Finite Group Databasebeta

← #1011233

Subgroups — #1011233

Order 512.

2546 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2492C212yesmaximal, complemented
2#17228C212yesmaximal, complemented
3#16652C212yesmaximal
4#16496C214yesmaximal, complemented
5#15978C212yesmaximal
6#15851C214yesmaximal, complemented
7#15180C211yesmaximal, characteristic, complemented
8#15178C212yesmaximal, complemented
9#15177C212yesmaximal
10#14467C212yesmaximal, complemented
11#14155C211yesmaximal, characteristic, complemented
12#14142C212yesmaximal, complemented
13#13997C212yesmaximal, complemented
14#13974C212yesmaximal, complemented
15C4×#394C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21#2492—216no—
22C2×#400C2214yes—
23C2×#394C2211yescharacteristic
24C2×#393C2214yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2214yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#389C2212yes—
33C2×#388C2214yes—
34C2×#388C2214yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2212yes—
42#1445C2212yes—
43#1445C2212yes—
44#1445C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1444C2214yes—
48#1443C2214yes—
49#1443C2214yes—
50#1442C2211yescharacteristic
51#1442C2214yes—
52#1442C2212yes—
53#1442C2212yes—
54#1441C2212yes—
55#1441C2211yescharacteristic
56#1441C2212yes—
57#1440C2214yes—
58#1440C2212yes—
59#1439C2214yes—
60#1439C2212yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2214yes—
66#1438C2211yescharacteristic
67#1437C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2214yes—
74#1434C2214yes—
75C4×#95C2211yescharacteristic
76C23×Q8C2312yes—
77C23×Q8C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4⋊Q8—42no—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8—44no—
83C2×C4⋊Q8—42no—
84C2×C4⋊Q8D418yescomplemented
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—44no—
88C2×C22⋊Q8—44no—
89C2×C22⋊Q8—44no—
90C2×C22⋊Q8—44no—
91C2×C4×D4—42no—
92C2×C4×D4—44no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2314yes—
119C22×C42C2312yes—
120#400—216no—
121#394—24no—
122#393—216no—
123#391—216no—
124#391—28no—
125#391—28no—
126#391—28no—
127#391—28no—
128#391—44no—
129#391—44no—
130#389—44no—
131#388—216no—
132#388—48no—
133#380—28no—
134#380—28no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—42no—
137C4×C4⋊C4—42no—
138C4×C22⋊C4—28no—
139C4×C22⋊C4—44no—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2311yescharacteristic
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2311yescharacteristic
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C25C2412yes—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—42no—
179C22×Q8—42no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×Q8C2×D418yes—
183C22×D4—24no—
184C22×D4—22no—
185C22×D4—42no—
186C22×D4—41no—
187C22×D4—41no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2411yescharacteristic
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C4⋊Q8—416no—
210C4⋊Q8—416no—
211C22⋊Q8—432no—
212C22⋊Q8—432no—
213C4×D4—416no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C2×D418yes—
244C2×C4⋊C4C2×D418yes—
245C2×C4⋊C4#4118yes—
246C2×C4⋊C4#4114yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—28no—
284C2×C42—24no—
285C2×C42—42no—
286C2×C42—44no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42C2×D414yes—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—22no—
304#95—24no—
305#95—28no—
306#95—24no—
307#95—42no—
308#95—41no—
309#95—42no—
310#95—42no—
311#95—41no—
312#95—41no—
313#95—42no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C242−1+414yes—
333C24C2511yescharacteristic
334C2×Q8—216no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—84no—
345C2×Q8—84no—
346C2×Q8—84no—
347C2×Q8—84no—
348C2×D4—48no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—48no—
352C2×D4—44no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4C22×D412yes—
432C22×C4C22×D418yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4118yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4114yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C4H(5,𝔽2)14yes—
444C22×C42−1+414yes—
445C22×C42−1+412yes—
446C4⋊C4—216no—
447C4⋊C4—216no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—416no—
452C4⋊C4—416no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—88no—
458C22⋊C4—216no—
459C22⋊C4—48no—
460C22⋊C4—416no—
461C22⋊C4—48no—
462C22⋊C4—416no—
463C22⋊C4—416no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—88no—
467C42—28no—
468C42—48no—
469C42—44no—
470C42—44no—
471C42—84no—
472C42—88no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23C23×D411yescharacteristic
509C23C22×#4112yes—
510C23C22×#4111yescharacteristic
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×H(5,𝔽2)12yes—
514C23C2×H(5,𝔽2)11yescharacteristic
515C23C2×2−1+411yescharacteristic
516C23C2×2−1+412yes—
517C23C2×2−1+412yes—
518C23#58511yescharacteristic
519Q8—816no—
520Q8—816no—
521D4—416no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—816no—
603C2×C4#53414yes—
604C2×C4#53818yes—
605C2×C4#54714yes—
606C2×C4#55018yes—
607C2×C4#55514yes—
608C2×C4H(3,𝔽4)14yes—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22C2×#53411yescharacteristic
625C22C2×#53812yes—
626C22C2×#54711yescharacteristic
627C22C2×#55012yes—
628C22C2×#55511yescharacteristic
629C22#348212yes—
630C22#348612yes—
631C22#349611yescharacteristic
632C22#350511yescharacteristic
633C22#350811yescharacteristic
634C22#351612yes—
635C22#352012yes—
636C22#352012yes—
637C22#352312yes—
638C22C2×H(3,𝔽4)11yescharacteristic
639C22#353612yes—
640C22#353611yescharacteristic
641C22#353811yescharacteristic
642C22#354112yes—
643C22#355011yescharacteristic
644C22#355012yes—
645C22#355212yes—
646C22#356111yescharacteristic
647C4—24no—
648C4—42no—
649C4—44no—
650C4—48no—
651C4—42no—
652C4—42no—
653C4—48no—
654C4—42no—
655C4—84no—
656C4—82no—
657C4—82no—
658C4—84no—
659C4—81no—
660C4#273318yes—
661C2—84no—
662C2#3948611yescharacteristic
663C2#4007612yes—
664C2#4280711yescharacteristic
665C2#4282812yes—
666C2#4511212yes—
667C2#4538411yescharacteristic
668C2#4566711yescharacteristic
669C2C2×#273311yescharacteristic
670C2#5295811yescharacteristic
671C2#5905512yes—
672C2#5995111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2492C2semidirect
2. #17228C2semidirect
4. #16496C2semidirect
6. #15851C2semidirect
7. #15180C2semidirect
8. #15178C2semidirect
10. #14467C2semidirect
11. #14155C2semidirect
12. #14142C2semidirect
13. #13997C2semidirect
14. #13974C2semidirect
84. C2×C4⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24922
#172282
#166522
#164962
#159782
#158512
#151802
#151782
#151772
#144672
#141552
#141422
#139972
#139742
C4×#3942