Finite Group Databasebeta

← #6345249

Subgroups — #6345249

Order 512.

1488 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60952C211yesmaximal, characteristic
2#60294C211yesmaximal, characteristic
3#60250C211yesmaximal, characteristic, complemented
4#17986C212yesmaximal, complemented
5#17984C212yesmaximal, complemented
6#17967C212yesmaximal, complemented
7#17965C212yesmaximal, complemented
8#17953C212yesmaximal, complemented
9#17951C212yesmaximal, complemented
10#17949C214yesmaximal, complemented
11#17925C212yesmaximal, complemented
12#17923C212yesmaximal, complemented
13#17921C214yesmaximal, complemented
14#17821C211yesmaximal, characteristic, complemented
15#17496C211yesmaximal, characteristic, complemented
16#17493C211yesmaximal, characteristic
17#17324C211yesmaximal, characteristic, complemented
18#3444C2211yescharacteristic
19C2×#533C2211yescharacteristic
20#3255C412yes—
21#3254C412yes—
22#3239C412yes—
23#3237C412yes—
24#3218C412yes—
25#3216C412yes—
26#3196C414yes—
27#3172C2212yes—
28#3172C2212yes—
29#3107C2214yes—
30#3104C2212yes—
31#3103C2212yes—
32#2990—22no—
33#2989C2212yes—
34#2988C2212yes—
35#2986—22no—
36#2985C2212yes—
37#2985C2211yescharacteristic
38#2985C2212yes—
39#2985C2211yescharacteristic
40#2984—22no—
41#2983C2212yes—
42#2982C2212yes—
43#2981C2211yescharacteristic
44#2981C2211yescharacteristic
45#2977—22no—
46#2968—22no—
47#2966C2212yes—
48#2966C2212yes—
49#2962—22no—
50#2960C2212yes—
51#2960C2212yes—
52#2925—22no—
53#2924—22no—
54#2920C2211yescharacteristic
55#2920C2211yescharacteristic
56#2920C2211yescharacteristic
57C2×#432C2211yescharacteristic
58C2×#431C2211yescharacteristic
59#1593C2214yescomplemented
60#1592C2214yescomplemented
61#1591C2214yescomplemented
62#1590C2214yescomplemented
63#1588C2214yescomplemented
64#1587C2214yescomplemented
65#1587C2214yescomplemented
66#1585C2214yescomplemented
67#1584C2214yescomplemented
68#1584C2214yescomplemented
69#1583C2214yescomplemented
70#1582C2214yescomplemented
71#1579C2214yescomplemented
72SD16⋊C8C2214yescomplemented
73C8×Q16C2214yescomplemented
74C8×SD16C2214yescomplemented
75#1573C2214yes—
76#1572C2212yes—
77#1572C2212yes—
78#1569C2212yes—
79#1569C2212yes—
80#1497C2212yes—
81#1497C2212yes—
82#1495C2212yes—
83#1495C2212yes—
84#1491C2212yes—
85#1491C2212yes—
86#1488C2214yes—
87#1485C2212yes—
88#1484C2212yes—
89#1481C2214yes—
90#1454C2212yes—
91#1454C2212yes—
92#1453C2212yes—
93#1453C2212yes—
94#1451C2211yescharacteristic
95#1451C2211yescharacteristic
96C8×M16C2211yescharacteristic
97C8×M16C2211yescharacteristic
98#533—21no—
99#533—21no—
100#533—21no—
101#533—21no—
102#533C2×C412yes—
103#533C2×C412yes—
104#533C2×C411yescharacteristic
105#533C2×C411yescharacteristic
106#533C2311yescharacteristic
107#533C2312yes—
108#533C2312yes—
109#532C2×C411yescharacteristic
110#532C2×C411yescharacteristic
111#532C2311yescharacteristic
112C2×C4⋊Q8C2×C411yescharacteristic
