Finite Group Databasebeta

← #392971

Subgroups — #392971

Order 512.

2310 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#405C211yesmaximal, characteristic, complemented
2C2×#2619C211yesmaximal, characteristic
3C2×#1961C211yesmaximal, characteristic, complemented
4C2×#1957C211yesmaximal, characteristic, complemented
5C2×#1895C212yesmaximal, complemented
6C2×#1893C212yesmaximal, complemented
7#14340C211yesmaximal, characteristic, complemented
8#10927C212yesmaximal, complemented
9#10925C212yesmaximal, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10842C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#127C2211yescharacteristic
15C22×#121C2212yes—
16C22×#117C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C2×#405—24no—
22C2×#405—24no—
23C2×#405—24no—
24C22×#97C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2619—24no—
29#2619C418yescomplemented
30C2×#394—22no—
31C2×#392—22no—
32C2×#390C2212yes—
33C2×#384C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37#1961—24no—
38#1957—24no—
39#1895—28no—
40#1893—28no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45#1441—22no—
46#1439—22no—
47C4⋊#95C2212yes—
48C23×D4—22no—
49C23×D4—22no—
50C23×D4C2312yes—
51C24×C4C2311yescharacteristic
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—22no—
58C22×M16D414yescomplemented
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—22no—
63C23×C8—22no—
64C23×C8—22no—
65C23×C8D414yescomplemented
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C2×#127—22no—
71C2×#127—22no—
72C2×#127—22no—
73C2×#127—22no—
74C2×#127—22no—
75C2×#127C2×C414yes—
76C2×#121—24no—
77C2×#121—24no—
78C2×#121—24no—
79C2×#121—24no—
80C2×#121—24no—
81C2×#121C2×C418yes—
82C2×#117—22no—
83C2×#117—22no—
84C2×#117—22no—
85C2×#117—22no—
86C2×#117C2×C414yes—
87C2×#117D414yes—
88C22×C4⋊C4—22no—
89C22×C4⋊C4D412yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—24no—
95C22×C22⋊C4—22no—
96C22×C22⋊C4D412yes—
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102—24no—
109C2×#102—24no—
110C2×#102—24no—
111#405—232no—
112C2×#97—22no—
113C2×#97—22no—
114C2×#97—22no—
115C2×#97—22no—
116C2×#97—22no—
117C2×#97D414yescomplemented
118C2×C4×C8—22no—
119C2×C4×C8—22no—
120C2×C4×C8—22no—
121C2×C4×C8—22no—
122C2×C4×C8—22no—
123C2×C4×C8D414yescomplemented
124#394—216no—
125#392—216no—
126#390—24no—
127#390C2×C418yes—
128#384—22no—
129#384—22no—
130#382—22no—
131#382—22no—
132#382—22no—
133#382C2×C414yes—
134#380—24no—
135#380—24no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—24no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—24no—
144C25—22no—
145C25C2×D411yescharacteristic
146C22×D4—28no—
147C22×D4—28no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—44no—
162C22×D4—44no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4—84no—
166C22×D4—84no—
167C22×D4—84no—
168C22×D4—84no—
169C22×D4C22×C414yes—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4C22×C412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4#4112yes—
189C23×C4#4112yes—
190C23×C4C2411yescharacteristic
191C2×M16—28no—
192C2×M16—28no—
193C2×M16—28no—
194C2×M16—28no—
195C2×M16—28no—
196C2×M16—28no—
197C2×M16—28no—
198C22×C8—28no—
199C22×C8—28no—
200C22×C8—28no—
201C22×C8—28no—
202C22×C8—28no—
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—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—28no—
220C22×C8—28no—
221C22×C8C2×D412yes—
222C22×C8C2×D412yes—
223C22×C8C2×D412yes—
224C22×C8C2×D412yes—
225C22×C8C2×D412yes—
226C22×C8C2×D412yes—
227C22×C8C2×D412yes—
228C22×C8C2×D412yes—
229C22×C8#4112yes—
230C22×C8#4112yes—
231C22×C8#4112yes—
232C22×C8#4112yes—
233#127—28no—
234#127—28no—
235#127—44no—
236#127—44no—
237#121—216no—
238#121—216no—
239#121—48no—
240#121—48no—
241#117—28no—
242#117—28no—
243#117—28no—
244#117—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—84no—
266C2×C4⋊C4C22⋊C418yes—
267C2×C4⋊C4C22×C414yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—216no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—84no—
276C2×C22⋊C4—84no—
277C2×C22⋊C4—84no—
278C2×C22⋊C4C22⋊C418yes—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42C22×C412yes—
284C2×C42C22×C412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42#4112yes—
290C2×C42#4112yes—
291#102—48no—
292#102—48no—
293#102—48no—
294#102—48no—
295#97—28no—
296#97—28no—
297C4×C8—28no—
298C4×C8—28no—
299#95—28no—
300#95—44no—
301#95—44no—
302#95—84no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—44no—
317C24—44no—
318C24—44no—
319C24—44no—
320C24C2×C22⋊C412yes—
321C24#12412yes—
322C24#12712yes—
323C24C22×D411yescharacteristic
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—416no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—416no—
354C2×D4—816no—
355C2×D4—88no—
356C2×D4—88no—
357C2×D4—816no—
