Finite Group Databasebeta

← #324807

Subgroups — #324807

Order 512.

548 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic
2#29226C211yesmaximal, characteristic
3#28694C211yesmaximal, characteristic
4C8×#125C211yesmaximal, characteristic
5#10597C212yesmaximal
6#10589C212yesmaximal
7#9905C211yesmaximal, characteristic
8#9702C211yesmaximal, characteristic
9#9568C212yesmaximal
10#9567C212yesmaximal
11#8273C211yesmaximal, characteristic
12C2×#501—21no—
13#2868C2211yescharacteristic
14#2868C2211yescharacteristic
15C4×#125C2211yescharacteristic
16#2099—21no—
17#2099—21no—
18#2084—21no—
19#2083—21no—
20#1993—21no—
21#1992—21no—
22#1990—21no—
23#1990—21no—
24#1990C2212yes—
25#1990C2212yes—
26#1987C2212yes—
27#1987C2212yes—
28#1948C2212yes—
29#1947C2212yes—
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1916—21no—
33#1916—21no—
34#1836—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×#333C2212yes—
47C2×#333C2212yes—
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#501—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×#97—21no—
68C2×#97—21no—
69C2×C4×C8—21no—
70C2×C4×C8—21no—
71C2×C4×C8D411yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#389D411yescharacteristic
78#389D411yescharacteristic
79#387—21no—
80#387—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384C2312yes—
89#382D411yescharacteristic
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2311yescharacteristic
93C4×C4⋊C4C2311yescharacteristic
94C43C2311yescharacteristic
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
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#333—24no—
115#333—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—44no—
169#106—44no—
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#97—42no—
179#97—42no—
180C4×C8—22no—
181C4×C8—22no—
182C4×C8—22no—
183C4×C8—22no—
184C4×C8—41no—
185C4×C8—41no—
186C4×C8—42no—
187C4×C8—42no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—21no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—82no—
197C2×Q8—82no—
198C2×Q8—82no—
199C2×Q8—82no—
200C2×Q8—82no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4C22×D411yescharacteristic
212C22×C4C22×D411yescharacteristic
213C22×C4C2×#4111yescharacteristic
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—24no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—42no—
236C2×C8#13712yes—
237C2×C8#13712yes—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4C2×SD1612yes—
263C4⋊C4C2×SD1612yes—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—41no—
272C42—41no—
273C42—41no—
274C42—41no—
275C42C2×SD1612yes—
276C42#13512yes—
277C42C2×#4112yes—
278C422−1+412yes—
279C23C2×#12111yescharacteristic
280Q8—162no—
281Q8—162no—
282Q8—164no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4#46512yes—
312C2×C4#46512yes—
313C2×C4#47112yes—
314C2×C4#47212yes—
315C2×C4#53211yescharacteristic
316C2×C4#53611yescharacteristic
317C2×C4#54812yes—
318C2×C4#55612yes—
319C2×C4C22×SD1611yescharacteristic
320C2×C4C2×#13511yescharacteristic
321C2×C4C2×#13711yescharacteristic
322C2×C4C2×#13711yescharacteristic
323C2×C4#57511yescharacteristic
324C2×C4#57511yescharacteristic
325C2×C4#57812yes—
326C8—22no—
327C8—22no—
328C8—44no—
329C8—41no—
330C8—41no—
331C8—82no—
332C8—81no—
333C8—81no—
334C22#254511yescharacteristic
335C22#303911yescharacteristic
336C22#304011yescharacteristic
337C22C2×#46511yescharacteristic
338C22#305911yescharacteristic
339C22C2×#47111yescharacteristic
340C22C2×#47211yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—42no—
347C4—42no—
348C4—81no—
349C4—81no—
350C4—162no—
351C4—162no—
352C4#317211yescharacteristic
353C4#318111yescharacteristic
354C4#323611yescharacteristic
355C4#323711yescharacteristic
356C4#335912yes—
357C4#336412yes—
358C2#1927011yescharacteristic
359C2#1927411yescharacteristic
360C2#1935911yescharacteristic
361C2#1936511yescharacteristic
362C2#2863911yescharacteristic
363C2#2866011yescharacteristic
364C2#2867011yescharacteristic
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
#292262
#286942
C8×#1252
#105972
#105892
#99052
#97022
#95682
#95672
#82732