113C2×C4⋊Q8C2×C411yescharacteristic
114#529—21no—
115#529—21no—
116#529—21no—
117#529—21no—
118#529C2×C412yes—
119#529C2312yes—
120C2×#125—21no—
121#525—21no—
122#525—21no—
123#525C2×C412yes—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8C2×C412yes—
126C4×#41—21no—
127C4×#41—21no—
128C4×#41C2312yes—
129C2×C4×Q8—21no—
130C2×C4×Q8C2312yes—
131C2×#117—21no—
132C22×C42C2311yescharacteristic
133#501C2×C412yes—
134#501C2×C412yes—
135#497C2×C412yes—
136#497C2×C412yes—
137#491C2×C412yes—
138#491C2×C412yes—
139#489C2×C412yes—
140#489C2×C412yes—
141#485C2×C414yes—
142#485C2×C414yes—
143#484C2×C414yes—
144#481C2×C414yes—
145#477C2×C414yes—
146#475C2×C414yes—
147#475C2×C414yes—
148#474C2×C414yes—
149#470C2×C414yes—
150#470C2×C414yes—
151#468C2×C414yes—
152#468C2×C414yes—
153#462C2×C414yes—
154#462C2×C414yes—
155#461C2×C414yes—
156#460C2×C414yes—
157#452C2×C414yes—
158#452C2×C414yes—
159#451C2×C414yes—
160#448C2×C414yes—
161#446—24no—
162#446—24no—
163#446—24no—
164#446—22no—
165#446—22no—
166#446—22no—
167#446—22no—
168#446C2314yes—
169#446C2312yes—
170#446C2314yes—
171#446C2312yes—
172C8×Q8—24no—
173C8×Q8—22no—
174C8×Q8—22no—
175C8×Q8C2314yes—
176C8×Q8C2312yes—
177C8×Q8C2314yes—
178C8×Q8C2312yes—
179C4×Q16C2314yes—
180C4×Q16C2314yes—
181C4×SD16C2314yes—
182C4×SD16C2314yes—
183#436C2312yes—
184#436C2312yes—
185#435—22no—
186#435—22no—
187#435—22no—
188#435—22no—
189#435—22no—
190#435—22no—
191C8×D4—22no—
192C8×D4—22no—
193C8×D4—22no—
194C8×D4—22no—
195C8×D4C2312yes—
196C8×D4C2312yes—
197#433—24no—
198#433—24no—
199#433—22no—
200#433—22no—
201#433—22no—
202#433—22no—
203#432—21no—
204#432—21no—
205#432—21no—
206#432—21no—
207#432—21no—
208#432—22no—
209#432—24no—
210#432C2×C412yes—
211#432C2×C412yes—
212#432C2312yes—
213#432C2311yescharacteristic
214#432C2311yescharacteristic
215#432C2312yes—
216#431—21no—
217#431—22no—
218#431—24no—
219#431D412yes—
220#431D412yes—
221#431C2312yes—
222#431C2311yescharacteristic
223#431C2311yescharacteristic
224#431C2312yes—
225#431C2312yes—
226#431C2312yes—
227#431C2311yescharacteristic
228#431C2311yescharacteristic
229#431C2311yescharacteristic
230#431C2311yescharacteristic
231#424—22no—
232#424—22no—
233#423—21no—
234#423—22no—
235#423C2×C412yes—
236#423C2312yes—
237C2×C4⋊C8—22no—
238C2×C4⋊C8D412yes—
239C2×C4⋊C8C2312yes—
240#408—22no—
241#408—22no—
242#407—21no—
243#407—21no—
244#407C2×C412yes—
245C2×C22⋊C8—21no—
246C2×C22⋊C8D412yes—
247#405—22no—
248#405—22no—
249#405—22no—
250#405—22no—
251C4×M16—21no—
252C4×M16C2312yes—
253C4×M16C2311yescharacteristic
254C4×M16C2311yescharacteristic
255C2×#97—21no—
256C2×C4×C8C2311yescharacteristic
257C2×C4×C8C2311yescharacteristic
258#335C2314yes—
259#335C2314yes—
260#334C2314yes—
261#334C2314yes—
262Q8⋊C8C2314yes—
263Q8⋊C8C2314yes—
264Q8⋊C8C2314yes—
265Q8⋊C8C2314yes—
266Q8⋊C8C2314yes—
267Q8⋊C8C2314yes—
268D4⋊C8C2314yes—
269D4⋊C8C2314yes—
270#322C2312yes—
271#322C2312yes—
272#322C2312yes—
