Finite Group Databasebeta

← #242803

Subgroups — #242803

Order 512.

740 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21373C211yesmaximal, characteristic
2#20760C211yesmaximal, characteristic, complemented
3#20743C212yesmaximal
4#10626C212yesmaximal, complemented
5#10625C212yesmaximal, complemented
6#10624C211yesmaximal, characteristic, complemented
7#10623C211yesmaximal, characteristic, complemented
8#10622C211yesmaximal, characteristic, complemented
9#10621C211yesmaximal, characteristic, complemented
10#8046C212yesmaximal, complemented
11#7875C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#432C2212yes—
14C2×#431C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2285C2212yes—
17#1984C2211yescharacteristic
18#1984C2211yescharacteristic
19#1954C2211yescharacteristic
20#1954C2211yescharacteristic
21#1869—22no—
22#1835—22no—
23#1834C2211yescharacteristic
24#1834C2211yescharacteristic
25#1834C2212yes—
26#1825—22no—
27#1803C2212yes—
28#1803C2212yes—
29#1802C2212yes—
30#1802C2212yes—
31#1788—22no—
32#1763—22no—
33#1762—22no—
34C4×C22⋊C8—22no—
35#1735—22no—
36#1296C2212yes—
37#1296C2212yes—
38#1296C2212yes—
39#1296C2212yes—
40#1277C2211yescharacteristic
41#1277C2211yescharacteristic
42#1277C2212yes—
43C22×#41D411yescharacteristic
44C23×C8D412yes—
45C4×#41—24no—
46C4×#41—24no—
47C2×C4×Q8D411yescharacteristic
48C2×C4×Q8C2311yescharacteristic
49C2×C4×D4—21no—
50C2×C4×D4—21no—
51C2×C4×D4D411yescharacteristic
52C2×C4×D4C2311yescharacteristic
53C2×#117—21no—
54C2×#117—22no—
55C2×#117D411yescharacteristic
56C22×C42—21no—
57C22×C42C2311yescharacteristic
58#432—24no—
59#432—24no—
60#431—24no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×#103—22no—
66C2×#103—22no—
67C2×#102—22no—
68C2×#102—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×#97—22no—
74C2×C4×C8—24no—
75C2×C4×C8—21no—
76C2×C4×C8D412yes—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79#384C2312yes—
80#384C2312yes—
81#382—22no—
82#380—22no—
83#380—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86#340—22no—
87#340—22no—
88#340—24no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#336—22no—
94#336—22no—
95C2×#41—22no—
96C2×#41—22no—
97C2×#41—24no—
98C2×#41—42no—
99C22×Q8C2×D411yescharacteristic
100C22×D4—21no—
101C22×D4C2×D411yescharacteristic
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4#4111yescharacteristic
108C22×C8—24no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—42no—
114C22×C8—42no—
115C22×C8C2×D412yes—
116C22×C8C2×D412yes—
117C4×Q8—24no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×D4—24no—
121C4×D4—22no—
122C4×D4—22no—
123C4×D4—42no—
124C4×D4—42no—
125C4×D4—44no—
126#117—24no—
127#117—44no—
128#117—44no—
129#117—44no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4C2×D411yescharacteristic
139C2×C4⋊C4C2×D411yescharacteristic
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—42no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—42no—
148C2×C42—42no—
149C2×C42—42no—
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42#4114yes—
153C2×C42#4114yes—
154C2×C42#4111yescharacteristic
155C2×C42C2411yescharacteristic
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162#103—44no—
163#103—44no—
164#102—44no—
165#102—44no—
166C22⋊C8—44no—
167C22⋊C8—44no—
168C22⋊C8—44no—
169C22⋊C8—42no—
170#97—44no—
171C4×C8—24no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—42no—
175#95—42no—
176#95—42no—
177C24—21no—
178C24#12111yescharacteristic
179#41—48no—
180#41—48no—
181C2×Q8—21no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8#13512yes—
185C2×D4—24no—
186C2×D4—22no—
187C2×D4—21no—
188C2×D4—22no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4#13512yes—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4#12111yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12314yes—
216C22×C4#12312yes—
217C22×C4#12412yes—
218C22×C4#12614yes—
219C22×C4C22×D411yescharacteristic
220C22×C4C22×D411yescharacteristic
221C22×C4C2×#4111yescharacteristic
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—48no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—84no—
257C42—41no—
258C42—41no—
259C42—42no—
260C42—42no—
261C42—84no—
262C42#12312yes—
263C42#12314yes—
264C42#12412yes—
265C42#12614yes—
266C42C2×#4112yes—
267C42C2×#4112yes—
268C23—22no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—22no—
273C23—41no—
274C23—42no—
275C23—41no—
276C23—42no—
277C23#39712yes—
278C23C2×#12111yescharacteristic
279Q8—24no—
280Q8—42no—
281D4—24no—
282D4—48no—
283D4—42no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—44no—
296C2×C4—48no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—84no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4#39712yes—
314C2×C4#39712yes—
315C2×C4#39712yes—
316C2×C4C2×#12312yes—
317C2×C4C2×#12311yescharacteristic
318C2×C4C2×#12411yescharacteristic
319C2×C4C2×#12612yes—
320C2×C4C2×#13511yescharacteristic
321C2×C4C2×#13511yescharacteristic
322C2×C4#57512yes—
323C2×C4#57512yes—
324C2×C4#57512yes—
325C8—42no—
326C8—42no—
327C8—82no—
328C8—82no—
329C8—81no—
330C8—81no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—42no—
335C22—42no—
336C22—44no—
337C22—42no—
338C22—81no—
339C22—81no—
340C22—82no—
341C22C2×#39711yescharacteristic
342C22#304012yes—
343C22#305011yescharacteristic
344C22#305411yescharacteristic
345C22#306712yes—
346C4—22no—
347C4—22no—
348C4—22no—
349C4—22no—
350C4—42no—
351C4—81no—
352C4—81no—
353C4—162no—
354C4—162no—
355C4#309112yes—
356C4#309112yes—
357C4#309312yes—
358C4#314211yescharacteristic
359C4#314211yescharacteristic
360C2—42no—
361C2—82no—
362C2#1927812yes—
363C2#1930312yes—
364C2#2363111yescharacteristic
365C2#2363811yescharacteristic
366C2#2364611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20760C2semidirect
4. #10626C2semidirect
5. #10625C2semidirect
6. #10624C2semidirect
7. #10623C2semidirect
8. #10622C2semidirect
9. #10621C2semidirect
10. #8046C2semidirect
11. #7875C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#213732
#207602
#207432
#106262
#106252
#106242
#106232
#106222
#106212
#80462
#78752