Finite Group Databasebeta

← #390406

Subgroups — #390406

Order 512.

1874 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1870C212yesmaximal, complemented
2C2×#1869C212yesmaximal, complemented
3C2×#1869C212yesmaximal, complemented
4C4×#392C211yesmaximal, characteristic
5#11298C212yesmaximal, complemented
6#11042C212yesmaximal, complemented
7#11042C211yesmaximal, characteristic, complemented
8#10715C211yesmaximal, characteristic, complemented
9#10714C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C24×C8C2212yes—
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C2×#392C411yescharacteristic
21C2×C4×C22⋊C4C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#1870—28no—
24#1869—28no—
25#1869—28no—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1444C411yescharacteristic
35#1439C412yes—
36#1439C411yescharacteristic
37#1438C411yescharacteristic
38#1438C412yes—
39C4×#95C2211yescharacteristic
40C23×D4C2×C412yes—
41C23×D4C2×C411yescharacteristic
42C24×C4C2312yes—
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45C22×M16—24no—
46C22×M16—42no—
47C22×M16—41no—
48C22×M16—41no—
49C23×C8—24no—
50C23×C8—24no—
51C23×C8—24no—
52C23×C8—42no—
53C23×C8—42no—
54C23×C8—42no—
55C23×C8—42no—
56C23×C8C2312yes—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C23×C8C2312yes—
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—41no—
67C2×C4×D4—42no—
68C2×C4×D4—41no—
69C2×C4×D4—42no—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2312yes—
82C2×C4⋊C8—41no—
83C2×C4⋊C8—42no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—41no—
91C2×C22⋊C8—41no—
92C2×C22⋊C8—42no—
93C2×C22⋊C8—42no—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8—42no—
97C2×C22⋊C8D414yescomplemented
98C2×C22⋊C8D418yescomplemented
99#392—24no—
100C4×C22⋊C4—42no—
101C4×C22⋊C4—41no—
102C4×C22⋊C4—42no—
103C4×C22⋊C4—41no—
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C412yes—
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C412yes—
109C2×#95C2×C412yes—
110C2×#95C2×C411yescharacteristic
111#336—24no—
112#336—24no—
113#336—24no—
114#336—24no—
115#336—24no—
116#336—24no—
117#336—41no—
118#336—42no—
119C25C22×C411yescharacteristic
120C25C22×C412yes—
121C22×D4—24no—
122C22×D4—24no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—24no—
126C22×D4—24no—
127C22×D4—24no—
128C22×D4—41no—
129C22×D4—42no—
130C22×D4—41no—
131C22×D4—42no—
132C22×D4M1614yescomplemented
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C412yes—
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D414yes—
163C23×C4#4114yes—
164C23×C4#4112yes—
165C23×C4C2411yescharacteristic
166C2×M16—44no—
167C2×M16—48no—
168C2×M16—48no—
169C2×M16—44no—
170C2×M16—44no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—48no—
197C22×C8—48no—
198C22×C8—48no—
199C22×C8—48no—
200C22×C8—48no—
201C22×C8—48no—
202C22×C8—48no—
203C22×C8C2×D414yes—
204C22×C8C2×D414yes—
205C22×C8C2×D414yes—
206C4×D4—44no—
207C4×D4—48no—
208C4×D4—48no—
209C4×D4—48no—
210C4×D4—48no—
211C4×D4—44no—
212C4×D4—48no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—21no—
252C2×C42—22no—
253C4⋊C8—44no—
254C22⋊C8—44no—
255C22⋊C8—48no—
256C22⋊C8—48no—
257C22⋊C8—48no—
258C22⋊C8—48no—
259#95—22no—
260#95—24no—
261#95—41no—
262#95—41no—
263#95—42no—
264#95—41no—
265#95—42no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—21no—
280C24—22no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—22no—
285C24—22no—
286C24—21no—
287C24—21no—
288C24C4×D412yes—
289C24C4×D414yes—
290C24C2×M1614yes—
291C24#13114yes—
292C24C23×C411yescharacteristic
293C2×D4—216no—
294C2×D4—216no—