273#322C2312yes—
274#322C2312yes—
275#322C2312yes—
276C82C2312yes—
277C82C2312yes—
278C2×#41—21no—
279C2×#41—21no—
280C2×#41C22×C412yes—
281C22×Q8—21no—
282C22×Q8C22×C412yes—
283C23×C4C22×C411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4#4111yescharacteristic
286C2×Q16C22×C414yes—
287C2×Q16C22×C414yes—
288C2×SD16C22×C414yes—
289C2×SD16C22×C414yes—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C2×M16—22no—
294C2×M16C22×C412yes—
295C2×M16C22×C412yes—
296C22×C8—22no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8C2×D412yes—
300C22×C8C2×D412yes—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8C22×C412yes—
306C4⋊Q8C22×C412yes—
307C4⋊Q8C22×C411yescharacteristic
308C4⋊Q8C22×C412yes—
309C4⋊Q8C22×C412yes—
310C4⋊Q8C22×C411yescharacteristic
311C4⋊Q8C22×C411yescharacteristic
312#127C22×C411yescharacteristic
313#126—22no—
314#126—22no—
315#125—22no—
316#125—22no—
317#125—21no—
318#125—22no—
319#125—22no—
320#125—21no—
321#125C22×C412yes—
322#124—21no—
323#124—21no—
324#124C22×C412yes—
325#123—22no—
326#123—22no—
327#123—22no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334C22⋊Q8—21no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—21no—
338C22⋊Q8C22×C414yes—
339C22⋊Q8C22×C414yes—
340C22⋊Q8C22×C412yes—
341C22⋊Q8C22×C412yes—
342#121—21no—
343#121—21no—
344#121—21no—
345#121—21no—
346#121C22×C412yes—
347#121C22×C412yes—
348#120—22no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—21no—
352C4×Q8—21no—
353C4×Q8—21no—
354C4×Q8—22no—
355C4×Q8—22no—
356C4×Q8—22no—
357C4×Q8—21no—
358C4×Q8—22no—
359C4×Q8C22×C414yes—
360C4×Q8C22×C412yes—
361C4×Q8C2414yes—
362C4×Q8C2412yes—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—21no—
370C4×D4—21no—
371C4×D4—21no—
372C4×D4—21no—
373C4×D4C22×C412yes—
374C4×D4C2412yes—
375#117—22no—
376#117—22no—
377#117—21no—
378#117—22no—
379#117—22no—
380#117—21no—
381#117—22no—
382#117—22no—
383#117—22no—
384#117C22×C412yes—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4C22×C412yes—
391C2×C22⋊C4—22no—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42C22×C411yescharacteristic
395C2×C42C22×C411yescharacteristic
396C2×C42C22×C411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42#4111yescharacteristic
401C2×C42#4111yescharacteristic
402C2×C42#4111yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406#107C22×C414yes—
407#107C22×C414yes—
408#106C22×C414yes—
409#106C22×C414yes—
410C4⋊C8—22no—
411C4⋊C8—22no—
412C4⋊C8—22no—
413C4⋊C8—22no—
414C4⋊C8—24no—
415C4⋊C8—24no—
416C4⋊C8—24no—
417C4⋊C8—24no—
418C4⋊C8—22no—
419C4⋊C8—22no—
420C4⋊C8—22no—
421C4⋊C8—21no—
422C4⋊C8—21no—
423C4⋊C8C22×C414yes—
424C4⋊C8C22×C414yes—
425C4⋊C8C2×D412yes—
426C4⋊C8C2×D412yes—
427C4⋊C8C2×D412yes—
428C4⋊C8C2×D412yes—
429C4⋊C8C2414yes—
430C4⋊C8C2412yes—
431C4⋊C8C2412yes—
432#103C22×C414yes—
433#103C22×C414yes—
434#103C22×C414yes—
435#103C22×C414yes—
436#103C22×C414yes—
437#103C22×C414yes—
438#102C22×C414yes—
