Finite Group Databasebeta

← #6796039

Subgroups — #6796039

Order 512.

1527 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#32422C214yesmaximal, complemented
5#31626C214yesmaximal, complemented
6#29937C212yesmaximal, complemented
7#29286C211yesmaximal, characteristic, complemented
8#28713C212yesmaximal, complemented
9#28687C211yesmaximal, characteristic, complemented
10#27627C212yesmaximal, complemented
11#26512C212yesmaximal, complemented
12#25451C212yesmaximal, complemented
13#25346C212yesmaximal
14#24844C212yesmaximal, complemented
15#22650C211yesmaximal, characteristic, complemented
16#21964C212yesmaximal, complemented
17C4×#474C211yesmaximal, characteristic, complemented
18C2×#529—21no—
19C2×#498C2212yes—
20C2×#476C2212yes—
21C2×#474C2211yescharacteristic
22C2×#436C2212yes—
23C2×#423C2212yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2571—21no—
27#2563C2212yes—
28#2553—22no—
29#2552C2211yescharacteristic
30#2547C2211yescharacteristic
31#2542C2211yescharacteristic
32#2538—21no—
33#2537C2211yescharacteristic
34#2497C2214yes—
35#2496C2212yes—
36#2492—22no—
37#2489—21no—
38#2439—22no—
39#2427—21no—
40#2422C2212yes—
41#2395—22no—
42#2369C2212yes—
43#2355—21no—
44#2354—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C22⋊Q8—21no—
48C4×#121C2212yes—
49#2285C2212yes—
50#2285C2212yes—
51C42×D4C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2100C2214yes—
54#2087C2214yes—
55#2086C2214yes—
56#2084C2214yes—
57#2084C2214yes—
58#2077C2214yes—
59#2077C2214yes—
60#2077C2214yes—
61#2065C2214yes—
62#2033C2214yes—
63#1990C2212yes—
64#1988C2212yes—
65#1988C2212yes—
66#1988C2211yescharacteristic
67#1987C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1948C2214yes—
71#1948C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1947C2211yescharacteristic
76#1947C2214yes—
77#1939C2212yes—
78#1926C2214yes—
79#1921C2214yes—
80#1918C2212yes—
81#1895C2214yes—
82#1862C2212yes—
83#1851C2214yes—
84#1842C2211yescharacteristic
85#1809C2214yes—
86#1809C2214yes—
87#1809C2214yes—
88#1802C2212yes—
89#1802C2211yescharacteristic
90#1802C2212yes—
91#1797C2212yes—
92C4×#107C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×M16C2314yes—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101#529—48no—
102C2×#126—22no—
103C2×#125—21no—
104C2×#124—21no—
105C2×#123—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×#121—21no—
110C2×#121C2312yes—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×D4—22no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C2×#117—22no—
122C22×C42—21no—
123C22×C42C2312yes—
124#498—44no—
125#476—44no—
126#474—24no—
127#436—44no—
128C2×#107C2312yes—
129C2×#107C2312yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134#423—28no—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C4×C8C2312yes—
146#400—24no—
147#400—22no—
148#399—22no—
149#398—22no—
150#397—22no—
151#397—22no—
152#397—22no—
153#396—24no—
154#394—22no—
155#394—22no—
156#393—24no—
157#391—21no—
158#391—21no—
159#391—21no—
160#391—22no—
161#391—24no—
162#390—22no—
163#390C2312yes—
164#389—22no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389C2312yes—
170#389C2312yes—
171#389C2314yes—
172#389C2312yes—
173#388—24no—
174#388—22no—
175#387—24no—
176#387—22no—
177#386—22no—
178#385—22no—
179#384—21no—
180#384—21no—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2312yes—
184#384C2312yes—
185#384C2312yes—
186#384C2312yes—
187#382—22no—
188#382—22no—
189#382—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—21no—
194#380C2314yes—
195#380C2314yes—
196#380C2312yes—
197#380C2312yes—
198#380C2312yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220C43C2311yescharacteristic
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×D4—22no—
238C22×D4C2×D411yescharacteristic
239C22×D4C2411yescharacteristic
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2412yes—
246C2×M16—28no—
247C2×M16—44no—
248C2×M16—44no—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—22no—
253C4⋊Q8—24no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—82no—
259C4⋊Q8—82no—
260#126—48no—
