Finite Group Databasebeta

← #6345523

Subgroups — #6345523

Order 512.

1527 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60947C211yesmaximal, characteristic
2#60265C211yesmaximal, characteristic
3#60251C211yesmaximal, characteristic, complemented
4#18072C212yesmaximal, complemented
5#18071C212yesmaximal, complemented
6#18068C212yesmaximal, complemented
7#18067C212yesmaximal, complemented
8#18023C214yesmaximal, complemented
9#18017C214yesmaximal, complemented
10#17949C214yesmaximal, complemented
11#17921C214yesmaximal, complemented
12C2×#1573C211yesmaximal, characteristic, complemented
13#17488C211yesmaximal, characteristic
14C2×#1481C211yesmaximal, characteristic, complemented
15#17326C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17C2×#533C2211yescharacteristic
18#3248C414yes—
19#3231C412yes—
20#3229C411yescharacteristic
21#3226C411yescharacteristic
22#3210C412yes—
23#3209C411yescharacteristic
24#3206C411yescharacteristic
25#3190C412yes—
26#3190C412yes—
27#3170C2212yes—
28#3167C2212yes—
29#3108C2212yes—
30#3108C2212yes—
31#3107C2214yes—
32#2994C2214yes—
33#2992—22no—
34#2990—22no—
35#2989C2212yes—
36#2988C2212yes—
37#2987—22no—
38#2985C2214yes—
39#2984C2214yes—
40#2983—22no—
41#2962C2212yes—
42#2962C2212yes—
43#2962C2212yes—
44#2961—21no—
45#2961—21no—
46#2935C2212yes—
47#2929—22no—
48#2925—21no—
49#2924—22no—
50#2920C2211yescharacteristic
51C2×#432C2211yescharacteristic
52C2×#431C2211yescharacteristic
53C2×#431C2211yescharacteristic
54#2906—21no—
55C22×C4⋊C8C2211yescharacteristic
56#1593C2214yescomplemented
57#1592C2214yescomplemented
58#1591C2214yescomplemented
59#1590C2214yescomplemented
60#1588C2214yescomplemented
61#1587C2214yescomplemented
62#1587C2214yescomplemented
63#1585C2214yescomplemented
64#1584C2214yescomplemented
65#1584C2214yescomplemented
66#1583C2214yescomplemented
67#1582C2214yescomplemented
68#1579C2214yescomplemented
69SD16⋊C8C2214yescomplemented
70C8×Q16C2214yescomplemented
71C8×SD16C2214yescomplemented
72#1573C2212yes—
73#1573C2212yes—
74#1573C2212yes—
75#1573C2212yes—
76C2×#335C2212yes—
77C2×#334C2212yes—
78#1494C2214yes—
79#1488C2214yes—
80#1481C2212yes—
81#1481C2212yes—
82#1481C2212yes—
83#1481C2212yes—
84#1481C2211yescharacteristic
85#1481C2211yescharacteristic
86#1480C2211yescharacteristic
87#1480C2211yescharacteristic
88C2×Q8⋊C8C2214yes—
89#1454C2214yes—
90#1452C2212yes—
91#1451C2212yes—
92#1450C2212yes—
93C8×M16C2212yes—
94C2×#131—22no—
95C23×C8—21no—
96#533—21no—
97#533—21no—
98#533—21no—
99#533—21no—
100#533C2×C412yes—
101#533C2×C412yes—
102#533C2×C411yescharacteristic
103#533C2×C411yescharacteristic
104#533C2312yes—
105#533C2312yes—
106#533C2311yescharacteristic
107#532C2×C411yescharacteristic
108#532C2×C411yescharacteristic
109#532C2311yescharacteristic
110C2×C4⋊Q8C2×C411yescharacteristic
111C2×C4⋊Q8C2×C411yescharacteristic
112#529—21no—
