Finite Group Databasebeta

← #6345654

Subgroups — #6345654

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60950C211yesmaximal, characteristic
2#60289C211yesmaximal, characteristic
3#60249C211yesmaximal, characteristic, complemented
4#18032C214yesmaximal, complemented
5#18028C214yesmaximal, complemented
6#18012C214yesmaximal, complemented
7#18010C214yesmaximal, complemented
8#17988C214yesmaximal, complemented
9#17969C214yesmaximal, complemented
10#17823C211yesmaximal, characteristic, complemented
11#17480C211yesmaximal, characteristic
12#17479C211yesmaximal, characteristic, complemented
13#17324C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C2×#529C2211yescharacteristic
16#3252C414yes—
17#3238C412yes—
18#3236C412yes—
19#3218C412yes—
20#3215C412yes—
21#3195C411yescharacteristic
22#3195C411yescharacteristic
23#3193C411yescharacteristic
24#3193C411yescharacteristic
25#3172C2214yes—
26#3108C2212yes—
27#3108C2212yes—
28#3104C2212yes—
29#3101C2212yes—
30#2995—22no—
31#2993C2214yes—
32#2987—22no—
33#2986C2214yes—
34#2985—22no—
35#2985C2214yescomplemented
36#2984C2212yes—
37#2975—21no—
38#2974C2212yes—
39#2973—21no—
40#2969C2214yes—
41#2967—22no—
42Q8×M16—22no—
43Q8×M16C2212yescomplemented
44Q8×M16C2212yescomplemented
45#2924—22no—
46#2921—21no—
47#2921—21no—
48#2920C2211yescharacteristic
49#2920C2211yescharacteristic
50#2920C2211yescharacteristic
51#2920C2211yescharacteristic
52C2×#431C2211yescharacteristic
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#1574C2214yescomplemented
70#1572C2214yes—
71#1569C2214yescomplemented
72#1497C2214yes—
73#1495C2214yes—
74#1494C2212yes—
75#1494C2212yes—
76#1493C2212yescomplemented
77#1493C2212yescomplemented
78#1479C2212yes—
79#1479C2212yescomplemented
80#1479C2212yescomplemented
81#1478C2212yes—
82#1454C2214yes—
83#1453C2214yescomplemented
84#1451C2212yescomplemented
85C8×M16C2212yescomplemented
86#533—21no—
87#533—21no—
88#533—21no—
89#533—21no—
90#533C2×C412yes—
91#533C2×C412yes—
92#533C2312yes—
93#533C2312yes—
94C2×C4⋊Q8C2×C411yescharacteristic
95C2×C4⋊Q8C2×C411yescharacteristic
96#529—21no—
97#529—21no—
98#529—21no—
99#529—21no—
100#529C2×C412yes—
101#529C2×C412yes—
102#529C2312yes—
103#529C2312yes—
104C2×#126—21no—
105C2×#125—21no—
106C2×#125C2×C411yescharacteristic
107C2×#124C2×C411yescharacteristic
108C2×#123—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8C2×C411yescharacteristic
111C2×C22⋊Q8C2×C411yescharacteristic
112C2×C22⋊Q8C2×C411yescharacteristic
113C2×#121C2×C411yescharacteristic
114C2×C4×Q8—21no—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4C2311yescharacteristic
120C2×#117—21no—
121C2×#117—21no—
122C22×C42C2311yescharacteristic
123#501C2×C414yes—
124#497C2×C414yes—
125#491C2×C414yes—
126#489C2×C414yes—
127#484C2×C412yes—
128#484C2×C412yes—
129#483C2×C412yes—
130#483C2×C412yes—
131#483C2×C412yes—
132#483C2×C412yes—
133#480C2×C412yes—
134#480C2×C412yes—
135#479C2×C414yes—
136#479C2×C414yes—
137#477C2×C414yes—
138#476C2×C414yes—
139#470C2×C414yes—
140#470C2×C414yes—
141#468C2×C414yes—
142#468C2×C414yes—
143#464C2×C414yes—
144#464C2×C414yes—
145#462C2×C414yes—
146#461C2×C414yes—
147#451C2×C412yes—
