Finite Group Databasebeta

← #324811

Subgroups — #324811

Order 512.

564 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic
2#29213C211yesmaximal, characteristic
3#28755C211yesmaximal, characteristic
4C8×#125C211yesmaximal, characteristic
5#10598C212yesmaximal
6#10590C212yesmaximal
7#9903C211yesmaximal, characteristic
8#9705C211yesmaximal, characteristic
9#9566C212yesmaximal
10#9565C212yesmaximal
11#8273C211yesmaximal, characteristic
12C2×#491—21no—
13#2868C2211yescharacteristic
14#2868C2211yescharacteristic
15C4×#125C2211yescharacteristic
16#2058—21no—
17#2057—21no—
18#2043—21no—
19#2043—21no—
20#1992—21no—
21#1990—21no—
22#1990—21no—
23#1990C2212yes—
24#1990C2212yes—
25#1981C2212yes—
26#1981C2212yes—
27#1960—21no—
28#1948C2212yes—
29#1947C2212yes—
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1916—21no—
33#1916—21no—
34#1847—21no—
35#1834—21no—
36#1834—21no—
37#1831C2211yescharacteristic
38C8×C4⋊C4—21no—
39C8×C4⋊C4—21no—
40C8×C4⋊C4C2211yescharacteristic
41C8×C4⋊C4C2211yescharacteristic
42#1767C2211yescharacteristic
43#1767C2211yescharacteristic
44C42×C8C2211yescharacteristic
45C2×#334C2212yes—
46C2×#332C2212yes—
47C2×#332C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1277C2212yes—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2312yes—
57C2×#125D411yescharacteristic
58#491—44no—
59C2×#107—22no—
60C2×#106—22no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#97—21no—
72C2×#97—21no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8D411yescharacteristic
76C2×C4×C8D411yescharacteristic
77C2×C4×C8D411yescharacteristic
78C2×C4×C8D411yescharacteristic
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81#389D411yescharacteristic
82#389D411yescharacteristic
83#387—21no—
84#387—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384C2312yes—
93#382D411yescharacteristic
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4C2311yescharacteristic
97C4×C4⋊C4C2311yescharacteristic
98C43C2311yescharacteristic
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#336—21no—
106#336—21no—
107#336—21no—
108#336—21no—
109#336—21no—
110#336—21no—
111#336—21no—
112#336—21no—
113#334—24no—
114#332—24no—
115#332—24no—
116C22×Q8—22no—
117C22×Q8—41no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D411yescharacteristic
125C22×C8C2×D411yescharacteristic
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—42no—
131C4⋊Q8—42no—
132C4⋊Q8—42no—
133C4⋊Q8—42no—
134C4⋊Q8—82no—
135C4⋊Q8—82no—
136C4⋊Q8—82no—
137#125—22no—
138#125SD1614yes—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4C2×D411yescharacteristic
156C2×C4⋊C4C2×D411yescharacteristic
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C42—21no—
160C2×C42—21no—
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42C2411yescharacteristic
168#107—82no—
169#106—82no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—42no—
176C4⋊C8—42no—
177C4⋊C8#4114yes—
178#103—44no—
179#103—44no—
180#103—82no—
181#103—82no—
182#97—42no—
183#97—42no—
184C4×C8—22no—
185C4×C8—22no—
186C4×C8—22no—
187C4×C8—22no—
188C4×C8—41no—
189C4×C8—41no—
190C4×C8—42no—
191C4×C8—42no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—21no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—82no—
205C2×Q8—82no—
206C2×Q8—82no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D411yescharacteristic
219C22×C4C2×#4111yescharacteristic
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—24no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—42no—
242C2×C8#13712yes—
243C2×C8#13712yes—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C4⋊C4C2×SD1612yes—
269C4⋊C4C2×SD1612yes—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—41no—
278C42—41no—
279C42—41no—
280C42—41no—
281C42C2×SD1612yes—
282C42#13512yes—
283C42C2×#4112yes—
284C422−1+412yes—
285C23C2×#12111yescharacteristic
286Q8—82no—
287Q8—82no—
288Q8—82no—
289Q8—82no—
290Q8—164no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4#46512yes—
320C2×C4#46512yes—
321C2×C4#47112yes—
322C2×C4#47312yes—
323C2×C4#53211yescharacteristic
324C2×C4#53611yescharacteristic
325C2×C4#54812yes—
326C2×C4#55612yes—
327C2×C4C22×SD1611yescharacteristic
328C2×C4C2×#13511yescharacteristic
329C2×C4C2×#13711yescharacteristic
330C2×C4C2×#13711yescharacteristic
331C2×C4#57511yescharacteristic
332C2×C4#57511yescharacteristic
333C2×C4#57612yes—
334C8—22no—
335C8—22no—
336C8—44no—
337C8—41no—
338C8—41no—
339C8—82no—
340C8—81no—
341C8—81no—
342C22#254511yescharacteristic
343C22#303911yescharacteristic
344C22#304011yescharacteristic
345C22C2×#46511yescharacteristic
346C22#305811yescharacteristic
347C22C2×#47111yescharacteristic
348C22C2×#47311yescharacteristic
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—42no—
355C4—42no—
356C4—81no—
357C4—81no—
358C4—162no—
359C4—162no—
360C4#317211yescharacteristic
361C4#318111yescharacteristic
362C4#323611yescharacteristic
363C4#323711yescharacteristic
364C4#336112yes—
365C4#336212yes—
366C2#1927011yescharacteristic
367C2#1927511yescharacteristic
368C2#1935811yescharacteristic
369C2#1936411yescharacteristic
370C2#2864011yescharacteristic
371C2#2866011yescharacteristic
372C2#2866911yescharacteristic
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
#332892
#292132
#287552
C8×#1252
#105982
#105902
#99032
#97052
#95662
#95652
#82732