Finite Group Databasebeta

← #6345377

Subgroups — #6345377

Order 512.

1560 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60560C211yesmaximal, characteristic
2#60287C211yesmaximal, characteristic
3#60248C211yesmaximal, characteristic, complemented
4#18049C214yesmaximal, complemented
5#18047C214yesmaximal, complemented
6#18032C214yesmaximal, complemented
7#18030C214yesmaximal, complemented
8#17947C214yesmaximal, complemented
9#17919C214yesmaximal, complemented
10#17823C211yesmaximal, characteristic, complemented
11#17499C211yesmaximal, characteristic
12#17449C211yesmaximal, characteristic, complemented
13#17339C211yesmaximal, characteristic, complemented
14#3444C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16#3117C414yes—
17#3113C411yescharacteristic
18#3113C411yescharacteristic
19#3112C411yescharacteristic
20#3112C411yescharacteristic
21#3108C2212yes—
22#3108C2212yes—
23#3104C2212yes—
24#3103C2212yes—
25#3092C412yes—
26#3078C412yes—
27#3037C412yes—
28#3004C412yes—
29#2993—22no—
30#2993C2214yes—
31#2986C2212yes—
32#2984—22no—
33#2984C2214yes—
34#2983—21no—
35#2983—21no—
36#2983C2212yes—
37#2982C2212yes—
38#2979C2212yes—
39#2978—22no—
40#2971—21no—
41#2971C2212yes—
42#2970—21no—
43#2970C2212yes—
44Q8×M16—22no—
45Q8×M16C2214yes—
46C4×#137C2214yes—
47#2923—22no—
48#2920—22no—
49#2920C2211yescharacteristic
50#2920C2211yescharacteristic
51#2920C2211yescharacteristic
52#2920C2211yescharacteristic
53#2920C2211yescharacteristic
54#1593C2214yescomplemented
55#1592C2214yescomplemented
56#1591C2214yescomplemented
57#1590C2214yescomplemented
58#1588C2214yescomplemented
59#1587C2214yescomplemented
60#1587C2214yescomplemented
61#1585C2214yescomplemented
62#1584C2214yescomplemented
63#1584C2214yescomplemented
64#1583C2214yescomplemented
65#1582C2214yescomplemented
66#1579C2214yescomplemented
67SD16⋊C8C2214yescomplemented
68C8×Q16C2214yescomplemented
69C8×SD16C2214yescomplemented
70#1574C2214yes—
71#1572C2214yes—
72#1569C2212yes—
73#1569C2212yes—
74#1497C2212yes—
75#1496C2212yes—
76#1491C2212yes—
77#1491C2212yes—
78#1489C2214yes—
79#1487C2212yes—
80#1487C2212yes—
81#1482C2214yes—
82#1479C2214yes—
83#1455C2214yes—
84#1454C2214yes—
85#1454C2214yes—
86C23×Q8C2×C411yescharacteristic
87C2×#137C2×C414yes—
88#533C2×C411yescharacteristic
89#533C2×C411yescharacteristic
90#533C2311yescharacteristic
91#532C2×C411yescharacteristic
92#532C2×C411yescharacteristic
93#532C2311yescharacteristic
94#529—21no—
95#529—21no—
96#529—21no—
97#529—21no—
98#529C2×C412yes—
99#529C2312yes—
100#525—21no—
101#525—21no—
102#525C2×C412yes—
103C4×#41—21no—
104C4×#41—21no—
105C4×#41C2312yes—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8C2×C412yes—
113C2×C4×Q8C2×C411yescharacteristic
114C2×C4×Q8C2×C411yescharacteristic
115C2×C4×Q8C2×C412yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C22×C4⋊C4—21no—
121C22×C4⋊C4C2×C411yescharacteristic
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124#484C2×C414yes—
125#483C2×C412yes—
126#483C2×C412yes—
127#481C2×C414yes—
128#480C2×C412yes—
129#480C2×C412yes—
130#479C2×C414yes—
131#479C2×C414yes—
132#477C2×C412yes—
133#477C2×C412yes—
134#475C2×C412yes—
135#475C2×C412yes—
136#464C2×C414yes—
