Finite Group Databasebeta

← #6568261

Subgroups — #6568261

Order 512.

1595 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic, complemented
2#34316C211yesmaximal, characteristic
3#30169C212yesmaximal, complemented
4#29864C211yesmaximal, characteristic, complemented
5#29255C211yesmaximal, characteristic, complemented
6#29113C212yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#27587C212yesmaximal, complemented
9#27546C214yesmaximal, complemented
10#27118C214yesmaximal, complemented
11#26303C212yesmaximal, complemented
12#25882C212yesmaximal
13#23774C211yesmaximal, characteristic, complemented
14#23763C212yesmaximal, complemented
15#23222C212yesmaximal, complemented
16#23066C212yesmaximal, complemented
17#22839C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#547C2211yescharacteristic
20C2×#487C2212yes—
21C2×#474C2212yes—
22#2868C2211yescharacteristic
23#2857C2212yes—
24#2568—21no—
25#2547—21no—
26#2547C2212yes—
27#2499—22no—
28#2496—22no—
29#2496C2214yes—
30#2374—22no—
31#2365—21no—
32#2365C2212yes—
33#2365C2211yescharacteristic
34#2363—21no—
35#2362C2212yes—
36#2354C2211yescharacteristic
37#2354C2211yescharacteristic
38#2345—21no—
39#2341C2212yes—
40#2335—22no—
41#2324—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C22⋊Q8C2212yes—
45#2297C2212yes—
46#2293C2211yescharacteristic
47#2285C2212yes—
48#2274—21no—
49C42×D4C2211yescharacteristic
50#2099C2211yescharacteristic
51#2096C2212yes—
52#2093C2211yescharacteristic
53#2086C2212yes—
54#2083C2212yes—
55#2083C2212yes—
56#2077C2214yes—
57#2076C2214yes—
58#2072C2212yes—
59#2072C2212yes—
60#2071C2212yes—
61#2071C2212yes—
62#2067C2212yes—
63#2057C2211yescharacteristic
64#2036C2214yes—
65#2036C2214yes—
66#2024C2212yes—
67#2023C2211yescharacteristic
68#1990C2212yes—
69#1988C2212yes—
70#1981C2212yes—
71#1954C2212yes—
72#1948C2212yes—
73#1936C2214yes—
74#1935C2214yes—
75#1928C2214yes—
76#1926C2214yes—
77#1924C2214yes—
78#1921C2214yes—
79#1865C2214yes—
80#1863C2214yes—
81#1854C2214yes—
82#1831C2211yescharacteristic
83#1825C2214yes—
84#1817C2212yes—
85#1803C2214yes—
86#1788C2214yes—
87#1788C2214yes—
88#1776C2214yes—
89C4×#107C2212yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#102C2212yes—
94#1749C2212yes—
95C22×#41C2312yes—
96#556—48no—
97#547—24no—
98#547—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#126—24no—
105C2×#124—22no—
106C2×#123—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8C2312yes—
109C2×#121—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×#117—22no—
119C2×#117—21no—
120C2×#117—21no—
121C2×#117C2312yes—
122C22×C4⋊C4—21no—
123C22×C42C2312yes—
124#487—28no—
125#474—28no—
126C2×#107C2314yes—
127C2×#107C2312yes—
128C2×#107C2312yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141C2×#97C2314yes—
142C2×C4×C8C2312yes—
143#400—24no—
144#397—24no—
145#397—24no—
146#391—22no—
147#391—22no—
148#391—21no—
149#391—22no—
150#391C2312yes—
151#391C2312yes—
152#390—21no—
153#390—22no—
154#390—22no—
155#390C2311yescharacteristic
156#390C2311yescharacteristic
157#390C2312yes—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389C2312yes—
164#389C2311yescharacteristic
165#389C2311yescharacteristic
166#389C2314yes—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—21no—
171#388—21no—
172#388—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—24no—
177#387—24no—
178#386—22no—
179#386—22no—
180#385—22no—
181#384—21no—
182#384—21no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2312yes—
187#382—22no—
188#380—22no—
189#380—22no—
190#380—21no—
191#380—22no—
192#380C2312yes—
193#380C2314yes—
194#380C2314yes—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218C43C2311yescharacteristic
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2314yes—
233C2×#41—24no—
234C2×#41—24no—
235C2×#41—44no—
236C2×#41—44no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2412yes—
240C22×D4—22no—
241C22×D4C2×D411yescharacteristic
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—22no—
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2412yes—
