Finite Group Databasebeta

← #505202

Subgroups — #505202

Order 512.

808 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2243C211yesmaximal, characteristic, complemented
2C22×#357C211yesmaximal, characteristic
3#26065C211yesmaximal, characteristic, complemented
4#25781C211yesmaximal, characteristic
5#13480C211yesmaximal, characteristic, complemented
6#13408C211yesmaximal, characteristic, complemented
7#13408C211yesmaximal, characteristic, complemented
8#13363C211yesmaximal, characteristic, complemented
9#13205C211yesmaximal, characteristic, complemented
10#13149C211yesmaximal, characteristic, complemented
11#13149C211yesmaximal, characteristic, complemented
12#13060C211yesmaximal, characteristic, complemented
13#12993C211yesmaximal, characteristic, complemented
14#12731C211yesmaximal, characteristic
15#12611C211yesmaximal, characteristic, complemented
16C23×D8C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#500C2211yescharacteristic, complemented
19C2×#495C2211yescharacteristic
20C2×#495C2211yescharacteristic
21C2×#467C2211yescharacteristic
22C2×#450—21no—
23C22×#107C2211yescharacteristic
24#2440—21no—
25#2435—21no—
26#2243—24no—
27C2×#358C2211yescharacteristic, complemented
28C2×#358C2211yescharacteristic, complemented
29C2×#358C2211yescharacteristic, complemented
30C2×#358C2211yescharacteristic, complemented
31C2×#357—24no—
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×#346C2211yescharacteristic, complemented
42C2×#346C2211yescharacteristic, complemented
43#2083—21no—
44#2057—21no—
45#2012—21no—
46#2012—21no—
47#1970C2211yescharacteristic
48#1970C2211yescharacteristic
49#1942C2211yescharacteristic, complemented
50#1942C2211yescharacteristic, complemented
51#1852—21no—
52#1845C2211yescharacteristic, complemented
53#1808C2211yescharacteristic
54C8×C22⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic, complemented
56#1380C2211yescharacteristic, complemented
57#1380C2211yescharacteristic, complemented
58#1380C2211yescharacteristic, complemented
59#1374C2211yescharacteristic, complemented
60#1374C2211yescharacteristic, complemented
61C23×D4—21no—
62C22×Q16C2311yescharacteristic
63C22×SD16—21no—
64C22×SD16—21no—
65C22×D8—22no—
66C22×D8—22no—
67C22×D8—21no—
68C22×D8—22no—
69C22×D8—22no—
70C22×D8—41no—
71C22×D8—41no—
72C22×D8C2311yescharacteristic
73C22×D8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C2×C4⋊Q8—21no—
76C2×#124—21no—
77C2×#124—21no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—41no—
80C22×C4⋊C4—21no—
81C22×C16—22no—
82C22×C16—22no—
83C22×C16—41no—
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86C22×C16C2311yescharacteristic
87C22×C16C2311yescharacteristic
88#500—22no—
89#500—41no—
90#495—22no—
91#495—22no—
92#495—41no—
93#495—41no—
94#467—22no—
95#467—22no—
96#450—44no—
97C2×#107—22no—
98C2×#107—21no—
99C2×#107—21no—
100C2×#107—21no—
101C2×#107—41no—
102C2×#107C2311yescharacteristic
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×#103—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×C22⋊C8D411yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#398—41no—
114#395—41no—
115#394—41no—
116#394—41no—
117#390—21no—
118#390—21no—
119#389—21no—
120#389—21no—
121#385—41no—
122#384—21no—
123#380—21no—
124#379—41no—
125#379—41no—
126C4×C22⋊C4D411yescharacteristic
127#358—22no—
128#358—22no—
129#358—41no—
130#358—41no—
131#357—22no—
132#357—22no—
133#357—22no—
134#357—22no—
135#357—22no—
136#357—41no—
137#357—41no—
138#357D414yes—
139C22⋊C16—22no—
140#346—22no—
141#346—41no—
142#340—21no—
143#340—21no—
144#340—21no—
145#336D411yescharacteristic
146#336D411yescharacteristic
147C25—41no—
148C22×Q8—21no—
149C22×Q8—41no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4—41no—
156C22×D4—42no—
157C22×D4—81no—
158C22×D4—81no—
159C23×C4—41no—
160C23×C4C2×D411yescharacteristic
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×SD16—42no—
168C2×SD16—42no—
169C2×SD16—42no—
170C2×SD16—42no—
171C2×SD16—81no—
172C2×SD16—81no—
173C2×D8—28no—
174C2×D8—21no—
175C2×D8—21no—
176C2×D8—21no—
177C2×D8—21no—
178C2×D8—21no—
179C2×D8—21no—
180C2×D8—21no—
181C2×D8—21no—
182C2×D8—21no—
183C2×D8—21no—
184C2×D8—44no—
185C2×D8—44no—
186C2×D8—42no—
187C2×D8—42no—
188C2×D8—44no—
189C2×D8—42no—
190C2×D8C2×D412yes—
191C2×D8#4112yes—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8#4112yes—
200C22×C8#4112yes—
201C22×C8C2411yescharacteristic
202C4⋊Q8—42no—
203C4⋊Q8—81no—
204#124—42no—
205#124—42no—
206#124—81no—
207#124—81no—
208C22⋊Q8—42no—
209C22⋊Q8—42no—
210C22⋊Q8—84no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—81no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4D811yescharacteristic
236C2×C22⋊C4D811yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—21no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—44no—
251C2×C16C2×D412yes—
252#107—21no—