113#529—21no—
114#529—21no—
115#529—21no—
116#529C2×C412yes—
117#529C2312yes—
118C2×#125—21no—
119#525—21no—
120#525—21no—
121#525C2×C412yes—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8C2×C412yes—
124C4×#41—21no—
125C4×#41—21no—
126C4×#41C2312yes—
127C2×C4×Q8—21no—
128C2×C4×Q8C2312yes—
129C2×#117—21no—
130C22×C42C2311yescharacteristic
131#500C2×C412yes—
132#498C2×C412yes—
133#494C2×C412yes—
134#492C2×C412yes—
135#487C2×C412yes—
136#487C2×C412yes—
137#487C2×C412yes—
138#486C2×C412yes—
139#485C2×C414yes—
140#485C2×C414yes—
141#483C2×C412yes—
142#483C2×C412yes—
143#480C2×C412yes—
144#480C2×C412yes—
145#479C2×C414yes—
146#479C2×C414yes—
147#475C2×C414yes—
148#474C2×C414yes—
149#467C2×C414yes—
150#467C2×C414yes—
151#465C2×C414yes—
152#465C2×C414yes—
153#464C2×C414yes—
154#464C2×C414yes—
155#462C2×C414yes—
156#461C2×C414yes—
157#452C2×C414yes—
158#452C2×C414yes—
159#451C2×C412yes—
160#451C2×C412yes—
161#448C2×C412yes—
162#448C2×C412yes—
163#446—24no—
164#446—24no—
165#446—24no—
166#446—24no—
167#446—24no—
168#446—24no—
169#446—24no—
170#446C2314yes—
171#446C2314yes—
172#446C2314yes—
173C8×Q8C2314yes—
174C8×Q8C2314yes—
175C8×Q8C2314yes—
176C4×Q16C2314yes—
177C4×Q16C2314yes—
178C4×SD16C2314yes—
179C4×SD16C2314yes—
180#436C2314yes—
181#435—24no—
182#435—24no—
183#435—24no—
184C8×D4—24no—
185C8×D4—24no—
186C8×D4C2314yes—
187#433—24no—
188#433—24no—
189#433—24no—
190#432—21no—
191#432—21no—
192#432—22no—
193#432—24no—
194#432D412yes—
195#432D412yes—
196#432C2312yes—
197#432C2312yes—
198#431—21no—
199#431—21no—
200#431—22no—
201#431—24no—
202#431C2×C411yescharacteristic
203#431C2×C411yescharacteristic
204#431C2×C412yes—
205#431C2311yescharacteristic
206#431C2311yescharacteristic
207#431C2312yes—
208#431C2312yes—
209#431C2311yescharacteristic
210#431C2311yescharacteristic
211#431C2312yes—
212#431C2312yes—
213#424—22no—
214#424—22no—
215#424—22no—
216#424—24no—
217#423—22no—
218#423D411yescharacteristic
219#423D411yescharacteristic
220C2×C4⋊C8—21no—
221C2×C4⋊C8—21no—
222C2×C4⋊C8—21no—
223C2×C4⋊C8—21no—
224C2×C4⋊C8—22no—
225C2×C4⋊C8C2×C412yes—
226C2×C4⋊C8C2311yescharacteristic
227C2×C4⋊C8C2312yes—
228C2×C4⋊C8C2311yescharacteristic
229C2×C4⋊C8C2311yescharacteristic
230C2×C4⋊C8C2311yescharacteristic
231C2×C4⋊C8C2312yes—
232#408—22no—
233#407D412yes—
234C2×C22⋊C8—21no—
235C2×C22⋊C8—21no—
236C2×C22⋊C8C2×C411yescharacteristic
237C2×C22⋊C8C2×C411yescharacteristic
238#405—22no—
239#405—22no—
240#405—22no—
241C4×M16C2312yes—
242C2×#97—21no—
243C2×C4×C8C2312yes—
244C2×C4×C8C2312yes—
245#335C2312yes—
246#335C2312yes—
247#335C2312yes—
248#335C2312yes—
249#334C2312yes—
250#334C2312yes—
251#334C2312yes—
252#334C2312yes—
253Q8⋊C8C2314yes—
254Q8⋊C8C2314yes—
255Q8⋊C8C2314yes—
256Q8⋊C8C2314yes—
257Q8⋊C8C2314yes—
258Q8⋊C8C2314yes—
259D4⋊C8C2314yes—
260D4⋊C8C2314yes—
261#322C2314yes—
262#322C2314yes—
263#322C2314yes—
