Finite Group Databasebeta

← #6345536

Subgroups — #6345536

Order 512.

1519 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60952C211yesmaximal, characteristic
2#60265C211yesmaximal, characteristic
3#60249C211yesmaximal, characteristic, complemented
4#18083C214yesmaximal, complemented
5#18081C214yesmaximal, complemented
6#18004C212yesmaximal, complemented
7#18003C212yesmaximal, complemented
8#17998C212yesmaximal, complemented
9#17997C212yesmaximal, complemented
10#17949C214yesmaximal, complemented
11#17921C214yesmaximal, complemented
12#17807C211yesmaximal, characteristic, complemented
13#17489C211yesmaximal, characteristic
14C2×#1481C211yesmaximal, characteristic, complemented
15#17324C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17C2×#533C2211yescharacteristic
18#3255C412yes—
19#3254C412yes—
20#3239C412yes—
21#3237C411yescharacteristic
22#3237C411yescharacteristic
23#3218C412yes—
24#3216C411yescharacteristic
25#3216C411yescharacteristic
26#3196C414yes—
27#3172C2212yes—
28#3172C2212yes—
29#3107C2214yes—
30#3104C2212yes—
31#3103C2212yes—
32#2994C2214yes—
33#2992—22no—
34#2990—22no—
35#2989C2212yes—
36#2988C2212yes—
37#2987—22no—
38#2986C2212yes—
39#2986C2212yes—
40#2985—22no—
41#2985C2214yes—
42Q8×M16—21no—
43Q8×M16—21no—
44Q8×M16C2212yes—
45Q8×M16C2212yes—
46#2962C2212yes—
47#2935C2212yes—
48#2929—22no—
49#2925—21no—
50#2921—22no—
51#2920C2211yescharacteristic
52#2920C2211yescharacteristic
53#2920C2211yescharacteristic
54C2×#431C2211yescharacteristic
55#2906—21no—
56C22×C4⋊C8C2211yescharacteristic
57#1593C2214yescomplemented
58#1592C2214yescomplemented
59#1591C2214yescomplemented
60#1590C2214yescomplemented
61#1588C2214yescomplemented
62#1587C2214yescomplemented
63#1587C2214yescomplemented
64#1585C2214yescomplemented
65#1584C2214yescomplemented
66#1584C2214yescomplemented
67#1583C2214yescomplemented
68#1582C2214yescomplemented
69#1579C2214yescomplemented
70SD16⋊C8C2214yescomplemented
71C8×Q16C2214yescomplemented
72C8×SD16C2214yescomplemented
73#1573C2212yes—
74#1573C2212yes—
75#1571C2212yes—
76#1570C2212yes—
77#1565C2214yes—
78#1493C2211yescharacteristic
79#1493C2211yescharacteristic
80#1492C2211yescharacteristic
81#1492C2211yescharacteristic
82#1488C2214yes—
83#1482C2214yes—
84#1481C2212yes—
85#1481C2212yes—
86#1481C2212yes—
87#1481C2212yes—
88C2×Q8⋊C8C2212yes—
89C2×Q8⋊C8C2212yes—
90#1454C2214yes—
91#1453C2214yes—
92#1451C2212yes—
93C8×M16C2212yes—
94C2×#131—22no—
95C23×C8—21no—
96#533—21no—
97#533—21no—
98#533—21no—
99#533—21no—
100#533C2×C411yescharacteristic
101#533C2×C411yescharacteristic
102#533C2×C412yes—
103#533C2×C411yescharacteristic
104#533C2×C411yescharacteristic
105#533C2311yescharacteristic
106#533C2312yes—
107#533C2311yescharacteristic
108#533C2311yescharacteristic
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×C412yes—
146#477C2×C412yes—
147#475C2×C412yes—
148#475C2×C412yes—
149#475C2×C414yes—
150#474C2×C414yes—
151#470C2×C414yes—
152#470C2×C414yes—
153#468C2×C414yes—
154#468C2×C414yes—
155#462C2×C412yes—
156#462C2×C412yes—
157#462C2×C414yes—
158#461C2×C414yes—
159#460C2×C412yes—
160#460C2×C412yes—
161#452C2×C414yes—
162#452C2×C414yes—
163#451C2×C414yes—
