Finite Group Databasebeta

← #414559

Subgroups — #414559

Order 512.

1576 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×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2057C211yesmaximal, characteristic, complemented
3C2×#2023C211yesmaximal, characteristic
4C2×#1925C212yesmaximal
5#14700C211yesmaximal, characteristic
6#11219C212yesmaximal, complemented
7#11207C211yesmaximal, characteristic, complemented
8#11066C212yesmaximal
9#11049C211yesmaximal, characteristic
10#10806C212yesmaximal, complemented
11#10795C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×#107C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C4×C4⋊Q8—28no—
18C2×#391—21no—
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389—21no—
23C2×#389C2212yes—
24C2×#389C2211yescharacteristic
25C2×#388—21no—
26C2×#388—21no—
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2212yes—
30C2×C43C2211yescharacteristic
31#2057—24no—
32#2023—24no—
33#1925—28no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43#1437—21no—
44#1437—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C23×D4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8D414yes—
55C2×C4×Q8—28no—
56C2×C4×Q8—44no—
57C2×C4×Q8—44no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C42—22no—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—24no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91#391—28no—
92#390—22no—
93#390—22no—
94#390—24no—
95#390—22no—
96#390—22no—
97#389—28no—
98#389—24no—
99#389—24no—
100#389—22no—
101#389—22no—
102#388—44no—
103#388—44no—
104#384—24no—
105C4×C4⋊C4—28no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4D418yes—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C43—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#336—24no—
122#336—22no—
123#336—24no—
124#336—22no—
125C25—21no—
126C22×Q8—24no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—22no—
141C22×D4—21no—
142C22×D4#4112yes—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8#4114yes—
169C4⋊Q8—216no—
170C4⋊Q8—216no—
171C4×Q8—432no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4#4112yes—
212C2×C4⋊C4#4114yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—28no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—41no—
251C2×C42—41no—
252C2×C42—44no—
253C2×C42—44no—
254C2×C42D814yes—
255C2×C42C2×D414yes—
256C2×C42C2×D412yes—
257C2×C42#4112yes—
258#107—48no—
259#102—216no—
260#102—48no—
261#95—28no—
262#95—28no—
263#95—42no—
264#95—42no—
265#95—42no—
266#95—42no—
267#95—42no—
268C24—22no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24C22×D411yescharacteristic
284C2×Q8—216no—
285C2×Q8—48no—
286C2×Q8—48no—
287C2×Q8—48no—
288C2×Q8—48no—
289C2×Q8—48no—
290C2×Q8—48no—
291C2×D4—24no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—24no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—24no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—21no—
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—24no—
341C22×C4—22no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
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#12012yes—
373C22×C4#12012yes—
374C22×C4#12112yes—
375C22×C4#12112yes—
376C22×C4C2×D812yes—
377C22×C4C2×D812yes—
378C22×C4C2×D812yes—
379C22×C4#13512yes—
380C22×C4AGL(1,ℤ/8)12yes—
381C22×C4AGL(1,ℤ/8)12yes—
382C22×C4#13712yes—
383C22×C4#13712yes—
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D412yes—
386C22×C4C2×#4112yes—
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4112yes—
390C22×C4C2×#4111yescharacteristic
391C22×C42−1+411yescharacteristic
392C22×C42−1+411yescharacteristic
393C22×C42−1+412yes—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C4⋊C4—216no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—416no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—84no—
428C4⋊C4#12118yes—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—84no—
