Finite Group Databasebeta

← #6449861

Subgroups — #6449861

Order 512.

1970 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3076C211yesmaximal, characteristic
2C2×C4×#137C211yesmaximal, characteristic
3#35179C211yesmaximal, characteristic, complemented
4#34189C211yesmaximal, characteristic
5#24699C212yesmaximal, complemented
6#24698C212yesmaximal, complemented
7#24697C212yesmaximal, complemented
8#24696C212yesmaximal, complemented
9#24687C214yesmaximal, complemented
10#24686C214yesmaximal, complemented
11#23634C212yesmaximal
12#23545C212yesmaximal
13#21323C211yesmaximal, characteristic, complemented
14#21175C212yesmaximal, complemented
15#21145C211yesmaximal, characteristic, complemented
16#21066C212yesmaximal, complemented
17C2×#1805C211yesmaximal, characteristic, complemented
18C22×#137C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22#3076—28no—
23C2×#479C2212yes—
24C2×#477C2211yescharacteristic
25C2×#475C2211yescharacteristic
26C2×#478C2212yes—
27C2×#476C2211yescharacteristic
28C2×#474C2211yescharacteristic
29C4×#137—28no—
30C2×#441C2212yes—
31C2×#440C2212yes—
32C2×C4×Q16C2212yes—
33C2×C4×SD16C2212yes—
34C2×#428C2211yescharacteristic
35C2×#427C2212yes—
36C2×#424C2212yes—
37C22×C4⋊C8C2211yescharacteristic
38C2×#417C2211yescharacteristic
39C2×#419C2211yescharacteristic
40C2×#416C2212yes—
41C22×#103C2212yes—
42C2×C4×M16C2211yescharacteristic
43#2855—22no—
44#2627—22no—
45#2582—22no—
46#2581—21no—
47#2581—21no—
48#2568C2212yes—
49#2567C2212yes—
50#2565C2214yes—
51#2357C2212yes—
52#2357C2212yes—
53#2356C2212yes—
54#2356C2212yes—
55#2353—22no—
56#2353—22no—
57#2317C2212yes—
58#2314—22no—
59#2311C2212yes—
60#2310C2212yes—
61#2285C2211yescharacteristic
62#2283—21no—
63#2283—21no—
64C2×#382C2211yescharacteristic
65#2274C2212yes—
66#2074C2214yes—
67#2073C2214yes—
68#2072C2214yes—
69#2071C2214yes—
70#2056C2214yes—
71#2056C2214yes—
72#2055C2214yes—
73#2055C2214yes—
74#1993C2212yes—
75#1993C2212yes—
76#1991C2214yes—
77#1949C2214yes—
78#1949C2214yes—
79#1948C2214yes—
80#1947C2214yes—
81#1940C2212yes—
82#1940C2212yes—
83#1939C2214yes—
84#1928C2214yes—
85#1928C2214yes—
86#1927C2214yes—
87#1927C2214yes—
88#1805C2214yes—
89#1805C2212yes—
90#1805C2212yes—
91#1736C2212yes—
92#1736C2212yes—
93C2×#340C2212yes—
94#1729C2212yes—
95C22×#41C2311yescharacteristic
96C23×Q8C2311yescharacteristic
97C2×#137—24no—
98C2×#137—24no—
99C2×#137—24no—
100C22×Q16C2312yes—
101C22×SD16C2312yes—
102C22×M16C2311yescharacteristic
103C22×M16C2311yescharacteristic
104C23×C8C2312yes—
105#533—24no—
106#533—24no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×#125C2312yes—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C4×#41—24no—
113C4×#41D418yes—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—24no—
119C2×C4×Q8—24no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117C2312yes—
130C2×#117C2311yescharacteristic
131C2×#117C2312yes—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C42—21no—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138#479—28no—
139#478—28no—
140#477—24no—
141#476—24no—
142#475—24no—
143#474—24no—
144#441—28no—
145#440—28no—
146C4×Q16—28no—
147C4×SD16—28no—
148#428—24no—
149#427—28no—
150C2×#107C2312yes—
151C2×#107C2312yes—
152C2×#107C2312yes—
153C2×#106C2312yes—
154C2×#106C2312yes—
155C2×#106C2312yes—
156#424—28no—
157C2×C4⋊C8—24no—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2311yescharacteristic
160C2×C4⋊C8C2311yescharacteristic
161C2×C4⋊C8C2314yes—
162C2×C4⋊C8C2314yes—
163#419—24no—
164#417—24no—
165#416—28no—
166C2×#103—28no—
167C2×#103C2312yes—
