Finite Group Databasebeta

← #382155

Subgroups — #382155

Order 512.

1662 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1955C212yesmaximal, complemented
2C2×#1954C211yesmaximal, characteristic, complemented
3C2×#1924C212yesmaximal, complemented
4C2×#1923C211yesmaximal, characteristic, complemented
5C4×#392C211yesmaximal, characteristic
6#11322C211yesmaximal, characteristic, complemented
7#11042C212yesmaximal, complemented
8#11042C211yesmaximal, characteristic, complemented
9#10964C212yesmaximal, complemented
10#10964C211yesmaximal, characteristic, complemented
11#10696C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2212yes—
24C22×#97C2212yes—
25C22×C4×C8C2211yescharacteristic
26C2×#392C411yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2212yes—
29#1955—28no—
30#1954—24no—
31#1924—28no—
32#1923—24no—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1444C411yescharacteristic
40#1439C411yescharacteristic
41#1439C412yes—
42#1438C412yes—
43#1438C411yescharacteristic
44C4×#95C2211yescharacteristic
45C23×D4C2×C412yes—
46C23×D4C2×C411yescharacteristic
47C24×C4C2312yes—
48C24×C4C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C2×C4×D4—22no—
56C2×C4×D4—24no—
57C2×C4×D4—22no—
58C2×C4×D4—24no—
59C2×C4×D4—42no—
60C2×C4×D4—42no—
61C2×C4×D4—42no—
62C2×C4×D4—42no—
63C2×C4×D4—42no—
64C2×C4×D4—42no—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2312yes—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—42no—
82C2×C4⋊C8—42no—
83C2×C4⋊C8—42no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8—42no—
97C2×C22⋊C8—42no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8—42no—
100C2×C22⋊C8—42no—
101C2×C22⋊C8—42no—
102C2×C22⋊C8—42no—
103C2×#97—42no—
104C2×#97—42no—
105C2×#97D418yescomplemented
106C2×C4×C8—42no—
107C2×C4×C8D414yescomplemented
108#392—24no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—24no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—24no—
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115C2×#95C2×C412yes—
116C2×#95C2×C411yescharacteristic
117C2×#95C2×C411yescharacteristic
118C2×#95C2×C412yes—
119C2×#95C2×C412yes—
120#336—22no—
121#336—24no—
122#336—22no—
123#336—24no—
124#336—24no—
125#336—22no—
126C25C22×C412yes—
127C25C22×C411yescharacteristic
128C22×D4—24no—
129C22×D4—24no—
130C22×D4—22no—
131C22×D4—24no—
132C22×D4—22no—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4M1614yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C412yes—
167C23×C4C22×C412yes—
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C412yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8C2×D412yes—
204C22×C8C2×D414yes—
205C22×C8C2×D412yes—
206C22×C8C2×D414yes—
207C4×D4—44no—
208C4×D4—48no—
209C4×D4—44no—
210C4×D4—48no—
211C4×D4—48no—
212C4×D4—48no—
213C4×D4—48no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42C2×D412yes—
251C2×C42C2×D414yes—
252C2×C42#4114yes—
253C2×C42#4112yes—
254C4⋊C8—44no—
255C4⋊C8—48no—
256C4⋊C8—44no—
257C22⋊C8—44no—
258C22⋊C8—48no—
259C22⋊C8—48no—
260C22⋊C8—48no—
261C22⋊C8—48no—
262C22⋊C8—44no—
263C22⋊C8—48no—
264#97—48no—
265C4×C8—44no—
266#95—24no—
267#95—22no—
268#95—22no—
269#95—24no—
270#95—22no—
271#95—22no—
272#95—42no—
273C24—21no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—21no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—22no—
288C24—21no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24C2×M1614yes—
298C24#13112yes—
299C24C23×C411yescharacteristic
300C2×D4—28no—
301C2×D4—48no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4—44no—
305C2×D4—48no—
306C2×D4—44no—
307C2×D4—48no—
308C2×D4—48no—
309C2×D4—48no—
310C2×D4—48no—
311C2×D4—48no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4C4×D412yes—
398C22×C4C4×D414yes—
399C22×C4C4×D414yes—
400C22×C4C4×D412yes—
401C22×C4C2×M1612yes—
402C22×C4#13112yes—
