Finite Group Databasebeta

← #4311790

Subgroups — #4311790

Order 512.

3370 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic, complemented
2C2×#2311C211yesmaximal, characteristic
3#16989C212yesmaximal, complemented
4#16630C212yesmaximal, complemented
5#16301C214yesmaximal, complemented
6#15951C214yesmaximal, complemented
7#15439C214yesmaximal, complemented
8#14893C212yesmaximal, complemented
9#14758C214yesmaximal, complemented
10#14720C212yesmaximal, complemented
11#14486C211yesmaximal, characteristic, complemented
12#14401C212yesmaximal, complemented
13#13956C212yesmaximal
14C23×#41C2211yescharacteristic
15C22×C22⋊Q8C2214yes—
16C22×C22⋊Q8C2212yes—
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#2683—28no—
20C2×#400C2212yes—
21C2×#399C2212yes—
22C2×#396C2214yes—
23C2×#395C2212yes—
24C2×#395C2212yes—
25#2311—28no—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#389C2212yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2214yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#385C2214yes—
37C2×#385C2214yes—
38C2×#381C2212yes—
39C2×#380C2214yes—
40C2×#380C2214yes—
41C22×#95C2212yes—
42#1446C2214yes—
43#1446C2214yes—
44#1445C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1442C2214yes—
50#1441C2212yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2212yes—
56#1441C2212yes—
57#1439C2214yes—
58#1439C2212yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2214yes—
63#1436C2214yes—
64#1434C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22×#41—24no—
70C22×#41—24no—
71C22×#41—42no—
72C22×#41—42no—
73C22×#41—42no—
74C22×#41—42no—
75C22×#41Q818yescomplemented
76C23×Q8C2312yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C2×C22⋊Q8—216no—
86C2×C22⋊Q8—44no—
87C2×C22⋊Q8—48no—
88C2×C22⋊Q8—48no—
89C2×C22⋊Q8—48no—
90C2×#117—24no—
91C2×#117—42no—
92C2×#117—42no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4—42no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#400—44no—
120#399—44no—
121#396—216no—
122#395—28no—
123#395—44no—
124#391—24no—
125#391—24no—
126#390—24no—
127#390D418yes—
128#389—28no—
129#386—28no—
130#386—28no—
131#386—216no—
132#386—44no—
133#386—44no—
134#385—48no—
135#385—48no—
136#381—44no—
137#380—216no—
138#380—48no—
139C2×#95—44no—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C25C2412yes—
162C25C2411yescharacteristic
163C2×#41—232no—
164C2×#41—416no—
165C2×#41—416no—
166C2×#41—416no—
167C2×#41—416no—
168C2×#41—416no—
169C2×#41—416no—
170C22×Q8—22no—
171C22×Q8—24no—
172C22×Q8—22no—
173C22×Q8—28no—
174C22×Q8—41no—
175C22×Q8—41no—
176C22×Q8—41no—
177C22×Q8—41no—
178C22×Q8—42no—
179C22×Q8—44no—
180C22×Q8—42no—
181C22×Q8C2×Q814yes—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—41no—
187C22×D4—41no—
188C22×D4—41no—
189C22×D4—41no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4C2×D414yes—
193C22×D4C2×Q818yes—
194C22×D4C2×Q814yes—
195C22×D4#4114yes—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4C2×Q814yes—
224C23×C4C2×Q818yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C22⋊Q8—464no—
237C22⋊Q8—432no—
238#117—416no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—416no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4C2×D414yes—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4C2×D4116yes—
301C2×C42—22no—
302C2×C42—41no—
303C2×C42—41no—
304C2×C42—41no—
305C2×C42—41no—
306C2×C42C2×D414yes—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95—42no—
326#95—42no—
327#95—44no—
328C24—21no—
329C24—21no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—21no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—21no—
338C24—22no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24C22×D414yes—
343C24C22×Q814yes—