439#102C22×C414yes—
440C22⋊C8—21no—
441C22⋊C8—21no—
442C22⋊C8—22no—
443C22⋊C8—22no—
444C22⋊C8—22no—
445C22⋊C8—22no—
446C22⋊C8—21no—
447C22⋊C8—21no—
448C22⋊C8—22no—
449C22⋊C8—22no—
450C22⋊C8—22no—
451C22⋊C8—22no—
452C22⋊C8C22×C414yes—
453C22⋊C8C22×C414yes—
454C22⋊C8C2×D412yes—
455C22⋊C8C2×D412yes—
456C22⋊C8C2×D412yes—
457C22⋊C8C2×D412yes—
458#97—22no—
459#97—22no—
460#97—24no—
461#97—24no—
462#97—22no—
463#97—21no—
464#97—21no—
465#97—22no—
466#97—22no—
467#97C22×C412yes—
468#97C22×C412yes—
469C4×C8—22no—
470C4×C8—22no—
471C4×C8—24no—
472C4×C8—24no—
473C4×C8—22no—
474C4×C8C2×D412yes—
475C4×C8C2×D412yes—
476C4×C8C2412yes—
477C4×C8C2412yes—
478C4×C8C2412yes—
479C4×C8C2412yes—
480C4×C8C2411yescharacteristic
481C4×C8C2411yescharacteristic
482C4×C8C2411yescharacteristic
483C4×C8C2411yescharacteristic
484C24C4×D411yescharacteristic
485#41—44no—
486#41—82no—
487#41—82no—
488C2×Q8—22no—
489C2×Q8—22no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—22no—
493C2×Q8—44no—
494C2×Q8—82no—
495C2×Q8C23×C412yes—
496C2×Q8C23×C414yes—
497C2×D4—22no—
498C2×D4—22no—
499C2×D4—22no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4C23×C412yes—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—81no—
511C22×C4C4×D411yescharacteristic
512C22×C4C4×D411yescharacteristic
513C22×C4C4×D411yescharacteristic
514C22×C4C4×D411yescharacteristic
515C22×C4C4×D411yescharacteristic
516C22×C4C4×D411yescharacteristic
517C22×C4C4×D411yescharacteristic
518C22×C4C4×D411yescharacteristic
519C22×C4C4×D411yescharacteristic
520C22×C4C23×C411yescharacteristic
521C22×C4C23×C411yescharacteristic
522C22×C4C23×C411yescharacteristic
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D411yescharacteristic
526C22×C4C2×#4111yescharacteristic
527C22×C4C2×#4111yescharacteristic
528C22×C4C2×#4111yescharacteristic
529Q16—44no—
530Q16—44no—
531SD16—44no—
532SD16—44no—
533M16—42no—
534M16—42no—
535M16—44no—
536M16—82no—
537M16—82no—
538M16—82no—
539C2×C8—24no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—42no—
545C2×C8—42no—
546C2×C8—82no—
547C2×C8—82no—
548C2×C8—82no—
549C2×C8C23×C414yes—
550C2×C8C23×C414yes—
551C2×C8C22×D412yes—
552C2×C8C22×D412yes—
553C2×C8C22×D412yes—
554C2×C8C22×D412yes—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—24no—
562C4⋊C4—24no—
563C4⋊C4—24no—
564C4⋊C4—22no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—21no—
569C4⋊C4—22no—
570C4⋊C4—22no—
571C4⋊C4—22no—
572C4⋊C4—22no—
573C4⋊C4—22no—
574C4⋊C4—24no—
575C4⋊C4—22no—
576C4⋊C4C23×C414yes—
577C4⋊C4C23×C412yes—
578C4⋊C4C23×C412yes—
579C22⋊C4—24no—
580C22⋊C4—24no—
581C22⋊C4—22no—
582C22⋊C4—22no—
583C22⋊C4—22no—
584C22⋊C4—22no—
585C22⋊C4—22no—
586C22⋊C4—22no—
587C22⋊C4—22no—
588C22⋊C4—22no—
589C22⋊C4—22no—
590C42—22no—
591C42—24no—
592C42—22no—
593C42—22no—
594C42—24no—
595C42C4×D411yescharacteristic
596C42C4×D411yescharacteristic
597C42C4×D411yescharacteristic
598C42C4×D411yescharacteristic
599C42C4×D411yescharacteristic