295C2×D4—44no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4—44no—
299C2×D4—48no—
300C2×D4—48no—
301C2×D4—48no—
302C2×D4—44no—
303C2×D4—48no—
304C2×D4—48no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4C4×D414yes—
393C22×C4C4×D412yes—
394C22×C4C2×M1612yes—
395C22×C4#13114yes—
396C22×C4#13112yes—
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4C2×#4111yescharacteristic
401C22×C4H(5,𝔽2)11yescharacteristic
402C22×C4H(5,𝔽2)11yescharacteristic
403C22×C4H(5,𝔽2)12yes—
404C22×C4H(5,𝔽2)11yescharacteristic
405C22×C4H(5,𝔽2)12yes—
406C22×C42−1+411yescharacteristic
407M16—416no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—432no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C42—41no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—41no—
452C42—42no—
453C42—41no—
454C42—42no—
455C42—41no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—42no—
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—41no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—42no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—41no—
513C23#43518yes—
514C23C2×C4×D412yes—
515C23C2×C4×D411yescharacteristic
516C23#51811yescharacteristic
517C23#51812yes—
518C23#52011yescharacteristic
519C23C22×M1611yescharacteristic
520C23C2×#13112yes—
521C23C2×#13112yes—
522C23C2×#13111yescharacteristic
523C23#56811yescharacteristic
524C23#56811yescharacteristic
525D4—416no—
526D4—432no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4#53411yescharacteristic
588C2×C4#53412yes—
589C2×C4#53711yescharacteristic
590C2×C4#53712yes—
591C2×C4#53811yescharacteristic
592C2×C4#53812yes—
593C2×C4#53911yescharacteristic
594C2×C4#53912yes—
595C2×C4D4211yescharacteristic
596C2×C4D4212yes—
597C2×C4#54612yes—
598C2×C4#54612yes—
599C2×C4#54612yes—
600C2×C4#55111yescharacteristic
601C2×C4#55112yes—
602C2×C4#56012yes—
603C2×C4#56011yescharacteristic
604C2×C4#56011yescharacteristic
605C8—44no—
606C8—44no—
607C8—44no—
608C8—82no—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—82no—
634C22—82no—
635C22#232511yescharacteristic
636C22#232512yes—
637C22#235811yescharacteristic
638C22#235812yes—
639C22#238011yescharacteristic
640C22C2×#43512yes—
641C22#292912yes—
642C22#293112yes—
643C22#293312yes—
644C22#293312yes—
645C22D4×M1611yescharacteristic
646C22#293511yescharacteristic
647C22#297311yescharacteristic
648C22#297412yes—
649C22#297612yes—
650C22#297611yescharacteristic
651C22#297711yescharacteristic
652C22#297712yes—
653C22#297911yescharacteristic
654C22#298112yes—
655C22#299011yescharacteristic
656C22#299412yes—
657C22#299511yescharacteristic
658C4—42no—
659C4—44no—
660C4—42no—
661C4—44no—
662C4—42no—
663C4—81no—
664C4—82no—
665C4—81no—
666C4#266812yes—
667C4#266811yescharacteristic
668C4#267511yescharacteristic
669C4#269312yes—
670C4#269311yescharacteristic
671C4#276011yescharacteristic
672C2—48no—
673C2—82no—
674C2#1457211yescharacteristic
675C2#2216911yescharacteristic
676C2#2217011yescharacteristic
677C2#2217212yes—
678C2#2217512yes—
679C2#2284411yescharacteristic
680C2#2284612yes—
681C2#2284712yes—
682C2#2285311yescharacteristic
683C2#2335611yescharacteristic
684C2#2335811yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC2×#392C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1870C2semidirect
2. C2×#1869C2semidirect
3. C2×#1869C2semidirect
5. #11298C2semidirect
6. #11042C2semidirect
7. #11042C2semidirect
8. #10715C2semidirect
9. #10714C2semidirect
97. C2×C22⋊C8D4semidirect
98. C2×C22⋊C8D4semidirect
132. C22×D4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#18702
C2×#18692
C4×#3922
#112982
#110422
#107152
#107142