Finite Group Databasebeta

← #324832

Subgroups — #324832

Order 512.

516 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30220C211yesmaximal, characteristic
2#29225C211yesmaximal, characteristic
3#29210C211yesmaximal, characteristic
4#22299C211yesmaximal, characteristic
5#10600C212yesmaximal
6#10592C212yesmaximal
7#9902C211yesmaximal, characteristic
8#9692C211yesmaximal, characteristic
9#9562C212yesmaximal
10#9561C212yesmaximal
11#8278C211yesmaximal, characteristic
12#2868C2211yescharacteristic
13#2859C2211yescharacteristic
14C4×#125C2211yescharacteristic
15#2100—22no—
16#2059—22no—
17#2043—22no—
18#2043—22no—
19#1992—21no—
20#1991—21no—
21#1988C2212yes—
22#1988C2212yes—
23#1956C2212yes—
24#1956C2212yes—
25#1948C2212yes—
26#1947C2212yes—
27#1939—22no—
28#1939C2212yes—
29#1916—22no—
30#1846—21no—
31#1842C2211yescharacteristic
32#1836—21no—
33C8×C4⋊C4C2212yes—
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
49C2×#107—22no—
50C2×#106—22no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2311yescharacteristic
55C2×C4⋊C8C2311yescharacteristic
56C2×C4⋊C8C2311yescharacteristic
57C2×C4⋊C8C2311yescharacteristic
58C2×#103—22no—
59C2×#103—22no—
60C2×#103—22no—
61C2×#103—22no—
62C2×#97—21no—
63C2×C4×C8—21no—
64C2×C4×C8—22no—
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67#389—22no—
68#389—22no—
69#389D412yes—
70#387—22no—
71#384—21no—
72#384—21no—
73#384—22no—
74#384C2312yes—
75#382—21no—
76#382D411yescharacteristic
77C4×C4⋊C4—22no—
78C4×C4⋊C4C2312yes—
79C43C2311yescharacteristic
80#340—22no—
81#340—22no—
82#340—22no—
83#340—22no—
84#340—22no—
85#340—22no—
86#336—22no—
87#336—22no—
88#336—21no—
89#336—21no—
90#336D412yes—
91#336D412yes—
92#335—22no—
93#335—22no—
94#332—24no—
95#332—24no—
96C22×Q8—22no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—22no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D412yes—
104C22×C8C2×D411yescharacteristic
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—82no—
110C4⋊Q8—82no—
111#125—22no—
112#125—81no—
113#125—81no—
114#125Q1614yes—
115C2×C4⋊C4—22no—
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—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4C2×D412yes—
126C2×C4⋊C4C2×D412yes—
127C2×C42—22no—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D412yes—
130C2×C42C2×D411yescharacteristic
131C2×C42#4112yes—
132C2×C42C2411yescharacteristic
133#107—82no—
134#106—82no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—21no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142C4⋊C8—21no—
143C4⋊C8—82no—
144C4⋊C8—82no—
145C4⋊C8#4112yes—
146C4⋊C8#4112yes—
147#103—82no—
148#103—82no—
149#103—82no—
150#103—82no—
151#97—42no—
152C4×C8—21no—
153C4×C8—21no—
154C4×C8—21no—
155C4×C8—21no—
156C4×C8—42no—
157C4×C8—82no—
158#95—22no—
159#95—22no—
160#95—22no—
161#95—42no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4#12112yes—
173C22×C4#12112yes—
174C22×C4C22×D411yescharacteristic
175C22×C4C22×D411yescharacteristic
176C22×C4C2×#4111yescharacteristic
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—42no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—41no—
198C2×C8—41no—
199C4⋊C4—22no—
200C4⋊C4—28no—
201C4⋊C4—22no—
202C4⋊C4—22no—
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—82no—
212C4⋊C4—82no—
213C4⋊C4—84no—
214C4⋊C4—84no—
215C4⋊C4—84no—
216C4⋊C4C2×Q1614yes—
217C42—22no—
218C42—21no—
219C42—21no—
220C42—22no—
221C42—42no—
222C42—42no—
223C42—42no—
224C42C2×Q1612yes—
225C42#13512yes—
226C42C2×#4111yescharacteristic
227C42C2×#4111yescharacteristic
228C422−1+411yescharacteristic
229C422−1+411yescharacteristic
230C23C2×#12111yescharacteristic
231Q8—82no—
232Q8—82no—
233Q8—82no—
234Q8—82no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4#46714yes—
254C2×C4#53211yescharacteristic
255C2×C4#53611yescharacteristic
256C2×C4#54811yescharacteristic
257C2×C4#54811yescharacteristic
258C2×C4#55611yescharacteristic
259C2×C4#55611yescharacteristic
260C2×C4C22×Q1611yescharacteristic
261C2×C4C2×#13511yescharacteristic
262C2×C4#57612yes—
263C2×C4#57711yescharacteristic
264C2×C4#57711yescharacteristic
265C2×C4#57712yes—
266C8—42no—
267C8—42no—
268C8—41no—
269C8—41no—
270C8—42no—
271C8—81no—
272C8—81no—
273C8—82no—
274C22#254511yescharacteristic
275C22#304512yes—
276C22C2×#46711yescharacteristic
277C22#305711yescharacteristic
278C22#306912yes—
279C4—21no—
280C4—21no—
281C4—24no—
282C4—42no—
283C4—44no—
284C4—82no—
285C4—162no—
286C4—162no—
287C4#317411yescharacteristic
288C4#317611yescharacteristic
289C4#324112yes—
290C4#336011yescharacteristic
291C4#336011yescharacteristic
292C4#336311yescharacteristic
293C4#336311yescharacteristic
294C2#1941112yes—
295C2#1953612yes—
296C2#2863811yescharacteristic
297C2#2866511yescharacteristic
298C2#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
#292102
#222992
#106002
#105922
#99022
#96922
#95622
#95612
#82782