264C82C2314yes—
265C2×#41—21no—
266C2×#41—21no—
267C2×#41C22×C412yes—
268C22×Q8—21no—
269C22×Q8C22×C412yes—
270C23×C4C22×C411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4#4111yescharacteristic
273C2×Q16C22×C414yes—
274C2×Q16C22×C414yes—
275C2×SD16C22×C414yes—
276C2×SD16C22×C414yes—
277#131—84no—
278C2×M16—24no—
279C2×M16C2×D412yes—
280C2×M16C2×D412yes—
281C22×C8—22no—
282C22×C8—21no—
283C22×C8—21no—
284C22×C8—22no—
285C22×C8—21no—
286C22×C8—22no—
287C22×C8—21no—
288C22×C8—24no—
289C22×C8—82no—
290C22×C8C22×C412yes—
291C22×C8C22×C412yes—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8C22×C412yes—
297C4⋊Q8C22×C412yes—
298C4⋊Q8C22×C411yescharacteristic
299C4⋊Q8C22×C412yes—
300C4⋊Q8C22×C412yes—
301C4⋊Q8C22×C411yescharacteristic
302C4⋊Q8C22×C411yescharacteristic
303#127C22×C411yescharacteristic
304#126—22no—
305#126—22no—
306#125—22no—
307#125—22no—
308#125—21no—
309#125—22no—
310#125—22no—
311#125—21no—
312#125C22×C412yes—
313#124—21no—
314#124—21no—
315#124C22×C412yes—
316#123—22no—
317#123—22no—
318#123—22no—
319C22⋊Q8—22no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8—21no—
324C22⋊Q8—21no—
325C22⋊Q8—21no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C22⋊Q8—21no—
329C22⋊Q8C22×C414yes—
330C22⋊Q8C22×C414yes—
331C22⋊Q8C22×C412yes—
332C22⋊Q8C22×C412yes—
333#121—21no—
334#121—21no—
335#121—21no—
336#121—21no—
337#121C22×C412yes—
338#121C22×C412yes—
339#120—22no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8—21no—
343C4×Q8—21no—
344C4×Q8—21no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—21no—
349C4×Q8—22no—
350C4×Q8C22×C414yes—
351C4×Q8C22×C412yes—
352C4×Q8C2414yes—
353C4×Q8C2412yes—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4—21no—
364C4×D4C22×C412yes—
365C4×D4C2412yes—
366#117—22no—
367#117—22no—
368#117—21no—
369#117—22no—
370#117—22no—
371#117—21no—
372#117—22no—
373#117—22no—
374#117—22no—
375#117C22×C412yes—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4C22×C412yes—
382C2×C22⋊C4—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42C22×C411yescharacteristic
386C2×C42C22×C411yescharacteristic
387C2×C42C22×C411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42#4111yescharacteristic
392C2×C42#4111yescharacteristic
393C2×C42#4111yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397#107C22×C414yes—
398#107C22×C414yes—
399#106C22×C414yes—
400#106C22×C414yes—
401C4⋊C8—24no—
402C4⋊C8—24no—
403C4⋊C8—24no—
404C4⋊C8—24no—
405C4⋊C8—24no—
406C4⋊C8—24no—
407C4⋊C8—21no—
408C4⋊C8—21no—
409C4⋊C8—22no—
410C4⋊C8—22no—
411C4⋊C8—21no—
412C4⋊C8—21no—
413C4⋊C8—22no—
414C4⋊C8—21no—
415C4⋊C8—21no—
416C4⋊C8C22×C412yes—
417C4⋊C8C22×C412yes—
418C4⋊C8C22×C412yes—
419C4⋊C8C22×C412yes—
420C4⋊C8C2×D412yes—
421C4⋊C8C2×D412yes—
422C4⋊C8C2×D412yes—
423C4⋊C8C2×D412yes—
424C4⋊C8C2412yes—
425C4⋊C8C2412yes—
426C4⋊C8C2412yes—
427C4⋊C8C2412yes—