148#451C2×C412yes—
149#451C2×C412yes—
150#451C2×C412yes—
151#450C2×C412yes—
152#450C2×C412yes—
153#448C2×C412yes—
154#448C2×C412yes—
155#446—24no—
156#446—24no—
157#446—24no—
158#446—24no—
159#446—24no—
160#446—24no—
161#446C2314yes—
162#446C2314yes—
163#446C2314yes—
164C8×Q8—24no—
165C8×Q8C2314yes—
166C8×Q8C2314yes—
167C8×Q8C2314yes—
168C4×Q16C2314yes—
169C4×Q16C2314yes—
170C4×SD16C2314yes—
171C4×SD16C2314yes—
172#436—24no—
173#436—24no—
174#436C2314yes—
175#435—24no—
176#435—24no—
177#435—24no—
178C8×D4C2314yes—
179#433—24no—
180#433—24no—
181#433—24no—
182#433—24no—
183#433—24no—
184#433—24no—
185#432—21no—
186#432—21no—
187#432—22no—
188#432C2×C412yes—
189#432C2×C412yes—
190#432D412yes—
191#432D412yes—
192#432C2312yes—
193#432C2312yes—
194#431—21no—
195#431—21no—
196#431—21no—
197#431—21no—
198#431—22no—
199#431C2312yes—
200#431C2312yes—
201#431C2312yes—
202#431C2312yes—
203#431C2312yes—
204#431C2312yes—
205#431C2312yes—
206#431C2312yes—
207#424—24no—
208#423—22no—
209#423—22no—
210#423C2×C412yes—
211#423D412yes—
212#423C2312yes—
213#423C2312yes—
214C2×C4⋊C8—21no—
215#407—21no—
216#407—21no—
217#407—22no—
218#407C2×C411yescharacteristic
219#407C2×C411yescharacteristic
220#407D411yescharacteristic
221#407D411yescharacteristic
222C2×C22⋊C8—21no—
223C2×C22⋊C8—22no—
224#405—24no—
225C4×M16—22no—
226C4×M16—22no—
227C4×M16—21no—
228C4×M16—21no—
229C4×M16C2312yes—
230C4×M16C2312yes—
231C2×C4×C8C2311yescharacteristic
232C2×C4×C8C2311yescharacteristic
233#335C2314yes—
234#335C2314yes—
235#334C2314yes—
236#334C2314yes—
237Q8⋊C8C2314yes—
238Q8⋊C8C2314yes—
239Q8⋊C8C2314yes—
240Q8⋊C8C2314yes—
241Q8⋊C8C2314yes—
242Q8⋊C8C2314yes—
243D4⋊C8C2314yes—
244D4⋊C8C2314yes—
245#322C2314yes—
246#322C2314yes—
247#322C2314yes—
248C82C2314yes—
249C22×Q8—21no—
250C22×Q8C22×C411yescharacteristic
251C22×Q8C22×C411yescharacteristic
252C22×Q8C22×C411yescharacteristic
253C22×D4—21no—
254C22×D4C22×C411yescharacteristic
255C23×C4—21no—
256C23×C4C22×C411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4#4111yescharacteristic
259C2×Q16C22×C414yes—
260C2×Q16C22×C414yes—
261C2×SD16C22×C414yes—
262C2×SD16C22×C414yes—
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—24no—
266C2×M16C22×C412yes—
267C2×M16C22×C412yes—
268C2×M16C2×D412yes—
269C2×M16C2×D412yes—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—24no—
273C4⋊Q8—21no—
274C4⋊Q8—21no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8C22×C412yes—
278C4⋊Q8C22×C412yes—
279C4⋊Q8C22×C412yes—
280C4⋊Q8C22×C412yes—
281#126—22no—
282#126—22no—
283#126—22no—
284#126—22no—
285#125—22no—
286#125—22no—
287#125—21no—
288#125—21no—
289#125—22no—
290#125—22no—
291#125C22×C412yes—
292#125C22×C412yes—
293#124—21no—
294#124—21no—
295#124C22×C412yes—
296#124C22×C412yes—
297#123—22no—
298#123—22no—
299#123—22no—
300#123—22no—
301C22⋊Q8—22no—
302C22⋊Q8—22no—
303C22⋊Q8—21no—
304C22⋊Q8—21no—
305C22⋊Q8—21no—
306C22⋊Q8—21no—
307C22⋊Q8—21no—
308C22⋊Q8—21no—
309C22⋊Q8—22no—
310C22⋊Q8—22no—
311C22⋊Q8C22×C412yes—
312C22⋊Q8C22×C412yes—
313C22⋊Q8C22×C412yes—
314C22⋊Q8C22×C412yes—
315C22⋊Q8C22×C412yes—
316C22⋊Q8C22×C412yes—
317#121—21no—
318#121—21no—
319#121C22×C412yes—
