Finite Group Databasebeta

← #182874

Subgroups — #182874

Order 512.

1097 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60233C211yesmaximal, characteristic
2#20527C211yesmaximal, characteristic, complemented
3#20525C211yesmaximal, characteristic, complemented
4#17559C211yesmaximal, characteristic
5#17542C211yesmaximal, characteristic
6#8240C212yesmaximal, complemented
7#8240C212yesmaximal, complemented
8#8239C212yesmaximal, complemented
9#8238C212yesmaximal, complemented
10#8055C211yesmaximal, characteristic, complemented
11#8051C211yesmaximal, characteristic, complemented
12C2×#525C2211yescharacteristic
13#2977—22no—
14#2976—21no—
15#2976—21no—
16#2974—22no—
17#2973—21no—
18#2972—21no—
19#2933C412yescomplemented
20#2933C412yescomplemented
21#2929C2212yes—
22#2929C2212yes—
23#2924C412yescomplemented
24#2880C412yescomplemented
25C2×#407C2211yescharacteristic
26C2×#407C2211yescharacteristic
27C2×#407C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29#2280C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1753—21no—
32#1753—21no—
33#1753—22no—
34#1752C2211yescharacteristic, complemented
35#1752C2211yescharacteristic, complemented
36#1752C2212yescomplemented
37C8×C22⋊C4C2211yescharacteristic, complemented
38C8×C22⋊C4C2211yescharacteristic, complemented
39C8×C22⋊C4C2212yescomplemented
40#1750—22no—
41#1749—21no—
42C4×C22⋊C8—21no—
43#1513C414yescomplemented
44#1511C412yescomplemented
45#1510C412yescomplemented
46#1508C412yescomplemented
47#1507C412yescomplemented
48#1504C414yescomplemented
49#1501C2212yes—
50#1501C2212yes—
51#1501C2212yes—
52#1501C2212yes—
53#1298C2212yescomplemented
54#1298C2212yescomplemented
55#1297C2212yescomplemented
56#1297C2212yescomplemented
57C22×#41C2×C411yescharacteristic
58C24×C4C2311yescharacteristic
59C2×#131C2×C412yescomplemented
60C2×#131C2×C412yescomplemented
61C22×M16—21no—
62C22×M16—21no—
63C22×M16C2×C411yescharacteristic
64C22×M16C2×C411yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67#525—21no—
68#525—21no—
69#525—21no—
70#525—21no—
71#525C2×C412yes—
72#525C2×C412yes—
73#525C2×C412yes—
74#525C2312yes—
75C2×#123—21no—
76C2×C22⋊Q8C2×C411yescharacteristic
77C2×C22⋊Q8C2×C411yescharacteristic
78C2×#121C2×C411yescharacteristic
79C2×#121C2×C411yescharacteristic
80C2×#120—21no—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×#117—21no—
84C2×#117C2×C411yescharacteristic
85C22×C22⋊C4—21no—
86C22×C42—21no—
87#436—22no—
88#436—22no—
89#435—22no—
90#435—22no—
91#435—22no—
92#435—22no—
93#435—22no—
94#435—22no—
95#435—22no—
96#435—22no—
97C8×D4—22no—
98C8×D4—22no—
99C8×D4—22no—
100C8×D4—22no—
101C8×D4—22no—
102C8×D4—22no—
103#433—22no—
104#433—22no—
105#432—22no—
106#431—22no—
107#424C2×C412yescomplemented
108#424C2×C412yescomplemented
109#408—22no—
110#408—22no—
111#408C2×C412yescomplemented
112#408C2×C412yescomplemented
113#407—21no—
114#407—21no—
115#407—21no—
116#407—21no—
117#407—21no—
118#407—21no—
119#407—22no—
120#407C2×C412yes—
121#407C2×C412yes—
122#407C2×C412yes—
123#407C2×C412yes—
124#407C2×C412yes—
125#407C2312yes—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8—22no—
131C2×C22⋊C8C2×C412yes—
132C2×C22⋊C8C2×C412yes—
133C2×C22⋊C8D411yescharacteristic
134C2×C22⋊C8D411yescharacteristic
135C2×C22⋊C8D412yes—
136C2×C22⋊C8D411yescharacteristic
137C2×C22⋊C8D411yescharacteristic
138C2×C22⋊C8D412yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2311yescharacteristic
141C2×C22⋊C8C2311yescharacteristic
142#405C2×C412yescomplemented
143#405C2×C412yescomplemented
144C2×#97—21no—
145C2×#97—21no—
146C2×#97—22no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8—22no—
150#381C2×C412yes—
151#380—21no—
152#380C2×C412yes—
153#379—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4C2×C412yes—
158C4×C22⋊C4C2311yescharacteristic
159C4×C22⋊C4C2312yes—
160C4×C22⋊C4C2311yescharacteristic
161C2×#95C2×C412yes—
162#336—21no—
163#336—21no—
164#336—21no—
165#336—21no—
166#336—22no—
167#336—22no—
168#328C2×C414yes—
169#328C2×C414yes—
170#328C2×C414yescomplemented