428#103C22×C414yes—
429#103C22×C414yes—
430#103C22×C414yes—
431#103C22×C414yes—
432#103C22×C414yes—
433#103C22×C414yes—
434#102C22×C414yes—
435#102C22×C414yes—
436C22⋊C8—22no—
437C22⋊C8—22no—
438C22⋊C8—22no—
439C22⋊C8—21no—
440C22⋊C8—21no—
441C22⋊C8—24no—
442C22⋊C8—24no—
443C22⋊C8C22×C411yescharacteristic
444C22⋊C8C22×C411yescharacteristic
445C22⋊C8C22×C411yescharacteristic
446C22⋊C8C22×C411yescharacteristic
447C22⋊C8C22×C412yes—
448C22⋊C8C22×C412yes—
449C22⋊C8C2×D414yes—
450C22⋊C8C2×D414yes—
451#97—24no—
452#97—24no—
453#97—24no—
454#97—22no—
455#97—22no—
456#97C2×D414yes—
457C4×C8—24no—
458C4×C8—24no—
459C4×C8—24no—
460C4×C8—22no—
461C4×C8C22×C412yes—
462C4×C8C22×C412yes—
463C4×C8C2412yes—
464C4×C8C2412yes—
465C4×C8C2412yes—
466C4×C8C2412yes—
467C4×C8C2414yes—
468C24C4×D411yescharacteristic
469#41—44no—
470#41—42no—
471#41—42no—
472#41—82no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—22no—
476C2×Q8—22no—
477C2×Q8—22no—
478C2×Q8—44no—
479C2×Q8—82no—
480C2×Q8C23×C412yes—
481C2×Q8C23×C414yes—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4C23×C412yes—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4C4×D411yescharacteristic
498C22×C4C4×D411yescharacteristic
499C22×C4C4×D411yescharacteristic
500C22×C4C4×D411yescharacteristic
501C22×C4C4×D411yescharacteristic
502C22×C4C4×D411yescharacteristic
503C22×C4C4×D411yescharacteristic
504C22×C4C4×D411yescharacteristic
505C22×C4C4×D411yescharacteristic
506C22×C4C23×C411yescharacteristic
507C22×C4C23×C411yescharacteristic
508C22×C4C23×C411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C2×#4111yescharacteristic
513C22×C4C2×#4111yescharacteristic
514C22×C4C2×#4111yescharacteristic
515Q16—44no—
516Q16—44no—
517SD16—44no—
518SD16—44no—
519M16—24no—
520M16—24no—
521M16—84no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—42no—
529C2×C8—42no—
530C2×C8—44no—
531C2×C8—42no—
532C2×C8—44no—
533C2×C8—42no—
534C2×C8—84no—
535C2×C8—81no—
536C2×C8—82no—
537C2×C8—81no—
538C2×C8C23×C412yes—
539C2×C8C23×C412yes—
540C2×C8C23×C412yes—
541C2×C8C23×C412yes—
542C2×C8C22×D414yes—
543C2×C8C22×D414yes—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—24no—
551C4⋊C4—24no—
552C4⋊C4—24no—
553C4⋊C4—22no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—24no—
564C4⋊C4—22no—
565C4⋊C4C23×C414yes—
566C4⋊C4C23×C412yes—
567C4⋊C4C23×C412yes—
568C22⋊C4—24no—
569C22⋊C4—24no—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—22no—
573C22⋊C4—22no—
574C22⋊C4—22no—
575C22⋊C4—22no—
576C22⋊C4—22no—
577C22⋊C4—22no—
578C22⋊C4—22no—
579C42—22no—
580C42—24no—
581C42—22no—
582C42—22no—
583C42—24no—
584C42C4×D411yescharacteristic
585C42C4×D411yescharacteristic
586C42C4×D411yescharacteristic
587C42C4×D411yescharacteristic
588C42C4×D411yescharacteristic
589C42C4×D411yescharacteristic
590C42C23×C411yescharacteristic
591C42C23×C411yescharacteristic
592C42C23×C411yescharacteristic
