Finite Group Databasebeta

← #357814

Subgroups — #357814

Order 512.

2622 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#430C211yesmaximal, characteristic
2C22×#419C211yesmaximal, characteristic
3C22×#419C211yesmaximal, characteristic
4C23×#101C211yesmaximal, characteristic
5C2×#1742C211yesmaximal, characteristic
6#20347C211yesmaximal, characteristic
7#20335C211yesmaximal, characteristic
8#11128C218yesmaximal, complemented
9C24×Q8C2211yescharacteristic
10C23×M16C2211yescharacteristic
11C23×M16C2211yescharacteristic
12C23×M16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#430—24no—
17C2×#430—24no—
18C2×#430—24no—
19C22×#108C2214yes—
20C2×#419—24no—
21C2×#419—24no—
22C2×#419—24no—
23C2×#419—22no—
24C2×#419—22no—
25C2×#419—22no—
26C2×#419C418yescomplemented
27C2×#419C2214yes—
28C22×#103—22no—
29C22×#103C2214yes—
30C22×#101—24no—
31C22×#101—24no—
32C22×#101—24no—
33C22×#101C2214yes—
34#1742—24no—
35C2×#343C2214yes—
36#1736—22no—
37#1736C2214yes—
38#1735—22no—
39#1729—22no—
40#1729C2214yes—
41#1728—22no—
42C23×Q8—22no—
43C23×Q8—22no—
44C23×Q8—22no—
45C23×Q8—21no—
46C23×Q8—21no—
47C23×Q8—21no—
48C23×Q8—21no—
49C23×Q8—21no—
50C23×Q8—22no—
51C23×Q8—22no—
52C23×Q8—22no—
53C23×Q8—22no—
54C23×Q8—22no—
55C23×Q8—22no—
56C23×Q8C2×C414yes—
57C23×Q8D414yes—
58C23×Q8Q814yes—
59C23×Q8C2312yes—
60C24×C4—21no—
61C24×C4C2311yescharacteristic
62C22×M16—22no—
63C22×M16—22no—
64C22×M16—22no—
65C22×M16—22no—
66C22×M16—22no—
67C22×M16—21no—
68C22×M16—21no—
69C22×M16—21no—
70C22×M16—22no—
71C22×M16—22no—
72C22×M16—22no—
73C22×M16—22no—
74C22×M16—22no—
75C22×M16—22no—
76C22×M16—22no—
77C22×M16—22no—
78C22×M16—22no—
79C22×M16—22no—
80C22×M16—22no—
81C22×M16—22no—
82C22×M16—22no—
83C22×M16—22no—
84C22×M16C2×C414yes—
85C22×M16D414yes—
86C22×M16D414yes—
87C22×M16D414yes—
88C22×M16D414yes—
89C22×M16D412yes—
90C22×M16D412yes—
91C22×M16C2312yes—
92C22×M16C2312yes—
93C22×M16C2312yes—
94C23×C8—21no—
95C23×C8—21no—
96C23×C8—21no—
97C23×C8C2312yes—
98C2×#117—21no—
99C2×#117—22no—
100C2×#117—21no—
101C2×#117—21no—
102C2×#117—21no—
103C2×#117—22no—
104C2×#117—22no—
105C2×#117—22no—
106C2×#117—21no—
107C2×#117—21no—
108C2×#117—22no—
109C2×#117—22no—
110C2×#117—22no—
111C2×#117C2×C414yes—
112C2×#117D412yes—
113C2×#117C2312yes—
114C22×C4⋊C4D412yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C42—21no—
119C22×C42—21no—
120#430—232no—
121C2×#108—28no—
122C2×#108—28no—
123C2×#108—28no—
124#419—216no—
125#419—28no—
126#419—216no—
127#419—216no—
128#419—28no—
129C2×#103—216no—
130C2×#103—28no—
131C2×#103—28no—
132C2×#103—28no—
133C2×#103C2×C4116yescomplemented
134C2×#101—232no—
135C2×#101—28no—
136C2×#101—28no—
137C2×#101—28no—
138#343—28no—
139#340—28no—
140#336—28no—
141C25#4111yescharacteristic
142C22×Q8—24no—
143C22×Q8—28no—
144C22×Q8—28no—
145C22×Q8—24no—
146C22×Q8—28no—
147C22×Q8—24no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—24no—
152C22×Q8—24no—
153C22×Q8—28no—
154C22×Q8—28no—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—28no—
162C22×Q8—28no—
