Finite Group Databasebeta

← #8097178

Subgroups — #8097178

Order 512.

5977 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#53065C212yesmaximal, complemented
2#53015C211yesmaximal, characteristic, complemented
3#53005C213yesmaximal, complemented
4#52925C214yesmaximal, complemented
5#52914C212yesmaximal, complemented
6#52367C216yesmaximal, complemented
7#52307C211yesmaximal, characteristic, complemented
8C2×#2785C212yesmaximal, complemented
9#46073C213yesmaximal, complemented
10#46043C216yesmaximal, complemented
11#46031C2112yesmaximal, complemented
12#45991C216yesmaximal, complemented
13#36238C216yesmaximal, complemented
14#36164C213yesmaximal, complemented
15#36085C211yesmaximal, characteristic, complemented
16#36078C212yesmaximal, complemented
17#35750C213yesmaximal, complemented
18C22×2−1+4C2211yescharacteristic, complemented
19C22×H(5,𝔽2)C2211yescharacteristic
20C2×#554C2213yescomplemented
21C2×#547C2213yes—
22C2×#546C2216yes—
23#2862C2213yescomplemented
24#2861C2216yescomplemented
25#2850C2216yescomplemented
26#2850C22112yescomplemented
27#2844C2216yescomplemented
28#2840C2212yescomplemented
29#2835C2214yescomplemented
30#2832C22112yescomplemented
31#2832C2216yescomplemented
32#2832C22112yescomplemented
33#2828C22112yescomplemented
34#2824C2216yescomplemented
35#2819C2216yescomplemented
36#2811C2211yescharacteristic, complemented
37#2785C2211yescharacteristic, complemented
38#2785C2216yescomplemented
39#2785C2212yescomplemented
40#2785C2214yescomplemented
41#2785C2216yescomplemented
42#2785C22112yescomplemented
43#2785C2213yescomplemented
44#2785C2212yescomplemented
45#2782C22112yescomplemented
46#2763C2216yescomplemented
47#2763C22112yescomplemented
48#2763C22112yescomplemented
49#2753C22112yescomplemented
50#2753C22112yescomplemented
51#2753C22112yescomplemented
52#2753C22112yescomplemented
53#2736C22112yescomplemented
54#2733C2213yescomplemented
55#2733C2216yescomplemented
56#2712C22112yescomplemented
57#2684C2216yescomplemented
58#2670C2216yescomplemented
59#2616C2213yes—
60#2616C2211yescharacteristic, complemented
61#2615C2212yescomplemented
62#2614C2214yescomplemented
63#2614C2212yescomplemented
64#2613C2216yes—
65#2592C2216yescomplemented
66#2591C22112yescomplemented
67#2568C2213yescomplemented
68#2563C2213yescomplemented
69#2562C2216yescomplemented
70#2561C22112yescomplemented
71#2504C2216yescomplemented
72#2503C2216yescomplemented
73#2492C2216yescomplemented
74#2491C2216yescomplemented
75#2490C2216yescomplemented
76#2489C2216yescomplemented
77#2489C2216yescomplemented
78#2489C2216yescomplemented
79#2488C22112yescomplemented
80#2488C22112yescomplemented
81#2488C22112yescomplemented
82#2488C22112yescomplemented
83#2472C22112yescomplemented
84#2472C22112yescomplemented
