Finite Group Databasebeta

← #308913

Subgroups — #308913

Order 512.

534 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28828C211yesmaximal, characteristic
2#26527C212yesmaximal
3#23157C211yesmaximal, characteristic
4#10600C212yesmaximal
5#10353C212yesmaximal
6#9823C211yesmaximal, characteristic
7#9731C211yesmaximal, characteristic
8#9615C211yesmaximal, characteristic
9#9594C211yesmaximal, characteristic
10#8925C212yesmaximal
11C8×#107C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13#2571C2212yes—
14#2365C2212yes—
15#2097—22no—
16#2085—22no—
17#2066—22no—
18#1990C2212yes—
19#1988C2212yes—
20#1981C2212yes—
21#1956C2212yes—
22#1948C2212yes—
23#1948C2212yes—
24#1942—21no—
25#1942—21no—
26#1939C2212yes—
27#1918—22no—
28#1863—22no—
29#1846—21no—
30#1845C2211yescharacteristic
31#1805—22no—
32C4×#107—21no—
33C4×#107C2211yescharacteristic
34C4×#107C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C8×C4⋊C4C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#335C2211yescharacteristic
39C2×#332C2212yes—
40C2×C82C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2211yescharacteristic
44#1279C2212yes—
45C2×C4⋊Q8C2312yes—
46C2×C4×Q8—22no—
47#500—22no—
48#500Q814yes—
49C2×#107—21no—
50C2×#107—21no—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107D411yescharacteristic
54C2×#107D411yescharacteristic
55C2×#107D411yescharacteristic
56C2×#107D411yescharacteristic
57C2×#106—22no—
58C2×#106—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×#103—22no—
64C2×#103—22no—
65C2×#103—22no—
66C2×#103—22no—
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#400—42no—
73#391—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#387—42no—
78#384C2312yes—
79#382—22no—
80#382—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4C2311yescharacteristic
83C4×C4⋊C4C2312yes—
84C4×C4⋊C4C2311yescharacteristic
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—42no—
90#340D411yescharacteristic
91#340D411yescharacteristic
92#340D412yes—
93#336—21no—
94#336—22no—
95#336—21no—
96#335—22no—
97#335—22no—
98#332—24no—
99C82—22no—
100C22×Q8—22no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—22no—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8C2×Q814yes—
113C4×Q8—44no—
114C4×Q8—82no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4C2×D412yes—
123C2×C4⋊C4C2×D412yes—
124C2×C4⋊C4C2×D411yescharacteristic
125C2×C4⋊C4C2×D411yescharacteristic
126C2×C4⋊C4C2×D411yescharacteristic
127C2×C4⋊C4C2×D411yescharacteristic
128C2×C42—22no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—42no—
132C2×C42C2411yescharacteristic
133#107—24no—
134#107—24no—
135#107—22no—
136#107—22no—
137#107—22no—
138#107—22no—
139#107—42no—
140#107—81no—
141#106—82no—
142#106—82no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—21no—
146C4⋊C8—21no—
147C4⋊C8—21no—
148C4⋊C8—82no—
149C4⋊C8#4112yes—
150#103—44no—
151#103—82no—
152#103—82no—
153#103—82no—
154C4×C8—21no—
155C4×C8—21no—
156C4×C8—21no—
157C4×C8—21no—
158C4×C8—42no—
159C4×C8—42no—
160C4×C8C2×Q812yes—
161C4×C8#4112yes—
162#95—21no—
163#95—22no—
164#95—21no—
165#95—42no—
166#95—42no—
167#95—42no—
168#95—42no—
169#95—42no—
170C2×Q8—24no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4C22×D411yescharacteristic
185C22×C4C22×D411yescharacteristic
186C22×C4H(5,𝔽2)11yescharacteristic
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—22no—
192C2×C8—21no—
193C2×C8—22no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—41no—
206C2×C8C22⋊Q812yes—
207C2×C8C22⋊Q812yes—
208C2×C8#13512yes—
209C2×C8#13512yes—
210C2×C8#13712yes—
211C4⋊C4—21no—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—21no—
215C4⋊C4—22no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—22no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—84no—
231C4⋊C4—82no—
232C4⋊C4—162no—
233C4⋊C4#13512yes—
234C42—41no—
235C42—41no—
236C42—42no—
237C42—42no—
238C42—41no—
239C42—41no—
240C42—84no—
241C42C22×Q811yescharacteristic
242C42C2×#4111yescharacteristic
243C42C2×#4111yescharacteristic
244C422−1+411yescharacteristic
245C23D4211yescharacteristic
246Q8—44no—
247Q8—44no—
248Q8—82no—
249Q8—82no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4C2×C22⋊Q811yescharacteristic
277C2×C4#53211yescharacteristic
278C2×C4#54211yescharacteristic
279C2×C4#54811yescharacteristic
280C2×C4D4×Q811yescharacteristic
281C2×C4#55411yescharacteristic
282C2×C4C2×#13511yescharacteristic
283C2×C4C2×#13511yescharacteristic
284C2×C4C2×#13511yescharacteristic
285C2×C4C2×#13711yescharacteristic
286C2×C4#57611yescharacteristic
287C2×C4#57611yescharacteristic
288C2×C4#57711yescharacteristic
289C2×C4#57811yescharacteristic
290C8—22no—
291C8—22no—
292C8—42no—
293C8—42no—
294C8—41no—
295C8—81no—
296C8—82no—
297C8#47914yes—
298C22#245211yescharacteristic
299C22#327811yescharacteristic
300C22#328011yescharacteristic
301C22#328311yescharacteristic
302C22#328711yescharacteristic
303C22#329011yescharacteristic
304C22#329311yescharacteristic
305C4—21no—
306C4—21no—
307C4—42no—
308C4—82no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4—84no—
313C4—82no—
314C4—162no—
315C4C2×#47911yescharacteristic
316C4#308211yescharacteristic
317C4#316811yescharacteristic
318C4#317611yescharacteristic
319C4#336011yescharacteristic
320C4#337411yescharacteristic
321C4#338011yescharacteristic
322C4#338011yescharacteristic
323C2#1990711yescharacteristic
324C2#1996611yescharacteristic
325C2#1998911yescharacteristic
326C2#2002011yescharacteristic
327C2#2646111yescharacteristic
328C2#2648211yescharacteristic
329C2#2650211yescharacteristic
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
#265272
#231572
#106002
#103532
#98232
#97312
#96152
#95942
#89252
C8×#1072