Finite Group Databasebeta

← #6345651

Subgroups — #6345651

Order 512.

1541 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#60950C211yesmaximal, characteristic
2#60264C211yesmaximal, characteristic
3#60251C211yesmaximal, characteristic, complemented
4#18023C214yesmaximal, complemented
5#18017C214yesmaximal, complemented
6#18003C212yesmaximal, complemented
7#18000C212yesmaximal, complemented
8#17998C212yesmaximal, complemented
9#17995C212yesmaximal, complemented
10#17988C214yesmaximal, complemented
11#17969C214yesmaximal, complemented
12#17822C211yesmaximal, characteristic, complemented
13C2×#1494C211yesmaximal, characteristic
14C2×#1493C211yesmaximal, characteristic, complemented
15#17323C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#529C2211yescharacteristic
18#3252C412yes—
19#3252C412yes—
20#3238C412yes—
21#3236C412yes—
22#3218C412yes—
23#3215C412yes—
24#3195C412yes—
25#3193C412yes—
26#3172C2214yes—
27#3108C2214yes—
28#3104C2212yes—
29#3101C2212yes—
30#2995—22no—
31#2994C2214yes—
32#2993C2214yes—
33#2992—22no—
34#2987—22no—
35#2985C2214yescomplemented
36#2984C2214yescomplemented
37#2983—22no—
38Q8×M16C2212yes—
39#2962C2212yes—
40#2962C2212yes—
41#2961—21no—
42#2961—21no—
43D4×M16—22no—
44D4×M16C2212yes—
45#2924—22no—
46#2921—21no—
47#2920C2211yescharacteristic
48#2920C2211yescharacteristic
49C2×#432C2211yescharacteristic
50#2905—21no—
51C2×#423C2211yescharacteristic
52C22×C4×C8C2211yescharacteristic
53#1593C2214yescomplemented
54#1592C2214yescomplemented
55#1591C2214yescomplemented
56#1590C2214yescomplemented
57#1588C2214yescomplemented
58#1587C2214yescomplemented
59#1587C2214yescomplemented
60#1585C2214yescomplemented
61#1584C2214yescomplemented
62#1584C2214yescomplemented
63#1583C2214yescomplemented
64#1582C2214yescomplemented
65#1579C2214yescomplemented
66SD16⋊C8C2214yescomplemented
67C8×Q16C2214yescomplemented
68C8×SD16C2214yescomplemented
69#1573C2214yescomplemented
70#1571C2212yes—
71#1570C2212yes—
72#1569C2214yescomplemented
73#1494C2212yes—
74#1494C2212yes—
75#1494C2212yes—
76#1494C2212yes—
77#1493C2212yes—
78#1493C2212yes—
79#1493C2212yescomplemented
80#1493C2212yescomplemented
81C2×Q8⋊C8C2211yescharacteristic
82C2×Q8⋊C8C2211yescharacteristic
83C2×Q8⋊C8C2212yescomplemented
84C2×Q8⋊C8C2212yescomplemented
85C2×D4⋊C8C2211yescharacteristic
86C2×D4⋊C8C2211yescharacteristic
87#1452C2212yescomplemented
88#1451C2212yes—
89#1451C2212yescomplemented
90#1450C2212yescomplemented
91C8×M16C2212yes—
92C8×M16C2212yescomplemented
93C22×M16—21no—
94C22×M16—22no—
95C23×C8—21no—
96#533—21no—
97#533—21no—
98#533—21no—
99#533—21no—
100#533C2×C412yes—
101#533C2×C412yes—
102#533C2312yes—
103#533C2312yes—
104C2×C4⋊Q8C2×C411yescharacteristic
105C2×C4⋊Q8C2×C411yescharacteristic
106#529—21no—
107#529—21no—
108#529—21no—
109#529—21no—
110#529C2×C412yes—
111#529C2×C412yes—
112#529C2312yes—
113#529C2312yes—
114C2×#126—21no—
115C2×#125—21no—
116C2×#125C2×C411yescharacteristic
