Finite Group Databasebeta

← #532298

Subgroups — #532298

Order 512.

2752 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2#14507C212yesmaximal, complemented
3#14506C212yesmaximal
4#14145C212yesmaximal
5#14093C214yesmaximal, complemented
6#14085C214yesmaximal, complemented
7#13905C211yesmaximal, characteristic
8#13885C212yesmaximal, complemented
9#13876C212yesmaximal, complemented
10#13876C212yesmaximal, complemented
11#13871C211yesmaximal, characteristic
12#13823C212yesmaximal, complemented
13C4⋊C42C212yesmaximal, complemented
14#13801C212yesmaximal, complemented
15C4×#380C212yesmaximal, complemented
16C22×#123C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C4×#123C4116yescomplemented
21C2×#388C2214yes—
22C2×#388C2214yes—
23C2×#388C2212yes—
24C2×#387C2212yes—
25C2×#386C2211yescharacteristic
26C2×#385C2211yescharacteristic
27C2×#384C2212yes—
28C2×#382C2211yescharacteristic
29C2×#381C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2211yescharacteristic
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2214yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C22⋊C4C2214yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2212yes—
43#1445C2212yes—
44#1445C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1440C2212yes—
49#1439C2214yes—
50#1439C2214yes—
51#1438C2212yes—
52#1438C2212yes—
53#1437C2212yes—
54#1437C2214yes—
55#1437C2214yes—
56#1437C2212yes—
57#1437C2212yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70#1434C2214yes—
71#1434C2214yes—
72#1434C2214yes—
73C22⋊#95C2214yes—
74C4×#95C2214yes—
75C4×#95C2214yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C2×#123—28no—
82C2×#123C2×C418yes—
83C2×C4×D4—42no—
84C2×C4×D4—42no—
85C2×C4×D4C2×C418yes—
86C2×#117—28no—
87C2×#117—42no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2311yescharacteristic
108C22×C42—42no—
109C22×C42—42no—
110C22×C42C2×C418yes—
111C22×C42C2312yes—
112C22×C42C2314yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2314yes—
117C22×C42C2314yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120#388—216no—
121#388—48no—
122#388C2×C4116yes—
123#387C2×C4116yes—
124#386C2×C418yes—
125#385C2×C418yes—
126#384—28no—
127#382—42no—
128#381C2×C418yes—
129#380—28no—
130#380—28no—
131#380—28no—
132#380—42no—
133C4×C4⋊C4—28no—
134C4×C4⋊C4—48no—
135C4×C4⋊C4—42no—
136C4×C4⋊C4—44no—
137C4×C4⋊C4—44no—
138C4×C4⋊C4C2×C4116yes—
139C4×C22⋊C4—48no—
140C4×C22⋊C4C2×C4116yes—
141C4×C22⋊C4C2×C418yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C25C2411yescharacteristic
168C25C2411yescharacteristic
169C22×D4—24no—
170C22×D4—41no—
171C22×D4—41no—
172C22×D4—42no—
173C22×D4C22×C414yes—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—42no—
186C23×C4—41no—
187C23×C4—41no—
188C23×C4—41no—
189C23×C4—41no—
190C23×C4C22×C414yes—
191C23×C4C22×C414yes—
192C23×C4C22×C414yes—
193C23×C4C22×C414yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207#123—232no—
208C4×D4—416no—
209#117—416no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C22×C418yes—
233C2×C4⋊C4C22×C418yes—
234C2×C4⋊C4C22×C414yes—
235C2×C4⋊C4#4118yes—
236C2×C4⋊C4#4118yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4C22×C418yes—
260C2×C22⋊C4C22×C418yes—
261C2×C22⋊C4C22×C414yes—
262C2×C22⋊C4C22×C414yes—
263C2×C22⋊C4C22×C414yes—
264C2×C22⋊C4C22×C414yes—
265C2×C22⋊C4#4118yes—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—28no—
273C2×C42—28no—
274C2×C42—28no—
275C2×C42—28no—
276C2×C42—28no—
277C2×C42—416no—
278C2×C42—42no—
279C2×C42—44no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42—42no—
288C2×C42C22×C414yes—
289C2×C42C22×C418yes—
290C2×C42C22×C418yes—
291C2×C42C22×C418yes—
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—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—42no—
308#95—42no—
309#95—42no—
310#95—42no—
311#95—44no—
312#95C22×C418yes—