403C22×C4#13112yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D412yes—
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4112yes—
408C22×C4H(5,𝔽2)12yes—
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C4H(5,𝔽2)12yes—
411C22×C4H(5,𝔽2)11yescharacteristic
412C22×C4H(5,𝔽2)11yescharacteristic
413C22×C42−1+411yescharacteristic
414C2×C8—22no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—41no—
431C2×C8—42no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—41no—
442C2×C8—42no—
443C2×C8—41no—
444C2×C8—42no—
445C2×C8—41no—
446C2×C8—42no—
447C2×C8#12714yes—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C22⋊C4—28no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—44no—
457C42—22no—
458C42—24no—
459C42—22no—
460C42—24no—
461C42—24no—
462C42—24no—
463C42—24no—
464C42—41no—
465C42—41no—
466C42—42no—
467C23—24no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23C2×C4×D411yescharacteristic
526C23C2×C4×D412yes—
527C23#51812yes—
528C23#51811yescharacteristic
529C23#52011yescharacteristic
530C23C22×M1611yescharacteristic
531C23C2×#13111yescharacteristic
532C23C2×#13111yescharacteristic
533C23C2×#13111yescharacteristic
534C23#56812yes—
535C23#56812yes—
536D4—416no—
537D4—432no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—82no—
584C2×C4—81no—
585C2×C4—82no—
586C2×C4—81no—
587C2×C4—82no—
588C2×C4—81no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—81no—
597C2×C4#43514yes—
598C2×C4C2×#12711yescharacteristic
599C2×C4#53211yescharacteristic
600C2×C4#53212yes—
601C2×C4#53411yescharacteristic
602C2×C4#53412yes—
603C2×C4#53511yescharacteristic
604C2×C4#53512yes—
605C2×C4#53511yescharacteristic
606C2×C4#53512yes—
607C2×C4#53711yescharacteristic
608C2×C4#53712yes—
609C2×C4#53812yes—
610C2×C4#53812yes—
611C2×C4#53812yes—
612C2×C4#54011yescharacteristic
613C2×C4#54012yes—
614C2×C4#54211yescharacteristic
615C2×C4#54212yes—
616C8—24no—
617C8—42no—
618C8—44no—
619C8—81no—
620C8—81no—
621C8—82no—
622C22—42no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—42no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—42no—
634C22—82no—
635C22—81no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—81no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—81no—
645C22—82no—
646C22—81no—
647C22—82no—
648C22C4×#12711yescharacteristic
649C22#232111yescharacteristic
650C22#232112yes—
651C22#232411yescharacteristic
652C22#232412yes—
653C22C2×#43511yescharacteristic
654C22#292911yescharacteristic
655C22#293011yescharacteristic
656C22#293211yescharacteristic
657C22#293312yes—
658C22#293312yes—
659C22D4×M1612yes—
660C22#293512yes—
661C22#297312yes—
662C22#297411yescharacteristic
663C22#297612yes—
664C22#297612yes—
665C22#297812yes—
666C22#297911yescharacteristic
667C22#298112yes—
668C22#299011yescharacteristic
669C22#299411yescharacteristic
670C22#299512yes—
671C4—24no—
672C4—42no—
673C4—44no—
674C4—44no—
675C4—42no—
676C4—81no—
677C4—82no—
678C4—81no—
679C4#265611yescharacteristic
680C4#265711yescharacteristic
681C4#266111yescharacteristic
682C4#266112yes—
683C4#266211yescharacteristic
684C4#266212yes—
685C2—82no—
686C2—84no—
687C2#1432111yescharacteristic
688C2#2168211yescharacteristic
689C2#2168511yescharacteristic
690C2#2203111yescharacteristic
691C2#2203411yescharacteristic
692C2#2203612yes—
693C2#2203912yes—
694C2#2213511yescharacteristic
695C2#2214011yescharacteristic
696C2#2214212yes—
697C2#2214312yes—
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×#1955C2semidirect
2. C2×#1954C2semidirect
3. C2×#1924C2semidirect
4. C2×#1923C2semidirect
6. #11322C2semidirect
7. #11042C2semidirect
8. #11042C2semidirect
9. #10964C2semidirect
10. #10964C2semidirect
11. #10696C2semidirect
105. C2×#97D4semidirect
107. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#19552
C2×#19542
C2×#19242
C2×#19232
C4×#3922
#113222
#110422
#109642
#106962