Finite Group Databasebeta

← #325814

Subgroups — #325814

Order 512.

506 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28828C211yesmaximal, characteristic
2#28827C211yesmaximal, characteristic
3#28792C212yesmaximal
4#10580C212yesmaximal
5#10577C212yesmaximal
6#9614C211yesmaximal, characteristic
7#9614C211yesmaximal, characteristic
8#9613C212yesmaximal
9#8814C211yesmaximal, characteristic
10C8×#107C211yesmaximal, characteristic
11C8×#106C211yesmaximal, characteristic
12#2571C2212yes—
13#2552C2212yes—
14#2085—22no—
15#2085—22no—
16#2083—22no—
17#2059—22no—
18#1991—22no—
19#1948C2212yes—
20#1947C2212yes—
21#1947C2212yes—
22#1947C2212yes—
23#1942—21no—
24#1942—21no—
25#1942—22no—
26#1846—21no—
27#1845—21no—
28#1845C2211yescharacteristic
29#1845C2211yescharacteristic
30#1844C2212yes—
31#1834C2212yes—
32C4×#107C2211yescharacteristic
33C4×#106C2211yescharacteristic
34C8×C4⋊C4C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C8×C4⋊C4C2212yes—
37C2×#335C2211yescharacteristic
38C2×#334C2211yescharacteristic
39C2×C82C2211yescharacteristic
40#1296C2212yes—
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2212yes—
44#1279C2211yescharacteristic
45C2×#125—22no—
46C2×#107—21no—
47C2×#107—21no—
48C2×#107—22no—
49C2×#107—21no—
50C2×#107—21no—
51C2×#107—22no—
52C2×#107—22no—
53C2×#107D411yescharacteristic
54C2×#107D411yescharacteristic
55C2×#106—22no—
56C2×#106—42no—
57C2×#106D411yescharacteristic
58C2×#106D411yescharacteristic
59C2×C4⋊C8C2311yescharacteristic
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2312yes—
62C2×C4×C8—21no—
63C2×C4×C8—21no—
64C2×C4×C8—22no—
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2311yescharacteristic
68#400—42no—
69#398—42no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#384—22no—
75#384C2312yes—
76#384C2312yes—
77#382—22no—
78#382—22no—
79C4×C4⋊C4C2311yescharacteristic
80C4×C4⋊C4C2312yes—
81C4×C4⋊C4C2311yescharacteristic
82#340—22no—
83#340—22no—
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#340D412yes—
89#340D411yescharacteristic
90#340D411yescharacteristic
91#336—22no—
92#336—21no—
93#336—21no—
94#335Q814yes—
95#334Q814yes—
96C82Q814yes—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—22no—
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105#125—44no—
106#125—82no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4C2×D411yescharacteristic
118C2×C4⋊C4C2×D412yes—
119C2×C4⋊C4C2×D412yes—
120C2×C4⋊C4C2×D411yescharacteristic
121C2×C4⋊C4C2×D411yescharacteristic
122C2×C4⋊C4C2×D411yescharacteristic
123C2×C42—22no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42C2411yescharacteristic
127#107—22no—
128#107—22no—
129#107—42no—
130#107—42no—
131#107—44no—
132#107—82no—
133#107—82no—
134#107—81no—
135#107—81no—
136#106—22no—
137#106—22no—
138#106—82no—
139#106—84no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142C4⋊C8—22no—
143C4⋊C8C2×Q812yes—
144C4⋊C8C2×Q812yes—
145C4⋊C8C2×Q814yes—
146C4×C8—21no—
147C4×C8—42no—
148C4×C8—42no—
149C4×C8—82no—
150C4×C8C2×Q812yes—
151C4×C8C2×Q812yes—
152C4×C8C2×Q812yes—
153C4×C8C2×Q812yes—
154C4×C8C2×Q812yes—
155#95—21no—
156#95—22no—
157#95—21no—
158#95—42no—
159#95—42no—
160#95—42no—
161#95—42no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4C22×D411yescharacteristic
171C22×C4C22×D411yescharacteristic
172C22×C4H(5,𝔽2)11yescharacteristic
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—24no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8C4⋊Q812yes—
192C2×C8C4⋊Q812yes—
193C2×C8C4⋊Q812yes—
194C2×C8C4⋊Q812yes—
195C2×C8#13512yes—
196C2×C8#13512yes—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—21no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—21no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—84no—
216C4⋊C4—84no—
217C4⋊C4—82no—
218C4⋊C4—82no—
219C4⋊C4—162no—
220C4⋊C4—162no—
221C4⋊C4#13512yes—
222C4⋊C4#13512yes—
223C42—41no—
224C42—41no—
225C42—42no—
226C42—42no—
227C42—41no—
228C42—41no—
229C42C22×Q811yescharacteristic
230C42C22×Q811yescharacteristic
231C42C22×Q811yescharacteristic
232C422−1+411yescharacteristic
233C23D4211yescharacteristic
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4C2×C4⋊Q811yescharacteristic
261C2×C4C2×C4⋊Q811yescharacteristic
262C2×C4#54411yescharacteristic
263C2×C4D4×Q811yescharacteristic
264C2×C4D4×Q811yescharacteristic
265C2×C4Q8211yescharacteristic
266C2×C4C2×#13511yescharacteristic
267C2×C4C2×#13511yescharacteristic
268C2×C4C2×#13511yescharacteristic
269C2×C4C2×#13511yescharacteristic
270C2×C4#57611yescharacteristic
271C2×C4#57711yescharacteristic
272C2×C4#57711yescharacteristic
273C2×C4#57811yescharacteristic
274C8—24no—
275C8—22no—
276C8—22no—
277C8—22no—
278C8—44no—
279C8—42no—
280C8—42no—
281C8#49914yes—
282C22#255111yescharacteristic
283C22#328011yescharacteristic
284C22#328411yescharacteristic
285C22#328411yescharacteristic
286C22#328511yescharacteristic
287C22#329211yescharacteristic
288C22#329311yescharacteristic
289C4—21no—
290C4—21no—
291C4—42no—
292C4—42no—
293C4—82no—
294C4—82no—
295C4—81no—
296C4—81no—
297C4—162no—
298C4—162no—
299C4C2×#49911yescharacteristic
300C4#316011yescharacteristic
301C4#316311yescharacteristic
302C4#316411yescharacteristic
303C4#338011yescharacteristic
304C4#338111yescharacteristic
305C4#338111yescharacteristic
306C4#338311yescharacteristic
307C2#1999011yescharacteristic
308C2#1999411yescharacteristic
309C2#2001211yescharacteristic
310C2#2001411yescharacteristic
311C2#2881111yescharacteristic
312C2#2881211yescharacteristic
313C2#2882511yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ 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
#288282
#288272
#287922
#105802
#105772
#96142
#96132
#88142
C8×#1072
C8×#1062