320#121C22×C412yes—
321C4×Q8—22no—
322C4×Q8—22no—
323C4×Q8—22no—
324C4×Q8—22no—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×Q8—21no—
331C4×Q8C22×C412yes—
332C4×Q8C22×C412yes—
333C4×Q8C22×C412yes—
334C4×Q8C2412yes—
335C4×Q8C2412yes—
336C4×Q8C2412yes—
337C4×D4—22no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—22no—
341C4×D4—21no—
342C4×D4—21no—
343C4×D4C22×C412yes—
344C4×D4C2412yes—
345#117—22no—
346#117—22no—
347#117—22no—
348#117—22no—
349#117—22no—
350#117—22no—
351#117—22no—
352#117—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4C22×C411yescharacteristic
365C2×C4⋊C4C22×C411yescharacteristic
366C2×C4⋊C4C22×C411yescharacteristic
367C2×C4⋊C4C22×C411yescharacteristic
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42C22×C411yescharacteristic
379C2×C42C22×C411yescharacteristic
380C2×C42C22×C411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42#4111yescharacteristic
385C2×C42#4111yescharacteristic
386C2×C42#4111yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390#107C22×C414yes—
391#107C22×C414yes—
392#106C22×C414yes—
393#106C22×C414yes—
394C4⋊C8—24no—
395C4⋊C8—24no—
396C4⋊C8—24no—
397C4⋊C8—24no—
398C4⋊C8—24no—
399C4⋊C8—24no—
400C4⋊C8—22no—
401C4⋊C8—22no—
402C4⋊C8—22no—
403C4⋊C8—22no—
404C4⋊C8C22×C414yes—
405C4⋊C8C22×C414yes—
406C4⋊C8C2×D414yes—
407C4⋊C8C2×D414yes—
408C4⋊C8C2414yes—
409C4⋊C8C2414yes—
410#103C22×C414yes—
411#103C22×C414yes—
412#103C22×C414yes—
413#103C22×C414yes—
414#103C22×C414yes—
415#103C22×C414yes—
416#102C22×C414yes—
417#102C22×C414yes—
418C22⋊C8—22no—
419C22⋊C8—22no—
420C22⋊C8—22no—
421C22⋊C8—22no—
422C22⋊C8—22no—
423C22⋊C8—22no—
424C22⋊C8—24no—
425C22⋊C8—24no—
426C22⋊C8C22×C412yes—
427C22⋊C8C22×C412yes—
428C22⋊C8C22×C412yes—
429C22⋊C8C22×C412yes—
430C22⋊C8C2×D412yes—
431C22⋊C8C2×D412yes—
432C22⋊C8C2×D412yes—
433C22⋊C8C2×D412yes—
434#97—24no—
435#97—24no—
436#97—24no—
437#97—22no—
438#97—22no—
439#97C22×C414yes—
440#97C2×D414yes—
441C4×C8—24no—
442C4×C8—24no—
443C4×C8—24no—
444C4×C8—22no—
445C4×C8—22no—
446C4×C8—21no—
447C4×C8—21no—
448C4×C8C2412yes—
449C4×C8C2412yes—
450C4×C8C2414yes—
451C4×C8C2414yes—
452C24—21no—
453C24C4×D411yescharacteristic
454C2×Q8—22no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—22no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—82no—
463C2×Q8C23×C412yes—
464C2×Q8C23×C412yes—
465C2×Q8C23×C412yes—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—42no—
470C2×D4—82no—
471C2×D4C23×C412yes—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—82no—
484C22×C4—81no—
485C22×C4C4×D411yescharacteristic
486C22×C4C4×D411yescharacteristic
487C22×C4C4×D411yescharacteristic
488C22×C4C4×D411yescharacteristic
489C22×C4C4×D411yescharacteristic
490C22×C4C4×D411yescharacteristic
491C22×C4C4×D411yescharacteristic
492C22×C4C4×D411yescharacteristic
493C22×C4C4×D411yescharacteristic
494C22×C4C23×C411yescharacteristic
495C22×C4C23×C411yescharacteristic
496C22×C4C23×C411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C2×#4111yescharacteristic
501C22×C4C2×#4111yescharacteristic
502C22×C4C2×#4111yescharacteristic
503Q16—44no—
504Q16—44no—
505SD16—44no—
506SD16—44no—
507M16—42no—
508M16—42no—
509M16—42no—