164#448C2×C414yes—
165#446—22no—
166#446—22no—
167#446—24no—
168#446—24no—
169#446—24no—
170#446—24no—
171#446—24no—
172#446C2314yes—
173#446C2314yes—
174#446C2314yes—
175C8×Q8—22no—
176C8×Q8—22no—
177C8×Q8C2314yes—
178C8×Q8C2314yes—
179C8×Q8C2314yes—
180C4×Q16C2314yes—
181C4×Q16C2314yes—
182C4×SD16C2314yes—
183C4×SD16C2314yes—
184#436C2314yes—
185#435—24no—
186#435—24no—
187#435—24no—
188C8×D4—24no—
189C8×D4—24no—
190C8×D4C2314yes—
191#433—24no—
192#433—24no—
193#433—24no—
194#433—24no—
195#433—24no—
196#432—21no—
197#432—22no—
198#432C2×C411yescharacteristic
199#432C2×C411yescharacteristic
200#432C2×C412yes—
201#432D412yes—
202#432D412yes—
203#432C2312yes—
204#431—21no—
205#431—21no—
206#431—21no—
207#431—22no—
208#431C2312yes—
209#431C2312yes—
210#431C2312yes—
211#431C2312yes—
212#431C2312yes—
213#431C2312yes—
214#431C2312yes—
215#424—22no—
216#424—24no—
217#423—22no—
218#423—22no—
219#423—22no—
220#423C2×C412yes—
221#423D411yescharacteristic
222#423D411yescharacteristic
223#423C2311yescharacteristic
224#423C2311yescharacteristic
225C2×C4⋊C8—21no—
226C2×C4⋊C8—21no—
227C2×C4⋊C8—21no—
228C2×C4⋊C8—21no—
229C2×C4⋊C8—22no—
230C2×C4⋊C8C2312yes—
231C2×C4⋊C8C2312yes—
232C2×C4⋊C8C2311yescharacteristic
233C2×C4⋊C8C2311yescharacteristic
234#408—22no—
235#407—21no—
236#407C2×C412yes—
237#407D412yes—
238C2×C22⋊C8—21no—
239#405—22no—
240C4×M16—22no—
241C4×M16—22no—
242C4×M16—21no—
243C4×M16C2312yes—
244C4×M16C2312yes—
245C2×C4×C8C2311yescharacteristic
246C2×C4×C8C2311yescharacteristic
247#335C2314yes—
248#335C2314yes—
249#334C2314yes—
250#334C2314yes—
251Q8⋊C8C2312yes—
252Q8⋊C8C2312yes—
253Q8⋊C8C2314yes—
254Q8⋊C8C2312yes—
255Q8⋊C8C2312yes—
256Q8⋊C8C2314yes—
257Q8⋊C8C2312yes—
258Q8⋊C8C2312yes—
259Q8⋊C8C2312yes—
260Q8⋊C8C2312yes—
261D4⋊C8C2314yes—
262D4⋊C8C2314yes—
263#322C2314yes—
264#322C2314yes—
265#322C2314yes—
266C82C2314yes—
267C2×#41—21no—
268C2×#41—21no—
269C2×#41C22×C412yes—
270C22×Q8—21no—
271C22×Q8C22×C412yes—
272C23×C4C22×C411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4#4111yescharacteristic
275C2×Q16C22×C414yes—
276C2×Q16C22×C414yes—
277C2×SD16C22×C414yes—
278C2×SD16C22×C414yes—
279#131—84no—
280C2×M16—21no—
281C2×M16—21no—
282C2×M16—24no—
283C2×M16C22×C414yes—
284C2×M16C2×D412yes—
285C2×M16C2×D412yes—
286C22×C8—22no—
287C22×C8—21no—
288C22×C8—22no—
289C22×C8—21no—
290C22×C8—21no—
291C22×C8—21no—
292C22×C8—24no—
293C22×C8—82no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8—21no—
298C4⋊Q8C22×C412yes—
299C4⋊Q8C22×C411yescharacteristic
300C4⋊Q8C22×C411yescharacteristic
301C4⋊Q8C22×C411yescharacteristic
302C4⋊Q8C22×C412yes—
303C4⋊Q8C22×C411yescharacteristic
304C4⋊Q8C22×C411yescharacteristic
305C4⋊Q8C22×C411yescharacteristic
306C4⋊Q8C22×C411yescharacteristic
307#127C22×C411yescharacteristic
308#126—22no—
309#126—22no—
310#125—21no—
311#125—21no—
312#125—22no—
313#125—21no—
314#125—22no—
315#125—21no—
316#125—21no—
317#125—21no—
318#125C22×C412yes—
319#124—21no—
320#124—21no—
321#124C22×C412yes—
322#123—22no—
323#123—22no—
324#123—22no—
325C22⋊Q8—22no—
326C22⋊Q8—21no—
327C22⋊Q8—21no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—21no—
333C22⋊Q8—22no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—21no—
