Finite Group Databasebeta

← #7104880

Subgroups — #7104880

Order 512.

1293 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59329C211yesmaximal, characteristic
2#39623C211yesmaximal, characteristic
3#32595C212yesmaximal, complemented
4#32341C212yesmaximal, complemented
5#32315C212yesmaximal, complemented
6#32228C212yesmaximal, complemented
7#29160C211yesmaximal, characteristic
8#28801C212yesmaximal, complemented
9#28048C212yesmaximal, complemented
10#27960C211yesmaximal, characteristic
11#27713C212yesmaximal
12#27377C212yesmaximal, complemented
13#26295C212yesmaximal, complemented
14#25988C212yesmaximal
15#24579C212yesmaximal
16C2×#2084C211yesmaximal, characteristic
17#23241C211yesmaximal, characteristic
18#22615C212yesmaximal
19#22079C211yesmaximal, characteristic
20C2×#484C2211yescharacteristic
21C22×#106C2212yes—
22#2857C2211yescharacteristic
23#2806—21no—
24#2794—22no—
25#2792—22no—
26#2734—21no—
27#2659—21no—
28#2582C2211yescharacteristic
29#2571C2211yescharacteristic
30#2570—21no—
31#2570C2212yes—
32#2567C2212yes—
33#2551C2212yes—
34#2511—21no—
35#2511C2212yes—
36#2507C2212yes—
37#2507C2212yes—
38#2503—21no—
39#2503C2212yes—
40#2484—22no—
41#2440—22no—
42#2440C2212yes—
43#2355C2211yescharacteristic
44#2355C2211yescharacteristic
45#2345C2212yes—
46#2344—21no—
47#2344C2212yes—
48#2329—21no—
49C4×C22⋊Q8C2211yescharacteristic
50C4×C22⋊Q8C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#384C2211yescharacteristic
53#2101C2212yes—
54#2101C2212yes—
55#2100C2211yescharacteristic
56#2099C2212yes—
57#2099C2212yes—
58#2099C2212yes—
59#2095C2211yescharacteristic
60#2086C2212yes—
61#2086C2212yes—
62#2085C2212yes—
63#2085C2212yes—
64#2084C2212yes—
65#2084C2212yes—
66#2084C2212yes—
67#2084C2212yes—
68#2072C2212yes—
69#2066C2212yes—
70#2066C2212yes—
71#2057C2212yes—
72#2057C2211yescharacteristic
73#2024C2212yes—
74#2024C2211yescharacteristic
75#1993C2212yes—
76#1990C2212yes—
77#1988C2212yes—
78#1985C2212yes—
79#1949C2212yes—
80#1949C2212yes—
81#1947C2212yes—
82#1944C2212yes—
83#1944C2212yes—
84#1942C2212yes—
85#1935C2212yes—
86#1935C2212yes—
87#1933C2212yes—
88#1927C2212yes—
89#1927C2212yes—
90#1921C2212yes—
91#1921C2212yes—
92#1920C2212yes—
93#1920C2212yes—
94#1919C2212yes—
95#1919C2212yes—
96#1918C2212yes—
97#1918C2212yes—
98#1917C2212yes—
99#1916C2212yes—
100#1892C2212yes—
101#1872C2211yescharacteristic
102#1863C2212yes—
103#1863C2212yes—
104#1852C2211yescharacteristic
105#1850C2212yes—
106#1836C2212yes—
107#1834C2212yes—
108#1824C2211yescharacteristic
109#1823C2212yes—
110#1823C2212yes—
111#1808C2212yes—
112#1776C2212yes—
113#1776C2212yes—
114#1771C2212yes—
115#1752C2212yes—
116C4×C22⋊C8C2211yescharacteristic
117C2×#340C2212yes—
118C2×#336C2211yescharacteristic
119C23×C8C2312yes—
120C2×C4⋊Q8—22no—
121C2×C4⋊Q8C2312yes—
122C2×C4⋊Q8C2311yescharacteristic
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8D411yescharacteristic
126C2×C22⋊Q8C2311yescharacteristic
127C2×C4×Q8C2311yescharacteristic
128C2×C4×Q8C2311yescharacteristic
129C2×#117—22no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#484—24no—
138C2×#107C2312yes—
139C2×#107C2312yes—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#106—28no—
143C2×#106C2312yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146C2×#106C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×C22⋊C8C2311yescharacteristic
155C2×#97C2312yes—
156C2×#97C2312yes—
157C2×C4×C8C2312yes—
158#400—22no—
159#400—22no—
160#400—22no—
161#400—22no—
162#400—22no—
163#400—22no—
164#398—22no—
165#398—22no—
166#398—22no—
167#398—22no—
168#398—22no—
169#398—22no—
170#398—22no—
171#395—22no—
172#395—22no—
173#395—22no—
174#395—22no—
175#395—22no—
176#391—22no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391D411yescharacteristic
181#391C2312yes—
182#391C2312yes—
183#391C2311yescharacteristic
184#389—21no—
185#389—21no—
186#389—22no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—22no—
191#389—22no—
192#389D411yescharacteristic
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2311yescharacteristic
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—21no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#387D412yes—
217#385—21no—
218#385—22no—
219#385D412yes—
220#384—22no—
221#384—21no—
222#384—21no—
223#384C2312yes—
224#384C2311yescharacteristic
225#384C2312yes—
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#382—21no—
229#382—21no—
230#382C2312yes—
231#382C2312yes—
232#381—22no—
233#381—22no—
234#381—22no—
235#380—21no—
236#380—22no—
237#380D411yescharacteristic
238#380C2311yescharacteristic
239#380C2312yes—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2311yescharacteristic
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2311yescharacteristic
250C4×C22⋊C4—21no—
251C4×C22⋊C4—22no—
