Finite Group Databasebeta

← #379744

Subgroups — #379744

Order 512.

2408 conjugacy classes of subgroups.

The subgroup lattice has 793 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2616C211yesmaximal, characteristic
2C2×#2311C211yesmaximal, characteristic
3C2×#1873C211yesmaximal, characteristic, complemented
4C2×#1872C211yesmaximal, characteristic, complemented
5C2×#1780C211yesmaximal, characteristic
6#11200C212yesmaximal, complemented
7#11199C212yesmaximal, complemented
8#10970C212yesmaximal, complemented
9#10970C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic, complemented
11#10730C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2212yes—
23C22×C22⋊C8C2212yes—
24#2616—24no—
25#2616C418yescomplemented
26C2×#395—21no—
27C2×#394—22no—
28C2×#394—21no—
29#2311—24no—
30#2311—24no—
31C2×#391—21no—
32C2×#390—21no—
33C2×#386—22no—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×#380C2212yes—
38#1873—24no—
39#1872—24no—
40#1780—24no—
41C2×#340C2212yes—
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C22×#41—22no—
49C22×#41—22no—
50C22×#41—22no—
51C22×#41—22no—
52C22×#41—42no—
53C22×#41—42no—
54C22×#41—42no—
55C22×#41—42no—
56C22×#41C2×C414yes—
57C22×#41D414yes—
58C23×Q8D411yescharacteristic
59C23×Q8C2311yescharacteristic
60C23×D4D412yes—
61C23×D4D411yescharacteristic
62C24×C4—21no—
63C24×C4C2312yes—
64C24×C4C2311yescharacteristic
65C23×C8C2312yes—
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—42no—
75C2×C22⋊Q8—42no—
76C2×C22⋊Q8—42no—
77C2×C22⋊Q8—42no—
78C2×C22⋊Q8C2×C414yes—
79C2×C22⋊Q8C2×C418yescomplemented
80C2×#117—22no—
81C2×#117—42no—
82C2×#117—42no—
83C2×#117D414yes—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4D414yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C22⋊C4—21no—
97C22×C22⋊C4—22no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—22no—
101C22×C22⋊C4—22no—
102C22×C22⋊C4—22no—
103C22×C42—21no—
104C2×#103—24no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—42no—
108C2×#103—42no—
109C2×#103—42no—
110C2×#103—42no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—24no—
114C2×C22⋊C8—24no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8—22no—
117C2×C22⋊C8—22no—
118C2×C22⋊C8—22no—
119C2×C22⋊C8—42no—
120C2×C22⋊C8—42no—
121C2×C22⋊C8—42no—
122C2×C22⋊C8—42no—
123#395—28no—
124#394—28no—
125#394—216no—
126#391—28no—
127#390—28no—
128#386—216no—
129#380—22no—
130#380—24no—
131#380—24no—
132#380—24no—
133#380—24no—
134#380—24no—
135#380C2×C414yes—
136#380C2×C418yescomplemented
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—22no—
141#340—24no—
142#340—22no—
143#340—22no—
144#340—42no—
145#340—42no—
146#340—42no—
147#340—42no—
148C25C2×D412yes—
149C25C2×D411yescharacteristic
150C2×#41—28no—
151C2×#41—28no—
152C2×#41—28no—
153C2×#41—28no—
154C2×#41—48no—
155C2×#41—48no—
156C2×#41—48no—
157C2×#41—48no—
158C2×#41—48no—
159C2×#41—44no—
160C2×#41—44no—
161C2×#41—44no—
162C2×#41—44no—
163C2×#41—48no—
164C2×#41—48no—
165C2×#41—48no—
166C2×#41—48no—
167C2×#41—48no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—42no—
180C22×Q8—42no—
181C22×Q8C22⋊C414yes—
182C22×Q8C22×C412yes—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4—42no—
193C22×D4C22⋊C418yes—
194C22×D4C22⋊C414yes—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—22no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—44no—
219C23×C4—81no—