85#2436C2216yescomplemented
86#2435C2213yes—
87#2434C2213yescomplemented
88#2432C2216yes—
89#2430C2216yes—
90#2430C22112yescomplemented
91#2430C2216yescomplemented
92#2410C22112yes—
93#2409C2216yescomplemented
94#2409C2213yescomplemented
95#2408C2213yes—
96#2407C2216yes—
97#2406C2216yes—
98#2406C2216yescomplemented
99#2405C22112yes—
100#2404C22112yescomplemented
101#2388C2213yes—
102#2388C2211yescharacteristic, complemented
103#2388C2212yescomplemented
104C2×#395C2212yescomplemented
105C2×#394C2216yes—
106C2×#393C2214yescomplemented
107#2379C22112yescomplemented
108#2375C2216yescomplemented
109#2339C2213yes—
110#2339C2216yescomplemented
111#2339C2213yescomplemented
112D4×C22⋊C4C2216yes—
113C4×#124C2213yescomplemented
114C4×#124C2216yescomplemented
115C2×#388C22112yescomplemented
116C2×#382C2216yescomplemented
117C42×D4C2213yescomplemented
118C2×2−1+4—412no—
119C2×H(5,𝔽2)—412no—
120C22×#41C2311yescharacteristic
121C22×#41C2313yes—
122C22×#41C2311yescharacteristic
123C22×#41C2312yes—
124C22×#41C2312yes—
125C22×#41C2311yescharacteristic
126C22×#41C2316yes—
127C23×Q8C2312yes—
128C23×Q8C2313yes—
129C23×D4C2312yes—
130C23×D4C2312yes—
131C23×D4C2314yes—
132C23×D4C2311yescharacteristic
133#554—412no—
134#547—412no—
135#546—424no—
136C2×C4⋊Q8C2313yes—
137C2×C4⋊Q8C2313yes—
138C2×#125C2316yes—
139C2×#124C23112yes—
140C2×#124C2316yes—
141C2×#124C2313yes—
142C2×#124C2316yes—
143C2×#123C2316yes—
144C2×#123C2316yes—
145C2×#123C23112yes—
146C2×C22⋊Q8C23112yes—
147C2×C22⋊Q8C2316yes—
148C2×C22⋊Q8C2316yes—
149C2×C22⋊Q8C2313yes—
150C2×C22⋊Q8C2313yes—
151C2×#121C23112yes—
152C2×#121C2316yes—
153C2×#121C2316yes—
154C2×#120C23112yes—
155C2×#120C2316yes—
156C2×C4×Q8C2313yes—
157C2×C4×D4C2313yes—
158C2×C4×D4C2316yes—
159C2×C4×D4C2316yes—
160C2×C4×D4C2313yes—
161C22×C4⋊C4C2316yes—
162C22×C4⋊C4C2316yes—
163C22×C22⋊C4C23112yes—
164C22×C22⋊C4C23112yes—
165C22×C42C2316yes—
166#400C23112yes—
167#400C23112yes—
168#400C23112yes—
169#400C23112yes—
170#400C23112yes—
171#400C23112yes—
172#397C23112yes—
173#397C23112yes—
174#397C23112yes—
175#397C23112yes—
176#397C23112yes—
177#395C2316yes—
178#395C23112yes—
179#395C2316yes—
180#395C2312yes—
181#395C2312yes—
182#395C2316yes—
183#395C2313yes—
184#395C2311yescharacteristic
185#395C2316yes—
186#394C23112yes—
187#394C2313yes—
188#394C2316yes—
189#394C23112yes—
190#394C2316yes—
191#394C2316yes—
192#394C23112yes—
193#394C23112yes—
194#394C23112yes—
195#394C2316yes—
196#394C23112yes—
197#394C2316yes—
198#394C2313yes—
199#394C23112yes—
200#394C2316yes—
201#394C2316yes—
202#393C23112yes—
203#393C23112yes—
204#393C2316yes—
205#393C2316yes—
206#393C23112yes—
207#393C2316yes—
208#393C23112yes—
209#393C23112yes—
210#393C2314yes—
211#393C2311yescharacteristic
212#393C2312yes—
213#393C23112yes—