253#107—21no—
254#107—21no—
255#107—21no—
256#107—21no—
257#107—21no—
258#107—21no—
259#107—21no—
260#107—21no—
261#107—42no—
262#107—42no—
263#107—81no—
264#107C2×D412yes—
265#103—42no—
266#102—42no—
267#102—42no—
268C22⋊C8SD1614yes—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273#95—41no—
274#95—41no—
275#95—41no—
276#95D811yescharacteristic
277#95D811yescharacteristic
278C24—41no—
279C24—41no—
280C24—82no—
281C24—82no—
282C24—82no—
283C24—82no—
284C24—82no—
285C24—161no—
286C24—161no—
287C24C2×D811yescharacteristic
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—82no—
295C2×Q8—82no—
296C2×Q8—161no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—48no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—84no—
311C2×D4—84no—
312C2×D4—82no—
313C2×D4—82no—
314C2×D4—84no—
315C2×D4—82no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—161no—
329C22×C4#12412yes—
330C22×C4C2×D811yescharacteristic
331C22×C4C2×D811yescharacteristic
332C22×C4C2×D811yescharacteristic
333C22×C4C2×D811yescharacteristic
334C22×C4C2×D811yescharacteristic
335C22×C4C2×SD1612yes—
336C22×C4C22×D411yescharacteristic
337Q16—41no—
338Q16—41no—
339Q16—41no—
340Q16—41no—
341SD16—84no—
342SD16—84no—
343D8—22no—
344D8—22no—
345D8—22no—
346D8—22no—
347D8—48no—
348D8—41no—
349D8—41no—
350D8—41no—
351D8—41no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8#12412yes—
363C2×C8C2×SD1612yes—
364C2×C8C2×SD1612yes—
365C2×C8C22×D411yescharacteristic
366C2×C8C2×#4111yescharacteristic
367C2×C8C2×#4111yescharacteristic
368C2×C8C2×#4111yescharacteristic
369C2×C8H(5,𝔽2)11yescharacteristic
370C2×C8H(5,𝔽2)11yescharacteristic
371C4⋊C4—22no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—161no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4—161no—
397C22⋊C4—161no—
398C22⋊C4—161no—
399C22⋊C4—161no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C16—22no—
405C16—41no—
406C16—41no—
407C16—41no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—42no—
413C23—42no—
414C23—88no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23—81no—
427C23—164no—
428C23—164no—
429C23—164no—
430C23—164no—
431C23—164no—
432C23—164no—
433C23#48612yes—
434C23#50912yes—
435C23#51012yes—
436C23C22×D811yescharacteristic
437Q8—81no—
438Q8—81no—
439Q8—81no—
440Q8—81no—
441Q8—164no—
442D4—42no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—88no—
447D4—81no—
448D4—81no—
449D4—81no—
450D4—81no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—42no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—164no—
488C2×C4#48612yes—
489C2×C4C2×#12411yescharacteristic
490C2×C4D4211yescharacteristic
491C2×C4#54611yescharacteristic
492C2×C4C22×SD1611yescharacteristic
493C2×C4#57611yescharacteristic
494C2×C4#57711yescharacteristic
495C8—22no—
496C8—22no—
497C8—22no—
498C8#52511yescharacteristic
499C8#53811yescharacteristic
500C8#54611yescharacteristic
501C8#55111yescharacteristic
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—1616no—
515C22—161no—
516C22—161no—
517C22—161no—
518C22—161no—
519C22—161no—
520C22—161no—
521C22—161no—
522C22—161no—
523C22#224314yescomplemented
524C22C2×#48611yescharacteristic
525C22D4×D811yescharacteristic
526C22#329411yescharacteristic
527C22C2×#50911yescharacteristic
528C22C2×#51011yescharacteristic
529C22#341511yescharacteristic
530C22#341611yescharacteristic
531C4—22no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—82no—
537C4—161no—
538C4—161no—
539C4—161no—
540C4—161no—
541C4—161no—
542C4#247211yescharacteristic
543C4#313611yescharacteristic
544C4D4×SD1611yescharacteristic
545C4#329511yescharacteristic
546C2—24no—
547C2—162no—
548C2—162no—
549C2#2725411yescharacteristic
550C2C2×#224311yescharacteristic
551C2#3370211yescharacteristic
552C2#3383011yescharacteristic
553C2#3383211yescharacteristic
554C2#3387311yescharacteristic
555C2#3388111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC23×D8C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2243C2semidirect
3. #26065C2semidirect
5. #13480C2semidirect
6. #13408C2semidirect
7. #13408C2semidirect
8. #13363C2semidirect
9. #13205C2semidirect
10. #13149C2semidirect
11. #13149C2semidirect
12. #13060C2semidirect
13. #12993C2semidirect
15. #12611C2semidirect
18. C2×#500C22semidirect
27. C2×#358C22semidirect
28. C2×#358C22semidirect
29. C2×#358C22semidirect
30. C2×#358C22semidirect
41. C2×#346C22semidirect
42. C2×#346C22semidirect
49. #1942C22semidirect
50. #1942C22semidirect
52. #1845C22semidirect
55. #1380C22semidirect
56. #1380C22semidirect
57. #1380C22semidirect
58. #1380C22semidirect
59. #1374C22semidirect
60. #1374C22semidirect
523. C22#2243semidirect

Maximal subgroups

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

subgroupindex
C2×#22432
C22×#3572
#260652
#257812
#134802
#134082
#133632
#132052
#131492
#130602
#129932
#127312
#126112