Finite Group Databasebeta

← #6354183

Subgroups — #6354183

Order 512.

1573 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60866C212yesmaximal
2C2×#3170C211yesmaximal, characteristic, complemented
3C2×#2966C211yesmaximal, characteristic
4#19365C214yesmaximal, complemented
5#19359C212yesmaximal, complemented
6#19354C212yesmaximal, complemented
7#19287C212yesmaximal, complemented
8#19281C212yesmaximal, complemented
9#19262C212yesmaximal, complemented
10#19256C212yesmaximal, complemented
11#19226C214yesmaximal, complemented
12#19220C214yesmaximal, complemented
13#17793C211yesmaximal, characteristic, complemented
14#17748C212yesmaximal, complemented
15C2×#533C2212yes—
16C2×#529C2211yescharacteristic
17#3240—24no—
18#3239C2214yes—
19#3238C2214yes—
20#3233C2218yes—
21#3232—24no—
22#3170—21no—
23#3170—21no—
24#3170—21no—
25#3170—21no—
26#3170C2212yes—
27#3170C2212yes—
28#3170C2212yescomplemented
29#3170C2212yescomplemented
30#3166—22no—
31C2×#500C2211yescharacteristic, complemented
32C2×#494C2211yescharacteristic, complemented
33#3140—22no—
34C2×#487C2212yescomplemented
35#3095—22no—
36#3094—22no—
37#3078C2214yes—
38#3077C2214yes—
39C2×#467—21no—
40C2×#467—21no—
41#2966—21no—
42#2966—21no—
43#2966—21no—
44#2966—21no—
45#2966C2212yes—
46#2966C2212yes—
47#2966C2212yes—
48#2966C2212yes—
49C2×#446C2211yescharacteristic
50C2×C8×Q8C2211yescharacteristic
51C2×C4×Q16—21no—
52C2×C4×Q16—21no—
53C2×#436C2211yescharacteristic
54C2×#435—21no—
55C2×C8×D4—21no—
56C2×C8×D4C2211yescharacteristic
57#2920C2212yes—
58C2×#433—21no—
59C2×#432C2211yescharacteristic
60C2×#405—21no—
61C22×C4×C8C2211yescharacteristic
62#1704C2218yescomplemented
63#1703C2218yescomplemented
64#1696C2214yescomplemented
65#1695C2214yescomplemented
66#1690C2214yescomplemented
67#1689C2214yescomplemented
68#1684C2218yescomplemented
69#1683C2218yescomplemented
70#1676C2214yescomplemented
71#1675C2214yescomplemented
72#1670C2214yescomplemented
73#1669C2214yescomplemented
74#1568C2212yescomplemented
75#1568C2212yescomplemented
76#1567C2212yes—
77#1566C2212yes—
78#1564C2212yescomplemented
79#1564C2212yescomplemented
80#1558C2214yescomplemented
81#1556C2214yescomplemented
82#1554C2218yescomplemented
83#1550C2214yes—
84#1550C2214yes—
85C22×Q16—21no—
86C22×Q16—21no—
87C22×M16—21no—
88C23×C8—21no—
89C23×C8D411yescharacteristic
90C23×C8D411yescharacteristic
91#533—22no—
92#533—22no—
93#533—22no—
94#533—22no—
95#533—22no—
96#533—22no—
97#533C2314yes—
98#533C2314yes—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101#529—21no—
102#529—21no—
103#529—21no—
104#529—21no—
105#529D412yes—
106#529D412yes—
107#529C2312yes—
108#529C2312yes—
109C2×#126—21no—
110C2×#125—22no—
111C2×#125D411yescharacteristic
112C2×#124D411yescharacteristic
113C2×#123—21no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8D411yescharacteristic
117C2×#121D411yescharacteristic
118C2×C4×Q8—22no—
119C2×C4×Q8—22no—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—21no—
122C2×C4×D4C2311yescharacteristic
123C2×#117—21no—
124C2×#117—22no—
125C22×C42C2311yescharacteristic
126#501—24no—
127#501—24no—
128#500—21no—
129#500—21no—
130#500C2312yes—
131#500C2312yes—
132#499—28no—
133#494—21no—
134#494—21no—
135#494C2312yes—
136#494C2312yes—
137#489—24no—
138#489—24no—
139#488—28no—
140#487—22no—
141#487—22no—
142#487C2314yes—
143#487C2314yes—
144#485—28no—
145#484—24no—
146#483—24no—
147#482—28no—
148#481—24no—
149#480—24no—
150#479—28no—
151#478—28no—