214#393C2313yes—
215#392C2316yes—
216#392C23112yes—
217#392C2313yes—
218#392C23112yes—
219#392C2316yes—
220#392C2316yes—
221#392C2313yes—
222#391C2316yes—
223#391C2316yes—
224#391C2316yes—
225#391C2313yes—
226#391C23112yes—
227#391C23112yes—
228#391C2316yes—
229#391C2316yes—
230#391C2316yes—
231#390C2316yes—
232#390C23112yes—
233#390C2316yes—
234#390C2316yes—
235#390C2313yes—
236#390C2316yes—
237#388C2316yes—
238#388C23112yes—
239#388C23112yes—
240#388C23112yes—
241#388C23112yes—
242#388C23112yes—
243#388C23112yes—
244#388C23112yes—
245#388C23112yes—
246#388C2316yes—
247#388C23112yes—
248#388C2316yes—
249#388C23112yes—
250#388C23112yes—
251#388C23112yes—
252#388C23112yes—
253#388C2316yes—
254#388C2316yes—
255#388C2316yes—
256#388C23112yes—
257#386C23112yes—
258#386C23112yes—
259#386C23112yes—
260#386C23112yes—
261#386C23112yes—
262#386C23112yes—
263#386C23112yes—
264#386C23112yes—
265#386C2316yes—
266#386C23112yes—
267#386C23112yes—
268#385C2316yes—
269#385C23112yes—
270#385C23112yes—
271#385C23112yes—
272#385C23112yes—
273#385C23112yes—
274#382C2316yes—
275#382C2316yes—
276#382C23112yes—
277#382C2316yes—
278#382C2316yes—
279#382C2316yes—
280#382C23112yes—
281#382C23112yes—
282#382C2313yes—
283#381C2316yes—
284#381C23112yes—
285#381C2316yes—
286#380C2316yes—
287#380C2313yes—
288#380C2316yes—
289#380C2313yes—
290#379C23112yes—
291#379C2316yes—
292#379C23112yes—
293C4×C4⋊C4C2316yes—
294C4×C4⋊C4C2313yes—
295C4×C22⋊C4C23112yes—
296C4×C22⋊C4C2316yes—
297C4×C22⋊C4C2313yes—
298C4×C22⋊C4C2313yes—
299C4×C22⋊C4C2316yes—
300C4×C22⋊C4C2316yes—
301C2×#95C2316yes—
302C2×#95C23112yes—
303C2×#95C2316yes—
304C43C2313yes—
305C25C2414yes—
306C25C2412yes—
3072−1+4—464no—
308H(5,𝔽2)—464no—
309C2×#41—224no—
310C2×#41—212no—
311C2×#41—212no—
312C2×#41—46no—
313C2×#41—46no—
314C2×#41—412no—
315C2×#41—424no—
316C2×#41—412no—
317C2×#41—412no—
318C22×Q8—412no—
319C22×Q8—412no—
320C22×Q8—46no—
321C22×Q8C2413yes—
322C22×Q8C2411yescharacteristic
323C22×Q8C2413yes—
324C22×Q8C2416yes—
325C22×Q8C2411yescharacteristic
326C22×Q8C2416yes—
327C22×Q8C2411yescharacteristic
328C22×D4—224no—
329C22×D4—46no—
330C22×D4—412no—
331C22×D4—424no—
332C22×D4C2416yes—
333C22×D4C2414yes—
334C22×D4C2412yes—
335C22×D4C2412yes—
336C22×D4C2413yes—
337C22×D4C2416yes—
338C22×D4C24112yes—
339C22×D4C2411yescharacteristic
340C22×D4C2414yes—
341C22×D4C2412yes—
342C22×D4C2412yes—
343C22×D4C2413yes—
344C22×D4C2411yescharacteristic
345C23×C4C2416yes—
346C23×C4C2412yes—
347C23×C4C2413yes—
348C23×C4C2412yes—
349C23×C4C2411yescharacteristic
350C23×C4C2416yes—
351C23×C4C24112yes—
352C23×C4C2414yes—
353C23×C4C2416yes—
354C4⋊Q8—46no—
355C4⋊Q8—46no—
356#125—412no—
357#124—424no—
358#124—46no—
359#124—412no—
360#124—412no—
361#123—424no—
362#123—412no—
363#123—412no—
