Finite Group Databasebeta

← #511721

Subgroups — #511721

Order 512.

396 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33258C211yesmaximal, characteristic
2#33257C211yesmaximal, characteristic
3#13539C212yesmaximal
4#13538C211yesmaximal, characteristic
5#13538C211yesmaximal, characteristic
6#13516C212yesmaximal
7#13514C212yesmaximal
8#13507C212yesmaximal
9#13506C211yesmaximal, characteristic
10#13505C211yesmaximal, characteristic
11#12879C211yesmaximal, characteristic
12#2864—21no—
13#2859—21no—
14C2×#367C2212yes—
15C2×#366C2212yes—
16C2×C4⋊C16C2212yes—
17#1942—21no—
18#1942—21no—
19#1942C2211yescharacteristic
20#1942C2211yescharacteristic
21#1942C2212yes—
22#1942C2211yescharacteristic
23#1942C2212yes—
24#1942C2211yescharacteristic
25#1941—21no—
26#1941—21no—
27#1846C2211yescharacteristic
28#1846C2211yescharacteristic
29#1845C2211yescharacteristic
30#1845C2211yescharacteristic
31#1835—21no—
32#1834—21no—
33C42×C8C2211yescharacteristic
34#1380C2212yes—
35#1380C2212yes—
36#1380C2212yes—
37#1380C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1374C2211yescharacteristic
41#1374C2211yescharacteristic
42#1374C2212yes—
43C2×#125—41no—
44C22×C16C2312yes—
45C22×C16C2312yes—
46C2×#107—21no—
47C2×#107—21no—
48C2×#107C2312yes—
49C2×#107C2312yes—
50C2×#107C2311yescharacteristic
51C2×#107C2311yescharacteristic
52C2×#107C2311yescharacteristic
53C2×#107C2311yescharacteristic
54C2×#106—21no—
55C2×#106—21no—
56C2×C4×C8D411yescharacteristic
57C2×C4×C8D411yescharacteristic
58C2×C4×C8D411yescharacteristic
59C2×C4×C8C2311yescharacteristic
60C2×C4×C8C2311yescharacteristic
61C2×C4×C8C2311yescharacteristic
62#389—22no—
63#389—22no—
64#389—21no—
65#389—21no—
66#389—21no—
67#389—21no—
68#389—41no—
69#389—41no—
70#387—41no—
71#387—41no—
72#387—41no—
73#387—41no—
74#384—21no—
75#384—21no—
76#382—21no—
77#382—21no—
78#382—41no—
79C43D411yescharacteristic
80#367—42no—
81#366—42no—
82C4⋊C16—42no—
83#340—22no—
84#340—22no—
85#340—21no—
86#340—21no—
87#340—21no—
88#340—21no—
89C22×C8C2×D411yescharacteristic
90C22×C8C2×D411yescharacteristic
91C22×C8C2×D411yescharacteristic
92C22×C8C2411yescharacteristic
93#125—84no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C42D811yescharacteristic
111C2×C42D811yescharacteristic
112C2×C42D811yescharacteristic
113C2×C42D811yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C16—22no—
118C2×C16—22no—
119C2×C16—22no—
120C2×C16—22no—
121C2×C16—22no—
122C2×C16—22no—
123#107—21no—
124#107—22no—
125#107—21no—
126#107—24no—
127#107—22no—
128#107—22no—
129#107—22no—
130#107—21no—
131#107—21no—
132#107—42no—
133#107—42no—
134#106—42no—
135#106—42no—
136C4×C8—22no—
137C4×C8—22no—
138C4×C8—21no—
139C4×C8—22no—
140C4×C8—22no—
141C4×C8—21no—
142C4×C8—41no—
143#95—41no—
144#95—41no—
145#95—41no—
146#95—41no—
147#95—42no—
148#95—42no—
149#95—41no—
150#95—41no—
151#95—41no—
152#95—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4C2×D811yescharacteristic
160C22×C4C2×D811yescharacteristic
161C22×C4C2×D811yescharacteristic
162C22×C4C2×D811yescharacteristic
163C22×C4C2×D811yescharacteristic
164C22×C4C2×D811yescharacteristic
165C22×C4C22×D411yescharacteristic
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8#13712yes—
174C2×C8#13712yes—
175C2×C8H(5,𝔽2)11yescharacteristic
176C2×C82−1+412yes—
177C2×C82−1+412yes—
178C2×C82−1+411yescharacteristic
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—42no—
182C4⋊C4—41no—
183C4⋊C4—42no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—42no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C4⋊C4—82no—
194C4⋊C4—82no—
195C4⋊C4—84no—
196C4⋊C4—82no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C4⋊C4—84no—
200C4⋊C4—82no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—22no—
208C42—22no—
209C42—22no—
210C42—41no—
211C16—42no—
212C16—42no—
213C23C22×D811yescharacteristic
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—82no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—82no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—82no—
246C2×C4—81no—
247C2×C4#50812yes—
248C2×C4#50912yes—
249C2×C4#51012yes—
250C2×C4#53611yescharacteristic
251C2×C4#53611yescharacteristic
252C2×C4#53711yescharacteristic
253C2×C4C2×#13711yescharacteristic
254C2×C4C2×#13711yescharacteristic
255C2×C4#57511yescharacteristic
256C8—22no—
257C8—22no—
258C8—41no—
259C8#56312yes—
260C8#56412yes—
261C22#322611yescharacteristic
262C22#322611yescharacteristic
263C22#324511yescharacteristic
264C22C2×#50811yescharacteristic
265C22C2×#50911yescharacteristic
266C22C2×#51011yescharacteristic
267C22#341411yescharacteristic
268C4—22no—
269C4—22no—
270C4—22no—
271C4—41no—
272C4—41no—
273C4—41no—
274C4—161no—
275C4—162no—
276C4—161no—
277C4—162no—
278C4—161no—
279C4—161no—
280C4#282211yescharacteristic
281C4#323611yescharacteristic
282C4#323611yescharacteristic
283C4#325511yescharacteristic
284C2#3284811yescharacteristic
285C2#3379711yescharacteristic
286C2#3379811yescharacteristic
287C2#3380111yescharacteristic
288C2#3380511yescharacteristic
289C2#3381311yescharacteristic
290C2#3381811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332582
#332572
#135392
#135382
#135162
#135142
#135072
#135062
#135052
#128792