Finite Group Databasebeta

← #6378570

Subgroups — #6378570

Order 512.

1563 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34967C211yesmaximal, characteristic
2#22363C214yesmaximal, complemented
3#22335C214yesmaximal, complemented
4#22300C211yesmaximal, characteristic, complemented
5#22265C211yesmaximal, characteristic, complemented
6#22138C212yesmaximal, complemented
7#22014C212yesmaximal, complemented
8#21927C211yesmaximal, characteristic, complemented
9#21684C211yesmaximal, characteristic, complemented
10#21676C211yesmaximal, characteristic, complemented
11#21565C214yesmaximal, complemented
12#21494C212yesmaximal, complemented
13#21488C214yesmaximal, complemented
14#20494C212yesmaximal, complemented
15#20475C211yesmaximal, characteristic, complemented
16C2×#436C2212yes—
17C2×#436C2212yes—
18C2×#436C2214yes—
19C2×#436C2214yes—
20C2×#433C2211yescharacteristic
21C2×#405C2211yescharacteristic
22C2×C4×M16C2214yes—
23#2522C414yes—
24#2516C414yes—
25#2464C414yes—
26#2457C414yes—
27#2402C2211yescharacteristic
28#2397C2211yescharacteristic
29#2392C2212yes—
30C4×#123C2214yes—
31C4×#121C2214yes—
32#2298C2211yescharacteristic
33C42×D4C2212yes—
34#1955C2214yes—
35#1955C2212yes—
36#1955C2212yes—
37#1954C2214yes—
38#1940C2212yes—
39#1940C2212yes—
40#1939C2214yes—
41#1924C2214yes—
42#1924C2214yes—
43#1924C2214yes—
44#1923C2214yes—
45#1923C2214yes—
46#1917C2214yes—
47#1917C2214yes—
48#1916C2214yes—
49#1870C2214yes—
50#1870C2214yes—
51#1870C2214yes—
52#1869C2214yes—
53#1851C2214yes—
54#1843C2211yescharacteristic
55#1832C2211yescharacteristic
56#1817C2214yes—
57#1798C2212yes—
58#1797C2214yes—
59#1771C2212yes—
60#1770C2212yes—
61C8×C4⋊C4C2212yes—
62C8×C4⋊C4C2212yes—
63C8×C4⋊C4C2214yes—
64#1767C2212yes—
65#1753C2214yes—
66#1753C2214yes—
67#1753C2214yes—
68#1753C2212yes—
69#1752C2212yes—
70#1749C2214yes—
71#1746C2211yescharacteristic
72#1745C2211yescharacteristic
73#1726C2211yescharacteristic
74C4×#97C2211yescharacteristic
75C22×M16C2314yes—
76C22×M16C2311yescharacteristic
77C22×M16C2314yes—
78C23×C8C2311yescharacteristic
79C2×#123C2×C414yes—
80C2×#121C2×C414yes—
81C2×C4×D4C2×C414yes—
82C2×C4×D4C2312yes—
83C2×C4×D4C2314yes—
84C2×C4×D4C2314yes—
85C2×#117C2311yescharacteristic
86C22×C42C2314yes—
87#436—28no—
88#436—28no—
89#436—48no—
90#436—48no—
91#433—24no—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2314yes—
96C2×C4⋊C8C2314yes—
97C2×C4⋊C8C2314yes—
98C2×C22⋊C8C2314yes—
99C2×C22⋊C8C2312yes—
100C2×C22⋊C8C2314yes—
101C2×C22⋊C8C2314yes—
102C2×C22⋊C8C2314yes—
103C2×C22⋊C8C2314yes—
104C2×C22⋊C8C2314yes—
105C2×C22⋊C8C2314yes—
106#405—24no—
107C4×M16—48no—
108C2×#97C2312yes—
109C2×#97C2311yescharacteristic
110C2×#97C2311yescharacteristic
111C2×#97C2311yescharacteristic
112C2×#97C2314yes—
113C2×#97C2312yes—
114C2×#97C2312yes—
115C2×C4×C8C2314yes—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2312yes—
118C2×C4×C8C2314yes—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#400C2×C414yes—
