Finite Group Databasebeta

← #6345538

Subgroups — #6345538

Order 512.

1491 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60947C211yesmaximal, characteristic
2#60287C211yesmaximal, characteristic
3#60250C211yesmaximal, characteristic, complemented
4#18090C212yesmaximal, complemented
5#18089C212yesmaximal, complemented
6#18086C212yesmaximal, complemented
7#18085C212yesmaximal, complemented
8#18013C214yesmaximal, complemented
9#18009C214yesmaximal, complemented
10#17947C214yesmaximal, complemented
11#17919C214yesmaximal, complemented
12#17806C211yesmaximal, characteristic, complemented
13#17491C211yesmaximal, characteristic, complemented
14#17467C211yesmaximal, characteristic
15#17324C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17C2×#533C2211yescharacteristic
18#3248C414yes—
19#3231C411yescharacteristic
20#3231C411yescharacteristic
21#3229C411yescharacteristic
22#3226C411yescharacteristic
23#3210C411yescharacteristic
24#3210C411yescharacteristic
25#3209C411yescharacteristic
26#3206C411yescharacteristic
27#3190C414yes—
28#3170C2211yescharacteristic
29#3170C2211yescharacteristic
30#3167C2211yescharacteristic
31#3167C2211yescharacteristic
32#3108C2214yes—
33#3107C2214yes—
34#2993—22no—
35#2993C2214yes—
36#2986C2212yes—
37#2986C2212yes—
38#2986C2212yes—
39#2985—22no—
40#2984—21no—
41#2984—21no—
42#2983C2212yes—
43#2982C2212yes—
44#2979C2212yes—
45#2978—22no—
46#2971—21no—
47#2971C2212yes—
48#2970—21no—
49#2970C2212yes—
50#2962C2214yes—
51#2960—22no—
52#2924—22no—
53#2923—22no—
54#2920C2211yescharacteristic
55#2920C2211yescharacteristic
56#2920C2211yescharacteristic
57C2×#431C2211yescharacteristic
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#1574C2212yes—
76#1574C2212yes—
77#1573C2212yes—
78#1573C2212yes—
79#1564C2212yes—
80#1564C2212yes—
81#1497C2212yes—
82#1497C2212yes—
83#1487C2214yes—
84#1485C2212yes—
85#1485C2212yes—
86#1485C2212yes—
87#1484C2212yes—
88#1482C2212yes—
89#1482C2212yes—
90#1477C2214yes—
91#1454C2214yes—
92#1453C2214yes—
93#1451C2212yes—
94C8×M16C2212yes—
95#533—21no—
96#533—21no—
97#533—21no—
98#533—21no—
99#533C2×C411yescharacteristic
100#533C2×C411yescharacteristic
101#533C2×C412yes—
102#533C2×C411yescharacteristic
103#533C2×C411yescharacteristic
104#533C2312yes—
105#533C2311yescharacteristic
106#533C2311yescharacteristic
107#533C2311yescharacteristic
108#532C2×C411yescharacteristic
109#532C2×C411yescharacteristic
110#532C2311yescharacteristic
111C2×C4⋊Q8C2×C411yescharacteristic
112C2×C4⋊Q8C2×C411yescharacteristic
113#529—21no—
114#529—21no—
115#529—21no—
116#529—21no—
117#529C2×C412yes—
118#529C2312yes—
119C2×#125—21no—
120#525—21no—
121#525—21no—
122#525C2×C412yes—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8C2×C412yes—
125C4×#41—21no—
126C4×#41—21no—
127C4×#41C2312yes—
128C2×C4×Q8—21no—
129C2×C4×Q8C2312yes—
130C2×#117—21no—