163C22×Q8—28no—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—28no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—44no—
178C22×Q8—41no—
179C22×Q8—41no—
180C22×Q8—44no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8—44no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×Q8—44no—
193C22×Q8—42no—
194C22×Q8—42no—
195C22×Q8—44no—
196C22×Q8—42no—
197C22×Q8—44no—
198C22×Q8—42no—
199C22×Q8—42no—
200C22×Q8C4⋊C4116yes—
201C22×Q8C22×C414yes—
202C22×Q8C2×D414yes—
203C22×Q8C2×Q814yes—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—41no—
211C23×C4—41no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4C22×C412yes—
216C23×C4C2×D412yes—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D412yes—
220C23×C4C2×D412yes—
221C23×C4C2×D412yes—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D412yes—
225C23×C4C2×Q812yes—
226C23×C4#4112yes—
227C23×C4#4112yes—
228C23×C4#4112yes—
229C23×C4#4112yes—
230C23×C4#4111yescharacteristic
231C23×C4#4112yes—
232C23×C4C2411yescharacteristic
233C2×M16—28no—
234C2×M16—28no—
235C2×M16—28no—
236C2×M16—24no—
237C2×M16—24no—
238C2×M16—24no—
239C2×M16—24no—
240C2×M16—28no—
241C2×M16—24no—
242C2×M16—24no—
243C2×M16—24no—
244C2×M16—22no—
245C2×M16—22no—
246C2×M16—22no—
247C2×M16—28no—
248C2×M16—28no—
249C2×M16—28no—
250C2×M16—28no—
251C2×M16—28no—
252C2×M16—28no—
253C2×M16—22no—
254C2×M16—22no—
255C2×M16—22no—
256C2×M16—22no—
257C2×M16—22no—
258C2×M16—22no—
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—42no—
266C2×M16—44no—
267C2×M16—42no—
268C2×M16—42no—
269C2×M16—42no—
270C2×M16—44no—
271C2×M16—44no—
272C2×M16—44no—
273C2×M16—44no—
274C2×M16—44no—
275C2×M16—42no—
276C2×M16—42no—
277C2×M16C22⋊C418yes—
278C2×M16C22⋊C418yes—
279C2×M16C2×D414yes—
280C2×M16C2×D414yes—
281C2×M16C2×D414yes—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8—22no—
285C22×C8—22no—
286C22×C8—24no—
287C22×C8—24no—
288C22×C8—24no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—42no—
293C22×C8—42no—
294C22×C8—42no—
295C22×C8—42no—
296C22×C8—42no—
297C22×C8—42no—
298C22×C8C22×C414yes—
299C22×C8C2×D414yes—
300C22×C8C2×D414yes—
301#117—24no—
302#117—24no—
303#117—24no—
304#117—24no—
305#117—24no—
306#117—24no—
307#117—22no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117—42no—
312#117—42no—
313#117—42no—
314#117—42no—
315#117—42no—
316#117—42no—
317#117—42no—
318#117—42no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4C22×C414yes—
328C2×C4⋊C4C2×D414yes—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—41no—
333C2×C22⋊C4—41no—
334C2×C22⋊C4—42no—
335C2×C22⋊C4—42no—
336C2×C22⋊C4—42no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—41no—
343C2×C42—41no—
344C2×C42—42no—
345#108—232no—
346#103—232no—
347#103—232no—
348#101—232no—
349C24—44no—
350C24C4×D412yes—
351C24#12111yescharacteristic
352C24#12111yescharacteristic
353C24C22⋊Q812yes—
354C24#12312yes—
355C24#12412yes—
356C24#12412yes—
357C24#12612yes—
358C24C2×#4111yescharacteristic
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—216no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—216no—
