Finite Group Databasebeta

← #6568264

Subgroups — #6568264

Order 512.

1651 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic, complemented
2#34330C211yesmaximal, characteristic
3#30166C212yesmaximal, complemented
4#29866C211yesmaximal, characteristic, complemented
5#29254C211yesmaximal, characteristic, complemented
6#29113C212yesmaximal, complemented
7#28716C211yesmaximal, characteristic, complemented
8#27587C212yesmaximal, complemented
9#27542C214yesmaximal, complemented
10#27122C214yesmaximal, complemented
11#26301C212yesmaximal, complemented
12#25887C212yesmaximal
13#23776C211yesmaximal, characteristic, complemented
14#23761C212yesmaximal, complemented
15#23220C212yesmaximal, complemented
16#23068C212yesmaximal, complemented
17#22839C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#550C2211yescharacteristic
20C2×#486C2212yes—
21C2×#475C2212yes—
22#2868C2211yescharacteristic
23#2857C2212yes—
24#2568—21no—
25#2547—21no—
26#2547C2212yes—
27#2499—22no—
28#2496—22no—
29#2496C2214yes—
30#2366—21no—
31#2366C2211yescharacteristic
32#2365—21no—
33#2365C2212yes—
34#2364—22no—
35#2363C2212yes—
36#2357—21no—
37#2355C2211yescharacteristic
38#2354C2211yescharacteristic
39#2339C2212yes—
40#2333—22no—
41#2328—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×#127C2211yescharacteristic
44C4×#121C2212yes—
45#2297C2212yes—
46#2295C2211yescharacteristic
47#2285C2212yes—
48#2275—21no—
49C42×D4C2211yescharacteristic
50#2099C2211yescharacteristic
51#2096C2211yescharacteristic
52#2093C2212yes—
53#2088C2212yes—
54#2084C2212yes—
55#2084C2212yes—
56#2077C2214yes—
57#2076C2214yes—
58#2072C2212yes—
59#2072C2212yes—
60#2071C2212yes—
61#2071C2212yes—
62#2066C2212yes—
63#2058C2211yescharacteristic
64#2034C2214yes—
65#2034C2214yes—
66#2024C2211yescharacteristic
67#2023C2212yes—
68#1990C2212yes—
69#1988C2212yes—
70#1982C2212yes—
71#1954C2212yes—
72#1947C2212yes—
73#1936C2214yes—
74#1935C2214yes—
75#1928C2214yes—
76#1926C2214yes—
77#1924C2214yes—
78#1920C2214yes—
79#1864C2214yes—
80#1862C2214yes—
81#1854C2214yes—
82#1831C2211yescharacteristic
83#1825C2214yes—
84#1817C2212yes—
85#1802C2214yes—
86#1788C2214yes—
87#1788C2214yes—
88#1776C2214yes—
89C4×#106C2212yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1762C2214yes—
93C4×#103C2212yes—
94#1749C2212yes—
95C22×#41C2312yes—
96#556—48no—
97#550—24no—
98#550—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#127—21no—
104C2×#127C2311yescharacteristic
105C2×#126—24no—
106C2×#124—22no—
107C2×C22⋊Q8—22no—
108C2×#121—22no—
109C2×#121C2312yes—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4—22no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C2×#117C2312yes—
125C22×C42C2312yes—
126#486—28no—
127#475—28no—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×#106C2312yes—
131C2×#106C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143C2×#97C2314yes—
144C2×C4×C8C2312yes—
145#400—24no—
146#397—24no—
147#397—24no—
148#391—22no—
149#391—22no—
150#391—21no—
151#391—22no—
152#391C2311yescharacteristic
153#391C2311yescharacteristic
154#391C2312yes—
155#390—21no—
156#390—21no—
157#390—22no—
158#390—21no—
159#390—21no—
160#390C2312yes—
161#390C2312yes—
162#389—21no—
163#389—22no—
164#389C2312yes—
165#389C2311yescharacteristic
166#389C2311yescharacteristic
167#389C2314yes—
168#388—22no—
169#388—22no—
170#388—22no—
171#387—21no—
172#387—21no—
173#387—22no—
174#387—24no—
175#387—24no—
176#386—22no—
177#386—22no—
178#386—22no—
179#386—22no—
180#386—22no—
181#386—22no—
182#386—22no—
183#384—21no—
184#384—21no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#382—21no—
190#382—22no—
191#381—22no—
192#380—22no—
193#380—22no—
194#380C2312yes—
195#380C2314yes—
196#380C2314yes—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216#376—21no—
217C2×#95—22no—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2312yes—
232#336C2314yes—
233C2×#41—24no—
234C2×#41—24no—
235C2×#41—44no—
236C2×#41—44no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2411yescharacteristic
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2×D411yescharacteristic