171#328C2×C414yescomplemented
172#327C2×C414yes—
173#327C2×C414yes—
174#327C2×C414yescomplemented
175#327C2×C414yescomplemented
176C25#4111yescharacteristic
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—81no—
180C2×#41—81no—
181C2×#41C4212yes—
182C2×#41C22×C412yes—
183C22×Q8C4211yescharacteristic
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4C4211yescharacteristic
188C23×C4—21no—
189C23×C4—22no—
190C23×C4—81no—
191C23×C4—81no—
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C22×C411yescharacteristic
196C23×C4C22×C411yescharacteristic
197C23×C4C22×C411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4#4111yescharacteristic
203C23×C4#4111yescharacteristic
204C23×C4#4111yescharacteristic
205C23×C4C2411yescharacteristic
206#131—82no—
207#131—82no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—22no—
214C2×M16—22no—
215C2×M16—22no—
216C2×M16—81no—
217C2×M16—81no—
218C2×M16—81no—
219C2×M16—81no—
220C2×M16—81no—
221C2×M16—81no—
222C2×M16C22×C412yes—
223C2×M16C22×C412yes—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—22no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—81no—
232C22×C8—81no—
233C22×C8C22×C412yes—
234C22×C8C22×C412yes—
235C22×C8C2×D411yescharacteristic
236C22×C8C2×D412yes—
237C22×C8C2×D411yescharacteristic
238C22×C8C2×D411yescharacteristic
239C22×C8C2×D412yes—
240C22×C8C2×D411yescharacteristic
241#123—22no—
242#123—22no—
243#123—22no—
244#123—22no—
245C22⋊Q8—21no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C22⋊Q8C4212yes—
250C22⋊Q8C4212yes—
251C22⋊Q8C22×C412yes—
252C22⋊Q8C22×C412yes—
253#121—21no—
254#121—21no—
255#121—21no—
256#121—21no—
257#121C4212yes—
258#121C4212yes—
259#121C22×C412yes—
260#121C22×C412yes—
261#120—22no—
262#120—22no—
263#120—22no—
264#120—22no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273#117—21no—
274#117—21no—
275#117—81no—
276#117—81no—
277#117C4212yes—
278#117C22×C412yes—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4C4211yescharacteristic
283C2×C4⋊C4C4211yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—44no—
296C2×C42—21no—
297C2×C42—22no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—22no—
301C2×C42—81no—
302C2×C42—81no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306C4⋊C8—22no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—22no—
310C22⋊C8—22no—
311C22⋊C8—22no—
312C22⋊C8—22no—
313C22⋊C8—22no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316C22⋊C8—21no—
317C22⋊C8—21no—
318C22⋊C8—22no—
319C22⋊C8—22no—
320C22⋊C8—22no—
321C22⋊C8—22no—
322C22⋊C8C22⋊C412yes—
323C22⋊C8C22⋊C412yes—
324C22⋊C8C22⋊C412yes—
325C22⋊C8C22⋊C412yes—
326C22⋊C8C22⋊C412yes—
327C22⋊C8C22⋊C412yes—
328C22⋊C8C22⋊C412yes—
329C22⋊C8C22⋊C412yes—
330C22⋊C8C22×C414yes—
331C22⋊C8C22×C412yes—
332C22⋊C8C22×C412yes—
333#97—22no—
334#97—22no—
335#97—81no—
336#97—81no—
337#97—82no—
338C4×C8—22no—
339C4×C8—22no—
340C4×C8—81no—
341C4×C8—81no—
342C4×C8—82no—
343#95—22no—
344#95—22no—
345#95C22×C412yes—
346#95C22×C412yes—
347#95C22×C414yes—
348C24—21no—
349C24—42no—
350C24—42no—
351C24—81no—
352C24#11711yescharacteristic
353C24#11711yescharacteristic
354C24C4×D411yescharacteristic
355C24C4×D411yescharacteristic
356C24C4×D411yescharacteristic
357C24C4×D411yescharacteristic
358C24C2×#4111yescharacteristic
359#41—42no—
360#41—42no—
361#41—81no—
362#41—81no—
363#41—82no—
364#41—82no—
365C2×Q8—21no—
366C2×Q8—81no—
367C2×Q8C2×C4212yes—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—44no—
376C2×D4—81no—
377C2×D4—82no—
378C2×D4—82no—
379C2×D4C2×C4212yes—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4—82no—
414C22×C4C2×C4211yescharacteristic
415C22×C4C2×C4211yescharacteristic
416C22×C4C2×C4211yescharacteristic
417C22×C4C2×C4211yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4#11711yescharacteristic
423C22×C4#11711yescharacteristic
424C22×C4C4×D411yescharacteristic
425C22×C4C4×D411yescharacteristic
426C22×C4C4×D411yescharacteristic
427C22×C4C4×D411yescharacteristic
428C22×C4C4×D411yescharacteristic
429C22×C4C4×D411yescharacteristic
