Finite Group Databasebeta

← #874496

Subgroups — #874496

Order 512.

4698 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2406C212yesmaximal, complemented
2C2×#2430C214yesmaximal, complemented
3#17072C211yesmaximal, characteristic, complemented
4#17068C211yesmaximal, characteristic, complemented
5#16878C212yesmaximal, complemented
6#15622C214yesmaximal, complemented
7#15498C214yesmaximal, complemented
8#14927C214yesmaximal, complemented
9#14719C212yesmaximal, complemented
10#14698C212yesmaximal, complemented
11#14535C212yesmaximal, complemented
12#14440C212yesmaximal, complemented
13C2×C4⋊#95C211yesmaximal, characteristic, complemented
14C23×#41C2212yes—
15C22×#123C2214yes—
16C22×#121C2212yes—
17C22×#121C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2430—232no—
22#2406—216no—
23C2×#395C2212yes—
24C2×#395C2214yes—
25C2×#395C2212yes—
26C2×#394C2212yes—
27C2×#394C2212yes—
28C2×#394C2214yes—
29C2×#394C2212yes—
30C2×#394C2212yes—
31C2×#393C2214yes—
32C2×#392C2214yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#386C2214yes—
36C2×#386C2214yes—
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#386C2214yes—
40C2×#385C2214yes—
41C2×#385C2214yes—
42C2×#385C2214yes—
43C2×#385C2214yes—
44C2×#381C2212yes—
45C2×#380C2212yes—
46C22×#95C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2212yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2212yes—
53#1440C2214yes—
54#1440C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2214yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2211yescharacteristic
68C22⋊#95C2212yes—
69C22×#41—216no—
70C22×#41—28no—
71C22×#41—44no—
72C22×#41—44no—
73C22×#41—48no—
74C23×Q8C2312yes—
75C23×D4C2314yes—
76C23×D4C2312yes—
77C23×D4C2312yes—
78C23×D4C2312yes—
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2314yes—
84C2×#123—216no—
85C2×#123—216no—
86C2×#123—48no—
87C2×#121—216no—
88C2×#121—44no—
89C2×#121—44no—
90C2×#121—44no—
91C2×#121D4116yescomplemented
92C22×C4⋊C4—24no—
93C22×C4⋊C4—42no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4—42no—
96C22×C4⋊C4—42no—
97C22×C4⋊C4D418yescomplemented
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C42—24no—
122C22×C42—42no—
123C22×C42—42no—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#395—28no—
128#395—28no—
129#395—216no—
130#394—28no—
131#394—28no—
132#394—28no—
133#394—216no—
134#394—44no—
135#393—216no—
136#392—216no—
137#390—28no—
138#390—44no—
139#386—216no—
140#386—216no—
141#386—216no—
142#386—216no—
143#386—216no—
144#385—216no—
145#385—216no—
146#385—216no—
147#385—216no—
148#381—28no—
149#380—44no—
150C2×#95—216no—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C25C2414yes—
169C25C2411yescharacteristic
170C2×#41—464no—
171C2×#41—432no—
172C2×#41—432no—
173C2×#41—464no—
174C2×#41—432no—
175C22×Q8—28no—
176C22×Q8—24no—
177C22×Q8—44no—
178C22×Q8—42no—
179C22×Q8—42no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—28no—
187C22×D4—28no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—44no—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4—42no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—44no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4C2×D4116yes—
203C22×D4C2×D418yes—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—28no—
219C23×C4—28no—
220C23×C4—24no—
221C23×C4—28no—
222C23×C4—28no—
223C23×C4—24no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—44no—
232C23×C4C2×D418yes—
233C23×C4C2×D414yes—
234C23×C4C2×D418yes—
235C23×C4C2414yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246#123—464no—
247#121—432no—
248#121—432no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—416no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—416no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D418yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D418yes—
285C2×C4⋊C4#4114yescomplemented
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4C2×D4116yes—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—41no—
323C2×C42—41no—
324C2×C42—42no—
325C2×C42—41no—
326C2×C42—41no—
327C2×C42—416no—
328C2×C42#4114yescomplemented
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—24no—
343#95—44no—
344#95—44no—
345#95—42no—
346C24—21no—
347C24—21no—
348C24—21no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24C22×D418yes—
367C24C22×D412yes—
368C24C22×D414yes—
369C24H(5,𝔽2)12yes—
370C24H(5,𝔽2)14yes—
371C24H(5,𝔽2)12yes—
372C24C2511yescharacteristic
373#41—4256no—
374C2×Q8—416no—
375C2×Q8—48no—
376C2×Q8—416no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×D4—232no—
380C2×D4—232no—
381C2×D4—232no—