168C2×#103C2312yes—
169C2×#103C2314yes—
170C2×#103C2314yes—
171C2×#103C2312yes—
172C2×#103C2312yes—
173C2×#103C2312yes—
174C2×#102C2312yes—
175C2×#102C2312yes—
176C2×#102C2312yes—
177C4×M16—24no—
178C2×#97C2312yes—
179C2×C4×C8C2312yes—
180#400—22no—
181#400—22no—
182#398—24no—
183#398—24no—
184#395—22no—
185#395—22no—
186#391—22no—
187#391—22no—
188#391—24no—
189#391C2314yes—
190#391C2314yes—
191#391C2314yes—
192#390C2314yes—
193#389—24no—
194#389—22no—
195#389—22no—
196#388—24no—
197#388—24no—
198#387—24no—
199#387—24no—
200#387—24no—
201#387—24no—
202#386—24no—
203#385—24no—
204#384D412yes—
205#384C2314yes—
206#383—22no—
207#383—22no—
208#382—21no—
209#382—21no—
210#382D411yescharacteristic
211#382D411yescharacteristic
212#382C2314yes—
213#382C2312yes—
214#382C2312yes—
215#381—24no—
216#381—22no—
217#381—22no—
218#380—22no—
219#380—22no—
220#380D412yes—
221#380D412yes—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4C2314yes—
225C4×C4⋊C4C2314yes—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C22⋊C4—24no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231#376—22no—
232#376—22no—
233C2×#95—21no—
234C2×#95—21no—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#336C2314yes—
242#336C2314yes—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41C2×D414yes—
249C22×Q8—21no—
250C22×Q8—21no—
251C22×Q8—21no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8#4114yes—
258C22×Q8C2411yescharacteristic
259C22×Q8C2412yes—
260C22×D4—21no—
261C22×D4C2411yescharacteristic
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—22no—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2411yescharacteristic
271#137—232no—
272C2×Q16—24no—
273C2×Q16—24no—
274C2×Q16—24no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—24no—
278C2×M16—22no—
279C2×M16—22no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16#4114yes—
284C22×C8—24no—
285C22×C8—24no—
286C22×C8—24no—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295#125—24no—
296#125—24no—
297C22⋊Q8—44no—
298C22⋊Q8—44no—
299C4×Q8—22no—
300C4×Q8—22no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—44no—
306C4×Q8—44no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×Q8C2×D414yes—
310C4×Q8C2×D414yes—
311C4×D4—28no—
312C4×D4—28no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4C2×D414yes—
316C4×D4C2×D414yes—
317#117—22no—
318#117—24no—
319#117—24no—
320#117—28no—
321#117—24no—
322#117—24no—
323#117C2×D414yes—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
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—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—44no—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D414yes—
344C2×C4⋊C4C2×D414yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2414yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4C2×D414yes—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—28no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—24no—
366C2×C42—24no—
367C2×C42—44no—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D414yes—
371C2×C42#4114yes—
372C2×C42C2414yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—24no—
376#107—24no—
377#107—24no—
378#106—24no—
379#106—24no—
380#106—24no—
381C4⋊C8—28no—
382C4⋊C8—28no—
383C4⋊C8—22no—
384C4⋊C8—22no—
385C4⋊C8—24no—
386#103—24no—
387#103—24no—
388#103—24no—
389#103—28no—
390#103—28no—
391#103—24no—
392#103—24no—
393#102—24no—
394#102—24no—
395#102—24no—
396#97—24no—
397C4×C8—24no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95—24no—
411#95—24no—
412#95C2×D412yes—
413#95C2×D412yes—
