Finite Group Databasebeta

← #317651

Subgroups — #317651

Order 512.

532 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29257C212yesmaximal
2C4×#500C211yesmaximal, characteristic
3#21673C211yesmaximal, characteristic
4#10651C211yesmaximal, characteristic
5#10649C211yesmaximal, characteristic
6#10545C211yesmaximal, characteristic
7#10545C211yesmaximal, characteristic
8#9574C212yesmaximal
9#9573C212yesmaximal
10#8785C211yesmaximal, characteristic
11#8593C212yesmaximal
12C2×#500C2211yescharacteristic
13C4×C4⋊Q8C2212yes—
14C4×#125C2211yescharacteristic
15#2097—22no—
16#2026—22no—
17#1990C2212yes—
18#1981C2212yes—
19#1981C2212yes—
20#1948C2212yes—
21#1947C2212yes—
22#1942—21no—
23#1942—21no—
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1916—21no—
27#1916—21no—
28#1865—22no—
29#1863—22no—
30#1845C2211yescharacteristic
31#1831C2212yes—
32#1831C2211yescharacteristic
33C4×#107—21no—
34C4×#107—21no—
35#1771—21no—
36#1771—21no—
37#1770C2211yescharacteristic
38#1770C2211yescharacteristic
39C42×C8C2211yescharacteristic
40C2×#332C2212yes—
41C2×#332C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2211yescharacteristic
45#1277C2211yescharacteristic
46#1277C2212yes—
47C2×C4⋊Q8C2312yes—
48C2×#125D411yescharacteristic
49C2×C4×Q8—22no—
50#500—24no—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107—21no—
55C2×#107—22no—
56C2×#106—22no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8C2311yescharacteristic
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×#103—22no—
64C2×#103—22no—
65C2×#103—22no—
66C2×#103—22no—
67C2×#97—21no—
68C2×#97—21no—
69C2×C4×C8D411yescharacteristic
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D411yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#391—22no—
76#389—22no—
77#389—22no—
78#389D411yescharacteristic
79#389D411yescharacteristic
80#387—21no—
81#387—21no—
82#384C2312yes—
83#382D411yescharacteristic
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4C2311yescharacteristic
89C4×C4⋊C4C2311yescharacteristic
90C43C2311yescharacteristic
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#336—21no—
96#336—21no—
97#336—21no—
98#336—21no—
99#336—22no—
100#336—22no—
101#336—21no—
102#336—21no—
103#332—24no—
104#332—24no—
105C22×Q8—22no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8#4114yes—
118#125—42no—
119C4×Q8—82no—
120C4×Q8—82no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4C2×D411yescharacteristic
132C2×C4⋊C4C2×D411yescharacteristic
133C2×C4⋊C4C2×D411yescharacteristic
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—42no—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42#4111yescharacteristic
144C2×C42C2411yescharacteristic
145#107—24no—
146#107—24no—
147#107—42no—
148#107—42no—
149#107—82no—
150#106—82no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—42no—
158C4⋊C8—42no—
159#103—44no—
160#103—44no—
161#103—82no—
162#103—82no—
163#97—42no—
164#97—42no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—41no—
171#95—21no—
172#95—21no—
173#95—21no—
174#95—21no—
175#95—21no—
176#95—21no—
177#95—42no—
178#95—42no—
179C2×Q8—24no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4#12111yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4C22×D411yescharacteristic
196C22×C4C22×D411yescharacteristic
197C22×C4C2×#4111yescharacteristic
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8#13512yes—
220C2×C8#13712yes—
221C2×C8#13712yes—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—82no—
241C4⋊C4—84no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—22no—
247C42—21no—
248C42—21no—
249C42—22no—
250C42—41no—
251C42—41no—
252C42—41no—
253C42—41no—
254C42—41no—
255C42—84no—
256C42C2×#4112yes—
257C422−1+412yes—
258C23C2×#12111yescharacteristic
259Q8—44no—
260Q8—44no—
261Q8—82no—
262Q8—82no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—44no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4#47312yes—
295C2×C4#52511yescharacteristic
296C2×C4#53611yescharacteristic
297C2×C4#54812yes—
298C2×C4#55212yes—
299C2×C4C2×#13511yescharacteristic
300C2×C4C2×#13711yescharacteristic
301C2×C4C2×#13711yescharacteristic
302C2×C4#57512yes—
303C2×C4#57511yescharacteristic
304C2×C4#57611yescharacteristic
305C2×C4#57611yescharacteristic
306C8—22no—
307C8—22no—
308C8—22no—
309C8—41no—
310C8—81no—
311C8—82no—
312C8—82no—
313C8—81no—
314C22#253611yescharacteristic
315C22#303511yescharacteristic
316C22#303911yescharacteristic
317C22#305411yescharacteristic
318C22#305811yescharacteristic
319C22C2×#47311yescharacteristic
320C22#306711yescharacteristic
321C4—22no—
322C4—22no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—81no—
327C4—84no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4—162no—
332C4#310811yescharacteristic
333C4#311711yescharacteristic
334C4#323111yescharacteristic
335C4#323611yescharacteristic
336C4#337012yes—
337C4#337012yes—
338C2#1916111yescharacteristic
339C2#1921911yescharacteristic
340C2#1929811yescharacteristic
341C2#1935811yescharacteristic
342C2#2819111yescharacteristic
343C2#2820911yescharacteristic
344C2#2821911yescharacteristic
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
#292572
C4×#5002
#216732
#106512
#106492
#105452
#95742
#95732
#87852
#85932