Finite Group Databasebeta

← #876610

Subgroups — #876610

Order 512.

3598 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic, complemented
2C2×#2406C212yesmaximal
3C2×#2297C211yesmaximal, characteristic
4#17078C212yesmaximal, complemented
5#16629C212yesmaximal, complemented
6#15390C214yesmaximal, complemented
7#15269C214yesmaximal, complemented
8#14907C212yesmaximal, complemented
9#14837C214yesmaximal, complemented
10#14697C212yesmaximal, complemented
11#14505C212yesmaximal, complemented
12#14440C211yesmaximal, characteristic, complemented
13#14043C214yesmaximal, complemented
14C23×#41C2211yescharacteristic
15C22×#123C2212yes—
16C22×#123C2212yes—
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19C23×C42C2211yescharacteristic
20#2724—28no—
21#2406—216no—
22C2×#398C2212yes—
23C2×#397C2214yes—
24C2×#397C2212yes—
25C2×#395C2214yes—
26C2×#394C2214yes—
27#2297—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#386C2212yes—
36C2×#386C2214yes—
37C2×#386C2212yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#385C2214yes—
41C2×#385C2214yes—
42C2×#381C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C22⋊C4C2214yes—
46#1445C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1443C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2212yes—
54#1441C2214yes—
55#1441C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66#1434C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22⋊#95C2214yes—
70C22×#41—24no—
71C22×#41—24no—
72C22×#41—42no—
73C22×#41—42no—
74C22×#41—42no—
75C22×#41—42no—
76C22×#41D418yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C2×#123—28no—
86C2×#123—28no—
87C2×#123—28no—
88C2×#123—28no—
89C2×#123—44no—
90C2×#123—44no—
91C2×#117—24no—
92C2×#117—42no—
93C2×#117—42no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C42—24no—
119C22×C42—42no—
120C22×C42—42no—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#398—28no—
126#397—216no—
127#397—44no—
128#395—216no—
129#394—216no—
130#391—24no—
131#391—24no—
132#390—24no—
133#390D418yes—
134#389—28no—
135#389—44no—
136#389—44no—
137#386—28no—
138#386—28no—
139#386—216no—
140#385—216no—
141#385—216no—
142#385—216no—
143#385—216no—
144#381—28no—
145#380—28no—
146#380—44no—
147C4×C22⋊C4—216no—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C25C2412yes—
169C25C2411yescharacteristic
170C2×#41—232no—
171C2×#41—416no—
172C2×#41—416no—
173C2×#41—416no—
174C2×#41—416no—
175C2×#41—416no—
176C2×#41—416no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—41no—
180C22×Q8—41no—
181C22×Q8—41no—
182C22×Q8—41no—
183C22×Q8C2×D414yes—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—41no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4—41no—
192C22×D4—42no—
193C22×D4—42no—
194C22×D4C2×D418yes—
195C22×D4C2×D414yes—
196C22×D4C2×D414yes—
197C22×D4#4114yes—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—28no—
217C23×C4—24no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—44no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4C2×D414yes—
230C23×C4C2×D418yes—
231C23×C4C2414yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242#123—432no—
243#123—432no—
244#117—416no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—432no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4#4118yescomplemented
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4C2×D4116yes—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—42no—
314C2×C42—41no—
315C2×C42—41no—
316C2×C42—42no—
317C2×C42—41no—
318C2×C42—41no—
319C2×C42—416no—
320C2×C42C2×D414yes—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—24no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—42no—
340#95—44no—
341C24—21no—
342C24—24no—
343C24—22no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—21no—
351C24—24no—
352C24—24no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24C22×D414yes—
358C24C22×D414yes—
359C24C22×D412yes—
360C24C2×#4114yes—
361C24H(5,𝔽2)12yes—