245C22×D4C2412yes—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2412yes—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—42no—
258C4⋊Q8—44no—
259C4⋊Q8—44no—
260#127—28no—
261#127—22no—
262#127#4114yes—
263#126—416no—
264#124—48no—
265C22⋊Q8—48no—
266#121—216no—
267#121—24no—
268#121—24no—
269C4×Q8—22no—
270C4×Q8—48no—
271C4×Q8—44no—
272C4×Q8#4114yes—
273C4×D4—28no—
274C4×D4—24no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—42no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—82no—
283C4×D4—84no—
284#117—48no—
285#117—44no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—21no—
326C2×C42—24no—
327C2×C42—21no—
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—44no—
343#106—28no—
344#106—42no—
345#106—42no—
346C4⋊C8—24no—
347C4⋊C8—42no—
348#103—28no—
349#103—42no—
350#103—42no—
351#102—28no—
352#102—28no—
353#102—44no—
354#102—44no—
355C22⋊C8—44no—
356C22⋊C8—44no—
357#97—44no—
358C4×C8—24no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—22no—
367#95—24no—
368#95—21no—
369#95—22no—
370#95—24no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—22no—
380C24C22×D412yes—
381#41—88no—
382#41—88no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8C2×#4112yes—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—81no—
406C2×D4—81no—
407C2×D4—82no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4—82no—
411C2×D4C2×#4114yes—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—28no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2511yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—48no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4C2×#4112yes—
487C22⋊C4—28no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C42—24no—
502C42—22no—
503C42—22no—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—42no—
508C42—42no—
509C42—41no—
510C42—48no—
511C42—44no—
512C42—42no—
513C42—82no—
514C42—82no—
515C42AGL(1,ℤ/8)14yes—
516C42C2×#4112yes—
517C42C2×#4112yes—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23C23×D411yescharacteristic
533Q8—24no—
534Q8—42no—
535Q8—84no—
536Q8—88no—
537D4—28no—
538D4—44no—
539D4—48no—
540D4—44no—
541D4—84no—
542D4—88no—
543D4—88no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—48no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—84no—
592C2×C4—88no—
593C2×C4—164no—
594C2×C4—164no—
595C2×C4#52512yes—
596C2×C4#52512yes—
597C2×C4C2×AGL(1,ℤ/8)12yes—
598C2×C4C2×AGL(1,ℤ/8)12yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4#57512yes—
601C2×C4#57712yes—
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4#58512yes—
605C2×C4#58512yes—
606C8—44no—
607C8—42no—
608C8—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—44no—
617C22—42no—
618C22—44no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22C2×#52511yescharacteristic
624C22#348511yescharacteristic
625C22#350011yescharacteristic
626C22C22×AGL(1,ℤ/8)11yescharacteristic
627C22C2×#57511yescharacteristic
628C22C2×#57711yescharacteristic
629C22#358611yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—42no—
635C4—42no—
636C4—41no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—82no—
641C4—81no—
642C4—82no—
643C4—82no—
644C4—84no—
645C4—81no—
646C4#311514yes—
647C4#351012yes—
648C4#352112yes—
649C2—82no—
650C2—84no—
651C2#3603311yescharacteristic
652C2C2×#311511yescharacteristic
653C2#6059111yescharacteristic
654C2#6171711yescharacteristic
655C2#6173911yescharacteristic
656C2#6254111yescharacteristic
657C2#6258711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
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. #30166C2semidirect
4. #29866C2semidirect
5. #29254C2semidirect
6. #29113C2semidirect
7. #28716C2semidirect
8. #27587C2semidirect
9. #27542C2semidirect
10. #27122C2semidirect
11. #26301C2semidirect
13. #23776C2semidirect
14. #23761C2semidirect
15. #23220C2semidirect
16. #23068C2semidirect
17. #22839C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#343302
#301662
#298662
#292542
#291132
#287162
#275872
#275422
#271222
#263012
#258872
#237762
#237612
#232202
#230682
#228392