Finite Group Databasebeta

← #464302

Subgroups — #464302

Order 512.

1704 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×#2542C211yesmaximal, characteristic
2C2×#2043C211yesmaximal, characteristic, complemented
3C2×#2035C212yesmaximal, complemented
4C2×#1920C211yesmaximal, characteristic, complemented
5#15181C211yesmaximal, characteristic
6#11010C211yesmaximal, characteristic, complemented
7#10970C212yesmaximal, complemented
8#10970C212yesmaximal, complemented
9#10970C212yesmaximal, complemented
10#10964C211yesmaximal, characteristic
11#10806C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×C22⋊Q8C2212yes—
15C22×C4×D4C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2212yes—
22C22×C22⋊C8C2212yes—
23#2542—24no—
24#2542—24no—
25C2×#399—22no—
26C2×#390—21no—
27C2×#389C2211yescharacteristic
28C2×#384—21no—
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32#2043—24no—
33#2035—28no—
34#1920—24no—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1442—21no—
43#1439—22no—
44#1438—21no—
45C4⋊#95C2211yescharacteristic
46C23×Q8C2312yes—
47C23×D4D411yescharacteristic
48C24×C4C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8D414yes—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—42no—
65C2×C22⋊Q8—42no—
66C2×C22⋊Q8—42no—
67C2×C22⋊Q8—42no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—42no—
71C2×C4×D4—42no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4D412yes—
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90C2×#106—22no—
91C2×#106—22no—
92C2×#106D414yescomplemented
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103—42no—
100C2×#103—42no—
101C2×#103—42no—
102C2×#103—42no—
103C2×#103—42no—
104C2×#103—42no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—42no—
107C2×C22⋊C8—42no—
108C2×C22⋊C8—42no—
109C2×C22⋊C8—42no—
110C2×C22⋊C8—42no—
111#399—216no—
112#390—28no—
113#389—22no—
114#389—22no—
115#384—28no—
116#380—24no—
117#380—42no—
118#380—42no—
119#380—42no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4D414yes—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—21no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—24no—
132#340—24no—
133#336—22no—
134#336—22no—
135C25C2×D412yes—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8C2×D414yes—
150C22×D4—24no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—42no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2411yescharacteristic
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
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×C8C2×D412yes—
196C4⋊Q8—48no—
197C4⋊Q8—44no—
198C4⋊Q8—44no—
199C22⋊Q8—48no—
200C22⋊Q8—48no—
201C22⋊Q8—48no—
202C22⋊Q8—48no—
203C22⋊Q8—48no—
204C22⋊Q8—48no—
205C22⋊Q8—48no—
206C22⋊Q8—48no—
207C4×D4—48no—
208C4×D4—44no—
209C4×D4—44no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—82no—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4#4112yes—
249C2×C4⋊C4#4114yes—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—82no—
263C2×C22⋊C4—82no—
264C2×C22⋊C4—82no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272#106—28no—
273#103—48no—
274#103—48no—
275#103—48no—
276#103—48no—
277C22⋊C8—48no—
278C22⋊C8—48no—
279#95—24no—
280#95—44no—
281#95—44no—
282#95—42no—
283#95—42no—
284#95—81no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24#13714yes—
300C24C22×D411yescharacteristic
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×D4—28no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—44no—
333C2×D4—44no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
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—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—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—41no—
411C22×C4#12012yes—
412C22×C4#12012yes—
413C22×C4#13512yes—
414C22×C4#13712yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4112yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C4H(5,𝔽2)11yescharacteristic
424C22×C4H(5,𝔽2)11yescharacteristic
425C2×C8—22no—
426C2×C8—24no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—44no—
450C4⋊C4—416no—
451C4⋊C4—416no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—84no—
464C4⋊C4#12114yes—
465C4⋊C4#12414yes—
466C22⋊C4—48no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C42—24no—
472C42—22no—
473C42—22no—
474C42—48no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
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—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—
510C23C2×#12011yescharacteristic
511C23#53412yes—
512C23D4211yescharacteristic
513C23#54611yescharacteristic
514C23#54612yes—
515C23C2×#13511yescharacteristic
516C23C2×#13711yescharacteristic
517C23C2×#13712yes—
518C23#57711yescharacteristic
519C23#57712yes—
520C23#57811yescharacteristic
521Q8—44no—
522Q8—44no—
523Q8—44no—
524Q8—44no—
525Q8—82no—
526Q8—82no—
527Q8—82no—
528Q8—82no—
529D4—416no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—81no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—81no—
585C2×C4—82no—
586C2×C4—81no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4#45214yes—
591C2×C4C2×#12111yescharacteristic
592C2×C4C2×#12411yescharacteristic
593C2×C4#53412yes—
594C2×C4#53511yescharacteristic
595C2×C4#53511yescharacteristic
596C2×C4#53511yescharacteristic
597C2×C4#53511yescharacteristic
598C2×C4#53512yes—
599C2×C4#54012yes—
600C2×C4#54012yes—
601C2×C4#54012yes—
602C2×C4D4211yescharacteristic
603C2×C4#54612yes—
604C2×C4#54611yescharacteristic
605C2×C4#55012yes—
606C2×C4#55912yes—
607C2×C4#56012yes—
608C2×C4#56012yes—
609C8—42no—
610C8—81no—
611C8—82no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22#240811yescharacteristic
628C22#243511yescharacteristic
629C22#261311yescharacteristic
630C22#267312yes—
631C22#270912yes—
632C22C2×#45211yescharacteristic
633C22#300411yescharacteristic
634C22#301011yescharacteristic
635C22#301311yescharacteristic
636C22#319012yes—
637C22#319512yes—
638C22#320012yes—
639C22#320112yes—
640C22#326612yes—
641C22#327611yescharacteristic
642C22#328411yescharacteristic
643C22#330211yescharacteristic
644C22#330312yes—
645C22#331212yes—
646C22#332012yes—
647C22#332012yes—
648C22#333112yes—
649C22#333111yescharacteristic
650C4—24no—
651C4—44no—
652C4—42no—
653C4—44no—
654C4—82no—
655C4—84no—
656C4—82no—
657C4—161no—
658C4—162no—
659C4#254811yescharacteristic
660C4#261911yescharacteristic
661C4#264111yescharacteristic
662C4#267312yes—
663C4#270912yes—
664C4#281811yescharacteristic
665C2—84no—
666C2#1650311yescharacteristic
667C2#2424111yescharacteristic
668C2#2425111yescharacteristic
669C2#2588611yescharacteristic
670C2#2589411yescharacteristic
671C2#2959911yescharacteristic
672C2#2960311yescharacteristic
673C2#3071012yes—
674C2#3072112yes—
675C2#3203412yes—
676C2#3203712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2043C2semidirect
3. C2×#2035C2semidirect
4. C2×#1920C2semidirect
6. #11010C2semidirect
7. #10970C2semidirect
8. #10970C2semidirect
9. #10970C2semidirect
11. #10806C2semidirect
92. C2×#106D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#20432
C2×#20352
C2×#19202
#151812
#110102
#109702
#109642
#108062