337C22⋊Q8C22×C412yes—
338C22⋊Q8C22×C412yes—
339C22⋊Q8C22×C412yes—
340C22⋊Q8C22×C412yes—
341C22⋊Q8C22×C412yes—
342C22⋊Q8C22×C412yes—
343#121—21no—
344#121—21no—
345#121—21no—
346#121—21no—
347#121C22×C412yes—
348#121C22×C412yes—
349#120—22no—
350C4×Q8—21no—
351C4×Q8—21no—
352C4×Q8—22no—
353C4×Q8—21no—
354C4×Q8—21no—
355C4×Q8—21no—
356C4×Q8—21no—
357C4×Q8—21no—
358C4×Q8—22no—
359C4×Q8—22no—
360C4×Q8—21no—
361C4×Q8—22no—
362C4×Q8C22×C412yes—
363C4×Q8C22×C412yes—
364C4×Q8C22×C412yes—
365C4×Q8C2412yes—
366C4×Q8C2412yes—
367C4×Q8C2412yes—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—21no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4C22×C412yes—
379C4×D4C2412yes—
380#117—21no—
381#117—21no—
382#117—22no—
383#117—21no—
384#117—22no—
385#117—21no—
386#117—21no—
387#117—21no—
388#117—22no—
389#117—22no—
390#117—22no—
391#117C22×C412yes—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4C22×C412yes—
398C2×C22⋊C4—22no—
399C2×C42—22no—
400C2×C42—22no—
401C2×C42C22×C411yescharacteristic
402C2×C42C22×C411yescharacteristic
403C2×C42C22×C411yescharacteristic
404C2×C42C2×D411yescharacteristic
405C2×C42C2×D411yescharacteristic
406C2×C42C2×D411yescharacteristic
407C2×C42#4111yescharacteristic
408C2×C42#4111yescharacteristic
409C2×C42#4111yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2411yescharacteristic
412C2×C42C2411yescharacteristic
413#107C22×C414yes—
414#107C22×C414yes—
415#106C22×C414yes—
416#106C22×C414yes—
417C4⋊C8—24no—
418C4⋊C8—24no—
419C4⋊C8—24no—
420C4⋊C8—24no—
421C4⋊C8—24no—
422C4⋊C8—24no—
423C4⋊C8—21no—
424C4⋊C8—21no—
425C4⋊C8—22no—
426C4⋊C8—22no—
427C4⋊C8—21no—
428C4⋊C8—21no—
429C4⋊C8—22no—
430C4⋊C8—22no—
431C4⋊C8C22×C412yes—
432C4⋊C8C22×C412yes—
433C4⋊C8C22×C414yes—
434C4⋊C8C2×D412yes—
435C4⋊C8C2×D412yes—
436C4⋊C8C2×D412yes—
437C4⋊C8C2×D412yes—
438C4⋊C8C2412yes—
439C4⋊C8C2412yes—
440C4⋊C8C2412yes—
441C4⋊C8C2412yes—
442#103C22×C414yes—
443#103C22×C414yes—
444#103C22×C414yes—
445#103C22×C414yes—
446#103C22×C414yes—
447#103C22×C414yes—
448#102C22×C414yes—
449#102C22×C414yes—
450C22⋊C8—22no—
451C22⋊C8—22no—
452C22⋊C8—22no—
453C22⋊C8—22no—
454C22⋊C8—24no—
455C22⋊C8—24no—
456C22⋊C8C22×C414yes—
457C22⋊C8C22×C412yes—
458C22⋊C8C22×C412yes—
459C22⋊C8C2×D414yes—
460C22⋊C8C2×D414yes—
461#97—24no—
462#97—24no—
463#97—24no—
464#97—22no—
465#97C22×C412yes—
466#97C22×C412yes—
467#97C2×D414yes—
468C4×C8—24no—
469C4×C8—24no—
470C4×C8—24no—
471C4×C8—22no—
472C4×C8—21no—
473C4×C8—21no—
474C4×C8C2412yes—
475C4×C8C2412yes—
476C4×C8C2414yes—
477C4×C8C2414yes—
478C24C4×D411yescharacteristic
479#41—44no—
480#41—42no—
481#41—42no—
482#41—82no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—22no—
488C2×Q8—22no—
489C2×Q8—42no—
490C2×Q8—42no—
491C2×Q8—82no—
492C2×Q8C23×C412yes—
493C2×Q8C23×C412yes—
494C2×Q8C23×C412yes—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4—22no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4C23×C412yes—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—42no—
509C22×C4—81no—
510C22×C4C4×D411yescharacteristic
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×C4C23×C411yescharacteristic
520C22×C4C23×C411yescharacteristic
