Finite Group Databasebeta

← #330541

Subgroups — #330541

Order 512.

508 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 150 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#33294C211yesmaximal, characteristic
2#29996C211yesmaximal, characteristic
3#29869C211yesmaximal, characteristic
4#21343C211yesmaximal, characteristic
5#10601C212yesmaximal
6#10597C212yesmaximal
7#10483C212yesmaximal
8#10482C212yesmaximal
9#9899C211yesmaximal, characteristic
10#9691C211yesmaximal, characteristic
11#8808C211yesmaximal, characteristic
12C2×#477C2211yescharacteristic
13C2×#477C2211yescharacteristic
14#2868C2211yescharacteristic
15#2859C2211yescharacteristic
16#2295C2211yescharacteristic
17#2088—22no—
18#2083—22no—
19#2072—22no—
20#2066—22no—
21#1992—21no—
22#1992—21no—
23#1990C2212yes—
24#1990C2212yes—
25#1987C2211yescharacteristic
26#1987C2211yescharacteristic
27#1984C2212yes—
28#1948C2212yes—
29#1939C2212yes—
30#1917—22no—
31#1842C2211yescharacteristic
32#1836—22no—
33#1831C2211yescharacteristic
34#1803C2212yes—
35#1768—21no—
36C4×C4⋊C8C2211yescharacteristic
37#1746—21no—
38C4×#97C2211yescharacteristic
39C2×#335C2212yes—
40C2×#334C2212yes—
41C2×#333C2212yes—
42C2×#333C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×#125—21no—
49C2×C4×Q8C2311yescharacteristic
50#477—42no—
51#477—42no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#106—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8C2311yescharacteristic
58C2×C4⋊C8C2311yescharacteristic
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×#103—21no—
62C2×#103—21no—
63C2×#103—22no—
64C2×#97D412yes—
65C2×#97D411yescharacteristic
66C2×#97D411yescharacteristic
67C2×C4×C8—21no—
68C2×C4×C8C2312yes—
69#391D412yes—
70#389—22no—
71#389—22no—
72#389D412yes—
73#387—22no—
74#384—22no—
75#384—21no—
76#384—21no—
77#384C2312yes—
78#382—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4C2311yescharacteristic
81#376—21no—
82C43C2311yescharacteristic
83#340—22no—
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#336—22no—
92#336—22no—
93#336—22no—
94#336—22no—
95#336—21no—
96#335—42no—
97#334—42no—
98#333—42no—
99#333—42no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8C2×D411yescharacteristic
103C22×C8—22no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—22no—
107C22×C8C2×D412yes—
108C22×C8C2×D412yes—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—84no—
114#125—42no—
115#125—42no—
116C4×Q8—22no—
117C4×Q8—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4C2×D411yescharacteristic
131C2×C4⋊C4C2×D412yes—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D412yes—
136C2×C42C2×D411yescharacteristic
137C2×C42#4112yes—
138C2×C42C2411yescharacteristic
139#107—42no—
140#107—42no—
141#106—82no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—44no—
149C4⋊C8—82no—
150#103—42no—
151#103—42no—
152#103—82no—
153#97—24no—
154#97—22no—
155#97—22no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—42no—
159#95—22no—
160#95—22no—
161#95—22no—
162#95—21no—
163#95—42no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—82no—
171C2×Q8—82no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4#12112yes—
179C22×C4#12112yes—
180C22×C4C22×D411yescharacteristic
181C22×C4C22×D411yescharacteristic
182C22×C4C2×#4111yescharacteristic
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8#13714yes—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—42no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—82no—
215C4⋊C4—82no—
216C4⋊C4—84no—
217C4⋊C4—84no—
218C4⋊C4—84no—
219C42—22no—
220C42—22no—
221C42—22no—
222C42—42no—
223C42—42no—
224C42—42no—
225C42#13512yes—
226C42#13712yes—
227C42H(5,𝔽2)12yes—
228C422−1+412yes—
229C23C2×#12111yescharacteristic
230Q8—41no—
231Q8—41no—
232Q8—42no—
233Q8—162no—
234Q8—162no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4#46414yes—
256C2×C4#53511yescharacteristic
257C2×C4#53611yescharacteristic
258C2×C4#53712yes—
259C2×C4#54212yes—
260C2×C4C2×#13511yescharacteristic
261C2×C4C2×#13712yes—
262C2×C4C2×#13711yescharacteristic
263C2×C4#57712yes—
264C2×C4#57812yes—
265C8—24no—
266C8—42no—
267C8—81no—
268C8—82no—
269C8—81no—
270C8—82no—
271C22#262011yescharacteristic
272C22C2×#46411yescharacteristic
273C22#303911yescharacteristic
274C22#306312yes—
275C22#306812yes—
276C4—24no—
277C4—22no—
278C4—42no—
279C4—81no—
280C4—81no—
281C4—82no—
282C4—162no—
283C4—161no—
284C4—161no—
285C4#322111yescharacteristic
286C4#322311yescharacteristic
287C4#324212yes—
288C4#324812yes—
289C4#325512yes—
290C2#1924412yes—
291C2#1937912yes—
292C2#2983111yescharacteristic
293C2#2985611yescharacteristic
294C2#2986111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#332942
#299962
#298692
#213432
#106012
#105972
#104832
#104822
#98992
#96912
#88082