Finite Group Databasebeta

← #6433534

Subgroups — #6433534

Order 512.

1720 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic
2#34268C211yesmaximal, characteristic
3#30184C214yesmaximal, complemented
4#30182C214yesmaximal, complemented
5#30176C212yesmaximal, complemented
6#30175C212yesmaximal, complemented
7#30169C212yesmaximal, complemented
8#30168C212yesmaximal, complemented
9#23658C211yesmaximal, characteristic, complemented
10#23641C212yesmaximal
11C2×#2072C211yesmaximal, characteristic, complemented
12#23551C211yesmaximal, characteristic, complemented
13#23484C212yesmaximal
14#23455C211yesmaximal, characteristic, complemented
15#21036C212yesmaximal, complemented
16#20880C212yesmaximal, complemented
17#20684C211yesmaximal, characteristic
18C2×#536C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×#519C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#428C2212yes—
23C2×#427C2212yes—
24C2×#419C2212yes—
25C22×#103C2212yes—
26C2×#408C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2629C2214yes—
30#2582—21no—
31#2582—21no—
32#2538—21no—
33#2538—21no—
34#2417—22no—
35#2417—22no—
36#2367—21no—
37#2367—21no—
38#2365—21no—
39#2365—21no—
40#2359—22no—
41#2359—22no—
42#2323C2212yes—
43#2323C2212yes—
44#2316C2212yes—
45#2311C2211yescharacteristic
46#2311C2212yes—
47C4×C22⋊Q8C2212yes—
48C4×C22⋊Q8C2212yes—
49#2296C2211yescharacteristic
50C2×#391C2211yescharacteristic
51C2×#384C2211yescharacteristic
52#2274C2211yescharacteristic
53#2084C2212yes—
54#2084C2212yes—
55#2083C2212yes—
56#2083C2212yes—
57#2077C2214yes—
58#2077C2214yes—
59#2076C2214yes—
60#2075C2214yes—
61#2072C2212yes—
62#2072C2212yes—
63#2072C2214yes—
64#2059C2212yes—
65#2059C2212yes—
66#2058C2211yescharacteristic
67#2058C2211yescharacteristic
68#2057C2211yescharacteristic
69#2057C2211yescharacteristic
70#2036C2214yes—
71#2036C2214yes—
72#2035C2214yes—
73#2035C2214yes—
74#2026C2214yes—
75#2024C2212yes—
76#2024C2212yes—
77#1921C2214yes—
78#1920C2214yes—
79#1919C2214yes—
80#1918C2214yes—
81#1865C2214yes—
82#1865C2214yes—
83#1863C2214yes—
84#1863C2214yes—
85#1818C2214yes—
86#1817C2212yes—
87#1817C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#536—24no—
100#536—24no—
101#533—44no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×#125C2312yes—
106C2×#124—21no—
107C2×#123—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114#519—44no—
115C4×#41—44no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×C4×D4D412yes—
125C2×C4×D4D412yes—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×#117—22no—
129C2×#117C2312yes—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2312yes—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140#428—28no—
141#427—44no—
142C2×#107C2314yes—
143C2×#107C2312yes—
144C2×#107C2312yes—
145C2×#106C2314yes—
146C2×#106C2312yes—
147C2×#106C2312yes—
148#419—44no—
149C2×#103—28no—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154#408—44no—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157#398—24no—
158#398—24no—
159#396—24no—
160#396—24no—
161#391—22no—
162#391—21no—
163#391—22no—
164#391—22no—
165#391—21no—
166#391—21no—
167#391C2312yes—
168#391C2312yes—
169#390—21no—
170#390—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389C2311yescharacteristic
178#389C2311yescharacteristic
179#389C2311yescharacteristic
180#389C2311yescharacteristic
181#389C2314yes—
182#389C2314yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#386—22no—
192#386—24no—
193#385—21no—
194#385—21no—
195#385—22no—
196#385—22no—
197#384—21no—
198#384—21no—
199#384C2312yes—
200#384C2312yes—
201#380—22no—
202#380—22no—
203#380—24no—
204#380C2312yes—
205#380C2312yes—
206#380C2314yes—
207#380C2314yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C2×#95—21no—
220C2×#95—21no—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—24no—
241C2×#41—24no—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—24no—
248C22×Q8C2×D411yescharacteristic
249C22×Q8#4114yes—
