Finite Group Databasebeta

← #411150

Subgroups — #411150

Order 512.

1512 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2099C211yesmaximal, characteristic, complemented
2C2×#2041C211yesmaximal, characteristic
3#14713C211yesmaximal, characteristic
4C4×#389C211yesmaximal, characteristic, complemented
5#11373C211yesmaximal, characteristic, complemented
6#11189C212yesmaximal, complemented
7#11066C212yesmaximal, complemented
8#11008C211yesmaximal, characteristic
9#10931C212yesmaximal, complemented
10#10931C212yesmaximal, complemented
11#10795C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×#102C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C2×#390—21no—
18C2×#390—21no—
19C2×#390—21no—
20C2×#390—21no—
21C2×#390C2212yes—
22C2×#390C2212yes—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C43C2211yescharacteristic
30#2099—24no—
31#2041—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1436—21no—
41#1436—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C4×#95—21no—
45C23×D4—21no—
46C23×D4—21no—
47C23×D4C2312yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#127—24no—
51C2×#127—22no—
52C2×#127—22no—
53C2×#127—42no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C42—21no—
68C22×C42—21no—
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—42no—
85C2×#102—42no—
86C2×#102—42no—
87C2×#102—42no—
88#390—24no—
89#390—24no—
90#390—42no—
91#390—42no—
92#390—44no—
93#390—44no—
94#390—44no—
95#390—44no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389D414yes—
100#384—22no—
101#384—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—42no—
105C4×C4⋊C4—44no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C43—42no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340—42no—
119#336—22no—
120#336—22no—
121#336—24no—
122#336—42no—
123C25—22no—
124C22×D4—24no—
125C22×D4—24no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4#4114yes—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2411yescharacteristic
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178#127—48no—
179#127—48no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4#4112yes—
213C2×C4⋊C4#4112yes—
214C2×C4⋊C4#4112yes—
215C2×C4⋊C4#4112yes—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42—41no—
243C2×C42—41no—
244C2×C42—41no—
245C2×C42—41no—
246C2×C42—41no—
247C2×C42—41no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42D814yes—
254C2×C42SD1614yes—
255C2×C42C2×D412yes—
256C2×C42#4112yes—
257C2×C42#4114yes—
258#102—48no—
259#102—48no—
260#102—48no—
261#102—48no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—42no—
266#95—42no—
267#95—42no—
268#95—42no—
269#95—42no—
270C24—24no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24C22×D411yescharacteristic
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—48no—
293C2×D4—48no—
294C2×D4—48no—
295C2×D4—48no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—84no—
321C2×D4—84no—
322C2×D4—84no—
323C2×D4—84no—
324C2×D4—84no—
325C2×D4—84no—
326C2×D4—84no—
327C2×D4—84no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4#12112yes—
394C22×C4#12112yes—
395C22×C4#12412yes—
396C22×C4#12412yes—
397C22×C4C2×D812yes—
398C22×C4C2×D812yes—
399C22×C4C2×D812yes—
400C22×C4C2×SD1612yes—
401C22×C4C2×SD1612yes—
402C22×C4C2×SD1612yes—
403C22×C4AGL(1,ℤ/8)12yes—
404C22×C4AGL(1,ℤ/8)12yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4H(5,𝔽2)12yes—
414C22×C4H(5,𝔽2)12yes—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
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—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—48no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—48no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4#12414yes—
447C22⋊C4—88no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C42—24no—
452C42—24no—
453C42—24no—
454C42—24no—
455C42—44no—
456C42—44no—
457C42—48no—
458C42—44no—
459C42—44no—
460C42—42no—
461C42—42no—
462C42—84no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23C2×#12111yescharacteristic
499C23#52511yescharacteristic
500C23#52511yescharacteristic
501C23#52511yescharacteristic
502C23C2×#12411yescharacteristic
503C23#53411yescharacteristic
504C23#53511yescharacteristic
505C23C22×D811yescharacteristic
506C23C22×SD1611yescharacteristic
507C23C2×AGL(1,ℤ/8)11yescharacteristic
508C23C2×AGL(1,ℤ/8)11yescharacteristic
509C23#57511yescharacteristic
510C23#57511yescharacteristic
511C23#57511yescharacteristic
512C23#57511yescharacteristic
513D4—44no—
514D4—44no—
515D4—44no—
516D4—44no—
517D4—816no—
518D4—816no—
519D4—82no—
520D4—82no—
521D4—82no—
522D4—82no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
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—42no—
577C2×C4—42no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4#45914yes—
598C2×C4#46014yes—
599C2×C4#48614yes—
600C2×C4C2×#12411yescharacteristic
601C2×C4#52911yescharacteristic
602C2×C4#52912yes—
603C2×C4#52912yes—
604C2×C4#52911yescharacteristic
605C2×C4#53211yescharacteristic
606C2×C4#53812yes—
607C2×C4#54012yes—
608C2×C4#54612yes—
609C2×C4#55012yes—
610C2×C4#55112yes—
611C2×C4#55112yes—
612C2×C4#55312yes—
613C2×C4#55312yes—
614C2×C4#55312yes—
615C2×C4#55912yes—
616C8—82no—
617C8—82no—
618C22—84no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—162no—
626C22—162no—
627C22—162no—
628C22—162no—
629C22—162no—
630C22—162no—
631C22—162no—
632C22—162no—
633C22#239811yescharacteristic
634C22#253411yescharacteristic
635C22#254811yescharacteristic
636C22#259111yescharacteristic
637C22#259811yescharacteristic
638C22#263511yescharacteristic
639C22#285011yescharacteristic
640C22C2×#45911yescharacteristic
641C22C2×#46011yescharacteristic
642C22#304011yescharacteristic
643C22#304011yescharacteristic
644C22#310011yescharacteristic
645C22#310011yescharacteristic
646C22#310011yescharacteristic
647C22#310111yescharacteristic
648C22#310311yescharacteristic
649C22#310311yescharacteristic
650C22#311411yescharacteristic
651C22#311411yescharacteristic
652C22#311511yescharacteristic
653C22#311511yescharacteristic
654C22#311511yescharacteristic
655C22#311511yescharacteristic
656C22C2×#48611yescharacteristic
657C22#313511yescharacteristic
658C22#313811yescharacteristic
659C22#314211yescharacteristic
660C22#318611yescharacteristic
661C22#318711yescharacteristic
662C22#319411yescharacteristic
663C22#319411yescharacteristic
664C22#320511yescharacteristic
665C22#320811yescharacteristic
666C22#321911yescharacteristic
667C22#321911yescharacteristic
668C4—24no—
669C4—24no—
670C4—24no—
671C4—44no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—162no—
679C4#249012yes—
680C4#251612yes—
681C4#255412yes—
682C4#259512yes—
683C2—164no—
684C2#1510711yescharacteristic
685C2#2381511yescharacteristic
686C2#2382411yescharacteristic
687C2#2811411yescharacteristic
688C2#2812011yescharacteristic
689C2#2874911yescharacteristic
690C2#2875711yescharacteristic
691C2#2932811yescharacteristic
692C2#2933711yescharacteristic
693C2#2936411yescharacteristic
694C2#2936611yescharacteristic
695C2#3010511yescharacteristic
696C2#3010811yescharacteristic
697C2#3302411yescharacteristic
698C2#3302511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2099C2semidirect
4. C4×#389C2semidirect
5. #11373C2semidirect
6. #11189C2semidirect
7. #11066C2semidirect
9. #10931C2semidirect
10. #10931C2semidirect
11. #10795C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20992
C2×#20412
#147132
C4×#3892
#113732
#111892
#110662
#110082
#109312
#107952