Finite Group Databasebeta

← #6519217

Subgroups — #6519217

Order 512.

1620 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#34955C211yesmaximal, characteristic
3#29140C212yesmaximal, complemented
4#29122C212yesmaximal, complemented
5#29122C212yesmaximal, complemented
6#28721C212yesmaximal, complemented
7#27816C212yesmaximal, complemented
8#27770C212yesmaximal, complemented
9#27750C212yesmaximal, complemented
10#27730C212yesmaximal, complemented
11#27701C214yesmaximal, complemented
12#27260C214yesmaximal, complemented
13#21884C211yesmaximal, characteristic
14#21859C211yesmaximal, characteristic
15#20734C211yesmaximal, characteristic, complemented
16C2×#1803C211yesmaximal, characteristic, complemented
17#20543C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#488C2214yes—
21C2×#431C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#103C2212yes—
24#2859C2212yes—
25#2859C2212yes—
26#2647—22no—
27#2592—21no—
28#2592—21no—
29#2591—22no—
30#2571—21no—
31#2571C2212yes—
32#2568C2211yescharacteristic
33#2568C2211yescharacteristic
34#2552—21no—
35#2552C2212yes—
36#2548C2211yescharacteristic
37#2548C2211yescharacteristic
38#2504C2212yes—
39#2442—22no—
40#2442C2214yes—
41#2436C2212yes—
42#2398—22no—
43#2317—21no—
44#2316—21no—
45C4×#124C2212yes—
46C4×#124C2212yes—
47#2297—21no—
48#2297—21no—
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2101C2214yes—
53#2101C2214yes—
54#2074C2214yes—
55#2073C2214yes—
56#2042C2214yes—
57#2042C2214yes—
58#2026C2214yes—
59#2025C2214yes—
60#1991C2214yes—
61#1990C2212yes—
62#1988C2212yes—
63#1982C2212yes—
64#1958C2214yes—
65#1957C2212yes—
66#1934C2214yes—
67#1934C2214yes—
68#1933C2214yes—
69#1933C2214yes—
70#1893C2214yes—
71#1874C2214yes—
72#1835C2214yes—
73#1824C2212yes—
74#1803C2211yescharacteristic
75#1803C2212yes—
76#1803C2212yes—
77#1803C2212yes—
78#1803C2212yes—
79#1803C2211yescharacteristic
80#1803C2211yescharacteristic
81#1803C2211yescharacteristic
82#1802C2212yes—
83#1802C2212yes—
84#1802C2212yes—
85#1802C2212yes—
86#1787C2212yes—
87C8×C4⋊C4C2212yes—
88C8×C4⋊C4C2212yes—
89C4×C4⋊C8C2212yes—
90C4×C4⋊C8C2212yes—
91C4×#103C2212yes—
92C4×#103C2212yes—
93C8×C22⋊C4C2214yes—
94C2×#340C2212yes—
95C22×#41—21no—
96C23×Q8C2311yescharacteristic
97C23×C8C2312yes—
98#532—24no—
99#532—24no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127C2311yescharacteristic
102C2×#125—22no—
103C2×#125C2314yes—
104C2×#124—21no—
105C2×#124C2312yes—
106C2×#124C2312yes—
107C2×#121—21no—
108C2×#121C2312yes—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—24no—
112C2×C4×Q8—24no—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4C2312yes—
119C2×#117—21no—
120C2×#117D411yescharacteristic
121C2×#117D411yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129#488—48no—
130#431—24no—
131C2×C4⋊C8—24no—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136C2×#103—44no—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×C22⋊C8C2312yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2314yes—
150C2×C4×C8C2314yes—
151#400—22no—
152#400—22no—
153#400—24no—
154#400—24no—
155#398—24no—
156#398—24no—
157#397—22no—
158#397—22no—
159#394—22no—
160#394—22no—
161#393—22no—
162#393—22no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391C2312yes—
168#391C2312yes—
169#390—21no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390C2312yes—
174#390C2312yes—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389C2314yes—
184#389C2314yes—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#385—24no—
194#384—22no—
195#384—21no—
196#384—21no—
197#384C2312yes—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2312yes—
201#382—22no—
202#382—22no—
203#382—21no—
204#382C2312yes—
205#382C2312yes—
206#382C2314yes—
207#380—21no—
208#380—24no—
209#380C2312yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222#376D412yes—
223C43C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236C2×#41—44no—
237C2×#41—44no—
238C2×#41—44no—
239C22×Q8—21no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—41no—
246C22×Q8—41no—
247C22×Q8C2411yescharacteristic
248C22×Q8C2411yescharacteristic
249C22×D4—22no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4C2412yes—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C22×C8—24no—