131C22×C42C2311yescharacteristic
132#500C2×C411yescharacteristic
133#500C2×C411yescharacteristic
134#498C2×C411yescharacteristic
135#498C2×C411yescharacteristic
136#494C2×C411yescharacteristic
137#494C2×C411yescharacteristic
138#492C2×C411yescharacteristic
139#492C2×C411yescharacteristic
140#487C2×C411yescharacteristic
141#487C2×C411yescharacteristic
142#487C2×C411yescharacteristic
143#487C2×C411yescharacteristic
144#487C2×C411yescharacteristic
145#487C2×C411yescharacteristic
146#486C2×C411yescharacteristic
147#486C2×C411yescharacteristic
148#485C2×C414yes—
149#485C2×C414yes—
150#483C2×C414yes—
151#480C2×C414yes—
152#479C2×C414yes—
153#479C2×C414yes—
154#475C2×C414yes—
155#474C2×C414yes—
156#467C2×C414yes—
157#467C2×C414yes—
158#465C2×C414yes—
159#465C2×C414yes—
160#464C2×C414yes—
161#464C2×C414yes—
162#462C2×C414yes—
163#461C2×C414yes—
164#452C2×C414yes—
165#452C2×C414yes—
166#451C2×C414yes—
167#448C2×C414yes—
168#446—24no—
169#446—22no—
170#446—22no—
171#446—22no—
172#446—24no—
173#446—24no—
174#446C2314yes—
175#446C2314yes—
176#446C2314yes—
177C8×Q8—24no—
178C8×Q8—22no—
179C8×Q8—22no—
180C8×Q8—22no—
181C8×Q8C2314yes—
182C8×Q8C2314yes—
183C8×Q8C2314yes—
184C4×Q16C2314yes—
185C4×Q16C2314yes—
186C4×SD16C2314yes—
187C4×SD16C2314yes—
188#436—24no—
189#436—22no—
190#436—24no—
191#436C2314yes—
192#435—24no—
193#435—24no—
194C8×D4—22no—
195C8×D4C2314yes—
196#433—24no—
197#433—24no—
198#433—24no—
199#433—24no—
200#432—21no—
201#432—21no—
202#432—21no—
203#432—22no—
204#432—22no—
205#432—22no—
206#432—22no—
207#432D412yes—
208#432D411yescharacteristic
209#432D411yescharacteristic
210#432C2311yescharacteristic
211#432C2311yescharacteristic
212#431—21no—
213#431—21no—
214#431—21no—
215#431—22no—
216#431—22no—
217#431C2×C411yescharacteristic
218#431C2×C411yescharacteristic
219#431C2×C411yescharacteristic
220#431C2×C411yescharacteristic
221#431C2312yes—
222#431C2312yes—
223#431C2311yescharacteristic
224#431C2311yescharacteristic
225#431C2312yes—
226#431C2312yes—
227#431C2312yes—
228#431C2312yes—
229#431C2312yes—
230#431C2312yes—
231#424—24no—
232#424—24no—
233#423—21no—
234#423D412yes—
235#423C2312yes—
236C2×C4⋊C8C2×C412yes—
237C2×C4⋊C8C2312yes—
238#408—24no—
239#407—21no—
240#407—21no—
241#407D412yes—
242C2×C22⋊C8—21no—
243C2×C22⋊C8C2×C412yes—
244#405—24no—
245C4×M16—21no—
246C4×M16—22no—
247C4×M16C2312yes—
248C4×M16C2312yes—
249C2×#97—22no—
250C2×C4×C8—21no—
251C2×C4×C8C2311yescharacteristic
252C2×C4×C8C2311yescharacteristic
253#335C2312yes—
254#335C2312yes—
255#335C2312yes—
256#335C2312yes—
257#334C2312yes—
258#334C2312yes—
259#334C2312yes—
260#334C2312yes—
261Q8⋊C8C2314yes—
262Q8⋊C8C2314yes—
263Q8⋊C8C2314yes—
264Q8⋊C8C2314yes—
265Q8⋊C8C2314yes—
266Q8⋊C8C2314yes—
267D4⋊C8C2314yes—
268D4⋊C8C2314yes—
269#322C2314yes—
270#322C2314yes—
271#322C2314yes—
272C82C2314yes—
273C2×#41—21no—
274C2×#41—21no—