122#399C2×C414yes—
123#398C2×C412yes—
124#398C2×C414yes—
125#397C2×C412yes—
126#396C2×C414yes—
127#394C2×C412yes—
128#394C2×C414yes—
129#392C2×C412yes—
130#392C2×C414yes—
131#390C2×C414yes—
132#389C2×C414yes—
133#388C2×C414yes—
134#388C2×C414yes—
135#388C2×C414yes—
136#388C2×C414yes—
137#388C2×C414yes—
138#387C2×C414yes—
139#387C2×C414yes—
140#387C2×C414yes—
141#386C2×C414yes—
142#386C2×C414yes—
143#386C2×C414yes—
144#385C2×C414yes—
145#381C2×C414yes—
146#380C2×C414yes—
147C4×C4⋊C4C2×C414yes—
148C4×C4⋊C4C2312yes—
149C4×C4⋊C4C2314yes—
150C4×C4⋊C4C2312yes—
151C4×C22⋊C4C2×C414yes—
152C4×C22⋊C4C2×C414yes—
153C4×C22⋊C4C2314yes—
154C4×C22⋊C4C2312yes—
155C4×C22⋊C4C2314yes—
156#376C2311yescharacteristic
157C43C2311yescharacteristic
158#336C2312yes—
159#336C2314yes—
160#336C2314yes—
161#336C2312yes—
162#336C2314yes—
163#336C2314yes—
164#336C2314yes—
165#336C2314yes—
166#336C2314yes—
167#336C2312yes—
168#336C2312yes—
169C22×D4C22×C412yes—
170C22×D4C22×C414yes—
171C22×D4C22×C414yes—
172C23×C4C22×C414yes—
173C23×C4C22×C414yes—
174C23×C4C2411yescharacteristic
175C23×C4C2414yes—
176C2×M16—22no—
177C2×M16—28no—
178C2×M16—28no—
179C2×M16—28no—
180C2×M16—28no—
181C2×M16—24no—
182C2×M16—44no—
183C2×M16—44no—
184C22×C8—22no—
185C22×C8—42no—
186C22×C8C2411yescharacteristic
187C22×C8C2414yes—
188C22×C8C2411yescharacteristic
189C22×C8C2414yes—
190C22×C8C2412yes—
191C22×C8C2411yescharacteristic
192C22×C8C2412yes—
193C22×C8C2411yescharacteristic
194#123—48no—
195#121—48no—
196C4×D4—24no—
197C4×D4—24no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4—44no—
201C4×D4—44no—
202C4×D4—48no—
203#117—22no—
204#117—22no—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C414yes—
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C414yes—
209C2×C4⋊C4C22×C414yes—
210C2×C4⋊C4C22×C414yes—
211C2×C4⋊C4C22×C414yes—
212C2×C4⋊C4C22×C414yes—
213C2×C22⋊C4C22×C412yes—
214C2×C22⋊C4C22×C414yes—
215C2×C22⋊C4C22×C414yes—
216C2×C22⋊C4C22×C414yes—
217C2×C22⋊C4C22×C414yes—
218C2×C22⋊C4C22×C414yes—
219C2×C22⋊C4C22×C414yes—
220C2×C22⋊C4C22×C414yes—
221C2×C22⋊C4C22×C414yes—
222C2×C22⋊C4C22×C414yes—
223C2×C22⋊C4C22×C414yes—
224C2×C42—44no—
225C2×C42—44no—
226C2×C42C22×C414yes—
227C2×C42C22×C414yes—
228C2×C42C22×C414yes—
229C2×C42C2411yescharacteristic
230C2×C42C2412yes—
231C2×C42C2414yes—
232C2×C42C2412yes—
233C2×C42C2411yescharacteristic
234C4⋊C8—24no—
235C4⋊C8—24no—
236C4⋊C8—24no—
237C4⋊C8—44no—
238C4⋊C8—44no—
239C4⋊C8—44no—
240C22⋊C8—24no—
241C22⋊C8—28no—
242C22⋊C8—28no—
243C22⋊C8—44no—
244C22⋊C8—44no—
245C22⋊C8—44no—
246C22⋊C8—44no—
247C22⋊C8—44no—
248#97—22no—
249#97—24no—
250#97—22no—
251#97—22no—
252#97—42no—
253#97—42no—
254#97—44no—
255C4×C8—22no—
256C4×C8—28no—
257C4×C8—24no—
258C4×C8—44no—
259C4×C8#4114yes—
260C4×C8#4114yes—
261#95C22×C414yes—
262#95C22×C412yes—
263#95C22×C412yes—
264#95C22×C414yes—
265#95C22×C414yes—
266#95C22×C414yes—
267#95C22×C414yes—
268#95C22×C414yes—
269#95C22×C414yes—
270#95C22×C414yes—