376C2×Q8—216no—
377C2×Q8—216no—
378C2×Q8—216no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—48no—
392C2×Q8—48no—
393C2×Q8—48no—
394C2×Q8—48no—
395C2×Q8—48no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—48no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—48no—
404C2×Q8—48no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—48no—
408C2×Q8—48no—
409C2×Q8—48no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8C2×C4⋊C418yes—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—216no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—48no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4C2×C22⋊C412yes—
438C22×C4C2×C22⋊C412yes—
439C22×C4C2×C4⋊C414yes—
440C22×C4#11714yes—
441C22×C4C4×D414yes—
442C22×C4C4×D412yes—
443C22×C4C4×D412yes—
444C22×C4C4×D414yes—
445C22×C4C4×D412yes—
446C22×C4#12012yes—
447C22×C4#12012yes—
448C22×C4#12014yes—
449C22×C4#12014yes—
450C22×C4#12114yes—
451C22×C4#12112yes—
452C22×C4#12112yes—
453C22×C4#12112yes—
454C22×C4#12112yes—
455C22×C4#12112yes—
456C22×C4#12112yes—
457C22×C4#12112yes—
458C22×C4#12112yes—
459C22×C4#12114yes—
460C22×C4#12112yes—
461C22×C4#12112yes—
462C22×C4#12111yescharacteristic
463C22×C4#12111yescharacteristic
464C22×C4C22⋊Q814yes—
465C22×C4C22⋊Q812yes—
466C22×C4C22⋊Q812yes—
467C22×C4C22⋊Q814yes—
468C22×C4C22⋊Q812yes—
469C22×C4C22⋊Q812yes—
470C22×C4C22⋊Q814yes—
471C22×C4C22⋊Q812yes—
472C22×C4#12314yes—
473C22×C4#12314yes—
474C22×C4#12312yes—
475C22×C4#12312yes—
476C22×C4#12314yes—
477C22×C4#12312yes—
478C22×C4#12414yes—
479C22×C4#12412yes—
480C22×C4#12412yes—
481C22×C4#12614yes—
482C22×C4#12612yes—
483C22×C4#12712yes—
484C22×C4#12712yes—
485C22×C4C23×C411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×Q811yescharacteristic
494C22×C4C2×#4111yescharacteristic
495C22×C4C2×#4111yescharacteristic
496C22×C4C2×#4111yescharacteristic
497C22×C4C2×#4111yescharacteristic
498C22×C4C2×#4111yescharacteristic
499M16—216no—
500M16—216no—
501M16—24no—
502M16—24no—
503M16—24no—
504M16—24no—
505M16—24no—
506M16—24no—
507M16—42no—
508M16—42no—
509M16—42no—
510M16—42no—
511M16—42no—
512M16—42no—
513M16—48no—
514M16—48no—
515M16—48no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8—24no—
524C2×C8—24no—
525C2×C8—42no—
526C2×C8—42no—
527C2×C8—42no—
528C2×C8—42no—
529C2×C8—42no—
530C2×C8—42no—
531C2×C8—48no—
532C2×C8—48no—
533C2×C8—48no—
534C2×C8C2×C22⋊C418yes—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C42—42no—
548C42—42no—
549C42—44no—
550C42—44no—
551C42—44no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23#38514yes—
562C23#38814yes—
563C23#39012yes—
564C23#39012yes—
565C23#39314yes—
566C23#39412yes—
567C23#39412yes—
568C23#39612yes—
569C23#39612yes—
570C23#39714yes—
571C23C2×C4×D411yescharacteristic
572C23C2×#12111yescharacteristic
573C23C2×C22⋊Q811yescharacteristic
574C23C2×#12311yescharacteristic
575C23C2×#12411yescharacteristic
576C23C2×#12411yescharacteristic
577C23C2×#12611yescharacteristic
578Q8—28no—
579Q8—28no—
580Q8—44no—
581Q8—44no—
582Q8—44no—
