Finite Group Databasebeta

← #367042

Subgroups — #367042

Order 512.

1742 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#1983C212yesmaximal
3C2×#1981C211yesmaximal, characteristic, complemented
4C2×#1960C211yesmaximal, characteristic, complemented
5C2×#1940C211yesmaximal, characteristic, complemented
6C2×#1804C211yesmaximal, characteristic
7C2×#1762C211yesmaximal, characteristic
8#13804C211yesmaximal, characteristic, complemented
9#11077C212yesmaximal, complemented
10#11009C212yesmaximal
11#10932C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2212yes—
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#102C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C22×#97C2211yescharacteristic
25#2547—24no—
26#2547C418yescomplemented
27C2×#397—21no—
28C2×#397—21no—
29C2×#391C2212yes—
30C2×#382C2211yescharacteristic
31C2×#380—22no—
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1983—28no—
36#1981—24no—
37#1960—24no—
38#1940—24no—
39#1804—24no—
40#1762—24no—
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#340C2211yescharacteristic
44C2×#336C2212yes—
45#1436—22no—
46C4⋊#95C2212yes—
47#1434—21no—
48C4×#95—21no—
49C23×Q8C2312yes—
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2×C414yes—
61C2×#124—24no—
62C2×#124—24no—
63C2×#124—24no—
64C2×#124—42no—
65C2×#124—42no—
66C2×#124C2×C418yes—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4C2×C414yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D412yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C42—21no—
84C22×C42—22no—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87C2×C4⋊C8—24no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—42no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—42no—
102C2×#103—42no—
103C2×#103—42no—
104C2×#103—42no—
105C2×#102—22no—
106C2×#102—22no—
107C2×#102—22no—
108C2×#102—22no—
109C2×#102—22no—
110C2×#102—24no—
111C2×#102—24no—
112C2×#102—42no—
113C2×#102D414yes—
114C2×#97—22no—
115C2×#97—22no—
116C2×#97—22no—
117#397—28no—
118#397—28no—
119#391—24no—
120#391C2×C418yes—
121#382—22no—
122#382D414yes—
123#380—216no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—42no—
127C4×C4⋊C4C2×C414yes—
128C4×C4⋊C4D414yes—
129C4×C4⋊C4D414yes—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95D411yescharacteristic
139C2×#95D411yescharacteristic
140#340—24no—
141#340—22no—
142#340—22no—
143#336—24no—
144C25—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8C22×C414yes—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4C22×C412yes—
171C22×D4C2×D412yes—
172C22×D4#4112yes—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—24no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2411yescharacteristic
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8C2×D412yes—
208C22×C8#4112yes—
209C4⋊Q8—28no—
210C4⋊Q8—28no—
211C4⋊Q8—44no—
212C4⋊Q8—44no—
213#124—216no—
214#124—48no—
215#124—48no—
216#124—48no—
217C4×D4—28no—
218C4×D4—28no—
219C4×D4—44no—
220C4×D4—44no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4C22⋊C418yes—
248C2×C4⋊C4C22×C412yes—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4#4112yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—44no—
269C2×C42—44no—
270C2×C42—82no—
271C2×C42—82no—
272C2×C42C22×C414yes—
273C2×C42C22×C412yes—
274C2×C42C2×D414yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42#4112yes—
279C2×C42#4112yes—
280C4⋊C8—48no—
281C4⋊C8—44no—
282#103—48no—
283#103—48no—
284#103—48no—
285#103—48no—
286#102—28no—
287#102—28no—
288#102—48no—
289#97—44no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—44no—
296#95—44no—
297#95—82no—
298#95C22⋊C414yes—
299#95C22⋊C414yes—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24C22×D411yescharacteristic
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
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—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4C4×D414yes—
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—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—416no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—48no—
415C22×C4—88no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4C2×C22⋊C414yes—