261#125—44no—
262#124—44no—
263#123—48no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C22⋊Q8—84no—
267#121—24no—
268#121—42no—
269#121—44no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×D4—24no—
273C4×D4—22no—
274C4×D4—24no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—41no—
282C4×D4—48no—
283#117—48no—
284#117—84no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
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—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—24no—
337C2×C42—44no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—24no—
349#107—42no—
350#107—44no—
351#106—44no—
352#106—44no—
353#106—44no—
354C4⋊C8—28no—
355C4⋊C8—24no—
356C4⋊C8—24no—
357C4⋊C8—22no—
358C4⋊C8—42no—
359C4⋊C8#4114yes—
360#102—24no—
361#102—24no—
362#102—44no—
363C22⋊C8—44no—
364C4×C8—24no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—22no—
379#95—24no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—21no—
384C24C22×D412yes—
385C2×Q8—42no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—82no—
393C2×Q8—82no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—82no—
405C2×D42−1+412yes—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—81no—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2511yescharacteristic
445M16—88no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8C2×#4114yes—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—24no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—81no—
488C4⋊C4—82no—
489C4⋊C4—84no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4#13714yes—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C42—24no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—21no—
514C42—21no—
515C42—41no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—44no—
520C42—44no—
521C42—84no—
522C42AGL(1,ℤ/8)14yes—
523C42C2×#4112yes—
524C42H(5,𝔽2)12yes—
525C422−1+412yes—
526C422−1+412yes—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C23×D411yescharacteristic
542Q8—84no—
543Q8—84no—
544Q8—164no—
545D4—22no—
546D4—22no—
547D4—44no—
548D4—41no—
549D4—41no—
550D4—88no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4#53712yes—
602C2×C4#53712yes—
603C2×C4#54712yes—
604C2×C4#54712yes—
605C2×C4C2×AGL(1,ℤ/8)12yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4C2×AGL(1,ℤ/8)12yes—
608C2×C4C2×#13714yes—
609C2×C4C2×#13712yes—
610C2×C4#57712yes—
611C2×C4C22×#4111yescharacteristic
612C2×C4C2×H(5,𝔽2)11yescharacteristic
613C2×C4C2×2−1+411yescharacteristic
614C2×C4C2×2−1+411yescharacteristic
615C2×C4C2×2−1+411yescharacteristic
616C2×C4#58511yescharacteristic
617C8—24no—
618C8—44no—
619C8—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22C2×#53711yescharacteristic
635C22C2×#54711yescharacteristic
636C22#348911yescharacteristic
637C22C22×AGL(1,ℤ/8)11yescharacteristic
638C22C22×#13711yescharacteristic
639C22C2×#57711yescharacteristic
640C22#358511yescharacteristic
641C4—22no—
642C4—24no—
643C4—22no—
644C4—22no—
645C4—42no—
646C4—42no—
647C4—41no—
648C4—81no—
649C4—82no—
650C4—82no—
651C4—81no—
652C4—82no—
653C4—81no—
654C4—82no—
655C4—84no—
656C4#325414yes—
657C4#335414yes—
658C4#351411yescharacteristic
659C4#352311yescharacteristic
660C4#355211yescharacteristic
661C4#355511yescharacteristic
662C2—82no—
663C2—82no—
664C2#3892311yescharacteristic
665C2C2×#325411yescharacteristic
666C2#6092411yescharacteristic
667C2C2×#335411yescharacteristic
668C2#6125411yescharacteristic
669C2#6211011yescharacteristic
670C2#6214711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
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. #36194C2semidirect
3. #33197C2semidirect
4. #32422C2semidirect
5. #31626C2semidirect
6. #29937C2semidirect
7. #29286C2semidirect
8. #28713C2semidirect
9. #28687C2semidirect
10. #27627C2semidirect
11. #26512C2semidirect
12. #25451C2semidirect
14. #24844C2semidirect
15. #22650C2semidirect
16. #21964C2semidirect
17. C4×#474C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#349322
#331972
#324222
#316262
#299372
#292862
#287132
#286872
#276272
#265122
#254512
#253462
#248442
#226502
#219642
C4×#4742