117C2×#124C2×C411yescharacteristic
118C2×#123—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8C2×C411yescharacteristic
121C2×C22⋊Q8C2×C412yes—
122C2×#121C2×C411yescharacteristic
123C2×C4×Q8—21no—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2312yes—
126C2×C4×D4—21no—
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117—21no—
130C22×C42C2311yescharacteristic
131#501C2×C414yes—
132#497C2×C414yes—
133#491C2×C414yes—
134#489C2×C414yes—
135#484C2×C414yes—
136#483C2×C414yes—
137#483C2×C414yes—
138#480C2×C414yes—
139#479C2×C414yes—
140#479C2×C414yes—
141#477C2×C412yes—
142#477C2×C412yes—
143#476C2×C412yes—
144#476C2×C412yes—
145#470C2×C414yes—
146#470C2×C414yes—
147#468C2×C414yes—
148#468C2×C414yes—
149#464C2×C414yes—
150#464C2×C414yes—
151#462C2×C412yes—
152#462C2×C412yes—
153#461C2×C412yes—
154#461C2×C412yes—
155#451C2×C414yes—
156#451C2×C414yes—
157#450C2×C414yes—
158#448C2×C414yes—
159#446—24no—
160#446—24no—
161#446—24no—
162#446—24no—
163#446—24no—
164#446—24no—
165#446—24no—
166#446C2314yes—
167#446C2314yes—
168#446C2314yes—
169C8×Q8C2314yes—
170C8×Q8C2314yes—
171C8×Q8C2314yes—
172C4×Q16C2314yes—
173C4×Q16C2314yes—
174C4×SD16C2314yes—
175C4×SD16C2314yes—
176#436—24no—
177#436C2314yes—
178#435—24no—
179#435—24no—
180#435—24no—
181C8×D4—24no—
182C8×D4C2314yes—
183#433—24no—
184#433—24no—
185#433—24no—
186#433—24no—
187#432—21no—
188#432—21no—
189#432—24no—
190#432C2×C412yes—
191#432C2×C412yes—
192#432D412yes—
193#432D412yes—
194#432C2312yes—
195#432C2312yes—
196#431—24no—
197#431C2312yes—
198#431C2312yes—
199#431C2312yes—
200#431C2312yes—
201#424—22no—
202#424—22no—
203#424—24no—
204#424—24no—
205#423—22no—
206#423—21no—
207#423—21no—
208#423C2×C411yescharacteristic
209#423C2×C411yescharacteristic
210#423D411yescharacteristic
211#423D411yescharacteristic
212#423C2312yes—
213#423C2312yes—
214C2×C4⋊C8—21no—
215C2×C4⋊C8C2311yescharacteristic
216C2×C4⋊C8C2311yescharacteristic
217C2×C4⋊C8C2312yes—
218#407—22no—
219#407C2×C412yes—
220#407D412yes—
221C2×C22⋊C8—21no—
222#405—22no—
223#405—22no—
224C4×M16—22no—
225C4×M16C2312yes—
226C4×M16C2312yes—
227C2×#97—21no—
228C2×C4×C8—21no—
229C2×C4×C8—21no—
230C2×C4×C8—21no—
231C2×C4×C8C2311yescharacteristic
232C2×C4×C8C2311yescharacteristic
233C2×C4×C8C2311yescharacteristic
234C2×C4×C8C2311yescharacteristic
235C2×C4×C8C2311yescharacteristic
236C2×C4×C8C2311yescharacteristic
237#335C2314yes—
238#335C2314yes—
239#334C2314yes—
240#334C2314yes—
241Q8⋊C8C2314yes—
242Q8⋊C8C2312yes—
243Q8⋊C8C2314yes—
244Q8⋊C8C2312yes—
245Q8⋊C8C2314yes—
246Q8⋊C8C2312yes—
247Q8⋊C8C2314yes—
248Q8⋊C8C2312yes—
249D4⋊C8C2312yes—
250D4⋊C8C2312yes—
251D4⋊C8C2312yes—
252D4⋊C8C2312yes—
253#322C2314yes—
254#322C2314yes—
255#322C2314yes—
256C82C2314yes—
257C22×Q8—21no—
258C22×Q8C22×C411yescharacteristic
259C22×Q8C22×C412yes—
260C22×D4—21no—
261C22×D4C22×C411yescharacteristic
262C23×C4—21no—
263C23×C4C22×C411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4#4111yescharacteristic
