Finite Group Databasebeta

← #364994

Subgroups — #364994

Order 512.

2174 conjugacy classes of subgroups.

The subgroup lattice has 778 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2535C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic, complemented
3C2×#1955C211yesmaximal, characteristic, complemented
4C2×#1875C212yesmaximal, complemented
5C2×#1802C211yesmaximal, characteristic, complemented
6C2×#1762C211yesmaximal, characteristic, complemented
7#11199C212yesmaximal, complemented
8#10969C212yesmaximal, complemented
9#10850C211yesmaximal, characteristic, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10731C212yesmaximal, complemented
12C22×#121C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#117C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22C22×#97C2211yescharacteristic
23#2535—24no—
24#2535C418yescomplemented
25C2×#397—21no—
26C2×#392—21no—
27#2354—24no—
28#2354—24no—
29C2×#390—21no—
30C2×#388—22no—
31C2×#386—22no—
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4—21no—
37C2×C4×C4⋊C4C2211yescharacteristic
38#1955—24no—
39#1875—28no—
40#1802—24no—
41#1762—24no—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2212yes—
48C2×#340C2212yes—
49C23×D4—22no—
50C23×D4D411yescharacteristic
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C24×C4C2312yes—
54C23×C8C2312yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C2×#121—24no—
58C2×#121—24no—
59C2×#121—24no—
60C2×#121—42no—
61C2×#121—42no—
62C2×#121C2×C418yes—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4C2×C414yes—
73C2×C4×D4C2×C414yes—
74C2×C4×D4D414yes—
75C2×#117—24no—
76C2×#117—24no—
77C2×#117—24no—
78C2×#117—24no—
79C2×#117—42no—
80C2×#117—42no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C22⋊C4—22no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4—22no—
93C22×C22⋊C4—22no—
94C22×C22⋊C4D412yes—
95C22×C42—22no—
96C22×C42—21no—
97C22×C42C2311yescharacteristic
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—22no—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102—42no—
109C2×#102D414yescomplemented
110C2×C22⋊C8—24no—
111C2×C22⋊C8—24no—
112C2×C22⋊C8—24no—
113C2×C22⋊C8—24no—
114C2×C22⋊C8—42no—
115C2×C22⋊C8—42no—
116C2×#97—22no—
117C2×#97—22no—
118C2×#97—22no—
119#397—28no—
120#392—28no—
121#390—28no—
122#388—216no—
123#386—216no—
124#384—22no—
125#384D414yescomplemented
126#382—22no—
127#382C2×C414yes—
128#380—24no—
129#380—24no—
130#380—24no—
131#380C2×C418yes—
132C4×C4⋊C4—28no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4D414yescomplemented
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—21no—
139C2×#95—21no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—22no—
144#340—22no—
145#340—42no—
146#340—42no—
147C25—21no—
148C25C2×D412yes—
149C22×D4—28no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4—82no—
170C22×D4—82no—
171C22×D4C22⋊C414yes—
172C22×D4C22×C412yes—
173C22×D4C2×D412yes—
174C22×D4#4112yes—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—21no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—21no—
192C23×C4—24no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—82no—
196C23×C4—82no—
197C23×C4C22×C414yes—
198C23×C4C2×D412yes—
199C23×C4C2×D414yes—
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2411yescharacteristic
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8C2×D412yes—
223C22×C8#4112yes—
224#121—216no—
225#121—48no—
226#121—48no—
227#121—48no—
228C4×D4—28no—
229C4×D4—28no—
230C4×D4—28no—
231C4×D4—28no—
232C4×D4—28no—
233C4×D4—28no—
234C4×D4—44no—
235C4×D4—44no—
236#117—48no—
237#117—48no—
238#117—48no—
239#117—48no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—82no—
269C2×C4⋊C4C22⋊C414yes—
270C2×C4⋊C4C22×C412yes—
271C2×C4⋊C4C22×C414yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4#4112yes—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—82no—
291C2×C22⋊C4—82no—
292C2×C22⋊C4—82no—
293C2×C22⋊C4—82no—
294C2×C22⋊C4C22⋊C418yes—
295C2×C42—28no—
296C2×C42—24no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42C22×C412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42#4112yes—
305C2×C42#4112yes—
306C4⋊C8—44no—
307#102—28no—
308#102—28no—
309#102—48no—
310C22⋊C8—48no—
311C22⋊C8—48no—
312#97—44no—
313#95—28no—
314#95—28no—
315#95—24no—
316#95—24no—
317#95—24no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—41no—
336C24—41no—
337C24—41no—
338C24—41no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24C2×C22⋊C414yes—
344C24C22×D411yescharacteristic
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—28no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—88no—
384C2×D4C4×D414yes—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—48no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4C2×C22⋊C414yes—
458C22×C4C2×C22⋊C412yes—
459C22×C4C2×C22⋊C412yes—
460C22×C4#12012yes—
461C22×C4#12012yes—
462C22×C4#12312yes—
463C22×C4#12312yes—