583Q8—44no—
584Q8—416no—
585Q8—416no—
586Q8—44no—
587Q8—44no—
588C2×C4—28no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—416no—
594C2×C4—416no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—84no—
601C2×C4—88no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4#38014yes—
605C2×C4#38014yes—
606C2×C4#38514yes—
607C2×C4#38514yes—
608C2×C4#38514yes—
609C2×C4#38614yes—
610C2×C4#38614yes—
611C2×C4#38814yes—
612C2×C4#38814yes—
613C2×C4#38814yes—
614C2×C4#39012yes—
615C2×C4#39012yes—
616C2×C4#39214yes—
617C2×C4#39214yes—
618C2×C4#39314yes—
619C2×C4#39414yes—
620C2×C4#39414yes—
621C2×C4#39412yes—
622C2×C4#39412yes—
623C2×C4#39614yes—
624C2×C4#39614yes—
625C2×C4#39612yes—
626C2×C4#39612yes—
627C2×C4#39714yes—
628C2×C4#39914yes—
629C2×C4#39914yes—
630C2×C4C22×C22⋊C411yescharacteristic
631C2×C4C22×C4⋊C411yescharacteristic
632C2×C4C2×#11711yescharacteristic
633C2×C4C2×C4×D411yescharacteristic
634C2×C4C2×C4×D411yescharacteristic
635C2×C4C2×C4×D411yescharacteristic
636C2×C4C2×#12011yescharacteristic
637C2×C4C2×#12011yescharacteristic
638C2×C4C2×#12011yescharacteristic
639C2×C4C2×#12111yescharacteristic
640C2×C4C2×#12111yescharacteristic
641C2×C4C2×#12111yescharacteristic
642C2×C4C2×#12111yescharacteristic
643C2×C4C2×#12111yescharacteristic
644C2×C4C2×#12111yescharacteristic
645C2×C4C2×#12111yescharacteristic
646C2×C4C2×C22⋊Q811yescharacteristic
647C2×C4C2×C22⋊Q811yescharacteristic
648C2×C4C2×C22⋊Q811yescharacteristic
649C2×C4C2×C22⋊Q811yescharacteristic
650C2×C4C2×C22⋊Q811yescharacteristic
651C2×C4C2×#12311yescharacteristic
652C2×C4C2×#12311yescharacteristic
653C2×C4C2×#12311yescharacteristic
654C2×C4C2×#12311yescharacteristic
655C2×C4C2×#12411yescharacteristic
656C2×C4C2×#12611yescharacteristic
657C2×C4C2×#12711yescharacteristic
658C8—28no—
659C8—44no—
660C8—44no—
661C8—44no—
662C22—22no—
663C22—22no—
664C22—24no—
665C22—24no—
666C22—22no—
667C22—22no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—22no—
672C22—22no—
673C22—24no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22#143914yes—
679C22#143914yes—
680C22C2×#38511yescharacteristic
681C22C2×#38811yescharacteristic
682C22C2×#39011yescharacteristic
683C22C2×#39311yescharacteristic
684C22C2×#39411yescharacteristic
685C22C2×#39611yescharacteristic
686C22C2×#39711yescharacteristic
687C22#306914yes—
688C4—48no—
689C4—84no—
690C4—84no—
691C4#143914yes—
692C4#143914yes—
693C4C2×#38011yescharacteristic
694C4C2×#38511yescharacteristic
695C4C2×#38611yescharacteristic
696C4C2×#38811yescharacteristic
697C4C2×#39211yescharacteristic
698C4C2×#39411yescharacteristic
699C4C2×#39611yescharacteristic
700C4C2×#39911yescharacteristic
701C2—21no—
702C2—21no—
703C2—24no—
704C2—24no—
705C2—21no—
706C2—21no—
707C2C2×#143911yescharacteristic
708C2#2125012yes—
709C2#2353212yes—
710C2#2360312yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11128C2semidirect
26. C2×#419C4semidirect
133. C2×#103C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4302
C22×#4192
C23×#1012
C2×#17422
#203472
#203352
#111282