430C22×C4C4×D411yescharacteristic
431C22×C4C4×D411yescharacteristic
432C22×C4C4×D411yescharacteristic
433C22×C4C4×D411yescharacteristic
434C22×C4C4×D411yescharacteristic
435C22×C4C4×D411yescharacteristic
436C22×C4C23×C411yescharacteristic
437C22×C4C23×C411yescharacteristic
438C22×C4C23×C411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4111yescharacteristic
442M16—42no—
443M16—42no—
444M16—82no—
445M16—82no—
446M16—81no—
447M16—81no—
448M16—82no—
449M16—82no—
450M16—82no—
451M16—82no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—41no—
458C2×C8—42no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—42no—
462C2×C8—41no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—42no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—42no—
471C2×C8—82no—
472C2×C8—82no—
473C2×C8—81no—
474C2×C8—81no—
475C2×C8C2×C22⋊C412yes—
476C2×C8C2×C22⋊C412yes—
477C2×C8C2×C22⋊C412yes—
478C2×C8C2×C22⋊C412yes—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—82no—
486C4⋊C4C2×C4212yes—
487C4⋊C4C2×C4212yes—
488C22⋊C4—24no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—42no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C42—22no—
505C42—22no—
506C42—41no—
507C42—41no—
508C42—42no—
509C42—41no—
510C42—41no—
511C42—42no—
512C42—82no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—42no—
536C23—81no—
537C23—81no—
538C23—88no—
539C23C4×C22⋊C411yescharacteristic
540C23C4×C22⋊C411yescharacteristic
541C23C4×C22⋊C411yescharacteristic
542C23C4×C22⋊C411yescharacteristic
543C23C2×#11711yescharacteristic
544C23C2×C4×D411yescharacteristic
545C23C2×C4×D411yescharacteristic
546Q8—42no—
547Q8—81no—
548D4—42no—
549D4—44no—
550D4—44no—
551D4—44no—
552D4—44no—
553D4—81no—
554D4—82no—
555D4—82no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—82no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—88no—
610C2×C4C4×C22⋊C411yescharacteristic
611C2×C4C4×C22⋊C411yescharacteristic
612C2×C4C4×C22⋊C411yescharacteristic
613C2×C4C4×C22⋊C411yescharacteristic
614C2×C4C22×C4211yescharacteristic
615C2×C4C22×C22⋊C411yescharacteristic
616C2×C4C2×C4×D411yescharacteristic
617C2×C4C2×C4×D411yescharacteristic
618C8—22no—
619C8—22no—
620C8—22no—
621C8—22no—
622C8—44no—
623C8—42no—
624C8—42no—
625C8—42no—
626C8—42no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—44no—
647C22—44no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—162no—
657C22C2×C4×C22⋊C411yescharacteristic
658C22#295512yes—
659C22#295612yes—
660C4—21no—
661C4—21no—
662C4—21no—
663C4—21no—
664C4—21no—
665C4—21no—
666C4—21no—
667C4—44no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—84no—
672C4—82no—
673C4#288311yescharacteristic
674C4#288311yescharacteristic
675C2—22no—
676C2—22no—
677C2—41no—
678C2—41no—
679C2—41no—
680C2—41no—
681C2—41no—
682C2—82no—
683C2#2039211yescharacteristic
684C2#2042011yescharacteristic
685C2#2042111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC22C2×C4×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20527C2semidirect
3. #20525C2semidirect
6. #8240C2semidirect
7. #8240C2semidirect
8. #8239C2semidirect
9. #8238C2semidirect
10. #8055C2semidirect
11. #8051C2semidirect
19. #2933C4semidirect
20. #2933C4semidirect
23. #2924C4semidirect
24. #2880C4semidirect
34. #1752C22semidirect
35. #1752C22semidirect
36. #1752C22semidirect
37. C8×C22⋊C4C22semidirect
38. C8×C22⋊C4C22semidirect
39. C8×C22⋊C4C22semidirect
43. #1513C4semidirect
44. #1511C4semidirect
45. #1510C4semidirect
46. #1508C4semidirect
47. #1507C4semidirect
48. #1504C4semidirect
53. #1298C22semidirect
54. #1298C22semidirect
55. #1297C22semidirect
56. #1297C22semidirect
59. C2×#131C2×C4semidirect
60. C2×#131C2×C4semidirect
107. #424C2×C4semidirect
108. #424C2×C4semidirect
111. #408C2×C4semidirect
112. #408C2×C4semidirect
142. #405C2×C4semidirect
143. #405C2×C4semidirect
170. #328C2×C4semidirect
171. #328C2×C4semidirect
174. #327C2×C4semidirect
175. #327C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602332
#205272
#205252
#175592
#175422
#82402
#82392
#82382
#80552
#80512