Finite Group Databasebeta

← #330732

Subgroups — #330732

Order 512.

483 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 483 conjugacy classes of subgroups — too many to draw. Only its 138 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#30206C211yesmaximal, characteristic
2#29996C211yesmaximal, characteristic
3#29867C211yesmaximal, characteristic
4#22019C211yesmaximal, characteristic
5#10600C212yesmaximal
6#10597C212yesmaximal
7#10486C212yesmaximal
8#10483C212yesmaximal
9#9897C211yesmaximal, characteristic
10#9697C211yesmaximal, characteristic
11#8812C211yesmaximal, characteristic
12C2×#477C2211yescharacteristic
13C2×#475C2211yescharacteristic
14#2862C2211yescharacteristic
15#2859C2211yescharacteristic
16#2295C2211yescharacteristic
17#2088—22no—
18#2085—22no—
19#2074—22no—
20#2074—22no—
21#2066—22no—
22#2026—21no—
23#2026—21no—
24#1990C2212yes—
25#1988C2211yescharacteristic
26#1987C2211yescharacteristic
27#1984C2212yes—
28#1956C2212yes—
29#1948C2212yes—
30#1939C2212yes—
31#1916—22no—
32#1842C2211yescharacteristic
33#1840—21no—
34#1832—21no—
35#1831C2211yescharacteristic
36#1803C2212yes—
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#335C2212yes—
40C2×#334C2212yes—
41C2×#333C2212yes—
42C2×#332C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×#125—21no—
48C2×C4×Q8C2311yescharacteristic
49#477—42no—
50#475—42no—
51C2×#107—21no—
52C2×#106—21no—
53C2×#106—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8C2311yescharacteristic
57C2×C4⋊C8C2311yescharacteristic
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×#103—21no—
61C2×#103—21no—
62C2×#103—22no—
63C2×#103—22no—
64C2×#103—22no—
65C2×C4×C8—21no—
66C2×C4×C8C2312yes—
67#391—21no—
68#391—21no—
69#391—22no—
70#391D412yes—
71#389—22no—
72#389—22no—
73#389D412yes—
74#387—22no—
75#384C2312yes—
76#382—21no—
77#382—22no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4C2311yescharacteristic
80#376—21no—
81C43C2311yescharacteristic
82#340—22no—
83#340—22no—
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#336—22no—
89#336—22no—
90#336—22no—
91#336—22no—
92#336—21no—
93#336D412yes—
94#336D411yescharacteristic
95#336D411yescharacteristic
96#335—42no—
97#334—42no—
98#333—42no—
99#332—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—
113#125—82no—
114C4×Q8—22no—
115C4×Q8—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4C2×D411yescharacteristic
127C2×C4⋊C4C2×D412yes—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D412yes—
132C2×C42C2×D411yescharacteristic
133C2×C42#4112yes—
134C2×C42C2411yescharacteristic
135#107—42no—
136#106—42no—
137#106—82no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—82no—
145C4⋊C8—82no—
146#103—42no—
147#103—42no—
148#103—82no—
149#103—82no—
150#103—82no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8—81no—
154#95—22no—
155#95—22no—
156#95—22no—
157#95—21no—
158#95—42no—
159#95—42no—
160#95—42no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—82no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4#12112yes—
178C22×C4#12112yes—
179C22×C4C22×D411yescharacteristic
180C22×C4C22×D411yescharacteristic
181C22×C4C2×#4111yescharacteristic
182C2×C8—22no—
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—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—44no—
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—84no—
216C42—21no—
217C42—22no—
218C42—22no—
219C42—21no—
220C42—22no—
221C42—42no—
222C42—42no—
223C42—41no—
224C42—41no—
225C42H(5,𝔽2)12yes—
226C422−1+412yes—
227C23C2×#12111yescharacteristic
228Q8—41no—
229Q8—41no—
230Q8—42no—
231Q8—82no—
232Q8—82no—
233Q8—162no—
234C2×C4—22no—
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—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4#53511yescharacteristic
257C2×C4#53611yescharacteristic
258C2×C4#53712yes—
259C2×C4#54212yes—
260C2×C4#57712yes—
261C2×C4#57712yes—
262C2×C4#57711yescharacteristic
263C2×C4#57812yes—
264C2×C4#57811yescharacteristic
265C8—42no—
266C8—42no—
267C8—81no—
268C8—82no—
269C8—81no—
270C8—82no—
271C22#262011yescharacteristic
272C22#304311yescharacteristic
273C22#304611yescharacteristic
274C22#306312yes—
275C22#306712yes—
276C4—22no—
277C4—42no—
278C4—42no—
279C4—82no—
280C4—81no—
281C4—81no—
282C4—162no—
283C4—161no—
284C4—161no—
285C4#322111yescharacteristic
286C4#322311yescharacteristic
287C4#324212yes—
288C4#325812yes—
289C4#326312yes—
290C2#1956812yes—
291C2#1958512yes—
292C2#2984111yescharacteristic
293C2#2985611yescharacteristic
294C2#2986011yescharacteristic
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
#302062
#299962
#298672
#220192
#106002
#105972
#104862
#104832
#98972
#96972
#88122