152#477C2318yes—
153#476C2318yes—
154#475C2318yes—
155#474C2318yes—
156#467—22no—
157#467—22no—
158#467—22no—
159#467—22no—
160#467—22no—
161#467—22no—
162#467—22no—
163#467—22no—
164#446—21no—
165#446—21no—
166#446C2312yes—
167#446C2312yes—
168C8×Q8—21no—
169C8×Q8—21no—
170C8×Q8C2312yes—
171C8×Q8C2312yes—
172C4×Q16—22no—
173C4×Q16—22no—
174C4×Q16—22no—
175C4×Q16—22no—
176C4×Q16—22no—
177C4×Q16—22no—
178C4×Q16—22no—
179C4×Q16—22no—
180#436—21no—
181#436—21no—
182#436C2312yes—
183#436C2312yes—
184#435—22no—
185#435—22no—
186#435—22no—
187#435—22no—
188C8×D4—22no—
189C8×D4—22no—
190C8×D4—22no—
191C8×D4—22no—
192C8×D4—21no—
193C8×D4—21no—
194C8×D4C2312yes—
195C8×D4C2312yes—
196#433—22no—
197#433—22no—
198#433—22no—
199#433—22no—
200#432—21no—
201#432—21no—
202#432—22no—
203#432C2312yes—
204#432C2312yes—
205#432C2314yes—
206#431—22no—
207#431C2314yes—
208#428—24no—
209#428—24no—
210C2×#107—21no—
211C2×#107—21no—
212#423C2314yes—
213C2×C4⋊C8—21no—
214C2×C4⋊C8—21no—
215C2×C4⋊C8C2311yescharacteristic
216C2×C4⋊C8C2311yescharacteristic
217#419—24no—
218#418—24no—
219C2×#103—22no—
220C2×#103—22no—
221#407—22no—
222C2×C22⋊C8—21no—
223C2×C22⋊C8—21no—
224C2×C22⋊C8—21no—
225#405—22no—
226#405—22no—
227#405—22no—
228#405—22no—
229C4×M16—22no—
230C2×#97—21no—
231C2×#97—21no—
232C2×C4×C8—21no—
233C2×C4×C8D411yescharacteristic
234C2×C4×C8D411yescharacteristic
235C2×C4×C8D411yescharacteristic
236C2×C4×C8D411yescharacteristic
237C2×C4×C8D411yescharacteristic
238C2×C4×C8D411yescharacteristic
239C2×C4×C8C2311yescharacteristic
240C2×C4×C8C2311yescharacteristic
241C2×C4×C8C2311yescharacteristic
242C2×C4×C8C2311yescharacteristic
243C2×C4×C8C2311yescharacteristic
244C2×C4×C8C2311yescharacteristic
245#335C2314yes—
246#335C2314yes—
247#334C2314yes—
248#334C2314yes—
249#333C2318yes—
250#333C2318yes—
251#332C2318yes—
252#332C2318yes—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8C2×D411yescharacteristic
256C22×D4—21no—
257C22×D4C2×D411yescharacteristic
258C23×C4—21no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4#4111yescharacteristic
262C2×Q16—22no—
263C2×Q16—22no—
264C2×Q16—22no—
265C2×Q16—22no—
266C2×Q16—44no—
267C2×Q16—44no—
268C2×M16—24no—
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—44no—
272C22×C8—21no—
273C22×C8—21no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8—42no—
277C22×C8—42no—
278C22×C8—44no—
279C22×C8C2×D411yescharacteristic
280C22×C8C2×D411yescharacteristic
281C22×C8C2×D411yescharacteristic
282C22×C8C2×D411yescharacteristic
283C22×C8C2×D411yescharacteristic
284C22×C8C2×D411yescharacteristic
285C22×C8C2×D411yescharacteristic
286C22×C8C2×D411yescharacteristic
287C22×C8C2×D411yescharacteristic
288C22×C8C2×D411yescharacteristic
289C22×C8C2×D411yescharacteristic
290C22×C8C2×D411yescharacteristic
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8C2414yes—
298C4⋊Q8C2414yes—
299#126—22no—
300#126—22no—
301#126—22no—
302#126—22no—
303#125—21no—
304#125—21no—
305#125—24no—
306#125—24no—
307#125—42no—
308#125—42no—
309#125C2×D412yes—
310#125C2×D412yes—
311#124—21no—
312#124—21no—
313#124C2×D412yes—
314#124C2×D412yes—
315#123—22no—
316#123—22no—
317#123—22no—
318#123—22no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C22⋊Q8—24no—
322C22⋊Q8—24no—
323C22⋊Q8—24no—
324C22⋊Q8—24no—
325C22⋊Q8—42no—
326C22⋊Q8—42no—
327C22⋊Q8—42no—
328C22⋊Q8—42no—
329C22⋊Q8C2×D412yes—
330C22⋊Q8C2×D412yes—
331#121—21no—
332#121—21no—
333#121C2×D412yes—
334#121C2×D412yes—