414#95C2×D412yes—
415#95C2×D412yes—
416C24—21no—
417C24C22×D411yescharacteristic
418#41—28no—
419#41—28no—
420#41—44no—
421#41—44no—
422C2×Q8—24no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—28no—
426C2×Q8—28no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—44no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8—82no—
443C2×Q8C22×D412yes—
444C2×Q8C2×#4114yes—
445C2×D4—24no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4C22×D412yes—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4#12114yes—
480C22×C4#12114yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C42−1+412yes—
493C22×C42−1+412yes—
494C22×C4C2511yescharacteristic
495Q16—216no—
496SD16—216no—
497M16—28no—
498M16—44no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—48no—
511C2×C8C2×#4114yes—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—24no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C4⋊C4C22×D412yes—
539C4⋊C4C22×D412yes—
540C22⋊C4—24no—
541C22⋊C4—28no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—44no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C42—24no—
548C42—24no—
549C42—24no—
550C42—24no—
551C42—24no—
552C42—44no—
553C42—44no—
554C42—44no—
555C42—84no—
556C42—84no—
557C42#12114yes—
558C42#12114yes—
559C42C22×D412yes—
560C42C22×D412yes—
561C42C2×#4112yes—
562C42C2×#4112yes—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—42no—
567C23—42no—
568C23—41no—
569C23—41no—
570C23C2×#12112yes—
571C23#54712yes—
572C23#54812yes—
573C23C23×D411yescharacteristic
574Q8—24no—
575Q8—22no—
576Q8—22no—
577Q8—24no—
578Q8—48no—
579Q8—42no—
580Q8—41no—
581Q8—42no—
582Q8—41no—
583Q8—84no—
584Q8—84no—
585D4—22no—
586D4—22no—
587D4—48no—
588D4—41no—
589D4—41no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—48no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—48no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—164no—
628C2×C4C2×#12112yes—
629C2×C4C2×#12112yes—
630C2×C4C2×#12112yes—
631C2×C4C2×#12112yes—
632C2×C4C2×#12112yes—
633C2×C4#52514yes—
634C2×C4#53614yes—
635C2×C4#54712yes—
636C2×C4#54812yes—
637C2×C4#55214yes—
638C2×C4#55614yes—
639C2×C4C23×D411yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C2×2−1+411yescharacteristic
644C2×C4C2×2−1+411yescharacteristic
645C8—24no—
646C8—42no—
647C8—42no—
648C8—42no—
649C22—22no—
650C22—22no—
651C22—24no—
652C22—24no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—82no—
657C22—82no—
658C22#241514yes—
659C22C22×#12111yescharacteristic
660C22C2×#54711yescharacteristic
661C22C2×#54811yescharacteristic
662C22#358412yes—
663C22#358612yes—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—42no—
669C4—42no—
670C4—44no—
671C4—44no—
672C4—42no—
673C4—44no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—84no—
678C4#241514yes—
679C4C2×#52511yescharacteristic
680C4C2×#53611yescharacteristic
681C4C2×#55211yescharacteristic
682C4C2×#55611yescharacteristic
683C2—24no—
684C2—82no—
685C2C2×#241511yescharacteristic
686C2#6042211yescharacteristic
687C2#6042711yescharacteristic
688C2#6128012yes—
689C2#6134412yes—
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
3. #35179C2semidirect
5. #24699C2semidirect
6. #24698C2semidirect
7. #24697C2semidirect
8. #24696C2semidirect
9. #24687C2semidirect
10. #24686C2semidirect
13. #21323C2semidirect
14. #21175C2semidirect
15. #21145C2semidirect
16. #21066C2semidirect
17. C2×#1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30762
C2×C4×#1372
#351792
#341892
#246992
#246982
#246972
#246962
#246872
#246862
#236342
#235452
#213232
#211752
#211452
#210662
C2×#18052