Finite Group Databasebeta

← #6458357

Subgroups — #6458357

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#35167C211yesmaximal, characteristic, complemented
3#27830C214yesmaximal, complemented
4#27828C214yesmaximal, complemented
5#27824C212yesmaximal, complemented
6#27821C212yesmaximal, complemented
7#27815C212yesmaximal, complemented
8#27814C212yesmaximal, complemented
9#23749C212yesmaximal, complemented
10#23458C212yesmaximal
11#21326C212yesmaximal, complemented
12#21324C211yesmaximal, characteristic, complemented
13C2×#1987C211yesmaximal, characteristic, complemented
14#21092C212yesmaximal
15#20645C211yesmaximal, characteristic
16#20441C211yesmaximal, characteristic, complemented
17C4×#417C211yesmaximal, characteristic
18C2×#544C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#417C2211yescharacteristic
23C2×#417C2211yescharacteristic
24C2×#416C2212yes—
25C2×#405C2212yes—
26C2×C4×M16C2211yescharacteristic
27#2857—21no—
28#2857—21no—
29#2857—22no—
30#2659—22no—
31#2628—22no—
32#2568C2212yes—
33#2567C2212yes—
34#2563C2212yes—
35#2552—21no—
36#2552C2212yes—
37#2551—21no—
38#2551C2212yes—
39#2550—22no—
40#2550C2214yes—
41#2542C2212yes—
42#2487—22no—
43#2409—22no—
44#2317C2212yes—
45#2316C2212yes—
46#2311C2212yes—
47#2285C2211yescharacteristic
48#2285C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2100C2214yes—
53#2099C2214yes—
54#2026C2214yes—
55#2025C2214yes—
56#2024C2214yes—
57#2023C2214yes—
58#1993C2214yes—
59#1993C2214yes—
60#1992C2214yes—
61#1991C2214yes—
62#1990C2212yes—
63#1990C2212yes—
64#1990C2214yes—
65#1987C2212yes—
66#1987C2212yes—
67#1987C2214yes—
68#1936C2214yes—
69#1935C2214yes—
70#1934C2214yes—
71#1933C2214yes—
72#1805C2212yes—
73#1804C2212yes—
74#1803C2211yescharacteristic
75#1803C2211yescharacteristic
76#1802C2211yescharacteristic
77#1802C2211yescharacteristic
78#1771C2214yes—
79#1770C2212yes—
80C8×C4⋊C4C2212yes—
81#1763C2212yes—
82#1762C2212yes—
83C4×#103C2212yes—
84C4×#102C2212yes—
85#1736C2212yes—
86#1736C2212yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1735C2212yes—
90C2×#340C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93C22×#41C2311yescharacteristic
94C23×Q8—21no—
95C22×M16C2311yescharacteristic
96C22×M16C2311yescharacteristic
97C23×C8C2312yes—
98#544—28no—
99#544—44no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8—24no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2314yes—
109C2×#125—22no—
110C2×#125C2314yes—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—24no—
114C4×#41—24no—
115C4×#41—24no—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×#117—21no—
123C2×#117C2312yes—
124C2×#117C2311yescharacteristic
125C2×#117C2312yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C42—21no—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C2×C4⋊C8—24no—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2312yes—
137#417—24no—
138#417Q818yescomplemented
139#416—44no—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148#405—44no—
149C4×M16—24no—
150C2×#97C2312yes—
151C2×#97C2314yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2314yes—
154#400—24no—
155#399—24no—
156#398—24no—
157#398—24no—
158#395—24no—
159#395—24no—
160#395—24no—
161#393—24no—
162#391—22no—
163#391—22no—
164#391—24no—
165#391C2314yes—
166#390C2314yes—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—24no—
176#389C2314yes—
177#389C2314yes—
178#386—24no—
179#385—24no—
180#384—22no—
181#384D411yescharacteristic
182#384D411yescharacteristic
183#384C2314yes—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#382D412yes—
187#382C2314yes—
188#382C2312yes—
189#380—21no—
190#380—21no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—24no—
196#380D412yes—
197#380D412yes—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C2×#95—21no—
207#340C2312yes—
208#340C2312yes—
209#340C2312yes—
210#340C2312yes—
211#340C2312yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#336C2314yes—
224#336C2314yes—
225C2×#41—22no—
226C2×#41—22no—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—22no—
230C2×#41C2×Q814yes—
231C22×Q8—22no—
232C22×Q8—24no—
233C22×Q8—22no—
234C22×Q8—44no—
235C22×Q8—82no—
236C22×Q8—82no—
237C22×Q8C2411yescharacteristic
238C22×D4—21no—
239C22×D4C2411yescharacteristic
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×M16—22no—
250C2×M16—22no—
251C2×M16—22no—
252C2×M16—22no—
253C2×M16—22no—
254C2×M16C2×Q814yes—
255C22×C8—24no—
256C22×C8—24no—