510M16—42no—
511M16—44no—
512M16—84no—
513M16—82no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—84no—
519C2×C8—81no—
520C2×C8—81no—
521C2×C8C23×C414yes—
522C2×C8C23×C414yes—
523C2×C8C22×D414yes—
524C2×C8C22×D414yes—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4C23×C412yes—
552C4⋊C4C23×C412yes—
553C4⋊C4C23×C412yes—
554C4⋊C4C23×C412yes—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42C4×D411yescharacteristic
576C42C4×D411yescharacteristic
577C42C4×D411yescharacteristic
578C42C4×D411yescharacteristic
579C42C4×D411yescharacteristic
580C42C4×D411yescharacteristic
581C42C23×C411yescharacteristic
582C42C23×C411yescharacteristic
583C42C23×C411yescharacteristic
584C42C22×D411yescharacteristic
585C42C22×D411yescharacteristic
586C42C22×D411yescharacteristic
587C42C2×#4111yescharacteristic
588C42C2×#4111yescharacteristic
589C42C2×#4111yescharacteristic
590C42C2511yescharacteristic
591C23—22no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—42no—
596C23—81no—
597C23—81no—
598C23C2×C4×D411yescharacteristic
599C23C2×C4×D411yescharacteristic
600C23C2×C4×D411yescharacteristic
601Q8—24no—
602Q8—24no—
603Q8—24no—
604Q8—44no—
605Q8—81no—
606Q8—81no—
607Q8—81no—
608Q8—82no—
609D4—24no—
610D4—81no—
611D4—82no—
612D4—82no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—82no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4C2×C4×D411yescharacteristic
648C2×C4C2×C4×D411yescharacteristic
649C2×C4C2×C4×D411yescharacteristic
650C2×C4C2×C4×D411yescharacteristic
651C2×C4C2×C4×D411yescharacteristic
652C2×C4C2×C4×D411yescharacteristic
653C2×C4C2×C4×D411yescharacteristic
654C2×C4C2×C4×D411yescharacteristic
655C2×C4C2×C4×D411yescharacteristic
656C2×C4C24×C411yescharacteristic
657C2×C4C23×D411yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C8—24no—
660C8—24no—
661C8—24no—
662C8—24no—
663C8—42no—
664C8—42no—
665C8—44no—
666C8—44no—
667C22—21no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—82no—
680C22—162no—
681C22C22×C4×D411yescharacteristic
682C4—21no—
683C4—21no—
684C4—21no—
685C4—21no—
686C4—21no—
687C4—21no—
688C4—21no—
689C4—21no—
690C4—21no—
691C4—21no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4#357312yes—
700C4#358612yes—
701C2—41no—
702C2—41no—
703C2—41no—
704C2—82no—
705C2#6014311yescharacteristic
706C2#6016911yescharacteristic
707C2#6018211yescharacteristic
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. #60249C2semidirect
4. #18032C2semidirect
5. #18028C2semidirect
6. #18012C2semidirect
7. #18010C2semidirect
8. #17988C2semidirect
9. #17969C2semidirect
10. #17823C2semidirect
12. #17479C2semidirect
13. #17324C2semidirect
35. #2985C22semidirect
43. Q8×M16C22semidirect
44. Q8×M16C22semidirect
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. #1574C22semidirect
71. #1569C22semidirect
76. #1493C22semidirect
77. #1493C22semidirect
79. #1479C22semidirect
80. #1479C22semidirect
83. #1453C22semidirect
84. #1451C22semidirect
85. C8×M16C22semidirect

Maximal subgroups

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

subgroupindex
#609502
#602892
#602492
#180322
#180282
#180122
#180102
#179882
#179692
#178232
#174802
#174792
#173242