521C22×C4C23×C411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D411yescharacteristic
525C22×C4C2×#4111yescharacteristic
526C22×C4C2×#4111yescharacteristic
527C22×C4C2×#4111yescharacteristic
528Q16—44no—
529Q16—44no—
530SD16—44no—
531SD16—44no—
532M16—24no—
533M16—24no—
534M16—44no—
535M16—42no—
536M16—42no—
537M16—84no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—44no—
545C2×C8—84no—
546C2×C8—81no—
547C2×C8—82no—
548C2×C8—81no—
549C2×C8—81no—
550C2×C8—81no—
551C2×C8C23×C414yes—
552C2×C8C23×C414yes—
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—22no—
625C23—41no—
626C23—41no—
627C23C2×C4×D411yescharacteristic
628C23C2×C4×D411yescharacteristic
629C23C2×C4×D411yescharacteristic
630Q8—22no—
631Q8—22no—
632Q8—22no—
633Q8—22no—
634Q8—24no—
635Q8—42no—
636Q8—42no—
637Q8—42no—
638Q8—82no—
639Q8—82no—
640Q8—81no—
641D4—24no—
642D4—44no—
643D4—44no—
644D4—44no—
645D4—42no—
646D4—81no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—82no—
677C2×C4—82no—
678C2×C4—82no—
679C2×C4C2×C4×D411yescharacteristic
680C2×C4C2×C4×D411yescharacteristic
681C2×C4C2×C4×D411yescharacteristic
682C2×C4C2×C4×D411yescharacteristic
683C2×C4C2×C4×D411yescharacteristic
684C2×C4C2×C4×D411yescharacteristic
685C2×C4C2×C4×D411yescharacteristic
686C2×C4C2×C4×D411yescharacteristic
687C2×C4C2×C4×D411yescharacteristic
688C2×C4#57712yes—
689C2×C4#57812yes—
690C2×C4C24×C411yescharacteristic
691C2×C4C23×D411yescharacteristic
692C2×C4C22×#4111yescharacteristic
693C8—24no—
694C8—24no—
695C8—24no—
696C8—24no—
697C8—42no—
698C8—42no—
699C8—42no—
700C8—42no—
701C8—42no—
702C8—42no—
703C22—21no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—21no—
709C22—21no—
710C22—21no—
711C22—44no—
712C22—42no—
713C22—42no—
714C22—42no—
715C22#294712yes—
716C22C22×C4×D411yescharacteristic
717C4—21no—
718C4—21no—
719C4—21no—
720C4—21no—
721C4—21no—
722C4—21no—
723C4—21no—
724C4—21no—
725C4—21no—
726C4—42no—
727C4—42no—
728C4—42no—
729C4—42no—
730C4—42no—
731C4—42no—
732C4—42no—
733C4#294712yes—
734C4#357312yes—
735C4C2×#57711yescharacteristic
736C4C2×#57811yescharacteristic
737C2—22no—
738C2—41no—
739C2—41no—
740C2—82no—
741C2#6014211yescharacteristic
742C2C2×#294711yescharacteristic
743C2#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 ◁ #2920 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #60249C2semidirect
4. #18083C2semidirect
5. #18081C2semidirect
6. #18004C2semidirect
7. #18003C2semidirect
8. #17998C2semidirect
9. #17997C2semidirect
10. #17949C2semidirect
11. #17921C2semidirect
12. #17807C2semidirect
14. C2×#1481C2semidirect
15. #17324C2semidirect
57. #1593C22semidirect
58. #1592C22semidirect
59. #1591C22semidirect
60. #1590C22semidirect
61. #1588C22semidirect
62. #1587C22semidirect
63. #1587C22semidirect
64. #1585C22semidirect
65. #1584C22semidirect
66. #1584C22semidirect
67. #1583C22semidirect
68. #1582C22semidirect
69. #1579C22semidirect
70. SD16⋊C8C22semidirect
71. C8×Q16C22semidirect
72. C8×SD16C22semidirect

Maximal subgroups

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

subgroupindex
#609522
#602652
#602492
#180832
#180812
#180042
#180032
#179982
#179972
#179492
#179212
#178072
#174892
C2×#14812
#173242