Finite Group Databasebeta

← #305831

Subgroups — #305831

Order 512.

520 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28828C211yesmaximal, characteristic
2#26093C212yesmaximal
3#22727C211yesmaximal, characteristic
4#10651C212yesmaximal
5#10576C212yesmaximal
6#9615C211yesmaximal, characteristic
7#9594C211yesmaximal, characteristic
8#8883C212yesmaximal
9#8818C211yesmaximal, characteristic
10#8645C211yesmaximal, characteristic
11C8×#103C211yesmaximal, characteristic
12C2×C8×Q8C2211yescharacteristic
13#2571C2212yes—
14#2355C2212yes—
15#2097—22no—
16#2085—22no—
17#2068—22no—
18#1949—22no—
19#1949—22no—
20#1948C2212yes—
21#1948C2212yes—
22#1948C2212yes—
23#1942—21no—
24#1939C2212yes—
25#1918—22no—
26#1846—21no—
27#1846—21no—
28#1845C2211yescharacteristic
29#1845C2212yes—
30#1845C2211yescharacteristic
31#1845C2212yes—
32#1803C2212yes—
33C4×#107—21no—
34C4×#107C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C4×#103C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×Q8⋊C8C2211yescharacteristic
39C2×C82C2211yescharacteristic
40#1296C2212yes—
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2212yes—
44#1277C2211yescharacteristic
45C2×C4×Q8C2311yescharacteristic
46C8×Q8—22no—
47C8×Q8Q814yes—
48C2×#107—22no—
49C2×#107—21no—
50C2×#107—21no—
51C2×#107—22no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107—42no—
55C2×#107D411yescharacteristic
56C2×#107D411yescharacteristic
57C2×#106—22no—
58C2×#106—22no—
59C2×#106—22no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×#103—22no—
66C2×#103D411yescharacteristic
67C2×#103D411yescharacteristic
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#400—42no—
74#391—22no—
75#389—22no—
76#389—22no—
77#387—42no—
78#384C2312yes—
79#384C2312yes—
80#382—22no—
81#382—22no—
82#382—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4C2311yescharacteristic
86C4×C4⋊C4C2312yes—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340D412yes—
92#340D411yescharacteristic
93#340D411yescharacteristic
94#336—21no—
95#336—22no—
96#335—22no—
97Q8⋊C8Q814yes—
98C82—22no—
99C22×Q8C2×D411yescharacteristic
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—22no—
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C4×Q8—21no—
109C4×Q8—21no—
110C4×Q8C2×Q812yes—
111C4×Q8C2×Q812yes—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4C2×D412yes—
121C2×C4⋊C4C2×D411yescharacteristic
122C2×C4⋊C4C2×D411yescharacteristic
123C2×C4⋊C4C2×D412yes—
124C2×C4⋊C4C2×D411yescharacteristic
125C2×C42—21no—
126C2×C42—22no—
127C2×C42—21no—
128C2×C42—42no—
129C2×C42C2411yescharacteristic
130#107—22no—
131#107—22no—
132#107—44no—
133#107—42no—
134#107—42no—
135#107—82no—
136#107—81no—
137#107—81no—
138#107—84no—
139#106—82no—
140#106—82no—
141#106—82no—
142C4⋊C8—24no—
143C4⋊C8—21no—
144C4⋊C8—22no—
145C4⋊C8—21no—
146C4⋊C8—22no—
147C4⋊C8—44no—
148C4⋊C8C2×Q812yes—
149C4⋊C8C2×Q812yes—
150#103—22no—
151#103—22no—
152#103—82no—
153C4×C8—24no—
154C4×C8—21no—
155C4×C8—21no—
156C4×C8—21no—
157C4×C8—42no—
158C4×C8C2×Q812yes—
159C4×C8C2×Q812yes—
160C4×C8#4112yes—
161#95—22no—
162#95—21no—
163#95—42no—
164#95—42no—
165#95—42no—
166#95—42no—
167#95—42no—
168#95—42no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8C4⋊Q812yes—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4C22×D411yescharacteristic
184C22×C4C22×D411yescharacteristic
185C22×C4H(5,𝔽2)11yescharacteristic
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—22no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8C22⋊Q812yes—
205C2×C8C22⋊Q812yes—
206C2×C8#13512yes—
207C2×C8#13512yes—
208C2×C8#13512yes—
209C2×C8#13712yes—
210C4⋊C4—22no—
211C4⋊C4—21no—
212C4⋊C4—22no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—21no—
219C4⋊C4—22no—
220C4⋊C4—21no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C4⋊C4—162no—
232C4⋊C4C4⋊Q812yes—
233C42—22no—
234C42—22no—
235C42—42no—
236C42—41no—
237C42—41no—
238C42—42no—
239C42—162no—
240C42C22×Q811yescharacteristic
241C42C22×Q811yescharacteristic
242C42C2×#4111yescharacteristic
243C422−1+411yescharacteristic
244C23D4211yescharacteristic
245Q8—22no—
246Q8—22no—
247Q8—41no—
248Q8—41no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4C2×C22⋊Q811yescharacteristic
276C2×C4C2×C4⋊Q811yescharacteristic
277C2×C4#54411yescharacteristic
278C2×C4#54711yescharacteristic
279C2×C4D4×Q811yescharacteristic
280C2×C4#55411yescharacteristic
281C2×C4C2×#13511yescharacteristic
282C2×C4C2×#13511yescharacteristic
283C2×C4C2×#13511yescharacteristic
284C2×C4C2×#13711yescharacteristic
285C2×C4#57611yescharacteristic
286C2×C4#57611yescharacteristic
287C2×C4#57811yescharacteristic
288C2×C4#57811yescharacteristic
289C8—24no—
290C8—22no—
291C8—22no—
292C8—22no—
293C8—42no—
294C8—42no—
295C8—82no—
296C8#47914yes—
297C22#244011yescharacteristic
298C22#327611yescharacteristic
299C22#328311yescharacteristic
300C22#328711yescharacteristic
301C22#328711yescharacteristic
302C22#329211yescharacteristic
303C22#329311yescharacteristic
304C4—21no—
305C4—21no—
306C4—42no—
307C4—42no—
308C4—82no—
309C4—81no—
310C4—82no—
311C4—81no—
312C4—162no—
313C4—162no—
314C4C2×#47911yescharacteristic
315C4#308311yescharacteristic
316C4#316011yescharacteristic
317C4#316411yescharacteristic
318C4#333611yescharacteristic
319C4#333911yescharacteristic
320C4#338311yescharacteristic
321C4#338511yescharacteristic
322C2#1995711yescharacteristic
323C2#1996411yescharacteristic
324C2#2002011yescharacteristic
325C2#2002111yescharacteristic
326C2#2604711yescharacteristic
327C2#2606611yescharacteristic
328C2#2608311yescharacteristic
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
#260932
#227272
#106512
#105762
#96152
#95942
#88832
#88182
#86452
C8×#1032