271#95C22×C414yes—
272C24C23×C411yescharacteristic
273C24C23×C414yes—
274C2×D4—24no—
275C2×D4—28no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—44no—
282C22×C4—21no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—28no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4C23×C414yes—
299C22×C4C23×C412yes—
300C22×C4C23×C414yes—
301C22×C4C23×C414yes—
302C22×C4C23×C414yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C23×C412yes—
305C22×C4C23×C414yes—
306C22×C4C2511yescharacteristic
307M16—416no—
308M16—44no—
309M16—416no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—22no—
320C2×C8—24no—
321C2×C8—22no—
322C2×C8—24no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—24no—
326C2×C8—44no—
327C2×C8C2×#4112yes—
328C2×C8C2×#4114yes—
329C2×C8C2×#4112yes—
330C2×C8H(5,𝔽2)14yes—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—28no—
334C4⋊C4—28no—
335C4⋊C4—28no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C22⋊C4—24no—
340C22⋊C4—28no—
341C22⋊C4—28no—
342C22⋊C4—28no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C42—21no—
351C42—21no—
352C42—24no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—24no—
358C42—44no—
359C42—44no—
360C42—44no—
361C42C2×#4112yes—
362C42C2×#4112yes—
363C23—21no—
364C23—24no—
365C23—24no—
366C23—22no—
367C23—24no—
368C23—24no—
369C23—24no—
370C23—24no—
371C23—24no—
372C23—24no—
373C23C24×C411yescharacteristic
374D4—48no—
375D4—88no—
376D4—88no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—21no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—88no—
404C2×C4—88no—
405C2×C4C4×#4114yes—
406C2×C4#56818yes—
407C2×C4C22×#4111yescharacteristic
408C2×C4C22×#4111yescharacteristic
409C2×C4C2×H(5,𝔽2)12yes—
410C2×C4#58512yes—
411C8—22no—
412C8—24no—
413C8—41no—
414C8—44no—
415C8—41no—
416C8—42no—
417C8—44no—
418C8#54014yes—
419C22—41no—
420C22—44no—
421C22—44no—
422C22—42no—
423C22—44no—
424C22—44no—
425C22—44no—
426C22—44no—
427C22—44no—
428C22—44no—
429C22C2×C4×#4111yescharacteristic
430C22C4×H(5,𝔽2)12yes—
431C22C2×#56814yes—
432C4—24no—
433C4—28no—
434C4—44no—
435C4—41no—
436C4—42no—
437C4—44no—
438C4—44no—
439C4—44no—
440C4C2×#54011yescharacteristic
441C4#345511yescharacteristic
442C4#346012yes—
443C2—84no—
444C2—81no—
445C2C4×#54011yescharacteristic
446C2#6020812yes—
447C2#6032414yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radical#34967C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22363C2semidirect
3. #22335C2semidirect
4. #22300C2semidirect
5. #22265C2semidirect
6. #22138C2semidirect
7. #22014C2semidirect
8. #21927C2semidirect
9. #21684C2semidirect
10. #21676C2semidirect
11. #21565C2semidirect
12. #21494C2semidirect
13. #21488C2semidirect
14. #20494C2semidirect
15. #20475C2semidirect

Maximal subgroups

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

subgroupindex
#349672
#223632
#223352
#223002
#222652
#221382
#220142
#219272
#216842
#216762
#215652
#214942
#214882
#204942
#204752