137#464C2×C414yes—
138#462C2×C412yes—
139#462C2×C412yes—
140#460C2×C412yes—
141#460C2×C412yes—
142#451C2×C414yes—
143#451C2×C412yes—
144#451C2×C412yes—
145#448C2×C414yes—
146#448C2×C412yes—
147#448C2×C412yes—
148#446—24no—
149#446—24no—
150#446—24no—
151#446—22no—
152#446—24no—
153#446—24no—
154#446C2314yes—
155#446C2314yes—
156#446C2314yes—
157C8×Q8—24no—
158C8×Q8—22no—
159C8×Q8C2314yes—
160C8×Q8C2314yes—
161C8×Q8C2314yes—
162#441C2×C414yes—
163#441C2×C414yes—
164#440C2×C414yes—
165#440C2×C414yes—
166C4×Q16C2314yes—
167C4×Q16C2314yes—
168C4×SD16C2314yes—
169C4×SD16C2314yes—
170#436—24no—
171#436—22no—
172#436—24no—
173#436C2314yes—
174#435—24no—
175#435—24no—
176C8×D4—22no—
177C8×D4C2314yes—
178#433—24no—
179#433—24no—
180#432—21no—
181#432—21no—
182#432—21no—
183#432—21no—
184#432—22no—
185#432—22no—
186#432—24no—
187#432—22no—
188#432—22no—
189#432C2×C411yescharacteristic
190#432C2×C411yescharacteristic
191#432C2×C412yes—
192#432D412yes—
193#432D411yescharacteristic
194#432D411yescharacteristic
195#432C2311yescharacteristic
196#432C2311yescharacteristic
197#432C2312yes—
198#431—21no—
199#431—21no—
200#431—22no—
201#431—22no—
202#431—24no—
203#431C2312yes—
204#431C2312yes—
205#431C2312yes—
206#431C2312yes—
207#431C2312yes—
208#431C2312yes—
209#431C2312yes—
210#431C2312yes—
211#428C2×C414yes—
212#424—24no—
213#423—24no—
214#423—21no—
215#423C2×C412yes—
216#423D412yes—
217#423C2312yes—
218#423C2312yes—
219#419C2×C414yes—
220#417C2×C414yes—
221#408—24no—
222#407—21no—
223#407—21no—
224#407—21no—
225#407C2×C412yes—
226#407D412yes—
227C4×M16—24no—
228C4×M16—21no—
229C4×M16—21no—
230C4×M16—22no—
231C4×M16C2312yes—
232C4×M16C2312yes—
233C4×M16C2312yes—
234C2×#97—22no—
235#335C2314yes—
236#335C2314yes—
237#334C2314yes—
238#334C2314yes—
239Q8⋊C8C2314yes—
240Q8⋊C8C2314yes—
241Q8⋊C8C2314yes—
242Q8⋊C8C2314yes—
243Q8⋊C8C2314yes—
244Q8⋊C8C2314yes—
245D4⋊C8C2314yes—
246D4⋊C8C2314yes—
247#322C2314yes—
248#322C2314yes—
249#322C2314yes—
250C82C2314yes—
251C2×#41—21no—
252C2×#41—21no—
253C2×#41C22×C412yes—
254C22×Q8—21no—
255C22×Q8—21no—
256C22×Q8—21no—
257C22×Q8—44no—
258C22×Q8C22×C411yescharacteristic
259C22×Q8C22×C412yes—
260C22×Q8C22×C412yes—
261C22×Q8C22×C411yescharacteristic
262C23×C4—21no—
263C23×C4—21no—
264C23×C4C22×C411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4#4111yescharacteristic
267#137—48no—
268C2×Q16C22×C414yes—
269C2×Q16C22×C414yes—
270C2×SD16C22×C414yes—
271C2×SD16C22×C414yes—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—24no—
276C2×M16C22×C414yes—
277C2×M16C2×D414yes—
278C22×C8—24no—
279C4⋊Q8C22×C411yescharacteristic
280C4⋊Q8C22×C411yescharacteristic
281C4⋊Q8C22×C411yescharacteristic
282#127C22×C411yescharacteristic
283#126—22no—
284#126—22no—
285#125—21no—
286#125—21no—
287#125C22×C412yes—
288#124—21no—
289#124—21no—
290#124C22×C412yes—
291#123—22no—
292#123—22no—
293#123—22no—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298C22⋊Q8C22×C412yes—
299C22⋊Q8C22×C412yes—
300#121—21no—
301#121—21no—
302#121—21no—
303#121—21no—
304#121C22×C412yes—
305#121C22×C412yes—
306#120—22no—
307C4×Q8—22no—