248C22×C8C2412yes—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C4⋊Q8—22no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—42no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257C4⋊Q8#4114yes—
258#126—416no—
259#124—48no—
260#123—216no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—48no—
264#121—28no—
265C4×Q8—44no—
266C4×Q8—44no—
267C4×Q8—48no—
268C4×Q8—44no—
269C4×D4—24no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—42no—
274C4×D4#4114yes—
275#117—44no—
276#117—48no—
277#117—44no—
278#117—44no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C22⋊C4—22no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—21no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—44no—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—28no—
343#107—42no—
344#107—42no—
345#106—44no—
346C4⋊C8—24no—
347C4⋊C8—42no—
348#103—28no—
349#103—28no—
350#103—44no—
351#103—44no—
352#102—28no—
353#102—42no—
354#102—42no—
355C22⋊C8—44no—
356C22⋊C8—44no—
357#97—44no—
358C4×C8—24no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—21no—
366#95—24no—
367#95—22no—
368#95—24no—
369#95—22no—
370#95—21no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24—21no—
378C24C22×D412yes—
379#41—88no—
380#41—88no—
381C2×Q8—22no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8C2×#4114yes—
391C2×D4—28no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—82no—
399C2×D4—82no—
400C2×D4C2×#4112yes—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C2511yescharacteristic
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C4⋊C4—28no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—48no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4C2×#4112yes—
477C22⋊C4—28no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—42no—
482C22⋊C4—44no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—81no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C42—22no—
493C42—22no—
494C42—24no—
495C42—22no—
496C42—22no—
497C42—42no—
498C42—42no—
499C42—41no—
500C42—44no—
501C42—48no—
502C42—42no—
503C42—44no—
504C42—82no—
505C42—82no—
506C42—82no—
507C42AGL(1,ℤ/8)14yes—
508C42C2×#4112yes—
509C42C2×#4112yes—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23C23×D411yescharacteristic
525Q8—28no—
526Q8—44no—
527Q8—84no—
528Q8—88no—
529D4—24no—
530D4—44no—
531D4—42no—
532D4—88no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—48no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—84no—
582C2×C4—88no—
583C2×C4#52512yes—
584C2×C4#52512yes—
585C2×C4C2×AGL(1,ℤ/8)12yes—
586C2×C4C2×AGL(1,ℤ/8)12yes—
587C2×C4C2×AGL(1,ℤ/8)12yes—
588C2×C4#57512yes—
589C2×C4#57812yes—
590C2×C4C22×#4111yescharacteristic
591C2×C4C22×#4111yescharacteristic
592C2×C4#58512yes—
593C2×C4#58512yes—
594C8—44no—
595C8—42no—
596C8—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22C2×#52511yescharacteristic
612C22#348511yescharacteristic
613C22#350011yescharacteristic
614C22C22×AGL(1,ℤ/8)11yescharacteristic
615C22C2×#57511yescharacteristic
616C22C2×#57811yescharacteristic
617C22#358611yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—41no—
626C4—42no—
627C4—44no—
628C4—81no—
629C4—82no—
630C4—81no—
631C4—82no—
632C4—84no—
633C4—82no—
634C4#311514yes—
635C4#351012yes—
636C4#352112yes—
637C2—82no—
638C2—82no—
639C2#3603311yescharacteristic
640C2C2×#311511yescharacteristic
641C2#6059411yescharacteristic
642C2#6171711yescharacteristic
643C2#6173011yescharacteristic
644C2#6253811yescharacteristic
645C2#6258711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36188C2semidirect
3. #30169C2semidirect
4. #29864C2semidirect
5. #29255C2semidirect
6. #29113C2semidirect
7. #28717C2semidirect
8. #27587C2semidirect
9. #27546C2semidirect
10. #27118C2semidirect
11. #26303C2semidirect
13. #23774C2semidirect
14. #23763C2semidirect
15. #23222C2semidirect
16. #23066C2semidirect
17. #22839C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#343162
#301692
#298642
#292552
#291132
#287172
#275872
#275462
#271182
#263032
#258822
#237742
#237632
#232222
#230662
#228392