Finite Group Databasebeta

← #558287

Subgroups — #558287

Order 512.

2760 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2#14702C212yesmaximal, complemented
3#14646C212yesmaximal, complemented
4#14618C212yesmaximal, complemented
5#14592C214yesmaximal, complemented
6#14454C212yesmaximal, complemented
7#14379C214yesmaximal
8#14138C212yesmaximal, complemented
9#14024C212yesmaximal, complemented
10#13969C212yesmaximal, complemented
11#13969C212yesmaximal, complemented
12#13921C212yesmaximal, complemented
13#13871C211yesmaximal, characteristic
14#13830C212yesmaximal, complemented
15C4×#386C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2542C4116yescomplemented
24C2×#399C2212yes—
25C2×#399C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#390C2212yes—
30C2×#386C2212yes—
31C2×#384C2212yes—
32C2×#383C2214yes—
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38#1443C2214yes—
39#1443C2214yes—
40#1441C2214yes—
41#1441C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1440C2214yes—
45#1440C2212yes—
46#1440C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1438C2212yes—
50#1437C2212yes—
51#1437C2214yes—
52#1437C2211yescharacteristic
53#1437C2214yes—
54#1437C2211yescharacteristic
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2211yescharacteristic
60#1436C2214yes—
61#1436C2214yes—
62#1436C2212yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C4×#95C2212yes—
76C23×Q8C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8—42no—
83C2×C4⋊Q8C2×C418yescomplemented
84C2×C22⋊Q8—44no—
85C2×C22⋊Q8—44no—
86C2×C22⋊Q8—44no—
87C2×C22⋊Q8—44no—
88C2×C22⋊Q8C2×C4116yes—
89C2×C22⋊Q8C2×C4116yescomplemented
90C2×C4×Q8—44no—
91C2×C4×Q8—44no—
92C2×C4×Q8—44no—
93C2×C4×D4—42no—
94C2×C4×D4—42no—
95C2×C4×D4C2×C418yes—
96C2×#117—44no—
97C2×#117—42no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C42—44no—
118C22×C42—42no—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2314yes—
127#399C2×C4116yes—
128#399C2×C4116yescomplemented
129#391—44no—
130#391—44no—
131#391—44no—
132#390C2×C4116yes—
133#386—28no—
134#384C2×C4116yescomplemented
135#383—216no—
136#380—44no—
137#380—42no—
138C4×C4⋊C4C2×C418yescomplemented
139C4×C22⋊C4—42no—
140C4×C22⋊C4—44no—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2311yescharacteristic
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2312yes—
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8C22×C418yes—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4C22×C414yes—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4C22×C414yes—
198C23×C4C22×C414yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C4⋊Q8—416no—
213C22⋊Q8—432no—
214C22⋊Q8—432no—
215C4×Q8—432no—
216C4×D4—416no—
217#117—416no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4C22×C418yes—
240C2×C4⋊C4C22×C414yes—
241C2×C4⋊C4C22×C4116yes—
242C2×C4⋊C4C22×C4116yes—
243C2×C4⋊C4C22×C418yes—
244C2×C4⋊C4C22×C418yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
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—42no—
269C2×C22⋊C4C22×C414yes—
270C2×C22⋊C4C22×C414yes—
271C2×C22⋊C4C22×C4116yes—
272C2×C22⋊C4C22×C4116yes—
273C2×C42—28no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—42no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42—41no—
288C2×C42—41no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—416no—
292C2×C42C22×C414yes—
293C2×C42C22×C418yes—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—44no—
303#95—42no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—41no—
308#95—41no—
309#95—42no—
310#95—42no—
311#95—42no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95C22×C418yes—
320#95#4118yes—
321C24—22no—
322C24—22no—
323C24—21no—
324C24—22no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—21no—
332C24—21no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24C23×C412yes—
337C24C23×C412yes—
338C242−1+414yes—
339C24C2511yescharacteristic
340C2×Q8—216no—
341C2×Q8—216no—
342C2×Q8—216no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×D4—28no—
348C2×D4—28no—
349C2×D4—44no—
350C2×D4—48no—
351C2×D4—48no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
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—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4C23×C412yes—
436C22×C4C23×C418yes—
437C22×C4C23×C418yes—
438C22×C4C23×C412yes—
439C22×C4C23×C412yes—
440C22×C4C23×C412yes—
441C22×C4C23×C412yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4114yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C4H(5,𝔽2)14yes—
448C22×C4H(5,𝔽2)14yes—
449C22×C42−1+414yes—
450C4⋊C4—216no—
451C4⋊C4—44no—
452C4⋊C4—416no—
453C4⋊C4—416no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—416no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—416no—
462C22⋊C4—416no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—88no—
468C22⋊C4—84no—
469C42—216no—
470C42—28no—
471C42—44no—
472C42—44no—
473C42—48no—
474C42—416no—
475C42—48no—
476C42—48no—
477C23—22no—
478C23—24no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23#51914yes—
519C23C24×C411yescharacteristic
520C23C22×#4112yes—
521C23C2×H(5,𝔽2)12yes—
522C23C2×H(5,𝔽2)12yes—
523C23C2×H(5,𝔽2)11yescharacteristic
524C23C2×H(5,𝔽2)11yescharacteristic
525C23C2×2−1+412yes—
526C23C2×2−1+412yes—
527C23#58512yes—
528Q8—432no—
529D4—416no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—28no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—432no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—416no—
586C2×C4—416no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4C4×#4114yes—
609C2×C4#51814yes—
610C2×C4#51818yes—
611C2×C4#56014yes—
612C2×C4#56018yes—
613C2×C4#56014yes—
614C22—28no—
615C22—24no—
616C22—44no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—42no—
625C22—42no—
626C22—44no—
627C22—42no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22C2×C4×#4111yescharacteristic
633C22C2×#51812yes—
634C22C2×#51811yescharacteristic
635C22C2×#51911yescharacteristic
636C22C4×2−1+412yes—
637C22#350612yes—
638C22#351112yes—
639C22#351712yes—
640C22#352012yes—
641C22#352012yes—
642C22#352012yes—
643C22C2×#56011yescharacteristic
644C22C2×#56011yescharacteristic
645C22C2×#56012yes—
646C22#353812yes—
647C22#354112yes—
648C22#354112yes—
649C22#355011yescharacteristic
650C22#355111yescharacteristic
651C22#355212yes—
652C22#356112yes—
653C4—24no—
654C4—48no—
655C4—42no—
656C4—42no—
657C4—48no—
658C4—44no—
659C4—44no—
660C4—42no—
661C4—44no—
662C4—84no—
663C4—82no—
664C4—84no—
665C4#238018yes—
666C2—44no—
667C2—82no—
668C2#3424511yescharacteristic
669C2#3431612yes—
670C2C2×#238011yescharacteristic
671C2#3441311yescharacteristic
672C2#3447512yes—
673C2#4562412yes—
674C2#4562812yes—
675C2#4567712yes—
676C2#4572112yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14702C2semidirect
3. #14646C2semidirect
4. #14618C2semidirect
5. #14592C2semidirect
6. #14454C2semidirect
8. #14138C2semidirect
9. #14024C2semidirect
10. #13969C2semidirect
11. #13969C2semidirect
12. #13921C2semidirect
14. #13830C2semidirect
23. #2542C4semidirect
83. C2×C4⋊Q8C2×C4semidirect
89. C2×C22⋊Q8C2×C4semidirect
128. #399C2×C4semidirect
134. #384C2×C4semidirect
138. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
#147022
#146462
#146182
#145922
#144542
#143792
#141382
#140242
#139692
#139212
#138712
#138302
C4×#3862