Finite Group Databasebeta

← #6461968

Subgroups — #6461968

Order 512.

1646 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic
3#29584C218yesmaximal, complemented
4#29578C214yesmaximal, complemented
5#29572C214yesmaximal, complemented
6#23745C211yesmaximal, characteristic, complemented
7C2×#2093C211yesmaximal, characteristic
8#23656C211yesmaximal, characteristic, complemented
9#23628C211yesmaximal, characteristic
10#21156C212yesmaximal, complemented
11#21091C212yesmaximal
12#21037C212yesmaximal, complemented
13#20879C212yesmaximal
14#20368C211yesmaximal, characteristic, complemented
15C2×#518C2211yescharacteristic
16C2×#427C2212yes—
17C22×#107C2212yes—
18C2×#418C2212yes—
19C22×#102C2212yes—
20C2×#405C2212yes—
21#2859C2214yescomplemented
22#2647C2214yescomplemented
23#2571—22no—
24#2552—22no—
25#2442—24no—
26#2357—22no—
27#2354—22no—
28#2342—24no—
29#2319C2214yes—
30#2317C2212yes—
31#2317C2211yescharacteristic
32#2316C2212yes—
33#2309C2211yescharacteristic
34#2296C2211yescharacteristic
35#2293C2214yes—
36C2×#390C2211yescharacteristic
37#2285C2211yescharacteristic
38C2×#384C2211yescharacteristic
39#2274C2211yescharacteristic
40C4×#117C2211yescharacteristic
41#2100C2214yescomplemented
42#2099C2214yescomplemented
43#2093C2212yes—
44#2093C2212yes—
45#2093C2214yes—
46#2085C2214yescomplemented
47#2084C2212yescomplemented
48#2083C2212yescomplemented
49#2073C2214yes—
50#2071C2214yes—
51#1944C2218yescomplemented
52#1942C2218yescomplemented
53#1936C2218yes—
54#1934C2218yes—
55#1921C2218yescomplemented
56#1918C2218yescomplemented
57#1864C2218yes—
58#1862C2218yes—
59#1746C2214yescomplemented
60#1745C2212yescomplemented
61#1744C2212yescomplemented
62#1736C2212yes—
63#1736C2212yes—
64#1736C2212yes—
65#1735C2212yes—
66#1729C2212yes—
67#1728C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C23×D4C2311yescharacteristic
70C22×M16C2312yes—
71C23×C8C2312yes—
72C2×#125—24no—
73#518—24no—
74#518—24no—
75C2×C4×D4—22no—
76C2×C4×D4C2314yes—
77C2×#117—24no—
78C2×#117C2311yescharacteristic
79C2×#117C2312yes—
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C22×C4⋊C4—22no—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—22no—
87C22×C22⋊C4—22no—
88C22×C42C2311yescharacteristic
89#427—28no—
90C2×#107—28no—
91C2×#107C2314yes—
92C2×#107C2314yes—
93C2×#107C2314yes—
94C2×#106C2314yes—
95#418—28no—
96C2×#102—28no—
97C2×#102C2318yes—
98C2×#102C2314yes—
99C2×#102C2314yes—
100#405—28no—
101C2×#97C2314yes—
102C2×C4×C8C2314yes—
103#400—28no—
104#398—28no—
105#390—21no—
106#390—21no—
107#390—22no—
108#390C2314yes—
109#390C2312yes—
110#390C2312yes—
111#389—24no—
112#389—24no—
113#389—24no—
114#389C2318yes—
115#389C2314yes—
116#389C2318yes—
117#389C2312yes—
118#389C2312yes—
119#388—24no—
120#388—24no—
121#388—24no—
122#388—24no—
123#385—22no—
124#385—22no—
125#385—24no—
126#384—21no—
127#384—22no—
128#384—22no—
129#384C2312yes—
130#384C2312yes—
131#384C2312yes—
132#382—22no—
133#382—21no—
134#382—24no—
135#382C2312yes—
136#381—24no—
137#380—22no—
138#380—24no—
139#380—24no—
140#379—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4D412yes—
145C4×C4⋊C4D412yes—
146C4×C4⋊C4C2314yes—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—24no—
149C4×C22⋊C4D414yes—
150#376C2312yes—
151C43C2312yes—
152#340C2318yes—
153#340C2314yes—
154#340C2314yes—
155#340C2318yes—
156#340C2318yes—
157#336C2312yes—
158#336C2312yes—
159#336C2311yescharacteristic
160#336C2311yescharacteristic
161#336C2312yes—
162#336C2314yes—
163#336C2314yes—
164C25—21no—
165C22×D4—21no—
166C22×D4—24no—
167C22×D4—22no—
168C22×D4—24no—
169C22×D4#4114yes—
170C22×D4C2412yes—
171C23×C4—24no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4C2×D412yes—
175C23×C4C2411yescharacteristic
176C2×M16—24no—
177C2×M16—24no—
178C2×M16—24no—
179C22×C8—24no—
180C22×C8—24no—
181C22×C8#4118yes—
182C22×C8C2414yes—
183C22×C8C2412yes—
184C22×C8C2412yes—
185#125—88no—
186C4×D4—216no—
187C4×D4—28no—
188C4×D4—28no—
189#117—22no—
190#117—22no—
191#117—22no—
192#117—24no—
193#117—24no—
194#117—24no—
195#117—24no—
196#117—88no—
197#117#4114yes—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—84no—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2412yes—