308C4×Q8—21no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8—22no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8—21no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8—22no—
324C4×Q8—22no—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×Q8C22×C412yes—
328C4×Q8C22×C412yes—
329C4×Q8C22×C412yes—
330C4×Q8C22×C412yes—
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—22no—
342C4×D4—22no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4C22×C412yes—
348C4×D4C2412yes—
349#117—21no—
350#117—21no—
351#117—22no—
352#117—22no—
353#117—22no—
354#117C22×C412yes—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4C22×C412yes—
367C2×C4⋊C4C22×C411yescharacteristic
368C2×C4⋊C4C22×C411yescharacteristic
369C2×C4⋊C4C22×C412yes—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—22no—
379C2×C42C22×C411yescharacteristic
380C2×C42C22×C411yescharacteristic
381C2×C42C22×C411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42#4111yescharacteristic
386C2×C42#4111yescharacteristic
387C2×C42#4111yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391#107C22×C414yes—
392#107C22×C414yes—
393#106C22×C414yes—
394#106C22×C414yes—
395C4⋊C8—24no—
396C4⋊C8—24no—
397C4⋊C8—24no—
398C4⋊C8—24no—
399C4⋊C8—24no—
400C4⋊C8—24no—
401C4⋊C8—22no—
402C4⋊C8—22no—
403C4⋊C8—22no—
404C4⋊C8—22no—
405C4⋊C8C22×C412yes—
406C4⋊C8C22×C412yes—
407C4⋊C8C22×C414yes—
408C4⋊C8C2×D414yes—
409C4⋊C8C2×D412yes—
410C4⋊C8C2×D412yes—
411C4⋊C8C2414yes—
412C4⋊C8C2412yes—
413C4⋊C8C2412yes—
414#103C22×C414yes—
415#103C22×C414yes—
416#103C22×C414yes—
417#103C22×C414yes—
418#103C22×C414yes—
419#103C22×C414yes—
420#102C22×C414yes—
421#102C22×C414yes—
422C22⋊C8—22no—
423C22⋊C8—22no—
424C22⋊C8—22no—
425C22⋊C8—21no—
426C22⋊C8—21no—
427C22⋊C8—22no—
428C22⋊C8—22no—
429C22⋊C8—24no—
430C22⋊C8—24no—
431C22⋊C8C22×C412yes—
432C22⋊C8C22×C412yes—
433C22⋊C8C22×C414yes—
434C22⋊C8C2×D414yes—
435C22⋊C8C2×D412yes—
436C22⋊C8C2×D412yes—
437#97—24no—
438#97—24no—
439#97—24no—
440#97—21no—
441#97—21no—
442#97—22no—
443#97—22no—
444#97C22×C412yes—
445#97C22×C412yes—
446#97C2×D412yes—
447#97C2×D412yes—
448C4×C8—24no—
449C4×C8—24no—
450C4×C8—24no—
451C4×C8—22no—
452C4×C8—22no—
453C4×C8C2414yes—
454C4×C8C2414yes—
455C4×C8C2414yes—
456C24C4×D411yescharacteristic
457#41—44no—
458#41—82no—
459#41—82no—
460C2×Q8—24no—
461C2×Q8—21no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—24no—
468C2×Q8—24no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8C23×C412yes—
476C2×Q8C23×C412yes—
477C2×Q8C23×C412yes—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4C23×C412yes—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—44no—
498C22×C4—81no—
499C22×C4—82no—
500C22×C4C4×D411yescharacteristic
501C22×C4C4×D411yescharacteristic
502C22×C4C4×D411yescharacteristic
503C22×C4C4×D411yescharacteristic
504C22×C4C4×D411yescharacteristic
505C22×C4C4×D411yescharacteristic
506C22×C4C4×D411yescharacteristic
507C22×C4C4×D411yescharacteristic
508C22×C4C4×D411yescharacteristic
509C22×C4C23×C411yescharacteristic
510C22×C4C23×C411yescharacteristic
511C22×C4C23×C411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C2×#4111yescharacteristic
516C22×C4C2×#4111yescharacteristic
517C22×C4C2×#4111yescharacteristic
518Q16—28no—
519Q16—44no—
520SD16—28no—
521SD16—44no—
522M16—44no—
