Finite Group Databasebeta

← #376115

Subgroups — #376115

Order 512.

2168 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D8C211yesmaximal, characteristic, complemented
2C2×#1999C211yesmaximal, characteristic
3C2×#1893C212yesmaximal, complemented
4C2×C8×C22⋊C4C211yesmaximal, characteristic, complemented
5#13970C212yesmaximal, complemented
6#11256C211yesmaximal, characteristic, complemented
7#11096C211yesmaximal, characteristic, complemented
8#11022C211yesmaximal, characteristic, complemented
9#10925C212yesmaximal, complemented
10#10820C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal, complemented
12C23×D8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#121C2212yes—
15C22×C4×D4C2212yes—
16C2×C4×D8—28no—
17C2×C4×D8—24no—
18C22×#107C2211yescharacteristic
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C2×#390C2212yes—
26C2×#386—22no—
27C2×#380C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29#1999C418yescomplemented
30#1893—28no—
31C8×C22⋊C4—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1440—22no—
40#1438—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C22⋊#95—22no—
44C23×D4—22no—
45C23×D4C2312yes—
46C24×C4—22no—
47C24×C4C2311yescharacteristic
48C22×D8—24no—
49C22×D8—24no—
50C22×D8—22no—
51C22×D8—22no—
52C22×D8C2×C414yes—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—24no—
57C23×C8—42no—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#121—24no—
63C2×#121—24no—
64C2×#121—24no—
65C2×#121—42no—
66C2×#121—42no—
67C2×#121C2×C418yescomplemented
68C2×C4×D4—24no—
69C2×C4×D4—24no—
70C2×C4×D4—24no—
71C2×C4×D4—24no—
72C2×C4×D4—42no—
73C2×C4×D4—42no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C4×D8—232no—
87C2×#107—22no—
88C2×#107—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—42no—
94C2×#102—42no—
95C2×#102—42no—
96C2×#102C2×C418yescomplemented
97C2×#102D418yescomplemented
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8C2×C414yescomplemented
101C2×C4×C8—24no—
102C2×C4×C8—22no—
103C2×C4×C8—42no—
104C2×C4×C8D414yescomplemented
105#390—24no—
106#390—42no—
107#386—48no—
108#380—24no—
109#380—42no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D411yescharacteristic
118C2×#95D411yescharacteristic
119#340—24no—
120#340—22no—
121#340—24no—
122#340—42no—
123#340C2×C414yescomplemented
124#336—22no—
125#336—22no—
126C25—22no—
127C25C2×D411yescharacteristic
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—44no—
141C22×D4—44no—
142C22×D4—44no—
143C22×D4—44no—
144C22×D4—82no—
145C22×D4—82no—
146C22×D4C22×C414yes—
147C22×D4C2×D414yes—
148C22×D4#4114yes—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—82no—
166C23×C4—82no—
167C23×C4C22×C412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4#4112yes—
174C23×C4C2411yescharacteristic
175C2×D8—28no—
176C2×D8—216no—
177C2×D8—28no—
178C2×D8—216no—
179C2×D8—28no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—28no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—48no—
195C22×C8—48no—
196C22×C8—44no—
197C22×C8—44no—
198C22×C8C22×C412yes—
199C22×C8C22×C412yes—
200C22×C8C2×D412yes—
201C22×C8C2×D414yes—
202C22×C8C2×D412yes—
203C22×C8#4112yes—
204C22×C8#4112yes—
205#121—216no—
206#121—48no—
207#121—48no—
208#121—48no—
209C4×D4—216no—
210C4×D4—48no—
211C4×D4—48no—
212C4×D4—48no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4C22×C414yes—
230C2×C4⋊C4C2×D414yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—82no—
249C2×C22⋊C4—82no—
250C2×C22⋊C4—82no—
251C2×C22⋊C4D814yescomplemented
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42C2×D412yes—
259C2×C42#4112yes—
260#107—28no—
261#102—216no—
262#102—48no—
263#102—48no—
264C22⋊C8—28no—
265C4×C8—28no—
266C4×C8—44no—
267#95—22no—
268#95—22no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—82no—
273#95—82no—
274C24—21no—
275C24—21no—
276C24—22no—
277C24—22no—
278C24—21no—
279C24—21no—
280C24—22no—
281C24—22no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24C4×D412yes—
298C24C2×D812yes—
299C24#13512yes—
300C24C22×D411yescharacteristic
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—416no—
317C2×D4—416no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—88no—
331C2×D4—88no—
332C2×D4—88no—
333C2×D4—88no—
334C2×D4—88no—
335C2×D4C4×D418yes—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—416no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—88no—