220C23×C4—82no—
221C23×C4—81no—
222C23×C4C22×C414yes—
223C23×C4C22×C412yes—
224C23×C4C2×D412yes—
225C23×C4C2×D414yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D412yes—
228C23×C4C2×D412yes—
229C23×C4C2×D412yes—
230C23×C4#4112yes—
231C23×C4#4112yes—
232C23×C4C2411yescharacteristic
233C22×C8—22no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—22no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—22no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22⋊Q8—28no—
251C22⋊Q8—28no—
252C22⋊Q8—48no—
253C22⋊Q8—48no—
254C22⋊Q8—48no—
255C22⋊Q8—48no—
256C22⋊Q8—44no—
257C22⋊Q8—44no—
258#117—44no—
259#117—44no—
260#117—48no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—48no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4C22×C412yes—
294C2×C4⋊C4C22×C414yes—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—81no—
311C2×C22⋊C4—82no—
312C2×C22⋊C4—81no—
313C2×C22⋊C4—82no—
314C2×C22⋊C4—82no—
315C2×C22⋊C4—82no—
316C2×C22⋊C4—82no—
317C2×C22⋊C4—82no—
318C2×C42—24no—
319C2×C42—44no—
320#103—48no—
321#103—44no—
322#103—44no—
323C22⋊C8—48no—
324C22⋊C8—48no—
325C22⋊C8—44no—
326C22⋊C8—44no—
327#95—28no—
328#95—28no—
329#95—24no—
330#95—28no—
331C24—22no—
332C24—22no—
333C24—21no—
334C24—21no—
335C24—22no—
336C24—21no—
337C24—22no—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24C2×C22⋊C414yes—
354C24C2×C22⋊C412yes—
355C24#12012yes—
356C24#12312yes—
357C24C22×D411yescharacteristic
358#41—432no—
359#41—416no—
360#41—416no—
361#41—432no—
362#41—416no—
363#41—416no—
364C2×Q8—28no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—48no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—48no—
373C2×Q8—41no—
374C2×Q8—41no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×D4—216no—
388C2×D4—28no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—42no—
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—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—48no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—41no—
469C22×C4—42no—
470C22×C4—41no—
471C22×C4—42no—
472C22×C4—41no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—41no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—41no—
480C22×C4—42no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—41no—
486C22×C4—84no—
487C22×C4—84no—
488C22×C4—88no—
489C22×C4—88no—
490C22×C4C2×C22⋊C414yes—
491C22×C4C2×C22⋊C412yes—
492C22×C4C4×D414yes—
493C22×C4C4×D414yes—
494C22×C4#12012yes—
495C22×C4#12114yes—
496C22×C4#12114yes—
497C22×C4#12312yes—
498C22×C4C23×C411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C2×#4111yescharacteristic
503C22×C4C2×#4111yescharacteristic
504C22×C4H(5,𝔽2)12yes—
505C22×C4H(5,𝔽2)12yes—
506C22×C4H(5,𝔽2)12yes—
507C22×C4H(5,𝔽2)12yes—
508C2×C8—42no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—42no—
512C2×C8—42no—
513C2×C8—42no—
514C2×C8—42no—
515C2×C8—42no—
516C2×C8—41no—
517C2×C8—41no—
518C2×C8—41no—
519C2×C8—41no—
520C2×C8—42no—
521C2×C8—42no—
522C2×C8—42no—
523C2×C8—42no—
524C2×C8—42no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—48no—
536C4⋊C4—416no—
537C4⋊C4—41no—
538C4⋊C4—41no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—416no—
548C22⋊C4—48no—
549C22⋊C4—84no—
550C22⋊C4—88no—
551C22⋊C4—88no—
552C22⋊C4—88no—
553C42—48no—
554C23—24no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—24no—
561C23—22no—
562C23—24no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23#38614yes—