419C22×C4C2×C22⋊C414yes—
420C22×C4C2×C22⋊C412yes—
421C22×C4C2×C22⋊C412yes—
422C22×C4#12112yes—
423C22×C4#12112yes—
424C22×C4#12112yes—
425C22×C4#12112yes—
426C22×C4C23×C411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4H(5,𝔽2)11yescharacteristic
437C22×C4H(5,𝔽2)11yescharacteristic
438C22×C4H(5,𝔽2)11yescharacteristic
439C22×C42−1+411yescharacteristic
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—48no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4#12114yes—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—48no—
483C42—24no—
484C42—24no—
485C42—24no—
486C42—24no—
487C42—84no—
488C42—88no—
489C42C4×D414yes—
490C42C4×D414yes—
491C42#12414yes—
492C42#12714yes—
493C23—24no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—81no—
512C23—81no—
513C23—81no—
514C23—81no—
515C23—82no—
516C23C22×C22⋊C411yescharacteristic
517C23C2×#12111yescharacteristic
518C23C2×#12111yescharacteristic
519C23D4211yescharacteristic
520C23#54611yescharacteristic
521C23#54811yescharacteristic
522C23#55011yescharacteristic
523C23#57512yes—
524C23#57512yes—
525C23#57712yes—
526C23#57812yes—
527Q8—44no—
528Q8—44no—
529Q8—44no—
530Q8—44no—
531Q8—82no—
532Q8—82no—
533Q8—82no—
534Q8—82no—
535D4—24no—
536D4—24no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—42no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—816no—
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—81no—
593C2×C4—81no—
594C2×C4—82no—
595C2×C4#39014yes—
596C2×C4#39014yes—
597C2×C4C2×C4×D411yescharacteristic
598C2×C4C2×C4×D411yescharacteristic
599C2×C4C2×C4×D411yescharacteristic
600C2×C4C4×#4111yescharacteristic
601C2×C4#51811yescharacteristic
602C2×C4#52011yescharacteristic
603C2×C4C2×#12111yescharacteristic
604C2×C4#52511yescharacteristic
605C2×C4C2×#12411yescharacteristic
606C2×C4#52911yescharacteristic
607C2×C4C2×#12711yescharacteristic
608C2×C4#53211yescharacteristic
609C2×C4#53511yescharacteristic
610C2×C4#53711yescharacteristic
611C2×C4#54011yescharacteristic
612C2×C4#54211yescharacteristic
613C2×C4D4211yescharacteristic
614C2×C4D4211yescharacteristic
615C2×C4#54611yescharacteristic
616C2×C4#54611yescharacteristic
617C2×C4#54711yescharacteristic
618C2×C4#54811yescharacteristic
619C2×C4#54811yescharacteristic
620C2×C4#55011yescharacteristic
621C2×C4#55011yescharacteristic
622C2×C4#55311yescharacteristic
623C2×C4#55311yescharacteristic
624C2×C4#55511yescharacteristic
625C8—42no—
626C8—42no—
627C8—82no—
628C22—48no—
629C22—44no—
630C22—82no—
631C22—82no—
632C22—84no—
633C22—84no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—84no—
639C22C2×#39011yescharacteristic
640C22D4×C22⋊C411yescharacteristic
641C22#234111yescharacteristic
642C22#241411yescharacteristic
643C22#241611yescharacteristic
644C22#242611yescharacteristic
645C22#242811yescharacteristic
646C22#289612yes—
647C22#289812yes—
648C22#304012yes—
649C22#304011yescharacteristic
650C22#304011yescharacteristic
651C22#304211yescharacteristic
652C22#304411yescharacteristic
653C22#304612yes—
654C22#326712yes—
655C22#326912yes—
656C22#330712yes—
657C22#331512yes—
658C22#337612yes—
659C22#337712yes—
660C22#340112yes—
661C22#340212yes—
662C4—24no—
663C4—24no—
664C4—44no—
665C4—44no—
666C4—44no—
667C4—82no—
668C4—84no—
669C4—82no—
670C4—84no—
671C4C4×#12111yescharacteristic
672C4#232411yescharacteristic
673C4#236111yescharacteristic
674C4#236311yescharacteristic
675C4#249011yescharacteristic
676C4#249111yescharacteristic
677C4#254011yescharacteristic
678C4#254111yescharacteristic
679C2—84no—
680C2#1399811yescharacteristic
681C2#2126111yescharacteristic
682C2#2126211yescharacteristic
683C2#2242012yes—
684C2#2254812yes—
685C2#2464311yescharacteristic
686C2#2464511yescharacteristic
687C2#2474311yescharacteristic
688C2#2474511yescharacteristic
689C2#2528912yes—
690C2#2538112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1981C2semidirect
4. C2×#1960C2semidirect
5. C2×#1940C2semidirect
8. #13804C2semidirect
9. #11077C2semidirect
11. #10932C2semidirect
26. #2547C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#19832
C2×#19812
C2×#19602
C2×#19402
C2×#18042
C2×#17622
#138042
#110772
#110092
#109322