257C22×C8—24no—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C4⋊Q8—28no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—44no—
268C4⋊Q8—48no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—84no—
272#125—28no—
273#125—44no—
274#125—48no—
275C22⋊Q8—416no—
276C22⋊Q8—84no—
277C22⋊Q8—84no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—44no—
287C4×D4—44no—
288#117—24no—
289#117—24no—
290#117—22no—
291#117—22no—
292#117—24no—
293#117—24no—
294#117—44no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×Q814yes—
325C2×C4⋊C4#4114yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4C2×D414yes—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—44no—
347C2×C42—82no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42#4114yes—
351C2×C42#4114yes—
352C2×C42C2414yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C4⋊C8—22no—
356C4⋊C8—22no—
357C4⋊C8—24no—
358#103—24no—
359#103—42no—
360#103—42no—
361#103C2×Q818yescomplemented
362#102—24no—
363#102—42no—
364#102—42no—
365#102C2×Q818yescomplemented
366#97—24no—
367#97—44no—
368C4×C8—24no—
369C4×C8—44no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95C2×D414yes—
381#95C2×D414yes—
382C24—21no—
383C24C22×D411yescharacteristic
384#41—28no—
385#41—28no—
386#41—44no—
387#41—44no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—48no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×Q8—84no—
401C2×Q8—84no—
402C2×Q8—816no—
403C2×Q8C22×Q812yes—
404C2×D4—24no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4C22×Q812yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4#12414yes—
444C22×C4#12414yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×Q812yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C42−1+412yes—
458C22×C4C2511yescharacteristic
459M16—28no—
460M16—44no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—48no—
473C2×C8C22×Q814yes—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4C22×Q814yes—
511C4⋊C4C2×#4114yes—
512C22⋊C4—28no—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C42—24no—
520C42—24no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—44no—
525C42—44no—
526C42—82no—
527C42#12414yes—
528C42#12414yes—
529C42C2×#4112yes—
530C42C2×#4112yes—
531C42C2×#4112yes—
532C42C2×#4112yes—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C2×#12412yes—
545C23#54612yes—
546C23D4×Q812yes—
547C23C23×D411yescharacteristic
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—88no—
553Q8—88no—
554Q8—168no—
555D4—22no—
556D4—22no—
557D4—48no—
558D4—41no—
559D4—41no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—48no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—88no—
603C2×C4—84no—
604C2×C4C2×#12412yes—
605C2×C4C2×#12412yes—
606C2×C4C2×#12412yes—
607C2×C4C2×#12412yes—
608C2×C4C2×#12412yes—
609C2×C4#53314yes—
610C2×C4#54214yes—
611C2×C4#54612yes—
612C2×C4D4×Q812yes—
613C2×C4#55414yes—
614C2×C4#55614yes—
615C2×C4#57712yes—
616C2×C4#57812yes—
617C2×C4C23×Q811yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C2×C4C2×2−1+411yescharacteristic
623C8—24no—
624C8—42no—
625C8—42no—
626C8—42no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22#249214yes—
641C22C22×#12411yescharacteristic
642C22C2×#54611yescharacteristic
643C22C2×D4×Q811yescharacteristic
644C22C2×#57711yescharacteristic
645C22C2×#57811yescharacteristic
646C22#358511yescharacteristic
647C22#358511yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—44no—
655C4—42no—
656C4—82no—
657C4—82no—
658C4—81no—
659C4—81no—
660C4—84no—
661C4#249214yes—
662C4#313714yes—
663C4C2×#53311yescharacteristic
664C4C2×#54211yescharacteristic
665C4C2×#55411yescharacteristic
666C4C2×#55611yescharacteristic
667C2—24no—
668C2—82no—
669C2C2×#249211yescharacteristic
670C2C2×#313711yescharacteristic
671C2#6061911yescharacteristic
672C2#6120011yescharacteristic
673C2#6120311yescharacteristic
674C2#6138211yescharacteristic
675C2#6138411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35167C2semidirect
3. #27830C2semidirect
4. #27828C2semidirect
5. #27824C2semidirect
6. #27821C2semidirect
7. #27815C2semidirect
8. #27814C2semidirect
9. #23749C2semidirect
11. #21326C2semidirect
12. #21324C2semidirect
13. C2×#1987C2semidirect
16. #20441C2semidirect
138. #417Q8semidirect
361. #103C2×Q8semidirect
365. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#352792
#351672
#278302
#278282
#278242
#278212
#278152
#278142
#237492
#234582
#213262
#213242
C2×#19872
#210922
#206452
#204412
C4×#4172