Finite Group Databasebeta

← #331150

Subgroups — #331150

Order 512.

694 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 162 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#33049C212yesmaximal
2#33043C212yesmaximal
3#29878C211yesmaximal, characteristic
4#29876C211yesmaximal, characteristic
5#10682C212yesmaximal, complemented
6#10682C212yesmaximal, complemented
7#10313C212yesmaximal, complemented
8#10313C212yesmaximal, complemented
9#7848C211yesmaximal, characteristic
10C2×#500C2211yescharacteristic
11C2×#500C2211yescharacteristic
12C2×#487C2212yes—
13C2×#487C2212yes—
14C2×#431C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2857C2212yes—
18#2857C2212yes—
19#2087—22no—
20#2086—22no—
21#2076—22no—
22#2075—22no—
23#1990C2212yes—
24#1990C2212yes—
25#1990C2212yes—
26#1990C2212yes—
27#1956C2212yes—
28#1956C2212yes—
29#1935—24no—
30#1933—24no—
31#1894—24no—
32#1892—24no—
33#1834C2212yes—
34#1834C2212yes—
35#1834C2211yescharacteristic
36#1834C2211yescharacteristic
37#1273C2212yes—
38#1273C2212yes—
39C22×M16D412yes—
40C23×C8D412yes—
41C2×C4⋊Q8C2312yes—
42C2×C4⋊Q8C2312yes—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C22×C42C2311yescharacteristic
46#500—41no—
47#500—41no—
48#500—41no—
49#500—41no—
50#487—42no—
51#487—42no—
52#487—42no—
53#487—42no—
54#431—41no—
55#431—41no—
56C2×#107—22no—
57C2×#107—22no—
58C2×C4⋊C8D412yes—
59C2×#103—24no—
60C2×#103—24no—
61C2×#103—24no—
62C2×#103—24no—
63C2×C22⋊C8D412yes—
64C4×M16—41no—
65C4×M16—41no—
66C2×#97D412yes—
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#391—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#384C2312yes—
81#384C2312yes—
82#380—22no—
83#380—22no—
84#380—22no—
85#380—22no—
86#340—22no—
87#340—24no—
88#340—22no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—24no—
94C22×Q8—22no—
95C22×Q8—22no—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C2×M16—24no—
100C2×M16—42no—
101C2×M16—42no—
102C22×C8—44no—
103C22×C8—42no—
104C22×C8C2×D414yes—
105C22×C8C2×D414yes—
106C22×C8C2×D412yes—
107C22×C8C2×D412yes—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C22⋊Q8—48no—
117C22⋊Q8—48no—
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—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2411yescharacteristic
140#107—44no—
141#107—44no—
142C4⋊C8—42no—
143#103—48no—
144#103—48no—
145#103—48no—
146#103—48no—
147C22⋊C8—42no—
148#97—42no—
149C4×C8—22no—
150C4×C8—22no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8—21no—
154C4×C8—21no—
155C4×C8—21no—
156C4×C8—21no—
157C4×C8—42no—
158#95—42no—
159#95—42no—
160#95—42no—
161#95—42no—
162C24#12711yescharacteristic
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—44no—
168C2×Q8—44no—
169C2×Q8—44no—
170C2×Q8—44no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4#12711yescharacteristic
185C22×C4#12711yescharacteristic
186C22×C4#12711yescharacteristic
187C22×C4#13514yes—
188C22×C4C22×D411yescharacteristic
189C22×C4C22×D411yescharacteristic
190C22×C4C22×D411yescharacteristic
191M16—48no—
192C2×C8—24no—
193C2×C8—24no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—24no—
197C2×C8—24no—
198C2×C8—24no—
199C2×C8—24no—
200C2×C8—24no—
201C2×C8—24no—
202C2×C8—48no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
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—84no—
218C4⋊C4—84no—
219C22⋊C4—84no—
220C22⋊C4—84no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42H(5,𝔽2)11yescharacteristic
228C42H(5,𝔽2)11yescharacteristic
229C42H(5,𝔽2)11yescharacteristic
230C422−1+411yescharacteristic
231C23—21no—
232C23—22no—
233C23—22no—
234C23—21no—
235C23#49512yes—
236C23C2×#12711yescharacteristic
237Q8—84no—
238Q8—84no—
239Q8—84no—
240Q8—84no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
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—82no—
265C2×C4#49512yes—
266C2×C4#53411yescharacteristic
267C2×C4#53411yescharacteristic
268C2×C4#53411yescharacteristic
269C2×C4#53711yescharacteristic
270C2×C4#53711yescharacteristic
271C2×C4#53711yescharacteristic
272C2×C4C2×#13512yes—
273C2×C4#57512yes—
274C2×C4#57514yes—
275C2×C4#57814yes—
276C8—44no—
277C8—44no—
278C8—42no—
279C8—42no—
280C22—22no—
281C22—24no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22#265611yescharacteristic
286C22C2×#49511yescharacteristic
287C22#315011yescharacteristic
288C22#315312yes—
289C22#315612yes—
290C4—22no—
291C4—41no—
292C4—41no—
293C4—162no—
294C4—162no—
295C4—162no—
296C4—162no—
297C4#319112yes—
298C4#319712yes—
299C4#320212yes—
300C4#325012yes—
301C4#325612yes—
302C4#326112yes—
303C2—42no—
304C2#1962812yes—
305C2#1969612yes—
306C2#3024111yescharacteristic
307C2#3024911yescharacteristic
308C2#3025711yescharacteristic
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. #10682C2semidirect
7. #10313C2semidirect
8. #10313C2semidirect

Maximal subgroups

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

subgroupindex
#330492
#330432
#298782
#298762
#106822
#103132
#78482