344C24C22×Q812yes—
345C24C2×#4114yes—
346C24H(5,𝔽2)12yes—
347C24H(5,𝔽2)12yes—
348C24C2511yescharacteristic
349#41—4128no—
350C2×Q8—28no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—416no—
354C2×Q8—48no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—84no—
360C2×Q8—88no—
361C2×Q8—84no—
362C2×D4—216no—
363C2×D4—216no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—48no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4C22⋊Q8116yescomplemented
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—216no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—216no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
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—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—48no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4—88no—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×Q814yes—
467C22×C4C22×Q812yes—
468C22×C4C22×Q812yes—
469C22×C4C2×#4112yes—
470C22×C4H(5,𝔽2)18yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C42−1+418yes—
474C4⋊C4—216no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—416no—
479C4⋊C4—44no—
480C4⋊C4—416no—
481C4⋊C4—416no—
482C4⋊C4—48no—
483C4⋊C4—84no—
484C4⋊C4—88no—
485C4⋊C4—88no—
486C22⋊C4—416no—
487C22⋊C4—416no—
488C22⋊C4—416no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C22⋊C4—416no—
492C22⋊C4—88no—
493C22⋊C4—84no—
494C22⋊C4—84no—
495C42—44no—
496C42—44no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—24no—
501C23—28no—
502C23—22no—
503C23—24no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—24no—
517C23—28no—
518C23—24no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23C2×C22⋊Q818yes—
536C23#54314yes—
537C23C23×D411yescharacteristic
538C23C23×Q811yescharacteristic
539C23C22×#4111yescharacteristic
540C23C2×H(5,𝔽2)11yescharacteristic
541C23C2×H(5,𝔽2)11yescharacteristic
542C23C2×H(5,𝔽2)11yescharacteristic
543C23C2×H(5,𝔽2)11yescharacteristic
544C23C2×H(5,𝔽2)14yes—
545C23C2×2−1+414yes—
546Q8—416no—
547Q8—816no—
548D4—416no—
549D4—432no—
550C2×C4—232no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—432no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—816no—
620C2×C4—816no—
621C2×C4—816no—
622C2×C4C2×C22⋊Q814yes—
623C2×C4#54314yes—
624C22—28no—
625C22—216no—
626C22—24no—
627C22—28no—
628C22—48no—
629C22—42no—
630C22—44no—
631C22—48no—
632C22—44no—
633C22—42no—
634C22—44no—
635C22—48no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22C22×C22⋊Q811yescharacteristic
645C22C2×#54311yescharacteristic
646C22C2×#54311yescharacteristic
647C22#348411yescharacteristic
648C22#348412yes—
649C22#348411yescharacteristic
650C22#349011yescharacteristic
651C22#349011yescharacteristic
652C22#349912yes—
653C22#350314yes—
654C22#352014yes—
655C22#353212yes—
656C22#353614yes—
657C22#353712yes—
658C22#353812yes—
659C22#354412yes—
660C22#354814yes—
661C4—24no—
662C4—24no—
663C4—48no—
664C4—44no—
665C4—44no—
666C4—44no—
667C4—48no—
668C4—82no—
669C4—84no—
670C4—82no—
671C4—84no—
672C2—44no—
673C2—48no—
674C2#3498211yescharacteristic
675C2#3573912yes—
676C2#3730712yes—
677C2#4199712yes—
678C2#4642312yes—
679C2#4766312yes—
680C2#5154211yescharacteristic
681C2#5162912yes—
682C2#5200411yescharacteristic
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×#2683C2semidirect
3. #16989C2semidirect
4. #16630C2semidirect
5. #16301C2semidirect
6. #15951C2semidirect
7. #15439C2semidirect
8. #14893C2semidirect
9. #14758C2semidirect
10. #14720C2semidirect
11. #14486C2semidirect
12. #14401C2semidirect
75. C22×#41Q8semidirect
375. C2×D4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#23112
#169892
#166302
#163012
#159512
#154392
#148932
#147582
#147202
#144862
#144012
#139562