275C2×#41C22×C412yes—
276C22×Q8—21no—
277C22×Q8C22×C412yes—
278C23×C4C22×C411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4#4111yescharacteristic
281C2×Q16C22×C414yes—
282C2×Q16C22×C414yes—
283C2×SD16C22×C414yes—
284C2×SD16C22×C414yes—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—24no—
288C2×M16C2×D414yes—
289C22×C8—22no—
290C22×C8—24no—
291C22×C8C22×C412yes—
292C22×C8C22×C412yes—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8C22×C412yes—
298C4⋊Q8C22×C411yescharacteristic
299C4⋊Q8C22×C411yescharacteristic
300C4⋊Q8C22×C411yescharacteristic
301C4⋊Q8C22×C412yes—
302C4⋊Q8C22×C411yescharacteristic
303C4⋊Q8C22×C411yescharacteristic
304C4⋊Q8C22×C411yescharacteristic
305C4⋊Q8C22×C411yescharacteristic
306#127C22×C411yescharacteristic
307#126—22no—
308#126—22no—
309#125—21no—
310#125—21no—
311#125—22no—
312#125—21no—
313#125—21no—
314#125—21no—
315#125—22no—
316#125—21no—
317#125C22×C412yes—
318#124—21no—
319#124—21no—
320#124C22×C412yes—
321#123—22no—
322#123—22no—
323#123—22no—
324C22⋊Q8—21no—
325C22⋊Q8—21no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—21no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334C22⋊Q8—22no—
335C22⋊Q8—21no—
336C22⋊Q8C22×C412yes—
337C22⋊Q8C22×C412yes—
338C22⋊Q8C22×C412yes—
339C22⋊Q8C22×C412yes—
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—21no—
351C4×Q8—21no—
352C4×Q8—21no—
353C4×Q8—21no—
354C4×Q8—21no—
355C4×Q8—21no—
356C4×Q8—21no—
357C4×Q8—22no—
358C4×Q8—22no—
359C4×Q8—21no—
360C4×Q8—22no—
361C4×Q8C22×C412yes—
362C4×Q8C22×C412yes—
363C4×Q8C22×C412yes—
364C4×Q8C2412yes—
365C4×Q8C2412yes—
366C4×Q8C2412yes—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—21no—
374C4×D4—21no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4C22×C412yes—
378C4×D4C2412yes—
379#117—21no—
380#117—21no—
381#117—22no—
382#117—21no—
383#117—22no—
384#117—21no—
385#117—21no—
386#117—21no—
387#117—22no—
388#117—22no—
389#117—22no—
390#117C22×C412yes—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4C22×C412yes—
397C2×C22⋊C4—22no—
398C2×C42—22no—
399C2×C42—22no—
400C2×C42C22×C411yescharacteristic
401C2×C42C22×C411yescharacteristic
402C2×C42C22×C411yescharacteristic
403C2×C42C2×D411yescharacteristic
404C2×C42C2×D411yescharacteristic
405C2×C42C2×D411yescharacteristic
406C2×C42#4111yescharacteristic
407C2×C42#4111yescharacteristic
408C2×C42#4111yescharacteristic
409C2×C42C2411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2411yescharacteristic
412#107C22×C414yes—
413#107C22×C414yes—
414#106C22×C414yes—
415#106C22×C414yes—
416C4⋊C8—24no—
417C4⋊C8—24no—
418C4⋊C8—24no—
419C4⋊C8—24no—
420C4⋊C8—24no—
421C4⋊C8—24no—
422C4⋊C8—22no—
423C4⋊C8—22no—
424C4⋊C8—22no—
425C4⋊C8—22no—
426C4⋊C8C22×C412yes—
427C4⋊C8C22×C412yes—
428C4⋊C8C22×C412yes—
429C4⋊C8C22×C412yes—
430C4⋊C8C2×D414yes—
431C4⋊C8C2×D412yes—
432C4⋊C8C2×D412yes—
433C4⋊C8C2412yes—
434C4⋊C8C2412yes—