335C4×Q8—21no—
336C4×Q8—21no—
337C4×Q8—24no—
338C4×Q8—24no—
339C4×Q8—24no—
340C4×Q8—24no—
341C4×Q8—42no—
342C4×Q8—42no—
343C4×Q8—42no—
344C4×Q8—42no—
345C4×Q8C2×D412yes—
346C4×Q8C2412yes—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—21no—
352C4×D4—21no—
353C4×D4C2×D412yes—
354C4×D4C2412yes—
355#117—22no—
356#117—22no—
357#117—22no—
358#117—22no—
359#117—24no—
360#117—24no—
361#117—42no—
362#117—42no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—42no—
372C2×C4⋊C4—42no—
373C2×C4⋊C4C2×D411yescharacteristic
374C2×C4⋊C4C2×D411yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—42no—
381C2×C42—21no—
382C2×C42—42no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42#4111yescharacteristic
390C2×C42#4111yescharacteristic
391C2×C42#4111yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395#107—22no—
396#107—22no—
397#107—22no—
398#107—22no—
399#107—28no—
400#107—28no—
401#106—28no—
402#106—28no—
403C4⋊C8—22no—
404C4⋊C8—22no—
405C4⋊C8—22no—
406C4⋊C8—22no—
407C4⋊C8—21no—
408C4⋊C8—21no—
409C4⋊C8—24no—
410C4⋊C8C2412yes—
411C4⋊C8C2412yes—
412C4⋊C8C2418yes—
413#103—28no—
414#103—28no—
415#103—24no—
416#103—24no—
417#103—24no—
418#103—24no—
419#102—28no—
420#102—28no—
421C22⋊C8—22no—
422C22⋊C8—22no—
423C22⋊C8—24no—
424C22⋊C8—24no—
425C22⋊C8—22no—
426C22⋊C8—22no—
427C22⋊C8—22no—
428C22⋊C8—22no—
429#97—22no—
430#97—22no—
431#97—22no—
432#97—22no—
433#97—24no—
434C4×C8—22no—
435C4×C8—22no—
436C4×C8—24no—
437C4×C8C2×D411yescharacteristic
438C4×C8C2×D411yescharacteristic
439C4×C8C2×D411yescharacteristic
440C4×C8C2×D411yescharacteristic
441C4×C8C2×D411yescharacteristic
442C4×C8C2×D411yescharacteristic
443C4×C8C2×D411yescharacteristic
444C4×C8C2×D411yescharacteristic
445C4×C8C2×D411yescharacteristic
446C4×C8C2×D411yescharacteristic
447C4×C8C2×D411yescharacteristic
448C4×C8C2×D411yescharacteristic
449C4×C8C2411yescharacteristic
450C4×C8C2411yescharacteristic
451C4×C8C2411yescharacteristic
452C4×C8C2411yescharacteristic
453C24—21no—
454C24#12111yescharacteristic
455C2×Q8—21no—
456C2×Q8—24no—
457C2×Q8—24no—
458C2×Q8—24no—
459C2×Q8—44no—
460C2×Q8—44no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—84no—
464C2×Q8C22×D412yes—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—21no—
468C2×D4—24no—
469C2×D4—44no—
470C2×D4C22×D412yes—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4#12111yescharacteristic
485C22×C4#12111yescharacteristic
486C22×C4#12111yescharacteristic
487C22×C4#12111yescharacteristic
488C22×C4#12111yescharacteristic
489C22×C4#12111yescharacteristic
490C22×C4#12111yescharacteristic
491C22×C4#12111yescharacteristic
492C22×C4#12111yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C2×#4111yescharacteristic
500C22×C4C2×#4111yescharacteristic
501C22×C4C2×#4111yescharacteristic
502Q16—44no—
503Q16—44no—
504Q16—44no—
505Q16—44no—
506M16—44no—
507M16—48no—
508M16—44no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—28no—
517C2×C8—22no—
518C2×C8—41no—
519C2×C8—41no—
520C2×C8—41no—
521C2×C8—41no—
522C2×C8—41no—
523C2×C8—41no—
524C2×C8—44no—
525C2×C8—42no—
526C2×C8—42no—
527C2×C8—44no—
528C2×C8C22×D411yescharacteristic
529C2×C8C22×D411yescharacteristic
530C2×C8C22×D411yescharacteristic
531C2×C8C22×D411yescharacteristic
532C2×C8C22×D411yescharacteristic
533C2×C8C22×D411yescharacteristic
534C2×C8C22×D411yescharacteristic
535C2×C8C22×D411yescharacteristic
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—22no—
546C4⋊C4—24no—