266C2×Q16C22×C414yes—
267C2×Q16C22×C414yes—
268C2×SD16C22×C414yes—
269C2×SD16C22×C414yes—
270C2×M16—22no—
271C2×M16—22no—
272C2×M16—24no—
273C2×M16—84no—
274C2×M16—82no—
275C2×M16C22×C412yes—
276C2×M16C22×C412yes—
277C2×M16C2×D412yes—
278C2×M16C2×D412yes—
279C22×C8—21no—
280C22×C8—21no—
281C22×C8—21no—
282C22×C8—21no—
283C22×C8—21no—
284C22×C8—21no—
285C22×C8—21no—
286C22×C8—21no—
287C22×C8—22no—
288C22×C8—24no—
289C22×C8—44no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8C22×C412yes—
295C4⋊Q8C22×C412yes—
296C4⋊Q8C22×C412yes—
297C4⋊Q8C22×C412yes—
298#126—22no—
299#126—22no—
300#126—22no—
301#126—22no—
302#125—22no—
303#125—22no—
304#125—21no—
305#125—21no—
306#125—22no—
307#125—22no—
308#125C22×C412yes—
309#125C22×C412yes—
310#124—21no—
311#124—21no—
312#124C22×C412yes—
313#124C22×C412yes—
314#123—22no—
315#123—22no—
316#123—22no—
317#123—22no—
318C22⋊Q8—22no—
319C22⋊Q8—22no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322C22⋊Q8—22no—
323C22⋊Q8—22no—
324C22⋊Q8—22no—
325C22⋊Q8—22no—
326C22⋊Q8C22×C412yes—
327C22⋊Q8C22×C412yes—
328C22⋊Q8C22×C414yes—
329C22⋊Q8C22×C414yes—
330#121—21no—
331#121—21no—
332#121C22×C412yes—
333#121C22×C412yes—
334C4×Q8—22no—
335C4×Q8—22no—
336C4×Q8—22no—
337C4×Q8—22no—
338C4×Q8—21no—
339C4×Q8—22no—
340C4×Q8—21no—
341C4×Q8—22no—
342C4×Q8C22×C412yes—
343C4×Q8C22×C414yes—
344C4×Q8C2412yes—
345C4×Q8C2414yes—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—22no—
350C4×D4—21no—
351C4×D4—21no—
352C4×D4C22×C412yes—
353C4×D4C2412yes—
354#117—22no—
355#117—22no—
356#117—22no—
357#117—22no—
358#117—22no—
359#117—22no—
360#117—22no—
361#117—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4C22×C411yescharacteristic
372C2×C4⋊C4C22×C411yescharacteristic
373C2×C4⋊C4C22×C412yes—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C42—21no—
381C2×C42—22no—
382C2×C42C22×C411yescharacteristic
383C2×C42C22×C411yescharacteristic
384C2×C42C22×C411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42#4111yescharacteristic
389C2×C42#4111yescharacteristic
390C2×C42#4111yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394#107C22×C414yes—
395#107C22×C414yes—
396#106C22×C414yes—
397#106C22×C414yes—
398C4⋊C8—24no—
399C4⋊C8—24no—
400C4⋊C8—24no—
401C4⋊C8—24no—
402C4⋊C8—24no—
403C4⋊C8—24no—
404C4⋊C8—22no—
405C4⋊C8—22no—
406C4⋊C8—22no—
407C4⋊C8—21no—
408C4⋊C8—21no—
409C4⋊C8C22×C412yes—
410C4⋊C8C22×C412yes—
411C4⋊C8C22×C412yes—
412C4⋊C8C22×C412yes—
413C4⋊C8C2×D412yes—
414C4⋊C8C2×D412yes—
415C4⋊C8C2×D412yes—
416C4⋊C8C2×D412yes—
417C4⋊C8C2412yes—
418C4⋊C8C2412yes—
419C4⋊C8C2414yes—
420#103C22×C414yes—
421#103C22×C414yes—
422#103C22×C414yes—
423#103C22×C414yes—
424#103C22×C414yes—
425#103C22×C414yes—
426#102C22×C414yes—
427#102C22×C414yes—
428C22⋊C8—22no—
429C22⋊C8—22no—
430C22⋊C8—24no—
431C22⋊C8—24no—
432C22⋊C8C22×C414yes—
433C22⋊C8C22×C414yes—
434C22⋊C8C2×D414yes—
435C22⋊C8C2×D414yes—
436#97—24no—
