Finite Group Databasebeta

← #324956

Subgroups — #324956

Order 512.

514 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30220C211yesmaximal, characteristic
2#29225C211yesmaximal, characteristic
3#28691C211yesmaximal, characteristic
4#22303C211yesmaximal, characteristic
5#10600C212yesmaximal
6#10592C212yesmaximal
7#9904C211yesmaximal, characteristic
8#9692C211yesmaximal, characteristic
9#9570C212yesmaximal
10#9569C212yesmaximal
11#8278C211yesmaximal, characteristic
12C2×#499—21no—
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15C4×#125C2211yescharacteristic
16#2100—22no—
17#2085—22no—
18#2043—22no—
19#2043—22no—
20#1992—21no—
21#1991—21no—
22#1988C2212yes—
23#1988C2212yes—
24#1956C2212yes—
25#1956C2212yes—
26#1948C2212yes—
27#1947C2212yes—
28#1940—22no—
29#1939C2212yes—
30#1917—22no—
31#1842C2211yescharacteristic
32#1836—21no—
33#1770C2212yes—
34#1767C2211yescharacteristic
35C4×C4⋊C8C2211yescharacteristic
36#1744C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#335C2211yescharacteristic
39C2×#332C2212yes—
40C2×#332C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1277C2211yescharacteristic
44#1277C2211yescharacteristic
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2312yes—
47C2×#125—21no—
48C2×#125D411yescharacteristic
49#499—44no—
50C2×#107—22no—
51C2×#106—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2311yescharacteristic
56C2×C4⋊C8C2311yescharacteristic
57C2×C4⋊C8C2311yescharacteristic
58C2×C4⋊C8C2311yescharacteristic
59C2×#103—22no—
60C2×#103—22no—
61C2×#103—22no—
62C2×#103—22no—
63C2×#97—21no—
64C2×#97—22no—
65C2×C4×C8—21no—
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2311yescharacteristic
68#389—22no—
69#389—22no—
70#389D412yes—
71#387—22no—
72#384—21no—
73#384—21no—
74#384—22no—
75#384C2312yes—
76#382—21no—
77#382D411yescharacteristic
78C4×C4⋊C4—22no—
79C4×C4⋊C4C2312yes—
80C43C2311yescharacteristic
81#340—22no—
82#340—22no—
83#340—22no—
84#340—22no—
85#340—22no—
86#340—22no—
87#336—22no—
88#336—22no—
89#336—21no—
90#336—21no—
91#336D412yes—
92#336D412yes—
93#335—22no—
94#335—22no—
95#332—24no—
96#332—24no—
97C22×Q8—22no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—22no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D412yes—
105C22×C8C2×D411yescharacteristic
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—82no—
111C4⋊Q8—82no—
112#125—22no—
113#125—22no—
114#125—42no—
115#125—42no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4C2×D412yes—
127C2×C4⋊C4C2×D412yes—
128C2×C42—22no—
129C2×C42C2×D412yes—
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42#4112yes—
133C2×C42C2411yescharacteristic
134#107—44no—
135#106—44no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—21no—
139C4⋊C8—21no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142C4⋊C8—21no—
143C4⋊C8—21no—
144C4⋊C8—82no—
145C4⋊C8—82no—
146C4⋊C8#4112yes—
147C4⋊C8#4112yes—
148#103—82no—
149#103—82no—
150#103—82no—
151#103—82no—
152#97—42no—
153#97—82no—
154C4×C8—21no—
155C4×C8—21no—
156C4×C8—21no—
157C4×C8—21no—
158C4×C8—42no—
159#95—22no—
160#95—22no—
161#95—22no—
162#95—42no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4#12112yes—
174C22×C4#12112yes—
175C22×C4C22×D411yescharacteristic
176C22×C4C22×D411yescharacteristic
177C22×C4C2×#4111yescharacteristic
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—42no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—41no—
199C2×C8—41no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—44no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—82no—
214C4⋊C4—82no—
215C4⋊C4—84no—
216C4⋊C4—84no—
217C4⋊C4—84no—
218C42—22no—
219C42—22no—
220C42—21no—
221C42—21no—
222C42—42no—
223C42—42no—
224C42—42no—
225C42AGL(1,ℤ/8)12yes—
226C42#13712yes—
227C42C2×#4111yescharacteristic
228C42C2×#4111yescharacteristic
229C422−1+411yescharacteristic
230C422−1+411yescharacteristic
231C23C2×#12111yescharacteristic
232Q8—82no—
233Q8—82no—
234Q8—82no—
235Q8—82no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4#46814yes—
255C2×C4#53211yescharacteristic
256C2×C4#53611yescharacteristic
257C2×C4#54811yescharacteristic
258C2×C4#54811yescharacteristic
259C2×C4#55611yescharacteristic
260C2×C4#55611yescharacteristic
261C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
262C2×C4C2×#13711yescharacteristic
263C2×C4#57612yes—
264C2×C4#57712yes—
265C2×C4#57811yescharacteristic
266C2×C4#57811yescharacteristic
267C8—42no—
268C8—42no—
269C8—41no—
270C8—41no—
271C8—42no—
272C8—81no—
273C8—81no—
274C8—82no—
275C22#254511yescharacteristic
276C22#304512yes—
277C22C2×#46811yescharacteristic
278C22#305911yescharacteristic
279C22#306912yes—
280C4—21no—
281C4—21no—
282C4—24no—
283C4—42no—
284C4—82no—
285C4—82no—
286C4—162no—
287C4—162no—
288C4#317411yescharacteristic
289C4#317611yescharacteristic
290C4#324112yes—
291C4#336711yescharacteristic
292C4#336711yescharacteristic
293C4#337411yescharacteristic
294C4#337411yescharacteristic
295C2#1944412yes—
296C2#1953912yes—
297C2#2864611yescharacteristic
298C2#2866511yescharacteristic
299C2#2867311yescharacteristic
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
#302202
#292252
#286912
#223032
#106002
#105922
#99042
#96922
#95702
#95692
#82782