432C22⋊C4—84no—
433C22⋊C4—84no—
434C22⋊C4—84no—
435C42—28no—
436C42—24no—
437C42—24no—
438C42—24no—
439C42—24no—
440C42—28no—
441C42—44no—
442C42—44no—
443C42—42no—
444C42—42no—
445C42—416no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—81no—
454C23—81no—
455C23—81no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23—81no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23—81no—
480C23—81no—
481C23C2×#12011yescharacteristic
482C23C2×#12111yescharacteristic
483C23#52511yescharacteristic
484C23#52511yescharacteristic
485C23#53211yescharacteristic
486C23#53611yescharacteristic
487C23#53611yescharacteristic
488C23C22×D811yescharacteristic
489C23C2×#13511yescharacteristic
490C23C2×AGL(1,ℤ/8)11yescharacteristic
491C23C2×AGL(1,ℤ/8)11yescharacteristic
492C23C2×#13711yescharacteristic
493C23C2×#13711yescharacteristic
494C23#57511yescharacteristic
495C23#57511yescharacteristic
496Q8—432no—
497D4—44no—
498D4—44no—
499D4—82no—
500D4—82no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—28no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—44no—
530C2×C4—41no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4#44714yes—
576C2×C4#45914yes—
577C2×C4#46414yes—
578C2×C4C2×#12112yes—
579C2×C4#52512yes—
580C2×C4#52512yes—
581C2×C4#53212yes—
582C2×C4#53612yes—
583C2×C4#53612yes—
584C2×C4#54811yescharacteristic
585C2×C4#54812yes—
586C2×C4#54811yescharacteristic
587C2×C4#55211yescharacteristic
588C2×C4#55212yes—
589C2×C4#55211yescharacteristic
590C2×C4#55212yes—
591C2×C4#55612yes—
592C2×C4#55612yes—
593C2×C4#55612yes—
594C8—44no—
595C8—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—161no—
604C22—161no—
605C22—161no—
606C22—161no—
607C22—161no—
608C22—161no—
609C22—161no—
610C22—161no—
611C22#238811yescharacteristic
612C22#253611yescharacteristic
613C22#254511yescharacteristic
614C22#259911yescharacteristic
615C22#260411yescharacteristic
616C22#261411yescharacteristic
617C22#284711yescharacteristic
618C22C2×#44711yescharacteristic
619C22#300211yescharacteristic
620C22#300411yescharacteristic
621C22#300611yescharacteristic
622C22C2×#45911yescharacteristic
623C22C2×#46411yescharacteristic
624C22#303611yescharacteristic
625C22#303711yescharacteristic
626C22#310011yescharacteristic
627C22#310011yescharacteristic
628C22#310611yescharacteristic
629C22#310811yescharacteristic
630C22#310911yescharacteristic
631C22#311011yescharacteristic
632C22#311611yescharacteristic
633C22#311711yescharacteristic
634C22#316911yescharacteristic
635C22#317111yescharacteristic
636C22#318111yescharacteristic
637C22#318211yescharacteristic
638C22#322611yescharacteristic
639C22#322611yescharacteristic
640C22#322811yescharacteristic
641C22#323311yescharacteristic
642C22#323511yescharacteristic
643C22#323811yescharacteristic
644C22#323811yescharacteristic
645C22#324011yescharacteristic
646C4—24no—
647C4—24no—
648C4—24no—
649C4—42no—
650C4—42no—
651C4—48no—
652C4—42no—
653C4—84no—
654C4—82no—
655C4—161no—
656C4—161no—
657C4#253612yes—
658C4#254512yes—
659C4#259912yes—
660C4#260412yes—
661C2—162no—
662C2#1517611yescharacteristic
663C2#2356711yescharacteristic
664C2#2357411yescharacteristic
665C2#2820111yescharacteristic
666C2#2820411yescharacteristic
667C2#2865511yescharacteristic
668C2#2865911yescharacteristic
669C2#2938811yescharacteristic
670C2#2939311yescharacteristic
671C2#2951411yescharacteristic
672C2#2952111yescharacteristic
673C2#2963811yescharacteristic
674C2#2964411yescharacteristic
675C2#3290211yescharacteristic
676C2#3290911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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×C4×C4⋊Q8C2semidirect
2. C2×#2057C2semidirect
6. #11219C2semidirect
7. #11207C2semidirect
10. #10806C2semidirect
11. #10795C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C2×#20572
C2×#20232
C2×#19252
#147002
#112192
#112072
#110662
#110492
#108062
#107952