Finite Group Databasebeta

← #414047

Subgroups — #414047

Order 512.

1446 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2093C211yesmaximal, characteristic
2C2×#2023C211yesmaximal, characteristic
3#14714C211yesmaximal, characteristic, complemented
4C4×#390C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11373C211yesmaximal, characteristic, complemented
7#11232C212yesmaximal, complemented
8#11076C212yesmaximal
9#10907C212yesmaximal, complemented
10#10832C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#107C2212yes—
15C22×C4⋊C8C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C2×#391—21no—
19C2×#391—21no—
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4—21no—
25C2×C4×C22⋊C4—21no—
26C2×C43C2211yescharacteristic
27#2093—24no—
28#2023—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1437—21no—
38#1437—21no—
39C4⋊#95—21no—
40C4⋊#95—21no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C2×C4×D4—22no—
55C2×C4×D4—42no—
56C2×C4×D4—42no—
57C2×C4×D4Q814yescomplemented
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C42—21no—
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#107—24no—
80C2×#107—24no—
81C2×#107—42no—
82C2×C4⋊C8—24no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—42no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—42no—
90C2×#102—42no—
91#391—44no—
92#391—44no—
93#390—22no—
94#390—22no—
95#390D414yes—
96#390Q814yescomplemented
97#384—22no—
98#384—22no—
99C4×C4⋊C4—42no—
100C4×C22⋊C4—44no—
101C4×C22⋊C4—44no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C43—42no—
112#340—24no—
113#340—24no—
114#340—42no—
115#340—42no—
116#340—42no—
117#336—22no—
118#336—22no—
119#336—24no—
120C25—21no—
121C22×Q8—24no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4C2×D412yes—
136C22×D4C2×Q812yes—
137C22×D4C2×Q812yes—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
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—22no—
169C22×C8—22no—
170C22×C8#4114yes—
171C4⋊Q8—48no—
172C4⋊Q8—48no—
173C4×D4—48no—
174C4×D4—48no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×Q812yes—
206C2×C4⋊C4C2×Q812yes—
207C2×C4⋊C4#4114yes—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
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×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—41no—
235C2×C42—41no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42D814yes—
244C2×C42C2×D412yes—
245C2×C42C2×Q812yes—
246C2×C42C2×Q812yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250#107—48no—
251C4⋊C8—48no—
252#102—48no—
253#102—48no—
254#95—42no—
255#95—42no—
256#95—44no—
257#95—44no—
258#95—42no—
259#95—42no—
260#95—44no—
261#95—44no—
262#95—42no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24C22×D411yescharacteristic
279C2×Q8—48no—
280C2×Q8—48no—
281C2×Q8—48no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×Q8—84no—
285C2×Q8—84no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4C4⋊Q814yes—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
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—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—48no—
335C22×C4—48no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—48no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4#12112yes—
373C22×C4#12112yes—
374C22×C4C22⋊Q812yes—
375C22×C4C22⋊Q812yes—
376C22×C4C22⋊Q812yes—
377C22×C4C22⋊Q812yes—
378C22×C4C2×D812yes—
379C22×C4C2×D812yes—
380C22×C4C2×D812yes—
381C22×C4#13512yes—
382C22×C4AGL(1,ℤ/8)12yes—
383C22×C4AGL(1,ℤ/8)12yes—
384C22×C4#13712yes—
385C22×C4#13712yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×Q811yescharacteristic
388C22×C4C22×Q811yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4112yes—
391C22×C4C2×#4112yes—
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4H(5,𝔽2)12yes—
395C22×C4H(5,𝔽2)12yes—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
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—