217C2×C4⋊C4C2414yes—
218C2×C4⋊C4C2412yes—
219C2×C4⋊C4C2414yes—
220C2×C4⋊C4C2412yes—
221C2×C22⋊C4—216no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—84no—
231C2×C22⋊C4C2×D412yes—
232C2×C22⋊C4C2×D412yes—
233C2×C42—21no—
234C2×C42—24no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42C2×D414yes—
239C2×C42C2×D412yes—
240C2×C42C2×D414yes—
241C2×C42C2412yes—
242C2×C42C2412yes—
243C2×C42C2411yescharacteristic
244C2×C42C2411yescharacteristic
245#107—28no—
246#107—28no—
247#107—28no—
248#106—28no—
249#102—216no—
250#102—28no—
251#102—28no—
252#97—28no—
253C4×C8—28no—
254#95—22no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—22no—
259#95—28no—
260#95—28no—
261#95C2×D414yes—
262#95C2×D414yes—
263C24—22no—
264C24—24no—
265C24—42no—
266C24—42no—
267C24—82no—
268C24—82no—
269C24C22×D411yescharacteristic
270C2×D4—24no—
271C2×D4—216no—
272C2×D4—24no—
273C2×D4—22no—
274C2×D4—24no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—48no—
278C2×D4C2×#4114yes—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—216no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4—82no—
293C22×C4—84no—
294C22×C4C22×D412yes—
295C22×C4C22×D412yes—
296C22×C4C22×D414yes—
297C22×C4C22×D412yes—
298C22×C4C22×D412yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C2×#4112yes—
301C22×C4C2×#4112yes—
302C22×C4C2×#4114yes—
303C22×C42−1+414yes—
304C22×C4C2511yescharacteristic
305M16—48no—
306C2×C8—216no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8C2×#4114yes—
315C2×C8C2×#4114yes—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—48no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—82no—
333C4⋊C4—84no—
334C4⋊C4—84no—
335C4⋊C4—88no—
336C4⋊C4—88no—
337C4⋊C4C2×#4114yes—
338C22⋊C4—216no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—44no—
343C22⋊C4—42no—
344C22⋊C4—48no—
345C22⋊C4—84no—
346C22⋊C4—84no—
347C22⋊C4—84no—
348C42—24no—
349C42—24no—
350C42—28no—
351C42—24no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—44no—
357C42—44no—
358C42—42no—
359C42—84no—
360C23—22no—
361C23—22no—
362C23—24no—
363C23—44no—
364C23—42no—
365C23—42no—
366C23—44no—
367C23—48no—
368C23—88no—
369C23—162no—
370C23—162no—
371C23—164no—
372C23#52512yes—
373C23#53212yes—
374C23#53612yes—
375C23C23×D411yescharacteristic
376D4—28no—
377D4—48no—
378D4—44no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—48no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—48no—
405C2×C4—816no—
406C2×C4—162no—
407C2×C4—164no—
408C2×C4#52512yes—
409C2×C4#53212yes—
410C2×C4#53612yes—
411C2×C4#55218yes—
412C2×C4#55218yes—
413C2×C4#57614yes—
414C2×C4C22×#4111yescharacteristic
415C2×C4C22×#4111yescharacteristic
416C2×C4C22×#4112yes—
417C2×C4C2×2−1+412yes—
418C8—24no—
419C8—24no—
420C8—44no—
421C22—22no—
422C22—22no—
423C22—24no—
424C22—24no—
425C22—44no—
426C22—48no—
427C22—82no—
428C22—82no—
429C22—84no—
430C22—168no—
431C22#261114yes—
432C22C2×#52511yescharacteristic
433C22C2×#53211yescharacteristic
434C22C2×#53611yescharacteristic
435C22C2×#57612yes—
436C22#358612yes—
437C4—24no—
438C4—42no—
439C4—42no—
440C4—44no—
441C4—44no—
442C4—42no—
443C4—82no—
444C4—84no—
445C4—84no—
446C4—82no—
447C4#261114yes—
448C4C2×#55212yes—
449C4C2×#55212yes—
450C2—24no—
451C2—84no—
452C2C2×#261111yescharacteristic
453C2#6059612yes—
454C2#6071411yescharacteristic
455C2#6072011yescharacteristic
456C2#6087112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35286C2semidirect
3. #29584C2semidirect
4. #29578C2semidirect
5. #29572C2semidirect
6. #23745C2semidirect
8. #23656C2semidirect
10. #21156C2semidirect
12. #21037C2semidirect
14. #20368C2semidirect
21. #2859C22semidirect
22. #2647C22semidirect
41. #2100C22semidirect
42. #2099C22semidirect
46. #2085C22semidirect
47. #2084C22semidirect
48. #2083C22semidirect
51. #1944C22semidirect
52. #1942C22semidirect
55. #1921C22semidirect
56. #1918C22semidirect
59. #1746C22semidirect
60. #1745C22semidirect
61. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#352862
#342552
#295842
#295782
#295722
#237452
C2×#20932
#236562
#236282
#211562
#210912
#210372
#208792
#203682