435C4⋊C8C2412yes—
436C4⋊C8C2412yes—
437#103C22×C414yes—
438#103C22×C414yes—
439#103C22×C414yes—
440#103C22×C414yes—
441#103C22×C414yes—
442#103C22×C414yes—
443#102C22×C414yes—
444#102C22×C414yes—
445C22⋊C8—21no—
446C22⋊C8—21no—
447C22⋊C8—22no—
448C22⋊C8—22no—
449C22⋊C8—21no—
450C22⋊C8—21no—
451C22⋊C8—22no—
452C22⋊C8—22no—
453C22⋊C8—24no—
454C22⋊C8—24no—
455C22⋊C8C22×C412yes—
456C22⋊C8C22×C412yes—
457C22⋊C8C22×C412yes—
458C22⋊C8C22×C412yes—
459C22⋊C8C2×D414yes—
460C22⋊C8C2×D412yes—
461C22⋊C8C2×D412yes—
462#97—24no—
463#97—24no—
464#97—24no—
465#97—21no—
466#97—21no—
467#97—22no—
468#97C2×D412yes—
469#97C2×D412yes—
470C4×C8—24no—
471C4×C8—24no—
472C4×C8—24no—
473C4×C8—21no—
474C4×C8—22no—
475C4×C8—21no—
476C4×C8—22no—
477C4×C8C22×C411yescharacteristic
478C4×C8C22×C411yescharacteristic
479C4×C8C22×C411yescharacteristic
480C4×C8C22×C411yescharacteristic
481C4×C8C2414yes—
482C4×C8C2412yes—
483C4×C8C2412yes—
484C4×C8C2414yes—
485C24C4×D411yescharacteristic
486#41—44no—
487#41—82no—
488#41—82no—
489C2×Q8—21no—
490C2×Q8—22no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—22no—
495C2×Q8—44no—
496C2×Q8—82no—
497C2×Q8C23×C412yes—
498C2×Q8C23×C412yes—
499C2×Q8C23×C412yes—
500C2×D4—22no—
501C2×D4—22no—
502C2×D4—22no—
503C2×D4—21no—
504C2×D4—21no—
505C2×D4C23×C412yes—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4—42no—
514C22×C4C4×D411yescharacteristic
515C22×C4C4×D411yescharacteristic
516C22×C4C4×D411yescharacteristic
517C22×C4C4×D411yescharacteristic
518C22×C4C4×D411yescharacteristic
519C22×C4C4×D411yescharacteristic
520C22×C4C4×D411yescharacteristic
521C22×C4C4×D411yescharacteristic
522C22×C4C4×D411yescharacteristic
523C22×C4C23×C411yescharacteristic
524C22×C4C23×C411yescharacteristic
525C22×C4C23×C411yescharacteristic
526C22×C4C22×D411yescharacteristic
527C22×C4C22×D411yescharacteristic
528C22×C4C22×D411yescharacteristic
529C22×C4C2×#4111yescharacteristic
530C22×C4C2×#4111yescharacteristic
531C22×C4C2×#4111yescharacteristic
532Q16—44no—
533Q16—44no—
534SD16—44no—
535SD16—44no—
536M16—44no—
537M16—44no—
538M16—84no—
539M16—82no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—24no—
545C2×C8—42no—
546C2×C8—42no—
547C2×C8—44no—
548C2×C8—84no—
549C2×C8C23×C412yes—
550C2×C8C23×C412yes—
551C2×C8C23×C412yes—
552C2×C8C23×C412yes—
553C2×C8C22×D414yes—
554C2×C8C22×D414yes—
555C4⋊C4—22no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—22no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—22no—
562C4⋊C4—21no—
563C4⋊C4—21no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—22no—
568C4⋊C4—22no—
569C4⋊C4—22no—
570C4⋊C4—22no—
571C4⋊C4—21no—
572C4⋊C4—21no—
573C4⋊C4—21no—
574C4⋊C4—21no—
575C4⋊C4—22no—
576C4⋊C4—22no—
577C4⋊C4—22no—
578C4⋊C4—22no—
579C4⋊C4—22no—
580C4⋊C4—22no—
581C4⋊C4—22no—
582C4⋊C4—22no—
583C4⋊C4C23×C412yes—
584C4⋊C4C23×C412yes—
585C4⋊C4C23×C412yes—
586C4⋊C4C23×C412yes—