313#95C22×C418yes—
314#95C22×C418yes—
315#95C22×C418yes—
316#95C22×C418yes—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—21no—
321C24—21no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—22no—
332C24C23×C412yes—
333C24C23×C412yes—
334C24C2×#4114yes—
335C24C2511yescharacteristic
336C2×D4—28no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—48no—
340C2×D4—48no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—28no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—216no—
391C22×C4—216no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—216no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—48no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4C23×C414yes—
424C22×C4C23×C412yes—
425C22×C4C23×C414yes—
426C22×C4C23×C412yes—
427C22×C4C23×C412yes—
428C22×C4C23×C412yes—
429C22×C4C23×C414yes—
430C22×C4C23×C414yes—
431C22×C4C23×C412yes—
432C22×C4C2×#4118yes—
433C22×C4C2×#4118yes—
434C22×C4C2×#4114yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4114yes—
437C22×C4C2×#4114yes—
438C22×C4H(5,𝔽2)14yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C42−1+414yes—
442C4⋊C4—216no—
443C4⋊C4—216no—
444C4⋊C4—216no—
445C4⋊C4—216no—
446C4⋊C4—44no—
447C4⋊C4—416no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—88no—
451C22⋊C4—216no—
452C22⋊C4—216no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C22⋊C4—416no—
459C22⋊C4—88no—
460C22⋊C4—84no—
461C22⋊C4#117116yescomplemented
462C42—216no—
463C42—216no—
464C42—28no—
465C42—48no—
466C42—416no—
467C42—48no—
468C42—88no—
469C42—88no—
470C42—84no—
471C23—24no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—41no—
487C23—42no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23C2×#11714yes—
513C23C24×C411yescharacteristic
514C23C22×#4112yes—
515C23C22×#4112yes—
516C23C22×#4112yes—
517C23C2×H(5,𝔽2)11yescharacteristic
518C23C2×H(5,𝔽2)11yescharacteristic
519C23C2×H(5,𝔽2)12yes—
520C23C2×2−1+412yes—
521C23#58512yes—
522D4—416no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—416no—
557C2×C4—416no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—416no—
586C2×C4—416no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4C2×#11718yes—
602C2×C4C4×#4118yes—
603C2×C4#51814yes—
604C2×C4#52018yes—
605C2×C4#55114yes—
606C2×C4#55318yes—
607C2×C4#55914yes—
608C22—28no—
609C22—24no—
610C22—44no—
611C22—44no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—42no—
618C22—44no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22C22×#11711yescharacteristic
627C22C2×C4×#4112yes—
628C22C2×#51811yescharacteristic
629C22C2×#52012yes—
630C22#342811yescharacteristic
631C22C2×#55111yescharacteristic
632C22C2×#55312yes—
633C22C2×#55911yescharacteristic
634C22#348112yes—
635C22#348112yes—
636C22#348212yes—
637C22#348212yes—
638C22#350612yes—
639C22#350612yes—
640C22#351412yes—
641C22#351512yes—
642C22#351712yes—
643C22#351812yes—
644C22#353612yes—
645C22#353711yescharacteristic
646C22#353811yescharacteristic
647C4—28no—
648C4—28no—
649C4—48no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—44no—
654C4—42no—
655C4—44no—
656C4—48no—
657C4—84no—
658C4—84no—
659C4#237418yes—
660C2—44no—
661C2—82no—
662C2#3412712yes—
663C2C2×#237411yescharacteristic
664C2#3418611yescharacteristic
665C2#3424212yes—
666C2#3436011yescharacteristic
667C2#5937712yes—
668C2#5939412yes—
669C2#5939512yes—
670C2#5963012yes—
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. #14507C2semidirect
5. #14093C2semidirect
6. #14085C2semidirect
8. #13885C2semidirect
9. #13876C2semidirect
10. #13876C2semidirect
12. #13823C2semidirect
13. C4⋊C42C2semidirect
14. #13801C2semidirect
15. C4×#380C2semidirect
20. C4×#123C4semidirect
461. C22⋊C4#117semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1232
#145072
#145062
#141452
#140932
#140852
#139052
#138852
#138762
#138712
#138232
C4⋊C422
#138012
C4×#3802