358C2×D4—816no—
359C2×D4—816no—
360C2×D4—88no—
361C2×D4—88no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
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—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4C2×C22⋊C414yes—
412C22×C4C2×C22⋊C414yes—
413C22×C4C2×C22⋊C412yes—
414C22×C4C4×D414yes—
415C22×C4C4×D414yes—
416C22×C4#12114yes—
417C22×C4#12114yes—
418C22×C4#12114yes—
419C22×C4#12114yes—
420C22×C4#12412yes—
421C22×C4#12712yes—
422C22×C4C23×C411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436M16—232no—
437C2×C8—232no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8#12114yes—
463C2×C8#12114yes—
464C2×C8#12114yes—
465C2×C8#12114yes—
466C4⋊C4—216no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—88no—
480C22⋊C4—216no—
481C22⋊C4—416no—
482C22⋊C4—816no—
483C22⋊C4—88no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42C4×D414yes—
489C42C4×D414yes—
490C42#12014yes—
491C42#12314yes—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—28no—
497C23—28no—
498C23—24no—
499C23—24no—
500C23—22no—
501C23—22no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—48no—
509C23—48no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—82no—
514C23—82no—
515C23—82no—
516C23—82no—
517C23—84no—
518C23—84no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23—82no—
523C23—84no—
524C23—84no—
525C23—84no—
526C23—84no—
527C23—84no—
528C23—84no—
529C23—84no—
530C23—84no—
531C23#39014yes—
532C23C22×C22⋊C411yescharacteristic
533C23C2×#12111yescharacteristic
534C23C2×#12111yescharacteristic
535C23C2×#12411yescharacteristic
536C23C2×#12711yescharacteristic
537C23#53211yescharacteristic
538C23#53211yescharacteristic
539C23#57614yes—
540C23#57714yes—
541D4—44no—
542D4—44no—
543D4—44no—
544D4—44no—
545D4—832no—
546D4—832no—
547D4—82no—
548D4—82no—
549D4—82no—
550D4—82no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—84no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4#38618yes—
590C2×C4#39014yes—
591C2×C4C2×C4×D411yescharacteristic
592C2×C4C2×C4×D411yescharacteristic
593C2×C4C2×C4×D411yescharacteristic
594C2×C4C2×C4×D411yescharacteristic
595C2×C4C4×#4111yescharacteristic
596C2×C4C4×#4111yescharacteristic
597C2×C4C2×#12011yescharacteristic
598C2×C4C2×#12111yescharacteristic
599C2×C4C2×#12111yescharacteristic
600C2×C4C2×#12111yescharacteristic
601C2×C4C2×#12111yescharacteristic
602C2×C4C2×#12111yescharacteristic
603C2×C4C2×#12111yescharacteristic
604C2×C4C2×#12311yescharacteristic
605C2×C4#52511yescharacteristic
606C2×C4#52511yescharacteristic
607C2×C4#52511yescharacteristic
608C2×C4#52511yescharacteristic
609C2×C4#52511yescharacteristic
610C2×C4#52511yescharacteristic
611C2×C4#52911yescharacteristic
612C2×C4#52911yescharacteristic
613C2×C4#52911yescharacteristic
614C2×C4#52911yescharacteristic
615C2×C4#53211yescharacteristic
616C2×C4#53211yescharacteristic
617C2×C4#53211yescharacteristic
618C2×C4#53211yescharacteristic
619C8—24no—
620C8—24no—
621C8—24no—
622C8—24no—
623C22—24no—
624C22—28no—
625C22—44no—
626C22—44no—
627C22—42no—
628C22—42no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—84no—
633C22—88no—
634C22—84no—
635C22—84no—
636C22—88no—
637C22—164no—
638C22—164no—
639C22—162no—
640C22—162no—
641C22—162no—
642C22—162no—
643C22C2×#38611yescharacteristic
644C22C2×#39011yescharacteristic
645C22#229711yescharacteristic
646C22#285011yescharacteristic
647C22#285011yescharacteristic
648C22#285611yescharacteristic
649C22#285611yescharacteristic
650C22#289714yes—
651C22#305712yes—
652C22#305812yes—
653C22#306214yes—
654C22#313614yes—
655C22#315412yes—
656C22#315512yes—
657C22#317412yes—
658C22#317612yes—
659C22#317914yes—
660C4—24no—
661C4—28no—
662C4—44no—
663C4—84no—
664C4—162no—
665C4—162no—
666C4C4×#12111yescharacteristic
667C4C4×#12111yescharacteristic
668C4C4×#12111yescharacteristic
669C4C4×#12111yescharacteristic
670C4#284711yescharacteristic
671C4#284711yescharacteristic
672C4#285411yescharacteristic
673C4#285411yescharacteristic
674C2—44no—
675C2—164no—
676C2#1469811yescharacteristic
677C2#2097512yes—
678C2#2120612yes—
679C2#2141412yes—
680C2#3304512yes—
681C2#3305512yes—
682C2#3317811yescharacteristic
683C2#3317911yescharacteristic
684C2#3318712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2619C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#405C2semidirect
3. C2×#1961C2semidirect
4. C2×#1957C2semidirect
5. C2×#1895C2semidirect
6. C2×#1893C2semidirect
7. #14340C2semidirect
8. #10927C2semidirect
9. #10925C2semidirect
10. #10843C2semidirect
11. #10842C2semidirect
29. #2619C4semidirect
58. C22×M16D4semidirect
65. C23×C8D4semidirect
117. C2×#97D4semidirect
123. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4052
C2×#26192
C2×#19612
C2×#19572
C2×#18952
C2×#18932
#143402
#109272
#109252
#108432
#108422