#81462