593C42C22×D411yescharacteristic
594C42C22×D411yescharacteristic
595C42C22×D411yescharacteristic
596C42C2×#4111yescharacteristic
597C42C2×#4111yescharacteristic
598C42C2×#4111yescharacteristic
599C42C2511yescharacteristic
600C23—22no—
601C23—22no—
602C23—22no—
603C23—41no—
604C23C2×C4×D411yescharacteristic
605C23C2×C4×D411yescharacteristic
606C23C2×C4×D411yescharacteristic
607Q8—24no—
608Q8—24no—
609Q8—24no—
610Q8—42no—
611Q8—82no—
612Q8—82no—
613Q8—82no—
614Q8—81no—
615D4—24no—
616D4—44no—
617D4—44no—
618D4—44no—
619D4—42no—
620D4—81no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4C2×C4×D411yescharacteristic
652C2×C4C2×C4×D411yescharacteristic
653C2×C4C2×C4×D411yescharacteristic
654C2×C4C2×C4×D411yescharacteristic
655C2×C4C2×C4×D411yescharacteristic
656C2×C4C2×C4×D411yescharacteristic
657C2×C4C2×C4×D411yescharacteristic
658C2×C4C2×C4×D411yescharacteristic
659C2×C4C2×C4×D411yescharacteristic
660C2×C4#57712yes—
661C2×C4#57812yes—
662C2×C4C24×C411yescharacteristic
663C2×C4C23×D411yescharacteristic
664C2×C4C22×#4111yescharacteristic
665C8—22no—
666C8—22no—
667C8—22no—
668C8—22no—
669C8—24no—
670C8—24no—
671C8—42no—
672C8—42no—
673C8—42no—
674C8—42no—
675C8—42no—
676C8—42no—
677C22—21no—
678C22—21no—
679C22—21no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—44no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22#294712yes—
689C22#295712yes—
690C22C22×C4×D411yescharacteristic
691C4—21no—
692C4—21no—
693C4—21no—
694C4—21no—
695C4—21no—
696C4—21no—
697C4—21no—
698C4—21no—
699C4—21no—
700C4—42no—
701C4—44no—
702C4—42no—
703C4—42no—
704C4—44no—
705C4#294712yes—
706C4#357312yes—
707C4C2×#57711yescharacteristic
708C4C2×#57811yescharacteristic
709C2—22no—
710C2—22no—
711C2—41no—
712C2—82no—
713C2#6014211yescharacteristic
714C2C2×#294711yescharacteristic
715C2C2×#295711yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#3444C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #2920 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #60251C2semidirect
4. #18072C2semidirect
5. #18071C2semidirect
6. #18068C2semidirect
7. #18067C2semidirect
8. #18023C2semidirect
9. #18017C2semidirect
10. #17949C2semidirect
11. #17921C2semidirect
12. C2×#1573C2semidirect
14. C2×#1481C2semidirect
15. #17326C2semidirect
56. #1593C22semidirect
57. #1592C22semidirect
58. #1591C22semidirect
59. #1590C22semidirect
60. #1588C22semidirect
61. #1587C22semidirect
62. #1587C22semidirect
63. #1585C22semidirect
64. #1584C22semidirect
65. #1584C22semidirect
66. #1583C22semidirect
67. #1582C22semidirect
68. #1579C22semidirect
69. SD16⋊C8C22semidirect
70. C8×Q16C22semidirect
71. C8×SD16C22semidirect

Maximal subgroups

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

subgroupindex
#609472
#602652
#602512
#180722
#180712
#180682
#180672
#180232
#180172
#179492
#179212
C2×#15732
#174882
C2×#14812
#173262