612C23C22×C22⋊C411yescharacteristic
613C23C2×#12011yescharacteristic
614C23C2×#12311yescharacteristic
615C23#53412yes—
616C23#53412yes—
617C23#57611yescharacteristic
618C23#57712yes—
619C23#57711yescharacteristic
620C23#57711yescharacteristic
621C23#57812yes—
622C23#57811yescharacteristic
623Q8—416no—
624Q8—44no—
625Q8—42no—
626Q8—42no—
627Q8—82no—
628Q8—81no—
629Q8—81no—
630D4—416no—
631D4—432no—
632C2×C4—24no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—28no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—28no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—816no—
675C2×C4—81no—
676C2×C4—81no—
677C2×C4—81no—
678C2×C4—81no—
679C2×C4—82no—
680C2×C4—82no—
681C2×C4—82no—
682C2×C4—82no—
683C2×C4—82no—
684C2×C4—82no—
685C2×C4—82no—
686C2×C4—82no—
687C2×C4—82no—
688C2×C4—82no—
689C2×C4—82no—
690C2×C4—82no—
691C2×C4—82no—
692C2×C4—81no—
693C2×C4—81no—
694C2×C4#38614yes—
695C2×C4C2×C4×D411yescharacteristic
696C2×C4C2×C4×D411yescharacteristic
697C2×C4#51812yes—
698C2×C4#51812yes—
699C2×C4C2×#12111yescharacteristic
700C2×C4C2×#12111yescharacteristic
701C2×C4#53412yes—
702C2×C4#53411yescharacteristic
703C2×C4#53411yescharacteristic
704C2×C4#53412yes—
705C2×C4#53412yes—
706C2×C4#53511yescharacteristic
707C2×C4#53511yescharacteristic
708C2×C4#53512yes—
709C2×C4#53812yes—
710C2×C4#53812yes—
711C2×C4#53812yes—
712C2×C4#53812yes—
713C8—82no—
714C8—81no—
715C8—81no—
716C22—24no—
717C22—24no—
718C22—24no—
719C22—44no—
720C22—44no—
721C22—42no—
722C22—42no—
723C22—42no—
724C22—42no—
725C22—44no—
726C22—44no—
727C22—44no—
728C22—44no—
729C22—44no—
730C22—44no—
731C22—44no—
732C22—44no—
733C22—44no—
734C22—82no—
735C22—82no—
736C22—82no—
737C22—82no—
738C22—82no—
739C22—82no—
740C22—82no—
741C22—82no—
742C22C2×#38611yescharacteristic
743C22#230812yes—
744C22#261311yescharacteristic
745C22#261311yescharacteristic
746C22#262212yes—
747C22#289711yescharacteristic
748C22#289811yescharacteristic
749C22#289812yes—
750C22#300911yescharacteristic
751C22#301111yescharacteristic
752C22#301211yescharacteristic
753C22#301311yescharacteristic
754C22#309712yes—
755C22#309811yescharacteristic
756C22#309911yescharacteristic
757C22#319812yes—
758C22#320012yes—
759C22#320112yes—
760C22#320112yes—
761C22#320112yes—
762C22#320212yes—
763C22#320312yes—
764C22#320412yes—
765C4—24no—
766C4—44no—
767C4—84no—
768C4—84no—
769C4—82no—
770C4—82no—
771C4—82no—
772C4—82no—
773C4#232112yes—
774C4#232111yescharacteristic
775C4#232111yescharacteristic
776C4#261712yes—
777C4#261711yescharacteristic
778C4#261711yescharacteristic
779C2—44no—
780C2—84no—
781C2#1425511yescharacteristic
782C2#2091311yescharacteristic
783C2#2091511yescharacteristic
784C2#2177112yes—
785C2#2177212yes—
786C2#2960411yescharacteristic
787C2#2960511yescharacteristic
788C2#2960811yescharacteristic
789C2#2960911yescharacteristic
790C2#2990012yes—
791C2#2990112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×#41 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1873C2semidirect
4. C2×#1872C2semidirect
6. #11200C2semidirect
7. #11199C2semidirect
8. #10970C2semidirect
9. #10970C2semidirect
10. #10731C2semidirect
11. #10730C2semidirect
25. #2616C4semidirect
79. C2×C22⋊Q8C2×C4semidirect
136. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26162
C2×#23112
C2×#18732
C2×#18722
C2×#17802
#112002
#111992
#109702
#107312
#107302