364C22⋊Q8—412no—
365C22⋊Q8—424no—
366C22⋊Q8—412no—
367C22⋊Q8—46no—
368C22⋊Q8—46no—
369#121—412no—
370#121—424no—
371#121—412no—
372#120—412no—
373#120—424no—
374C4×Q8—46no—
375C4×D4—224no—
376C4×D4—212no—
377C4×D4—412no—
378C4×D4—46no—
379C2×C4⋊C4—412no—
380C2×C4⋊C4—412no—
381C2×C4⋊C4C2416yes—
382C2×C4⋊C4C2413yes—
383C2×C4⋊C4C2413yes—
384C2×C4⋊C4C2413yes—
385C2×C4⋊C4C2416yes—
386C2×C4⋊C4C2416yes—
387C2×C4⋊C4C2416yes—
388C2×C4⋊C4C24112yes—
389C2×C4⋊C4C2416yes—
390C2×C4⋊C4C2416yes—
391C2×C4⋊C4C2416yes—
392C2×C4⋊C4C24112yes—
393C2×C4⋊C4C24112yes—
394C2×C4⋊C4C2416yes—
395C2×C22⋊C4—424no—
396C2×C22⋊C4—424no—
397C2×C22⋊C4C2416yes—
398C2×C22⋊C4C24112yes—
399C2×C22⋊C4C24112yes—
400C2×C22⋊C4C2416yes—
401C2×C22⋊C4C2416yes—
402C2×C22⋊C4C2416yes—
403C2×C22⋊C4C24112yes—
404C2×C22⋊C4C24112yes—
405C2×C22⋊C4C24112yes—
406C2×C22⋊C4C24112yes—
407C2×C22⋊C4C24112yes—
408C2×C22⋊C4C2416yes—
409C2×C22⋊C4C2413yes—
410C2×C22⋊C4C2413yes—
411C2×C22⋊C4C24112yes—
412C2×C22⋊C4C2416yes—
413C2×C22⋊C4C2416yes—
414C2×C22⋊C4C2416yes—
415C2×C22⋊C4C2416yes—
416C2×C22⋊C4C2416yes—
417C2×C22⋊C4C2416yes—
418C2×C22⋊C4C2416yes—
419C2×C22⋊C4C24112yes—
420C2×C22⋊C4C24112yes—
421C2×C22⋊C4C24112yes—
422C2×C22⋊C4C2416yes—
423C2×C42—412no—
424C2×C42C2416yes—
425C2×C42C2413yes—
426C2×C42C2416yes—
427C2×C42C2413yes—
428C2×C42C2416yes—
429C2×C42C2413yes—
430C2×C42C24112yes—
431#95C2413yes—
432#95C2416yes—
433#95C2413yes—
434#95C2416yes—
435#95C24112yes—
436#95C2416yes—
437#95C24112yes—
438#95C24112yes—
439#95C24112yes—
440#95C24112yes—
441#95C2416yes—
442#95C24112yes—
443#95C2413yes—
444#95C2416yes—
445#95C2413yes—
446#95C2416yes—
447#95C2416yes—
448#95C24112yes—
449#95C24112yes—
450#95C24112yes—
451#95C24112yes—
452#95C24112yes—
453C24—224no—
454C24—46no—
455C24—412no—
456C24—44no—
457C24—46no—
458C24—42no—
459C24C2511yescharacteristic
460C24C2512yes—
461C24C2514yes—
462C24C2513yes—
463C24C2512yes—
464#41—432no—
465#41—432no—
466#41—416no—
467#41—496no—
468#41—416no—
469#41—416no—
470#41—448no—
471C2×Q8—212no—
472C2×Q8—26no—
473C2×Q8—26no—
474C2×Q8—26no—
475C2×Q8—43no—
476C2×Q8—412no—
477C2×Q8—46no—
478C2×Q8—43no—
479C2×Q8—43no—
480C2×Q8—43no—
481C2×Q8—46no—
482C2×Q8—412no—
483C2×Q8—448no—
484C2×Q8—432no—
485C2×D4—224no—
486C2×D4—224no—
487C2×D4—212no—
488C2×D4—212no—
489C2×D4—212no—
490C2×D4—26no—
491C2×D4—43no—
492C2×D4—46no—
493C2×D4—46no—
494C2×D4—412no—
495C2×D4—46no—
496C2×D4—43no—
497C2×D4—46no—
498C2×D4—43no—
499C2×D4—46no—
500C2×D4—43no—
501C2×D4—46no—
502C2×D4—412no—
503C2×D4—412no—
504C2×D4—412no—
505C2×D4—464no—
506C2×D4—432no—
507C2×D4—416no—
508C2×D4—432no—
509C2×D4H(5,𝔽2)112yes—
510C22×C4—224no—
511C22×C4—212no—
512C22×C4—212no—
513C22×C4—212no—
514C22×C4—224no—
515C22×C4—212no—
516C22×C4—212no—
517C22×C4—26no—
518C22×C4—26no—