587C22⋊C4—22no—
588C22⋊C4—22no—
589C22⋊C4—22no—
590C22⋊C4—22no—
591C22⋊C4—22no—
592C22⋊C4—22no—
593C22⋊C4—22no—
594C22⋊C4—22no—
595C22⋊C4—22no—
596C22⋊C4—22no—
597C22⋊C4—22no—
598C22⋊C4—22no—
599C22⋊C4—22no—
600C42—22no—
601C42—22no—
602C42—22no—
603C42—22no—
604C42—22no—
605C42—22no—
606C42—22no—
607C42C4×D411yescharacteristic
608C42C4×D411yescharacteristic
609C42C4×D411yescharacteristic
610C42C4×D411yescharacteristic
611C42C4×D411yescharacteristic
612C42C4×D411yescharacteristic
613C42C23×C411yescharacteristic
614C42C23×C411yescharacteristic
615C42C23×C411yescharacteristic
616C42C22×D411yescharacteristic
617C42C22×D411yescharacteristic
618C42C22×D411yescharacteristic
619C42C2×#4111yescharacteristic
620C42C2×#4111yescharacteristic
621C42C2×#4111yescharacteristic
622C42C2511yescharacteristic
623C23—22no—
624C23—41no—
625C23—41no—
626C23—41no—
627C23C2×C4×D411yescharacteristic
628C23C2×C4×D411yescharacteristic
629C23C2×C4×D411yescharacteristic
630Q8—24no—
631Q8—24no—
632Q8—24no—
633Q8—44no—
634Q8—42no—
635Q8—42no—
636Q8—42no—
637Q8—82no—
638Q8—81no—
639D4—24no—
640D4—42no—
641D4—81no—
642D4—82no—
643D4—82no—
644D4—82no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—82no—
673C2×C4—82no—
674C2×C4—82no—
675C2×C4C2×C4×D411yescharacteristic
676C2×C4C2×C4×D411yescharacteristic
677C2×C4C2×C4×D411yescharacteristic
678C2×C4C2×C4×D411yescharacteristic
679C2×C4C2×C4×D411yescharacteristic
680C2×C4C2×C4×D411yescharacteristic
681C2×C4C2×C4×D411yescharacteristic
682C2×C4C2×C4×D411yescharacteristic
683C2×C4C2×C4×D411yescharacteristic
684C2×C4C24×C411yescharacteristic
685C2×C4C23×D411yescharacteristic
686C2×C4C22×#4111yescharacteristic
687C8—24no—
688C8—24no—
689C8—22no—
690C8—22no—
691C8—22no—
692C8—22no—
693C8—44no—
694C8—42no—
695C8—42no—
696C8—42no—
697C8—42no—
698C22—21no—
699C22—21no—
700C22—21no—
701C22—21no—
702C22—21no—
703C22—21no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—42no—
708C22—42no—
709C22—42no—
710C22—82no—
711C22C22×C4×D411yescharacteristic
712C4—21no—
713C4—21no—
714C4—21no—
715C4—21no—
716C4—21no—
717C4—21no—
718C4—21no—
719C4—21no—
720C4—21no—
721C4—42no—
722C4—42no—
723C4—42no—
724C4—42no—
725C4—42no—
726C4—42no—
727C4—42no—
728C4#295712yes—
729C4#357312yes—
730C4#358612yes—
731C2—41no—
732C2—41no—
733C2—41no—
734C2—82no—
735C2#6014211yescharacteristic
736C2#6016911yescharacteristic
737C2C2×#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. #60250C2semidirect
4. #18090C2semidirect
5. #18089C2semidirect
6. #18086C2semidirect
7. #18085C2semidirect
8. #18013C2semidirect
9. #18009C2semidirect
10. #17947C2semidirect
11. #17919C2semidirect
12. #17806C2semidirect
13. #17491C2semidirect
15. #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
#609472
#602872
#602502
#180902
#180892
#180862
#180852
#180132
#180092
#179472
#179192
#178062
#174912
#174672
#173242