547C4⋊C4—24no—
548C4⋊C4—24no—
549C4⋊C4—24no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4C22×D412yes—
559C4⋊C4C22×D412yes—
560C22⋊C4—22no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C42—22no—
573C42—22no—
574C42—44no—
575C42—44no—
576C42#12111yescharacteristic
577C42#12111yescharacteristic
578C42#12111yescharacteristic
579C42#12111yescharacteristic
580C42#12111yescharacteristic
581C42#12111yescharacteristic
582C42C22×D411yescharacteristic
583C42C22×D411yescharacteristic
584C42C22×D411yescharacteristic
585C42C22×D411yescharacteristic
586C42C22×D411yescharacteristic
587C42C22×D411yescharacteristic
588C42C2×#4111yescharacteristic
589C42C2×#4111yescharacteristic
590C42C2×#4111yescharacteristic
591C42C2511yescharacteristic
592C23—22no—
593C23—22no—
594C23—22no—
595C23—41no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23C2×#12111yescharacteristic
600C23C2×#12111yescharacteristic
601C23C2×#12111yescharacteristic
602Q8—24no—
603Q8—42no—
604Q8—44no—
605Q8—44no—
606Q8—84no—
607Q8—84no—
608Q8—84no—
609D4—24no—
610D4—44no—
611D4—44no—
612D4—42no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4C2×#12111yescharacteristic
647C2×C4C2×#12111yescharacteristic
648C2×C4C2×#12111yescharacteristic
649C2×C4C2×#12111yescharacteristic
650C2×C4C2×#12111yescharacteristic
651C2×C4C2×#12111yescharacteristic
652C2×C4C2×#12111yescharacteristic
653C2×C4C2×#12111yescharacteristic
654C2×C4C2×#12111yescharacteristic
655C2×C4#57814yes—
656C2×C4C23×D411yescharacteristic
657C2×C4C23×D411yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C8—21no—
660C8—21no—
661C8—21no—
662C8—21no—
663C8—21no—
664C8—21no—
665C8—42no—
666C8—42no—
667C8—48no—
668C8#57512yes—
669C8#57512yes—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—44no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—84no—
682C22#305912yes—
683C22#306712yes—
684C22C22×#12111yescharacteristic
685C4—21no—
686C4—21no—
687C4—21no—
688C4—21no—
689C4—21no—
690C4—21no—
691C4—21no—
692C4—21no—
693C4—21no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—84no—
698C4—84no—
699C4#305912yes—
700C4C2×#57511yescharacteristic
701C4C2×#57511yescharacteristic
702C4C2×#57812yes—
703C2—22no—
704C2—22no—
705C2—41no—
706C2—82no—
707C2#6041511yescharacteristic
708C2C2×#305911yescharacteristic
709C2C2×#306711yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radicalC2×#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3170C2semidirect
4. #19365C2semidirect
5. #19359C2semidirect
6. #19354C2semidirect
7. #19287C2semidirect
8. #19281C2semidirect
9. #19262C2semidirect
10. #19256C2semidirect
11. #19226C2semidirect
12. #19220C2semidirect
13. #17793C2semidirect
14. #17748C2semidirect
28. #3170C22semidirect
29. #3170C22semidirect
31. C2×#500C22semidirect
32. C2×#494C22semidirect
34. C2×#487C22semidirect
62. #1704C22semidirect
63. #1703C22semidirect
64. #1696C22semidirect
65. #1695C22semidirect
66. #1690C22semidirect
67. #1689C22semidirect
68. #1684C22semidirect
69. #1683C22semidirect
70. #1676C22semidirect
71. #1675C22semidirect
72. #1670C22semidirect
73. #1669C22semidirect
74. #1568C22semidirect
75. #1568C22semidirect
78. #1564C22semidirect
79. #1564C22semidirect
80. #1558C22semidirect
81. #1556C22semidirect
82. #1554C22semidirect

Maximal subgroups

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

subgroupindex
#608662
C2×#31702
C2×#29662
#193652
#193592
#193542
#192872
#192812
#192622
#192562
#192262
#192202
#177932
#177482