600C42C4×D411yescharacteristic
601C42C23×C411yescharacteristic
602C42C23×C411yescharacteristic
603C42C23×C411yescharacteristic
604C42C22×D411yescharacteristic
605C42C22×D411yescharacteristic
606C42C22×D411yescharacteristic
607C42C2×#4111yescharacteristic
608C42C2×#4111yescharacteristic
609C42C2×#4111yescharacteristic
610C42C2511yescharacteristic
611C23—22no—
612C23—41no—
613C23—41no—
614C23—41no—
615C23C2×C4×D411yescharacteristic
616C23C2×C4×D411yescharacteristic
617C23C2×C4×D411yescharacteristic
618Q8—24no—
619Q8—24no—
620Q8—24no—
621Q8—44no—
622Q8—44no—
623Q8—82no—
624Q8—81no—
625Q8—81no—
626D4—24no—
627D4—44no—
628D4—44no—
629D4—81no—
630D4—81no—
631D4—82no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—82no—
660C2×C4—82no—
661C2×C4C2×C4×D411yescharacteristic
662C2×C4C2×C4×D411yescharacteristic
663C2×C4C2×C4×D411yescharacteristic
664C2×C4C2×C4×D411yescharacteristic
665C2×C4C2×C4×D411yescharacteristic
666C2×C4C2×C4×D411yescharacteristic
667C2×C4C2×C4×D411yescharacteristic
668C2×C4C2×C4×D411yescharacteristic
669C2×C4C2×C4×D411yescharacteristic
670C2×C4C24×C411yescharacteristic
671C2×C4C23×D411yescharacteristic
672C2×C4C22×#4111yescharacteristic
673C8—22no—
674C8—22no—
675C8—22no—
676C8—22no—
677C8—24no—
678C8—24no—
679C8—42no—
680C8—42no—
681C8—42no—
682C8—42no—
683C8—44no—
684C22—21no—
685C22—21no—
686C22—21no—
687C22—21no—
688C22—21no—
689C22—21no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—82no—
697C22C22×C4×D411yescharacteristic
698C4—21no—
699C4—21no—
700C4—21no—
701C4—21no—
702C4—21no—
703C4—21no—
704C4—21no—
705C4—21no—
706C4—21no—
707C4—21no—
708C4—42no—
709C4—42no—
710C4—44no—
711C4—42no—
712C4—44no—
713C4#357312yes—
714C4#358612yes—
715C2—41no—
716C2—41no—
717C2—41no—
718C2—82no—
719C2#6014011yescharacteristic
720C2#6016911yescharacteristic
721C2#6018211yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#3444C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#431 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #60250C2semidirect
4. #17986C2semidirect
5. #17984C2semidirect
6. #17967C2semidirect
7. #17965C2semidirect
8. #17953C2semidirect
9. #17951C2semidirect
10. #17949C2semidirect
11. #17925C2semidirect
12. #17923C2semidirect
13. #17921C2semidirect
14. #17821C2semidirect
15. #17496C2semidirect
17. #17324C2semidirect
59. #1593C22semidirect
60. #1592C22semidirect
61. #1591C22semidirect
62. #1590C22semidirect
63. #1588C22semidirect
64. #1587C22semidirect
65. #1587C22semidirect
66. #1585C22semidirect
67. #1584C22semidirect
68. #1584C22semidirect
69. #1583C22semidirect
70. #1582C22semidirect
71. #1579C22semidirect
72. SD16⋊C8C22semidirect
73. C8×Q16C22semidirect
74. C8×SD16C22semidirect

Maximal subgroups

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

subgroupindex
#609522
#602942
#602502
#179862
#179842
#179672
#179652
#179532
#179512
#179492
#179252
#179232
#179212
#178212
#174962
#174932
#173242