261C22×C8—42no—
262C22×C8—42no—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268#127—22no—
269#127—22no—
270#125—44no—
271#125—44no—
272#125—84no—
273#124—24no—
274#124—24no—
275#124—42no—
276#124—42no—
277#124—44no—
278#121—24no—
279#121—24no—
280#121—44no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—44no—
294#117—24no—
295#117—24no—
296#117—44no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—44no—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42#4114yes—
359C2×C42#4114yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C4⋊C8—24no—
365C4⋊C8—24no—
366C4⋊C8—22no—
367C4⋊C8—22no—
368#103—28no—
369#103—24no—
370#103—24no—
371#103—44no—
372#103—42no—
373#103—42no—
374#102—44no—
375#102—44no—
376#102—44no—
377#102—44no—
378C22⋊C8—24no—
379C4×C8—24no—
380C4×C8—44no—
381C4×C8—44no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—22no—
401C24C22×D411yescharacteristic
402#41—816no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—28no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8H(5,𝔽2)12yes—
424C2×Q82−1+412yes—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—81no—
439C2×D4—81no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4#12414yes—
467C22×C4#12414yes—
468C22×C4#13514yes—
469C22×C4#13514yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2511yescharacteristic
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—84no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C42—24no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—42no—
547C42—42no—
548C42—42no—
549C42—42no—
550C42—42no—
551C42—42no—
552C42—44no—
553C42#12414yes—
554C42#12414yes—
555C42C2×#4112yes—
556C42C2×#4112yes—
557C42C2×#4112yes—
558C42C2×#4112yes—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23C2×#12412yes—
571C23C2×#13512yes—
572C23C2×#13512yes—
573C23C23×D411yescharacteristic
574Q8—22no—
575Q8—22no—
576Q8—22no—
577Q8—22no—
578Q8—48no—
579Q8—41no—
580Q8—41no—
581Q8—41no—
582Q8—41no—
583Q8—84no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—84no—
589D4—88no—
590D4—84no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—44no—
616C2×C4—48no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—84no—
640C2×C4—88no—
641C2×C4—164no—
642C2×C4#488116yescomplemented
643C2×C4C2×#12412yes—
644C2×C4C2×#12412yes—
645C2×C4C2×#12412yes—
646C2×C4C2×#12412yes—
647C2×C4C2×#12412yes—
648C2×C4C2×#13512yes—
649C2×C4C2×#13512yes—
650C2×C4C2×#13512yes—
651C2×C4C2×#13512yes—
652C2×C4C22×#4111yescharacteristic
653C2×C4C22×#4111yescharacteristic
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C2×C4C2×2−1+411yescharacteristic
656C2×C4#58511yescharacteristic
657C2×C4#58511yescharacteristic
658C8—42no—
659C8—42no—
660C8—44no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22C2×#48814yes—
675C22C22×#12411yescharacteristic
676C22#348811yescharacteristic
677C22#348911yescharacteristic
678C22C22×#13511yescharacteristic
679C22C22×#13511yescharacteristic
680C22#358511yescharacteristic
681C22#358611yescharacteristic
682C4—22no—
683C4—22no—
684C4—22no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—84no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—81no—
696C4—81no—
697C4C2×#48814yes—
698C4C2×#48814yes—
699C4#349211yescharacteristic
700C4#349311yescharacteristic
701C4#349411yescharacteristic
702C4#349511yescharacteristic
703C2—24no—
704C2—162no—
705C2#3554211yescharacteristic
706C2C22×#48811yescharacteristic
707C2#6062111yescharacteristic
708C2#6195311yescharacteristic
709C2#6195411yescharacteristic
710C2#6210111yescharacteristic
711C2#6210211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35286C2semidirect
3. #29140C2semidirect
4. #29122C2semidirect
5. #29122C2semidirect
6. #28721C2semidirect
7. #27816C2semidirect
8. #27770C2semidirect
9. #27750C2semidirect
10. #27730C2semidirect
11. #27701C2semidirect
12. #27260C2semidirect
15. #20734C2semidirect
16. C2×#1803C2semidirect
17. #20543C2semidirect
642. C2×C4#488semidirect

Maximal subgroups

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

subgroupindex
#352862
#349552
#291402
#291222
#287212
#278162
#277702
#277502
#277302
#277012
#272602
#218842
#218592
#207342
C2×#18032
#205432