362C24H(5,𝔽2)12yes—
363C24C2511yescharacteristic
364#41—4128no—
365C2×Q8—28no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×D4—216no—
373C2×D4—216no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—48no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4#121116yes—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—216no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—416no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—416no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D414yes—
485C22×C4C2×#4118yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4114yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+414yes—
491C22×C42−1+414yes—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—416no—
496C4⋊C4—44no—
497C4⋊C4—416no—
498C4⋊C4—416no—
499C4⋊C4—416no—
500C4⋊C4—48no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C22⋊C4—232no—
505C22⋊C4—416no—
506C22⋊C4—416no—
507C22⋊C4—416no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—416no—
512C22⋊C4—416no—
513C42—44no—
514C42—48no—
515C42—48no—
516C42—44no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—28no—
522C23—24no—
523C23—28no—
524C23—28no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—28no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—28no—
537C23—28no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—42no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—42no—
550C23—44no—
551C23—42no—
552C23—41no—
553C23—44no—
554C23—42no—
555C23—44no—
556C23—41no—
557C23—42no—
558C23—41no—
559C23—42no—
560C23—44no—
561C23C2×#12118yes—
562C23#53414yes—
563C23C23×D411yescharacteristic
564C23C23×D411yescharacteristic
565C23C22×#4111yescharacteristic
566C23C22×#4112yes—
567C23C2×H(5,𝔽2)11yescharacteristic
568C23C2×H(5,𝔽2)11yescharacteristic
569C23C2×H(5,𝔽2)11yescharacteristic
570C23C2×H(5,𝔽2)11yescharacteristic
571C23C2×2−1+412yes—
572C23C2×2−1+412yes—
573C23#58512yes—
574Q8—416no—
575D4—416no—
576D4—432no—
577C2×C4—232no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—432no—
608C2×C4—432no—
609C2×C4—432no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—816no—
652C2×C4C2×#12114yes—
653C2×C4#53414yes—
654C2×C4#54718yes—
655C22—28no—
656C22—216no—
657C22—24no—
658C22—28no—
659C22—44no—
660C22—42no—
661C22—48no—
662C22—44no—
663C22—48no—
664C22—44no—
665C22—42no—
666C22—48no—
667C22—44no—
668C22—42no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—42no—
673C22—44no—
674C22—42no—
675C22—42no—
676C22—44no—
677C22C22×#12111yescharacteristic
678C22C2×#53411yescharacteristic
679C22C2×#53411yescharacteristic
680C22#345912yes—
681C22C2×#54712yes—
682C22D4×#4112yes—
683C22#348111yescharacteristic
684C22#348212yes—
685C22#348411yescharacteristic
686C22#348411yescharacteristic
687C22#348912yes—
688C22#349011yescharacteristic
689C22#349011yescharacteristic
690C22#349212yes—
691C22#349912yes—
692C22#351512yes—
693C22#352212yes—
694C22#352911yescharacteristic
695C22#353512yes—
696C22#353812yes—
697C22#355011yescharacteristic
698C22#355011yescharacteristic
699C22#355912yes—
700C4—28no—
701C4—24no—
702C4—24no—
703C4—44no—
704C4—44no—
705C4—44no—
706C4—48no—
707C4—84no—
708C4—82no—
709C4—84no—
710C4—84no—
711C2—44no—
712C2—48no—
713C2#3498111yescharacteristic
714C2#3625912yes—
715C2#3738611yescharacteristic
716C2#3776012yes—
717C2#3842512yes—
718C2#3880012yes—
719C2#4622811yescharacteristic
720C2#4629311yescharacteristic
721C2#5216311yescharacteristic
722C2#5907011yescharacteristic
723C2#6004611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2724C2semidirect
4. #17078C2semidirect
5. #16629C2semidirect
6. #15390C2semidirect
7. #15269C2semidirect
8. #14907C2semidirect
9. #14837C2semidirect
10. #14697C2semidirect
11. #14505C2semidirect
12. #14440C2semidirect
13. #14043C2semidirect
280. C2×C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#24062
C2×#22972
#170782
#166292
#153902
#152692
#149072
#148372
#146972
#145052
#144402
#140432