464C22×C4C23×C411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2×#4111yescharacteristic
470C22×C4C2×#4111yescharacteristic
471C22×C4C2×#4111yescharacteristic
472C22×C4C2×#4111yescharacteristic
473C22×C4C2×#4111yescharacteristic
474C22×C4H(5,𝔽2)11yescharacteristic
475C22×C4H(5,𝔽2)11yescharacteristic
476C22×C4H(5,𝔽2)11yescharacteristic
477C22×C42−1+411yescharacteristic
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—42no—
493C2×C8—42no—
494C2×C8—42no—
495C2×C8—42no—
496C2×C8—42no—
497C2×C8—42no—
498C2×C8—42no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—28no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—48no—
512C4⋊C4—42no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—84no—
521C4⋊C4#12114yes—
522C22⋊C4—216no—
523C22⋊C4—416no—
524C22⋊C4—416no—
525C22⋊C4—48no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C42—48no—
529C42—416no—
530C42C4×D414yes—
531C42C4×D414yes—
532C42#12114yes—
533C42#12114yes—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—44no—
567C23—42no—
568C23—42no—
569C23—44no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—44no—
578C23—42no—
579C23—42no—
580C23—44no—
581C23—42no—
582C23—82no—
583C23—82no—
584C23—82no—
585C23—82no—
586C23—81no—
587C23—81no—
588C23—81no—
589C23—81no—
590C23—82no—
591C23—82no—
592C23C22×C22⋊C411yescharacteristic
593C23C2×#12011yescharacteristic
594C23C2×#12311yescharacteristic
595C23D4211yescharacteristic
596C23#54611yescharacteristic
597C23#54611yescharacteristic
598C23#54711yescharacteristic
599C23#57512yes—
600C23#57512yes—
601C23#57612yes—
602C23#57712yes—
603D4—24no—
604D4—24no—
605D4—416no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—42no—
610D4—42no—
611D4—42no—
612D4—832no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—816no—
653C2×C4—82no—
654C2×C4—82no—
655C2×C4—81no—
656C2×C4—81no—
657C2×C4—82no—
658C2×C4—82no—
659C2×C4—82no—
660C2×C4—82no—
661C2×C4—82no—
662C2×C4—82no—
663C2×C4—82no—
664C2×C4—82no—
665C2×C4—82no—
666C2×C4#38614yes—
667C2×C4#38614yes—
668C2×C4C2×C4×D411yescharacteristic
669C2×C4C2×C4×D411yescharacteristic
670C2×C4C2×C4×D411yescharacteristic
671C2×C4C4×#4111yescharacteristic
672C2×C4#51811yescharacteristic
673C2×C4#52011yescharacteristic
674C2×C4C2×#12111yescharacteristic
675C2×C4C2×#12111yescharacteristic
676C2×C4C2×#12111yescharacteristic
677C2×C4#52511yescharacteristic
678C2×C4#52911yescharacteristic
679C2×C4#53211yescharacteristic
680C2×C4#53511yescharacteristic
681C2×C4#53711yescharacteristic
682C2×C4#54011yescharacteristic
683C2×C4#54211yescharacteristic
684C2×C4D4211yescharacteristic
685C2×C4D4211yescharacteristic
686C2×C4#54611yescharacteristic
687C2×C4#54611yescharacteristic
688C2×C4#54611yescharacteristic
689C2×C4#54711yescharacteristic
690C2×C4#54711yescharacteristic
691C2×C4#54811yescharacteristic
692C2×C4#55011yescharacteristic
693C2×C4#55311yescharacteristic
694C2×C4#55311yescharacteristic
695C2×C4#55511yescharacteristic
696C8—42no—
697C8—42no—
698C8—82no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—48no—
703C22—44no—
704C22—44no—
705C22—44no—
706C22—44no—
707C22—44no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—84no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—84no—
724C22—84no—
725C22C2×#38611yescharacteristic
726C22D4×C22⋊C411yescharacteristic
727C22#233911yescharacteristic
728C22#240711yescharacteristic
729C22#240811yescharacteristic
730C22#242011yescharacteristic
731C22#242211yescharacteristic
732C22#289612yes—
733C22#289712yes—
734C22#300511yescharacteristic
735C22#300511yescharacteristic
736C22#300811yescharacteristic
737C22#301011yescharacteristic
738C22#309312yes—
739C22#309612yes—
740C22#326712yes—
741C22#326812yes—
742C22#330712yes—
743C22#330912yes—
744C22#331112yes—
745C22#331912yes—
746C22#334712yes—
747C22#335012yes—
748C4—24no—
749C4—24no—
750C4—44no—
751C4—44no—
752C4—82no—
753C4—84no—
754C4—82no—
755C4—84no—
756C4C4×#12111yescharacteristic
757C4#232411yescharacteristic
758C4#236111yescharacteristic
759C4#236311yescharacteristic
760C4#242711yescharacteristic
761C4#242811yescharacteristic
762C4#253511yescharacteristic
763C4#253711yescharacteristic
764C2—84no—
765C2—84no—
766C2#1397011yescharacteristic
767C2#2091011yescharacteristic
768C2#2091411yescharacteristic
769C2#2242412yes—
770C2#2249812yes—
771C2#2412911yescharacteristic
772C2#2413111yescharacteristic
773C2#2424911yescharacteristic
774C2#2425011yescharacteristic
775C2#2497812yes—
776C2#2502312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2354C2semidirect
3. C2×#1955C2semidirect
4. C2×#1875C2semidirect
5. C2×#1802C2semidirect
6. C2×#1762C2semidirect
7. #11199C2semidirect
8. #10969C2semidirect
9. #10850C2semidirect
10. #10843C2semidirect
11. #10731C2semidirect
24. #2535C4semidirect
109. C2×#102D4semidirect
125. #384D4semidirect
134. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25352
C2×#23542
C2×#19552
C2×#18752
C2×#18022
C2×#17622
#111992
#109692
#108502
#108432
#107312