395C22×C4—88no—
396C22×C4—88no—
397C22×C4—88no—
398C22×C4—88no—
399C22×C4—88no—
400C22×C4C4×D412yes—
401C22×C4#12112yes—
402C22×C4#12112yes—
403C22×C4C2×D812yes—
404C22×C4C2×D812yes—
405C22×C4#13512yes—
406C22×C4#13512yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D412yes—
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4112yes—
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4H(5,𝔽2)12yes—
416C22×C4H(5,𝔽2)12yes—
417D8—232no—
418C2×C8—24no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—22no—
429C2×C8—24no—
430C2×C8—416no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8C4×D414yes—
438C2×C8#12014yes—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C22⋊C4—28no—
452C22⋊C4—28no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—88no—
456C22⋊C4—88no—
457C22⋊C4—88no—
458C42—22no—
459C42—22no—
460C42—24no—
461C42—41no—
462C42—41no—
463C42—42no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—28no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23—82no—
515C23—82no—
516C23—82no—
517C23—82no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23C4×D814yescomplemented
522C23C2×C4×D411yescharacteristic
523C23C2×#12111yescharacteristic
524C23#53211yescharacteristic
525C23D4212yes—
526C23#54612yes—
527C23C22×D811yescharacteristic
528C23C2×#13511yescharacteristic
529C23C2×#13511yescharacteristic
530C23C2×#13511yescharacteristic
531C23#57612yes—
532C23#57712yes—
533D4—28no—
534D4—28no—
535D4—44no—
536D4—44no—
537D4—44no—
538D4—44no—
539D4—44no—
540D4—44no—
541D4—832no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—832no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4#46614yes—
595C2×C4C2×C4×D411yescharacteristic
596C2×C4C2×C4×D412yes—
597C2×C4C4×#4111yescharacteristic
598C2×C4#51812yes—
599C2×C4C2×#12011yescharacteristic
600C2×C4#52511yescharacteristic
601C2×C4#52511yescharacteristic
602C2×C4#52512yes—
603C2×C4#52911yescharacteristic
604C2×C4#53511yescharacteristic
605C2×C4#53511yescharacteristic
606C2×C4#54012yes—
607C2×C4#54612yes—
608C2×C4#54612yes—
609C2×C4#54612yes—
610C2×C4#55112yes—
611C2×C4#55312yes—
612C2×C4#55312yes—
613C8—24no—
614C8—24no—
615C8—42no—
616C8—42no—
617C22—24no—
618C22—28no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—48no—
629C22—48no—
630C22—48no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22—84no—
638C22—84no—
639C22—84no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22C4×#12111yescharacteristic
644C22#235812yes—
645C22#240811yescharacteristic
646C22#246412yes—
647C22#253511yescharacteristic
648C22C2×C4×D811yescharacteristic
649C22C4×#13511yescharacteristic
650C22#294612yes—
651C22C2×#46611yescharacteristic
652C22#305011yescharacteristic
653C22#305811yescharacteristic
654C22#306011yescharacteristic
655C22#316811yescharacteristic
656C22#316911yescharacteristic
657C22#317912yes—
658C22D4×D812yes—
659C22#328012yes—
660C22#328212yes—
661C22#328412yes—
662C22#329412yes—
663C22#329712yes—
664C22#329712yes—
665C22#330312yes—
666C4—24no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—84no—
672C4—84no—
673C4—162no—
674C4C4×#12011yescharacteristic
675C4#232811yescharacteristic
676C4#236412yes—
677C4#240411yescharacteristic
678C4#245712yes—
679C4#259811yescharacteristic
680C2—44no—
681C2—88no—
682C2#1417511yescharacteristic
683C2C4×#46611yescharacteristic
684C2#2153611yescharacteristic
685C2#2287012yes—
686C2#2287212yes—
687C2#2426511yescharacteristic
688C2#2426711yescharacteristic
689C2#2696012yes—
690C2#2696112yes—
691C2#2814011yescharacteristic
692C2#2814311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#1999C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C4×D8C2semidirect
3. C2×#1893C2semidirect
4. C2×C8×C22⋊C4C2semidirect
5. #13970C2semidirect
6. #11256C2semidirect
7. #11096C2semidirect
8. #11022C2semidirect
9. #10925C2semidirect
10. #10820C2semidirect
11. #10788C2semidirect
29. #1999C4semidirect
67. C2×#121C2×C4semidirect
96. C2×#102C2×C4semidirect
97. C2×#102D4semidirect
100. C2×C22⋊C8C2×C4semidirect
104. C2×C4×C8D4semidirect
123. #340C2×C4semidirect
251. C2×C22⋊C4D8semidirect
521. C23C4×D8semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D82
C2×#19992
C2×#18932
C2×C8×C22⋊C42
#139702
#112562
#110962
#110222
#109252
#108202
#107882