519C22×C4—212no—
520C22×C4—46no—
521C22×C4—412no—
522C22×C4—46no—
523C22×C4—412no—
524C22×C4—412no—
525C22×C4—412no—
526C22×C4C2516yes—
527C22×C4C25112yes—
528C22×C4C2513yes—
529C22×C4C2516yes—
530C22×C4C2511yescharacteristic
531C22×C4C2512yes—
532C22×C4C2512yes—
533C22×C4C2516yes—
534C22×C4C2513yes—
535C22×C4C2516yes—
536C22×C4C2513yes—
537C22×C4C2511yescharacteristic
538C4⋊C4—212no—
539C4⋊C4—26no—
540C4⋊C4—43no—
541C4⋊C4—43no—
542C4⋊C4—46no—
543C4⋊C4—46no—
544C4⋊C4—46no—
545C4⋊C4—412no—
546C4⋊C4—46no—
547C4⋊C4—46no—
548C4⋊C4—46no—
549C4⋊C4—412no—
550C4⋊C4—412no—
551C4⋊C4—46no—
552C22⋊C4—224no—
553C22⋊C4—212no—
554C22⋊C4—212no—
555C22⋊C4—224no—
556C22⋊C4—212no—
557C22⋊C4—46no—
558C22⋊C4—412no—
559C22⋊C4—412no—
560C22⋊C4—412no—
561C22⋊C4—412no—
562C22⋊C4—412no—
563C22⋊C4—412no—
564C22⋊C4—46no—
565C22⋊C4—46no—
566C22⋊C4—43no—
567C22⋊C4—43no—
568C22⋊C4—46no—
569C22⋊C4—412no—
570C22⋊C4—46no—
571C22⋊C4—46no—
572C22⋊C4—46no—
573C22⋊C4—46no—
574C22⋊C4—412no—
575C22⋊C4—412no—
576C22⋊C4—46no—
577C22⋊C4—46no—
578C42—212no—
579C42—26no—
580C42—26no—
581C42—212no—
582C42—412no—
583C42—43no—
584C42—46no—
585C23—26no—
586C23—22no—
587C23—23no—
588C23—23no—
589C23—26no—
590C23—212no—
591C23—26no—
592C23—26no—
593C23—26no—
594C23—212no—
595C23—23no—
596C23—23no—
597C23—24no—
598C23—23no—
599C23—22no—
600C23—21no—
601C23—48no—
602C23—416no—
603C23—824no—
604C23—824no—
605C23—166no—
606C23—166no—
607C23C2×H(5,𝔽2)112yes—
608C23C2611yescharacteristic
609Q8—44no—
610Q8—424no—
611Q8—44no—
612Q8—412no—
613Q8—44no—
614Q8—86no—
615Q8—166no—
616D4—48no—
617D4—44no—
618D4—48no—
619D4—416no—
620D4—412no—
621D4—416no—
622D4—48no—
623D4—424no—
624D4—48no—
625D4—86no—
626D4—812no—
627D4—1612no—
628D4H(3,𝔽4)116yes—
629C2×C4—23no—
630C2×C4—23no—
631C2×C4—26no—
632C2×C4—23no—
633C2×C4—26no—
634C2×C4—26no—
635C2×C4—212no—
636C2×C4—26no—
637C2×C4—212no—
638C2×C4—23no—
639C2×C4—23no—
640C2×C4—26no—
641C2×C4—26no—
642C2×C4—26no—
643C2×C4—26no—
644C2×C4—212no—
645C2×C4—26no—
646C2×C4—26no—
647C2×C4—26no—
648C2×C4—26no—
649C2×C4—212no—
650C2×C4—26no—
651C2×C4—26no—
652C2×C4—424no—
653C2×C4—416no—
654C2×C4—424no—
655C2×C4—48no—
656C2×C4—424no—
657C2×C4—48no—
658C2×C4—44no—
659C2×C4—412no—
660C2×C4—824no—
661C2×C4C2×H(5,𝔽2)16yes—
662C2×C4#58516yes—
663C22—212no—
664C22—212no—
665C22—46no—
666C22—412no—
667C22—43no—
668C22—46no—
669C22—43no—
670C22—43no—
671C22—46no—
672C22—46no—
673C22—43no—
674C22—412no—
675C22—43no—
676C22—44no—
677C22—42no—
678C22—41no—
679C22—1616no—
680C22—168no—
681C22C2×H(3,𝔽4)18yes—
682C22C22×H(5,𝔽2)13yes—
683C22C2×#58513yes—
684C222−1+611yescharacteristic
685C4—26no—
686C4—46no—
687C4—412no—
688C4—43no—
689C4—46no—
690C4—46no—
691C4—41no—
692C4—42no—
693C4—42no—
694C4—43no—
695C4—46no—
696C4C2×H(3,𝔽4)14yes—