382C2×D4—216no—
383C2×D4—416no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—416no—
388C2×D4—48no—
389C2×D4—416no—
390C2×D4—48no—
391C2×D4—416no—
392C2×D4—416no—
393C2×D4—416no—
394C2×D4—416no—
395C2×D4—48no—
396C2×D4—84no—
397C2×D4—84no—
398C2×D4—84no—
399C2×D4—84no—
400C2×D4—84no—
401C2×D4—88no—
402C2×D4—84no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—216no—
423C22×C4—216no—
424C22×C4—232no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—216no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—48no—
470C22×C4—416no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—416no—
476C22×C4—416no—
477C22×C4—416no—
478C22×C4—416no—
479C22×C4—48no—
480C22×C4—416no—
481C22×C4—416no—
482C22×C4—416no—
483C22×C4—416no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—88no—
488C22×C4—84no—
489C22×C4C22×D414yes—
490C22×C4C22×D414yes—
491C22×C4C22×D414yes—
492C22×C4C22×D418yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D414yes—
496C22×C4C2×#4114yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4114yes—
500C22×C4H(5,𝔽2)12yes—
501C22×C42−1+412yes—
502C22×C42−1+414yes—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—416no—
511C4⋊C4—416no—
512C4⋊C4—44no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C22⋊C4—416no—
516C22⋊C4—416no—
517C22⋊C4—416no—
518C22⋊C4—416no—
519C22⋊C4—416no—
520C22⋊C4—416no—
521C22⋊C4—416no—
522C22⋊C4—416no—
523C22⋊C4—416no—
524C22⋊C4—88no—
525C22⋊C4—88no—
526C42—44no—
527C42—48no—
528C42—44no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—28no—
548C23—28no—
549C23—28no—
550C23—28no—
551C23—28no—
552C23—28no—
553C23—28no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23#53414yes—
583C23#53418yes—
584C23D4218yes—
585C23D4214yes—
586C23H(3,𝔽4)14yes—
587C23C23×D411yescharacteristic
588C23C23×D412yes—
589C23C22×#4111yescharacteristic
590C23C22×#4111yescharacteristic
591C23C2×H(5,𝔽2)11yescharacteristic
592C23C2×H(5,𝔽2)11yescharacteristic
593C23C2×H(5,𝔽2)12yes—
594C23C2×H(5,𝔽2)11yescharacteristic
595C23C2×2−1+411yescharacteristic
596C23C2×2−1+412yes—
597C23#58512yes—
598Q8—432no—
599D4—432no—
600D4—464no—
601D4—816no—
602D4—816no—
603C2×C4—232no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—28no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—28no—
629C2×C4—28no—
630C2×C4—24no—
631C2×C4—28no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—432no—
635C2×C4—464no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—432no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—816no—
674C2×C4#53214yes—
675C2×C4#53714yes—
676C2×C4#54718yes—
677C22—24no—
678C22—28no—
679C22—28no—
680C22—24no—
681C22—24no—
682C22—24no—
683C22—24no—
684C22—28no—
685C22—28no—
686C22—24no—
687C22—48no—
688C22—48no—
689C22—48no—
690C22—48no—
691C22—48no—
692C22—48no—
693C22—48no—
694C22—84no—
695C22—84no—
696C22—84no—
697C22—84no—
698C22—84no—
699C22—84no—
700C22—84no—
701C22—84no—
702C22#267918yes—
703C22C2×#53211yescharacteristic
704C22C2×#53412yes—
705C22C2×#53411yescharacteristic
706C22C2×#53711yescharacteristic
707C22C2×D4212yes—
708C22C2×D4211yescharacteristic
709C22C2×#54712yes—
710C22D4×#4112yes—
711C22#349011yescharacteristic
712C22#349011yescharacteristic
713C22#349211yescharacteristic
714C22#349211yescharacteristic
715C22#349412yes—
716C22#349612yes—
717C22#349712yes—
718C22#349912yes—
719C22#352212yes—
720C22C2×H(3,𝔽4)11yescharacteristic
721C22#353512yes—
722C22#353612yes—
723C22#354411yescharacteristic
724C22#354511yescharacteristic
725C22#355912yes—
726C4—28no—
727C4—24no—
728C4—24no—
729C4—44no—
730C4—48no—
731C4—44no—
732C4—44no—
733C4—44no—
734C4—84no—
735C4—82no—
736C4—84no—
737C2—28no—
738C2—88no—
739C2#3505111yescharacteristic
740C2#3612612yes—
741C2#3719011yescharacteristic
742C2#3786012yes—
743C2#3842512yes—
744C2#3935912yes—
745C2#4599111yescharacteristic
746C2#4609911yescharacteristic
747C2C2×#267911yescharacteristic
748C2#5931311yescharacteristic
749C2#5981411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2406C2semidirect
2. C2×#2430C2semidirect
3. #17072C2semidirect
4. #17068C2semidirect
5. #16878C2semidirect
6. #15622C2semidirect
7. #15498C2semidirect
8. #14927C2semidirect
9. #14719C2semidirect
10. #14698C2semidirect
11. #14535C2semidirect
12. #14440C2semidirect
13. C2×C4⋊#95C2semidirect
91. C2×#121D4semidirect
97. C22×C4⋊C4D4semidirect
285. C2×C4⋊C4#41semidirect
328. C2×C42#41semidirect

Maximal subgroups

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

subgroupindex
C2×#24062
C2×#24302
#170722
#170682
#168782
#156222
#154982
#149272
#147192
#146982
#145352
#144402
C2×C4⋊#952