Finite Group Databasebeta

← #6475734

Subgroups — #6475734

Order 512.

1816 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C212yesmaximal
2#33212C211yesmaximal, characteristic, complemented
3#33176C211yesmaximal, characteristic, complemented
4#33049C212yesmaximal, complemented
5#29743C214yesmaximal
6#29478C218yesmaximal, complemented
7#27889C212yesmaximal, complemented
8#27875C212yesmaximal, complemented
9#25674C212yesmaximal, complemented
10#25600C212yesmaximal, complemented
11#22188C214yesmaximal, complemented
12C4×#407C211yesmaximal, characteristic
13C2×#533C2212yes—
14C23×C42C2211yescharacteristic
15C2×#500C2211yescharacteristic, complemented
16C2×#494C2211yescharacteristic, complemented
17C2×#487C2212yescomplemented
18C2×#470C2214yes—
19C2×#407C2212yes—
20C2×C4×M16C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2857C2212yes—
23#2857C2212yes—
24#2857C2212yes—
25#2857C2212yes—
26#2855—22no—
27#2855C2214yes—
28#2851—22no—
29#2851—22no—
30#2848—22no—
31#2848C2214yes—
32C4×C22⋊Q8—22no—
33C4×C22⋊Q8—22no—
34C4×C22⋊Q8C2214yes—
35C4×C22⋊Q8C2214yes—
36#2296—22no—
37#2285C2212yes—
38#2280—22no—
39#2077C2218yes—
40#2036C2218yes—
41#2035C2218yes—
42#2001C2218yes—
43#1990C2212yescomplemented
44#1990C2212yescomplemented
45#1971C2214yescomplemented
46#1956C2212yescomplemented
47#1956C2212yescomplemented
48#1944C2214yescomplemented
49#1935C2218yescomplemented
50#1927C2218yescomplemented
51#1920C2218yescomplemented
52#1894C2218yes—
53#1878C2218yescomplemented
54#1863C2218yescomplemented
55#1844C2211yescharacteristic, complemented
56#1844C2211yescharacteristic, complemented
57#1834C2212yescomplemented
58#1809C2214yes—
59#1749C2214yes—
60C4×C22⋊C8C2212yes—
61C4×C22⋊C8C2212yes—
62#1728C2212yes—
63C24×C4D411yescharacteristic
64C24×C4D411yescharacteristic
65C24×C4C2311yescharacteristic
66C22×Q16C2314yes—
67C22×M16C2312yes—
68C22×M16C2312yes—
69#533—44no—
70#533—44no—
71C2×C4⋊Q8C2312yes—
72C2×C4⋊Q8C2312yes—
73C2×#125—22no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8C2314yes—
78C2×C22⋊Q8C2314yes—
79C2×C22⋊Q8C2314yes—
80C2×C4×Q8—22no—
81C2×C4×Q8C2314yes—
82C2×#117—22no—
83C2×#117—22no—
84C22×C42—42no—
85C22×C42—42no—
86C22×C42D411yescharacteristic
87C22×C42D412yes—
88C22×C42D412yes—
89C22×C42D411yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2312yes—
92C22×C42C2312yes—
93C22×C42C2311yescharacteristic
94#500—42no—
95#494—42no—
96#487—44no—
97#470—216no—
98C2×#107C2314yes—
99C2×#106C2314yes—
100C2×#106C2314yes—
101C2×#103C2318yes—
102C2×#103C2318yes—
103C2×#103C2318yes—
104#407—28no—
105C2×C22⋊C8C2314yes—
106C2×C22⋊C8C2314yes—
107C4×M16—42no—
108C4×M16—42no—
109C2×#97C2312yes—
110C2×#97C2312yes—
111C2×C4×C8C2312yes—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—22no—
115#391—22no—
116#391C2314yes—
117#391C2314yes—
118#389—22no—
119#389—22no—
120#389—22no—
121#389—22no—
122#389—22no—
123#389—22no—
124#389C2314yes—
125#389C2314yes—
126#387—24no—
127#387—24no—
128#387—24no—
129#387—24no—
130#385—24no—
131#385—24no—
132#385—24no—
133#385—24no—
134#385—24no—
135#385—24no—
136#384C2312yes—
137#384C2312yes—
138#382—22no—
139#381—24no—
140#380—22no—
141#380—22no—
142#380—22no—
143#380—22no—
144#380—22no—
145#380—22no—
146#380—22no—
147#380C2314yes—
148#380C2314yes—
149#380C2314yes—
150#379—24no—
151C4×C4⋊C4—22no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4C2314yes—
155C4×C22⋊C4—24no—
156C4×C22⋊C4—24no—
157#340C2318yes—
158#340C2318yes—
159#340C2318yes—
160#336C2314yes—
161#336C2314yes—
162C25C2×D411yescharacteristic
163C22×Q8—22no—
164C22×Q8C2414yes—
165C23×C4—28no—
166C23×C4—24no—
167C23×C4—44no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—41no—
172C23×C4—41no—
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D412yes—
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185C2×Q16—28no—
186C2×Q16—44no—
187C2×Q16—44no—
188C2×M16—24no—
189C2×M16—42no—
190C2×M16—42no—
191C2×M16—42no—
192C2×M16—42no—
193C2×M16#4118yes—
194C22×C8C2414yes—
195C22×C8C2412yes—
196C22×C8C2412yes—
197C4⋊Q8—24no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200C4⋊Q8—24no—
201#125—48no—
202C22⋊Q8—216no—
203C22⋊Q8—48no—
204C22⋊Q8—44no—
205C22⋊Q8—44no—
206C22⋊Q8—48no—
207C22⋊Q8—48no—
208C22⋊Q8—48no—