697C2—28no—
698C2—83no—
699C2—84no—
700C2—82no—
701C2—81no—
702C2C22×H(3,𝔽4)11yescharacteristic
703C2#6289813yes—
704C2#6302413yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #53065C2semidirect
2. #53015C2semidirect
3. #53005C2semidirect
4. #52925C2semidirect
5. #52914C2semidirect
6. #52367C2semidirect
7. #52307C2semidirect
8. C2×#2785C2semidirect
9. #46073C2semidirect
10. #46043C2semidirect
11. #46031C2semidirect
12. #45991C2semidirect
13. #36238C2semidirect
14. #36164C2semidirect
15. #36085C2semidirect
16. #36078C2semidirect
17. #35750C2semidirect
18. C22×2−1+4C22semidirect
20. C2×#554C22semidirect
23. #2862C22semidirect
24. #2861C22semidirect
25. #2850C22semidirect
26. #2850C22semidirect
27. #2844C22semidirect
28. #2840C22semidirect
29. #2835C22semidirect
30. #2832C22semidirect
31. #2832C22semidirect
32. #2832C22semidirect
33. #2828C22semidirect
34. #2824C22semidirect
35. #2819C22semidirect
36. #2811C22semidirect
37. #2785C22semidirect
38. #2785C22semidirect
39. #2785C22semidirect
40. #2785C22semidirect
41. #2785C22semidirect
42. #2785C22semidirect
43. #2785C22semidirect
44. #2785C22semidirect
45. #2782C22semidirect
46. #2763C22semidirect
47. #2763C22semidirect
48. #2763C22semidirect
49. #2753C22semidirect
50. #2753C22semidirect
51. #2753C22semidirect
52. #2753C22semidirect
53. #2736C22semidirect
54. #2733C22semidirect
55. #2733C22semidirect
56. #2712C22semidirect
57. #2684C22semidirect
58. #2670C22semidirect
60. #2616C22semidirect
61. #2615C22semidirect
62. #2614C22semidirect
63. #2614C22semidirect
65. #2592C22semidirect
66. #2591C22semidirect
67. #2568C22semidirect
68. #2563C22semidirect
69. #2562C22semidirect
70. #2561C22semidirect
71. #2504C22semidirect
72. #2503C22semidirect
73. #2492C22semidirect
74. #2491C22semidirect
75. #2490C22semidirect
76. #2489C22semidirect
77. #2489C22semidirect
78. #2489C22semidirect
79. #2488C22semidirect
80. #2488C22semidirect
81. #2488C22semidirect
82. #2488C22semidirect
83. #2472C22semidirect
84. #2472C22semidirect
85. #2436C22semidirect
87. #2434C22semidirect
90. #2430C22semidirect
91. #2430C22semidirect
93. #2409C22semidirect
94. #2409C22semidirect
98. #2406C22semidirect
100. #2404C22semidirect
102. #2388C22semidirect
103. #2388C22semidirect
104. C2×#395C22semidirect
106. C2×#393C22semidirect
107. #2379C22semidirect
108. #2375C22semidirect
110. #2339C22semidirect
111. #2339C22semidirect
113. C4×#124C22semidirect
114. C4×#124C22semidirect
115. C2×#388C22semidirect
116. C2×#382C22semidirect
117. C42×D4C22semidirect

Maximal subgroups

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

subgroupindex
#530652
#530152
#530052
#529252
#529142
#523672
#523072
C2×#27852
#460732
#460432
#460312
#459912
#362382
#361642
#360852
#360782
#357502