252C4×C22⋊C4C2312yes—
253C2×#95—21no—
254#340C2312yes—
255#340C2312yes—
256#340C2312yes—
257#340C2312yes—
258#340C2312yes—
259#340C2312yes—
260#340C2312yes—
261#340C2312yes—
262#340C2312yes—
263#340C2312yes—
264#340C2312yes—
265#340C2312yes—
266#340C2312yes—
267#340C2312yes—
268#340C2312yes—
269#340C2312yes—
270#340C2312yes—
271#340C2312yes—
272#340C2312yes—
273#340C2312yes—
274#336C2311yescharacteristic
275#336C2312yes—
276#336C2312yes—
277#336C2311yescharacteristic
278#336C2312yes—
279#336C2312yes—
280#336C2312yes—
281#336C2312yes—
282C22×Q8—21no—
283C22×Q8—22no—
284C22×Q8C2×D411yescharacteristic
285C22×Q8C2411yescharacteristic
286C23×C4—21no—
287C23×C4—22no—
288C23×C4—21no—
289C23×C4C2×D411yescharacteristic
290C23×C4C2×D411yescharacteristic
291C23×C4C2411yescharacteristic
292C22×C8—24no—
293C22×C8—24no—
294C22×C8—42no—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C4⋊Q8—42no—
300C4⋊Q8—42no—
301C4⋊Q8—42no—
302C4⋊Q8—84no—
303C22⋊Q8—24no—
304C22⋊Q8—22no—
305C22⋊Q8—22no—
306C22⋊Q8—84no—
307C22⋊Q8—82no—
308C4×Q8—42no—
309C4×Q8—42no—
310#117—84no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—42no—
342C2×C4⋊C4—44no—
343C2×C4⋊C4—82no—
344C2×C4⋊C4—82no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4#4114yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—42no—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2412yes—
379#107—24no—
380#107—24no—
381#107—42no—
382#107—42no—
383#106—24no—
384#106—24no—
385#106—24no—
386#106—24no—
387C4⋊C8—24no—
388C4⋊C8—42no—
389#103—24no—
390#103—24no—
391#103—42no—
392#103—42no—
393C22⋊C8—22no—
394C22⋊C8—22no—
395#97—24no—
396#97—42no—
397C4×C8—42no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—21no—
402#95—21no—
403#95—22no—
404#95—21no—
405#95—22no—
406#95—22no—
407#95—21no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—21no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95C2×D412yes—
426#95C2×D412yes—
427#95C2×D412yes—
428#95C2×D412yes—
429#95C2×D412yes—
430C24C22×D411yescharacteristic
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—41no—
437C2×Q8—41no—
438C2×Q8—42no—
439C2×Q8—81no—
440C2×Q8—82no—
441C2×Q8—81no—
442C2×Q8—82no—
443C2×Q8—82no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—81no—
467C22×C4—81no—
468C22×C4—81no—
469C22×C4—81no—
470C22×C4#13714yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C2×#4112yes—
481C22×C42−1+412yes—
482C22×C4C2511yescharacteristic
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—48no—
495C2×C82−1+414yes—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—81no—
527C4⋊C4—82no—
528C4⋊C4—81no—
529C4⋊C4—82no—
530C4⋊C4—81no—
531C4⋊C4—81no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—81no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4—81no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4C2×#4114yes—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—44no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—41no—
557C42—41no—
558C42—42no—
559C42—42no—
560C42—82no—
561C42—82no—
562C42—82no—
563C42—82no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23#54812yes—
569C23C2×#13712yes—
570C23#57612yes—
571C23C23×D411yescharacteristic
572Q8—42no—
573Q8—42no—
574Q8—82no—
575Q8—162no—
576Q8—164no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—84no—
635C2×C4—88no—
636C2×C4—162no—
637C2×C4#54812yes—
638C2×C4C2×#13712yes—
639C2×C4C2×#13712yes—
640C2×C4#57712yes—
641C2×C4C22×#4111yescharacteristic
642C2×C4C2×2−1+412yes—
643C2×C4C2×2−1+411yescharacteristic
644C2×C4#58512yes—
645C8—24no—
646C8—42no—
647C8—42no—
648C8—42no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22#337514yes—
656C22C2×#54811yescharacteristic
657C22#348511yescharacteristic
658C22#349911yescharacteristic
659C22C22×#13711yescharacteristic
660C22C2×#57611yescharacteristic
661C22C2×#57711yescharacteristic
662C22#358611yescharacteristic
663C4—22no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—41no—
671C4—42no—
672C4—44no—
673C4—81no—
674C4—82no—
675C4—81no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—81no—
681C4—81no—
682C4#349512yes—
683C4#356312yes—
684C2—24no—
685C2#4444811yescharacteristic
686C2C2×#337511yescharacteristic
687C2#6134011yescharacteristic
688C2#6170511yescharacteristic
689C2#6176611yescharacteristic
690C2#6247811yescharacteristic
691C2#6251511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32595C2semidirect
4. #32341C2semidirect
5. #32315C2semidirect
6. #32228C2semidirect
8. #28801C2semidirect
9. #28048C2semidirect
12. #27377C2semidirect
13. #26295C2semidirect

Maximal subgroups

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

subgroupindex
#593292
#396232
#325952
#323412
#323152
#322282
#291602
#288012
#280482
#279602
#277132
#273772
#262952
#259882
#245792
C2×#20842
#232412
#226152
#220792