250C22×Q8C2412yes—
251C22×D4C2×D412yes—
252C22×D4C2×D411yescharacteristic
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C2×M16—24no—
263C2×M16—24no—
264C2×M16—24no—
265C22×C8—24no—
266C22×C8—24no—
267C22×C8—24no—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2414yes—
271C4⋊Q8—28no—
272C4⋊Q8—42no—
273C4⋊Q8—42no—
274#125—42no—
275#125—42no—
276#124—28no—
277#123—216no—
278C22⋊Q8—24no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C22⋊Q8—84no—
285C22⋊Q8—84no—
286C4×Q8—42no—
287C4×Q8—42no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×D4—42no—
295C4×D4—42no—
296C4×D4—44no—
297C4×D4—44no—
298#117—22no—
299#117—24no—
300#117—24no—
301#117—24no—
302#117—24no—
303#117—44no—
304#117—48no—
305#117#4114yes—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—82no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C42—21no—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—44no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107—28no—
372#107—42no—
373#107—42no—
374#106—28no—
375#106—42no—
376#106—42no—
377#103—28no—
378#103—28no—
379#103—44no—
380#103—44no—
381C22⋊C8—44no—
382C22⋊C8—44no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—21no—
394#95—21no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399C24C22×D412yes—
400C24C22×D411yescharacteristic
401#41—48no—
402#41—48no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—216no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—48no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8C2×#4114yes—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)14yes—
472C22×C42−1+414yes—
473C22×C4C2511yescharacteristic
474M16—48no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—48no—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—81no—
515C4⋊C4—81no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4C2×#4112yes—
521C4⋊C4C2×#4112yes—
522C22⋊C4—28no—
523C22⋊C4—28no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—24no—
540C42—42no—
541C42—42no—
542C42—44no—
543C42—44no—
544C42—42no—
545C42—44no—
546C42—82no—
547C42—82no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23#52512yes—
560C23#53412yes—
561C23#53612yes—
562C23C23×D411yescharacteristic
563Q8—28no—
564Q8—48no—
565Q8—44no—
566Q8—44no—
567Q8—84no—
568Q8—84no—
569D4—48no—
570D4—44no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—164no—
625C2×C4—162no—
626C2×C4#52512yes—
627C2×C4#53412yes—
628C2×C4#53612yes—
629C2×C4#53918yes—
630C2×C4#54218yes—
631C2×C4#57712yes—
632C2×C4#57812yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)12yes—
636C2×C4C2×2−1+412yes—
637C8—42no—
638C8—42no—
639C8—44no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22#263814yes—
654C22C2×#52511yescharacteristic
655C22C2×#53411yescharacteristic
656C22C2×#53611yescharacteristic
657C22C2×#57711yescharacteristic
658C22C2×#57811yescharacteristic
659C22#358511yescharacteristic
660C22#358611yescharacteristic
661C4—22no—
662C4—22no—
663C4—42no—
664C4—44no—
665C4—41no—
666C4—42no—
667C4—41no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—81no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4—84no—
676C4—82no—
677C4#263814yes—
678C4C2×#53912yes—
679C4C2×#54212yes—
680C2—24no—
681C2—82no—
682C2C2×#263811yescharacteristic
683C2#6059311yescharacteristic
684C2#6059411yescharacteristic
685C2#6076311yescharacteristic
686C2#6076411yescharacteristic
687C2#6087811yescharacteristic
688C2#6088511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30184C2semidirect
4. #30182C2semidirect
5. #30176C2semidirect
6. #30175C2semidirect
7. #30169C2semidirect
8. #30168C2semidirect
9. #23658C2semidirect
11. C2×#2072C2semidirect
12. #23551C2semidirect
14. #23455C2semidirect
15. #21036C2semidirect
16. #20880C2semidirect

Maximal subgroups

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

subgroupindex
#352802
#342682
#301842
#301822
#301762
#301752
#301692
#301682
#236582
#236412
C2×#20722
#235512
#234842
#234552
#210362
#208802
#206842