Finite Group Databasebeta

← #275901

Subgroups — #275901

Order 512.

637 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33201C211yesmaximal, characteristic
2#28656C211yesmaximal, characteristic, complemented
3#22795C211yesmaximal, characteristic
4#21614C211yesmaximal, characteristic
5#10485C211yesmaximal, characteristic
6#10481C211yesmaximal, characteristic
7#10278C211yesmaximal, characteristic
8#10275C211yesmaximal, characteristic
9#9737C211yesmaximal, characteristic, complemented
10#9730C211yesmaximal, characteristic, complemented
11#9701C211yesmaximal, characteristic, complemented
12#9604C211yesmaximal, characteristic, complemented
13#9568C211yesmaximal, characteristic, complemented
14#9561C211yesmaximal, characteristic, complemented
15#8587C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#487C2211yescharacteristic
18C2×#476C2211yescharacteristic
19C2×#476C2211yescharacteristic
20C2×#474C2211yescharacteristic
21C2×#431—21no—
22#2862C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24C42×D4C2211yescharacteristic
25#2068—21no—
26#2068—21no—
27#1990C2211yescharacteristic
28#1988C2211yescharacteristic
29#1987C2211yescharacteristic
30#1987C2211yescharacteristic
31#1956C2211yescharacteristic
32#1954C2211yescharacteristic
33#1954C2211yescharacteristic
34#1949—21no—
35#1949—21no—
36#1947C2211yescharacteristic
37#1928—21no—
38#1928—21no—
39#1923—21no—
40#1923—21no—
41#1919—21no—
42#1862—21no—
43#1842C2211yescharacteristic
44#1842C2211yescharacteristic
45#1804—21no—
46#1804—21no—
47#1802C2211yescharacteristic
48#1797—21no—
49#1776—21no—
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1762—21no—
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C22×C42—21no—
75#500—22no—
76#500—41no—
77#487—22no—
78#487—41no—
79#476—22no—
80#476—22no—
81#476—41no—
82#476—41no—
83#474—41no—
84#474—41no—
85#431—44no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8D411yescharacteristic
97C2×C4⋊C8D411yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×#97—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#391—21no—
122#391—21no—
123#391—21no—
124#391—21no—
125#391—21no—
126#389—21no—
127#389—21no—
128#384C2311yescharacteristic
129#382—21no—
130#382—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C43C2311yescharacteristic
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336D411yescharacteristic
154#336D411yescharacteristic
155#335—41no—
156#335—41no—
157#333—22no—
158#333—41no—
159#333—41no—
160#333—41no—
161#332—22no—
162#332—41no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×D4C2×D411yescharacteristic
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8#4112yes—
192C4×Q8—82no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—41no—
200C4×D4Q1614yes—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42—41no—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42C2411yescharacteristic
230#107—42no—
231#107—42no—
232#107—42no—
233#107—42no—
234#106—24no—
235#106—42no—
236C4⋊C8—22no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—41no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8—42no—
252C4⋊C8—81no—
253C4⋊C8#4112yes—
254#103—42no—
255#103—42no—
256#103—42no—
257#103—42no—
258#102—24no—
259#102—42no—
260#102—42no—
261#102—42no—
262C22⋊C8—42no—
263C22⋊C8—42no—
264C22⋊C8—81no—
265C22⋊C8—81no—
266#97—42no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—42no—
272C4×C8—42no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—41no—
277#95—41no—
278#95—41no—
279#95—41no—
280#95—41no—
281#95—41no—
282C24—21no—
283C2×Q8—22no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—81no—
294C2×Q8—81no—
295C2×Q8—81no—
296C2×Q8—81no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4C2×Q1612yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—24no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—81no—
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12111yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8AGL(1,ℤ/8)12yes—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—81no—
380C4⋊C4—82no—
381C4⋊C4C2×Q1612yes—
382C4⋊C4#13712yes—
383C22⋊C4—24no—
384C22⋊C4—42no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—41no—
394C42—41no—
395C42—42no—
396C42—42no—
397C42—41no—
398C42—41no—
399C42—82no—
400C42C2×Q1612yes—
401C42#13512yes—
402C42C2×#4111yescharacteristic
403C42C2×#4111yescharacteristic
404C42H(5,𝔽2)11yescharacteristic
405C42H(5,𝔽2)11yescharacteristic
406C23—22no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23C2×#12111yescharacteristic
414Q8—82no—
415Q8—82no—
416Q8—81no—
417Q8—81no—
418Q8—81no—
419Q8—81no—
420Q8—161no—
421Q8—161no—
422D4—22no—
423D4—22no—
424D4—41no—
425D4—41no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—48no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—84no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4#46712yes—
466C2×C4#46712yes—
467C2×C4#52511yescharacteristic
468C2×C4#53511yescharacteristic
469C2×C4#54611yescharacteristic
470C2×C4#55011yescharacteristic
471C2×C4#55311yescharacteristic
472C2×C4#55511yescharacteristic
473C2×C4C22×Q1611yescharacteristic
474C2×C4C2×#13511yescharacteristic
475C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
476C2×C4C2×#13711yescharacteristic
477C2×C4#57511yescharacteristic
478C2×C4#57511yescharacteristic
479C2×C4#57711yescharacteristic
480C2×C4#57811yescharacteristic
481C8—22no—
482C8—42no—
483C8—41no—
484C8—41no—
485C8—41no—
486C8—81no—
487C8—81no—
488C8—81no—
489C8—81no—
490C8—81no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22#241411yescharacteristic
499C22#303611yescharacteristic
500C22#304611yescharacteristic
501C22C2×#46711yescharacteristic
502C22#305611yescharacteristic
503C22#306711yescharacteristic
504C22#306811yescharacteristic
505C4—22no—
506C4—21no—
507C4—21no—
508C4—22no—
509C4—41no—
510C4—42no—
511C4—41no—
512C4—42no—
513C4—81no—
514C4—82no—
515C4—161no—
516C4—161no—
517C4—161no—
518C4#310411yescharacteristic
519C4#310711yescharacteristic
520C4#320911yescharacteristic
521C4#321211yescharacteristic
522C4#330811yescharacteristic
523C4#332111yescharacteristic
524C4#339711yescharacteristic
525C4#340211yescharacteristic
526C2—82no—
527C2#1925611yescharacteristic
528C2#1930511yescharacteristic
529C2#1941711yescharacteristic
530C2#1949911yescharacteristic
531C2#2464011yescharacteristic
532C2#2465211yescharacteristic
533C2#2467411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #28656C2semidirect
9. #9737C2semidirect
10. #9730C2semidirect
11. #9701C2semidirect
12. #9604C2semidirect
13. #9568C2semidirect
14. #9561C2semidirect
15. #8587C2semidirect

Maximal subgroups

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

subgroupindex
#332012
#286562
#227952
#216142
#104852
#104812
#102782
#102752
#97372
#97302
#97012
#96042
#95682
#95612
#85872