408C2×C8—42no—
409C2×C8—42no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—48no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—84no—
426C4⋊C4—84no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C42—24no—
433C42—24no—
434C42—24no—
435C42—24no—
436C42—24no—
437C42—24no—
438C42—44no—
439C42—44no—
440C42—44no—
441C42—44no—
442C42—84no—
443C23—24no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
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—
478C23C2×#12111yescharacteristic
479C23C2×C22⋊Q811yescharacteristic
480C23C2×C22⋊Q811yescharacteristic
481C23#52511yescharacteristic
482C23#53211yescharacteristic
483C23#53411yescharacteristic
484C23#53511yescharacteristic
485C23C22×D811yescharacteristic
486C23C2×#13511yescharacteristic
487C23C2×AGL(1,ℤ/8)11yescharacteristic
488C23C2×AGL(1,ℤ/8)11yescharacteristic
489C23C2×#13711yescharacteristic
490C23C2×#13711yescharacteristic
491C23#57511yescharacteristic
492C23#57511yescharacteristic
493Q8—816no—
494D4—28no—
495D4—44no—
496D4—44no—
497D4—44no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
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—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—816no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4#45914yes—
573C2×C4#46314yes—
574C2×C4#47414yes—
575C2×C4#47414yes—
576C2×C4#52912yes—
577C2×C4C2×C4⋊Q811yescharacteristic
578C2×C4#53211yescharacteristic
579C2×C4#53311yescharacteristic
580C2×C4#53311yescharacteristic
581C2×C4#53312yes—
582C2×C4#53812yes—
583C2×C4#54312yes—
584C2×C4#54612yes—
585C2×C4#54612yes—
586C2×C4#55112yes—
587C2×C4#55112yes—
588C2×C4#55412yes—
589C2×C4#55412yes—
590C2×C4#55912yes—
591C2×C4#55912yes—
592C8—44no—
593C8—82no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22#239611yescharacteristic
610C22#253411yescharacteristic
611C22#254411yescharacteristic
612C22#256411yescharacteristic
613C22#258011yescharacteristic
614C22#262611yescharacteristic
615C22#285711yescharacteristic
616C22C2×#45911yescharacteristic
617C22C2×#46311yescharacteristic
618C22#303911yescharacteristic
619C22#304011yescharacteristic
620C22C2×#47411yescharacteristic
621C22C2×#47411yescharacteristic
622C22#307611yescharacteristic
623C22#307611yescharacteristic
624C22#307711yescharacteristic
625C22#307711yescharacteristic
626C22#307911yescharacteristic
627C22#307911yescharacteristic
628C22#310011yescharacteristic
629C22#310811yescharacteristic
630C22#311211yescharacteristic
631C22#311411yescharacteristic
632C22#316911yescharacteristic
633C22#317211yescharacteristic
634C22#318111yescharacteristic
635C22#318411yescharacteristic
636C22#318611yescharacteristic
637C22#319011yescharacteristic
638C22#319311yescharacteristic
639C22#319411yescharacteristic
640C22#320511yescharacteristic
641C22#320711yescharacteristic
642C22#321511yescharacteristic
643C22#321611yescharacteristic
644C4—24no—
645C4—24no—
646C4—24no—
647C4—24no—
648C4—42no—
649C4—42no—
650C4—82no—
651C4—82no—
652C4—84no—
653C4—82no—
654C4—162no—
655C4#254912yes—
656C4#255712yes—
657C4#258912yes—
658C4#260712yes—
659C2—84no—
660C2#1516111yescharacteristic
661C2#2377411yescharacteristic
662C2#2378111yescharacteristic
663C2#2811411yescharacteristic
664C2#2811811yescharacteristic
665C2#2861511yescharacteristic
666C2#2862011yescharacteristic
667C2#2915711yescharacteristic
668C2#2916311yescharacteristic
669C2#2925411yescharacteristic
670C2#2926211yescharacteristic
671C2#2995011yescharacteristic
672C2#2995511yescharacteristic
673C2#3319511yescharacteristic
674C2#3320111yescharacteristic
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
3. #14714C2semidirect
5. #11384C2semidirect
6. #11373C2semidirect
7. #11232C2semidirect
9. #10907C2semidirect
10. #10832C2semidirect
57. C2×C4×D4Q8semidirect
96. #390Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20932
C2×#20232
#147142
C4×#3902
#113842
#113732
#112322
#110762
#109072
#108322
#107882