437#97—24no—
438#97—24no—
439#97—22no—
440#97—22no—
441#97C22×C414yes—
442#97C2×D414yes—
443C4×C8—24no—
444C4×C8—24no—
445C4×C8—24no—
446C4×C8—21no—
447C4×C8—21no—
448C4×C8—21no—
449C4×C8—21no—
450C4×C8—21no—
451C4×C8—21no—
452C4×C8C2411yescharacteristic
453C4×C8C2411yescharacteristic
454C4×C8C2411yescharacteristic
455C4×C8C2411yescharacteristic
456C4×C8C2414yes—
457C4×C8C2414yes—
458C24—21no—
459C24C4×D411yescharacteristic
460C2×Q8—21no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—41no—
465C2×Q8—41no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—82no—
469C2×Q8C23×C412yes—
470C2×Q8C23×C414yes—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—21no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4—82no—
477C2×D4C23×C412yes—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4C4×D411yescharacteristic
493C22×C4C4×D411yescharacteristic
494C22×C4C4×D411yescharacteristic
495C22×C4C4×D411yescharacteristic
496C22×C4C4×D411yescharacteristic
497C22×C4C4×D411yescharacteristic
498C22×C4C4×D411yescharacteristic
499C22×C4C4×D411yescharacteristic
500C22×C4C4×D411yescharacteristic
501C22×C4C23×C411yescharacteristic
502C22×C4C23×C411yescharacteristic
503C22×C4C23×C411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C2×#4111yescharacteristic
508C22×C4C2×#4111yescharacteristic
509C22×C4C2×#4111yescharacteristic
510Q16—44no—
511Q16—44no—
512SD16—44no—
513SD16—44no—
514M16—24no—
515M16—24no—
516M16—42no—
517M16—42no—
518M16—44no—
519M16—84no—
520M16—82no—
521C2×C8—24no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—24no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—42no—
530C2×C8—42no—
531C2×C8—42no—
532C2×C8—42no—
533C2×C8—42no—
534C2×C8—42no—
535C2×C8—42no—
536C2×C8—84no—
537C2×C8—82no—
538C2×C8C23×C414yes—
539C2×C8C23×C414yes—
540C2×C8C22×D414yes—
541C2×C8C22×D414yes—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—24no—
555C4⋊C4—24no—
556C4⋊C4—24no—
557C4⋊C4—22no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—24no—
562C4⋊C4—22no—
563C4⋊C4C23×C414yes—
564C4⋊C4C23×C412yes—
565C4⋊C4C23×C412yes—
566C22⋊C4—24no—
567C22⋊C4—24no—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—22no—
573C22⋊C4—22no—
574C22⋊C4—22no—
575C22⋊C4—22no—
576C22⋊C4—22no—
577C22⋊C4—22no—
578C42—22no—
579C42—24no—
580C42—22no—
581C42—24no—
582C42C4×D411yescharacteristic
583C42C4×D411yescharacteristic
584C42C4×D411yescharacteristic
585C42C4×D411yescharacteristic
586C42C4×D411yescharacteristic
587C42C4×D411yescharacteristic
588C42C23×C411yescharacteristic
589C42C23×C411yescharacteristic
590C42C23×C411yescharacteristic
591C42C22×D411yescharacteristic
592C42C22×D411yescharacteristic
593C42C22×D411yescharacteristic
594C42C2×#4111yescharacteristic
595C42C2×#4111yescharacteristic
596C42C2×#4111yescharacteristic
597C42C2511yescharacteristic
598C23—22no—
599C23—22no—
600C23—22no—
601C23—41no—
602C23—42no—
603C23—81no—
604C23—81no—
605C23C2×C4×D411yescharacteristic