209C4×Q8—44no—
210C4×Q8—44no—
211C4×Q8—48no—
212#117—48no—
213#117—48no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4C2414yes—
228C2×C4⋊C4C2414yes—
229C2×C4⋊C4C2414yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—44no—
243C2×C42—44no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—41no—
249C2×C42—41no—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2412yes—
258C2×C42C2411yescharacteristic
259C2×C42C2411yescharacteristic
260#107—44no—
261#106—28no—
262#106—44no—
263#103—216no—
264#103—48no—
265#103—48no—
266C22⋊C8—28no—
267C22⋊C8—28no—
268#97—42no—
269#97—42no—
270C4×C8—24no—
271C4×C8—22no—
272C4×C8—22no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283C24—22no—
284C24—24no—
285C24—41no—
286C24—41no—
287C24—42no—
288C24C22×D411yescharacteristic
289C24C22×D412yes—
290C2×Q8—28no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—28no—
294C2×Q8—44no—
295C2×Q8—44no—
296C2×Q8—44no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—416no—
334C22×C4—48no—
335C22×C4—48no—
336C22×C4—416no—
337C22×C4#13718yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D412yes—
340C22×C4C22×D412yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D412yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C2×#4114yes—
347C22×C4C2511yescharacteristic
348Q16—88no—
349M16—48no—
350M16—48no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8C2×#4114yes—
359C2×C8C2×#4114yes—
360C4⋊C4—28no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—28no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—48no—
369C4⋊C4—48no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C22⋊C4—48no—
378C22⋊C4—48no—
379C22⋊C4—48no—
380C22⋊C4—48no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C42—22no—
384C42—22no—
385C42—24no—
386C42—24no—
387C42—21no—
388C42—21no—
389C42—22no—
390C42—22no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—48no—
395C42—48no—
396C42H(5,𝔽2)12yes—
397C42H(5,𝔽2)12yes—
398C23—22no—
399C23—22no—
400C23—21no—
401C23—22no—
402C23—21no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—44no—
409C23—48no—
410C23—44no—
411C23—48no—
412C23—44no—
413C23#53212yes—
414C23C2×#13714yes—
415C23C23×D411yescharacteristic
416Q8—48no—
417Q8—48no—
418Q8—88no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—216no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—44no—
437C2×C4—48no—
438C2×C4—48no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—48no—
442C2×C4—44no—
443C2×C4—48no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4#53212yes—
460C2×C4#53412yes—
461C2×C4#53412yes—
462C2×C4C2×#13714yes—
463C2×C4C2×#13714yes—
464C2×C4#57514yes—
465C2×C4C22×#4112yes—
466C2×C4C2×H(5,𝔽2)11yescharacteristic
467C2×C4C2×H(5,𝔽2)11yescharacteristic
468C2×C4#58512yes—
469C8—24no—
470C8—24no—
471C8—44no—
472C22—24no—
473C22—24no—
474C22—24no—
475C22—24no—
476C22—22no—
477C22—416no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—41no—
482C22—42no—
483C22—41no—
484C22—42no—
485C22C2×#53211yescharacteristic
486C22C2×#53411yescharacteristic
487C22#344911yescharacteristic
488C22C22×#13712yes—
489C22C2×#57512yes—
490C4—24no—
491C4—22no—
492C4—41no—
493C4—42no—
494C4—41no—
495C4—42no—
496C4—84no—
497C4—84no—
498C4—84no—
499C4—84no—
500C4#319514yes—
501C4#319714yes—
502C4#350512yes—
503C4#350612yes—
504C2—42no—
505C2—44no—
506C2#3503211yescharacteristic
507C2#6070112yes—
508C2C2×#319511yescharacteristic
509C2C2×#319711yescharacteristic
510C2#6096612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33212C2semidirect
3. #33176C2semidirect
4. #33049C2semidirect
6. #29478C2semidirect
7. #27889C2semidirect
8. #27875C2semidirect
9. #25674C2semidirect
10. #25600C2semidirect
11. #22188C2semidirect
15. C2×#500C22semidirect
16. C2×#494C22semidirect
17. C2×#487C22semidirect
43. #1990C22semidirect
44. #1990C22semidirect
45. #1971C22semidirect
46. #1956C22semidirect
47. #1956C22semidirect
48. #1944C22semidirect
49. #1935C22semidirect
50. #1927C22semidirect
51. #1920C22semidirect
53. #1878C22semidirect
54. #1863C22semidirect
55. #1844C22semidirect
56. #1844C22semidirect
57. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#366252
#332122
#331762
#330492
#297432
#294782
#278892
#278752
#256742
#256002
#221882
C4×#4072