523M16—44no—
524M16—44no—
525M16—84no—
526M16—82no—
527M16—82no—
528C2×C8—24no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—24no—
532C2×C8—84no—
533C2×C8C23×C414yes—
534C2×C8C23×C414yes—
535C2×C8C22×D414yes—
536C2×C8C22×D414yes—
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—22no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4C23×C412yes—
560C4⋊C4C23×C412yes—
561C4⋊C4C23×C412yes—
562C4⋊C4C23×C412yes—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42C4×D411yescharacteristic
584C42C4×D411yescharacteristic
585C42C4×D411yescharacteristic
586C42C4×D411yescharacteristic
587C42C4×D411yescharacteristic
588C42C4×D411yescharacteristic
589C42C23×C411yescharacteristic
590C42C23×C411yescharacteristic
591C42C23×C411yescharacteristic
592C42C22×D411yescharacteristic
593C42C22×D411yescharacteristic
594C42C22×D411yescharacteristic
595C42C2×#4111yescharacteristic
596C42C2×#4111yescharacteristic
597C42C2×#4111yescharacteristic
598C42C2511yescharacteristic
599C23—22no—
600C23—41no—
601C23—41no—
602C23—41no—
603C23C2×C4×D411yescharacteristic
604C23C2×C4×D411yescharacteristic
605C23C2×C4×D411yescharacteristic
606Q8—24no—
607Q8—22no—
608Q8—22no—
609Q8—24no—
610Q8—22no—
611Q8—44no—
612Q8—44no—
613Q8—42no—
614Q8—42no—
615Q8—81no—
616Q8#56814yes—
617D4—24no—
618D4—42no—
619D4—81no—
620D4—82no—
621D4—82no—
622D4—82no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4—81no—
657C2×C4—81no—
658C2×C4—82no—
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×C4C2×C4×D411yescharacteristic
666C2×C4C2×C4×D411yescharacteristic
667C2×C4C2×C4×D411yescharacteristic
668C2×C4C24×C411yescharacteristic
669C2×C4C23×D411yescharacteristic
670C2×C4C22×#4111yescharacteristic
671C8—24no—
672C8—24no—
673C8—24no—
674C8—24no—
675C8—44no—
676C8—42no—
677C8—44no—
678C8—42no—
679C22—21no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—21no—
687C22—21no—
688C22—44no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22C22×C4×D411yescharacteristic
693C4—21no—
694C4—21no—
695C4—21no—
696C4—21no—
697C4—21no—
698C4—21no—
699C4—21no—
700C4—24no—
701C4—21no—
702C4—21no—
703C4—21no—
704C4—42no—
705C4—42no—
706C4—42no—
707C4—42no—
708C4—42no—
709C4—42no—
710C4C2×#56812yes—
711C4#358612yes—
712C2—41no—
713C2—41no—
714C2—41no—
715C2—82no—
716C2#6014111yescharacteristic
717C2#6016911yescharacteristic
718C2#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. #60248C2semidirect
4. #18049C2semidirect
5. #18047C2semidirect
6. #18032C2semidirect
7. #18030C2semidirect
8. #17947C2semidirect
9. #17919C2semidirect
10. #17823C2semidirect
12. #17449C2semidirect
13. #17339C2semidirect
54. #1593C22semidirect
55. #1592C22semidirect
56. #1591C22semidirect
57. #1590C22semidirect
58. #1588C22semidirect
59. #1587C22semidirect
60. #1587C22semidirect
61. #1585C22semidirect
62. #1584C22semidirect
63. #1584C22semidirect
64. #1583C22semidirect
65. #1582C22semidirect
66. #1579C22semidirect
67. SD16⋊C8C22semidirect
68. C8×Q16C22semidirect
69. C8×SD16C22semidirect

Maximal subgroups

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

subgroupindex
#605602
#602872
#602482
#180492
#180472
#180322
#180302
#179472
#179192
#178232
#174992
#174492
#173392