606C23C2×C4×D411yescharacteristic
607C23C2×C4×D411yescharacteristic
608Q8—24no—
609Q8—24no—
610Q8—22no—
611Q8—22no—
612Q8—81no—
613Q8—82no—
614Q8—82no—
615Q8—82no—
616D4—22no—
617D4—22no—
618D4—44no—
619D4—81no—
620D4—82no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—22no—
637C2×C4—24no—
638C2×C4—22no—
639C2×C4—24no—
640C2×C4—22no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—81no—
654C2×C4—82no—
655C2×C4—81no—
656C2×C4C2×C4×D411yescharacteristic
657C2×C4C2×C4×D411yescharacteristic
658C2×C4C2×C4×D411yescharacteristic
659C2×C4C2×C4×D411yescharacteristic
660C2×C4C2×C4×D411yescharacteristic
661C2×C4C2×C4×D411yescharacteristic
662C2×C4C2×C4×D411yescharacteristic
663C2×C4C2×C4×D411yescharacteristic
664C2×C4C2×C4×D411yescharacteristic
665C2×C4#57712yes—
666C2×C4#57812yes—
667C2×C4C24×C411yescharacteristic
668C2×C4C23×D411yescharacteristic
669C2×C4C22×#4111yescharacteristic
670C8—22no—
671C8—22no—
672C8—24no—
673C8—24no—
674C8—24no—
675C8—24no—
676C8—41no—
677C8—41no—
678C8—42no—
679C8—44no—
680C8—42no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—21no—
687C22—21no—
688C22—44no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—162no—
693C22#294712yes—
694C22#295612yes—
695C22C22×C4×D411yescharacteristic
696C4—21no—
697C4—21no—
698C4—21no—
699C4—21no—
700C4—21no—
701C4—21no—
702C4—21no—
703C4—21no—
704C4—21no—
705C4—42no—
706C4—44no—
707C4—42no—
708C4—42no—
709C4—44no—
710C4#294712yes—
711C4#357311yescharacteristic
712C4#357311yescharacteristic
713C4C2×#57711yescharacteristic
714C4C2×#57811yescharacteristic
715C2—22no—
716C2—22no—
717C2—41no—
718C2—82no—
719C2#6014311yescharacteristic
720C2C2×#294711yescharacteristic
721C2C2×#295611yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radicalC2×#121C2×C4
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. #18023C2semidirect
5. #18017C2semidirect
6. #18003C2semidirect
7. #18000C2semidirect
8. #17998C2semidirect
9. #17995C2semidirect
10. #17988C2semidirect
11. #17969C2semidirect
12. #17822C2semidirect
14. C2×#1493C2semidirect
15. #17323C2semidirect
35. #2985C22semidirect
36. #2984C22semidirect
53. #1593C22semidirect
54. #1592C22semidirect
55. #1591C22semidirect
56. #1590C22semidirect
57. #1588C22semidirect
58. #1587C22semidirect
59. #1587C22semidirect
60. #1585C22semidirect
61. #1584C22semidirect
62. #1584C22semidirect
63. #1583C22semidirect
64. #1582C22semidirect
65. #1579C22semidirect
66. SD16⋊C8C22semidirect
67. C8×Q16C22semidirect
68. C8×SD16C22semidirect
69. #1573C22semidirect
72. #1569C22semidirect
79. #1493C22semidirect
80. #1493C22semidirect
83. C2×Q8⋊C8C22semidirect
84. C2×Q8⋊C8C22semidirect
87. #1452C22semidirect
89. #1451C22semidirect
90. #1450C22semidirect
92. C8×M16C22semidirect

Maximal subgroups

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

subgroupindex
#609502
#602642
#602512
#180232
#180172
#180032
#180002
#179982
#179952
#179882
#179692
#178222
C2×#14942
C2×#14932
#173232