Finite Group Databasebeta

← #416185

Subgroups — #416185

Order 512.

1642 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#1955C211yesmaximal, characteristic
3C2×#1836C211yesmaximal, characteristic, complemented
4C2×#1805C211yesmaximal, characteristic, complemented
5C2×#1763C211yesmaximal, characteristic, complemented
6#13905C211yesmaximal, characteristic
7#11200C212yesmaximal, complemented
8#11199C212yesmaximal, complemented
9#10970C212yesmaximal, complemented
10#10970C212yesmaximal, complemented
11#10849C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21C22×#97C2211yescharacteristic
22#2538—24no—
23#2538—24no—
24C2×#396—22no—
25C2×#390—21no—
26C2×#389—21no—
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1955—24no—
34#1836—24no—
35#1805—24no—
36#1763—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44#1445—21no—
45#1438—21no—
46#1437—22no—
47C23×Q8C2311yescharacteristic
48C23×D4D411yescharacteristic
49C24×C4C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2312yes—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—42no—
58C2×C22⋊Q8—42no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8D414yes—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4D412yes—
86C22×C42—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—42no—
98C2×#103—42no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—42no—
102C2×C22⋊C8—42no—
103C2×C22⋊C8—42no—
104C2×C22⋊C8—42no—
105C2×#97—22no—
106C2×#97—22no—
107C2×#97D414yes—
108#396—216no—
109#390—28no—
110#389—28no—
111#384—22no—
112#384D414yes—
113#382—22no—
114#382—22no—
115#380—24no—
116#380—24no—
117#380—24no—
118#380—24no—
119#380—42no—
120#380—42no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—21no—
128C2×#95—21no—
129#340—24no—
130#340—22no—
131#340—22no—
132#340—24no—
133#340—24no—
134#340—42no—
135#340—42no—
136C25C2×D412yes—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8C2×D412yes—
146C22×Q8#4114yes—
147C22×D4—24no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—42no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
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—21no—
171C23×C4—21no—
172C23×C4C2×D412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2411yescharacteristic
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8C2×D414yes—
193C22⋊Q8—48no—
194C22⋊Q8—48no—
195C22⋊Q8—48no—
196C22⋊Q8—48no—
197C4×Q8—28no—
198C4×Q8—28no—
199C4×Q8—44no—
200C4×Q8—44no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4—44no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—81no—
241C2×C4⋊C4—81no—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4#4114yes—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—82no—
257C2×C22⋊C4—82no—
258C2×C22⋊C4—82no—
259C2×C22⋊C4—82no—
260C2×C42—24no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—81no—
265C2×C42—81no—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42#4112yes—
270C2×C42#4112yes—
271C2×C42#4112yes—
272C4⋊C8—44no—
273#103—216no—
274#103—44no—
275#103—44no—
276C22⋊C8—48no—
277C22⋊C8—48no—
278#97—28no—
279#95—28no—
280#95—44no—
281#95—44no—
282#95—42no—
283#95—42no—
284#95—42no—
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—22no—
302C2×Q8—24no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8#12414yes—
319C2×D4—28no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—44no—
323C2×D4—44no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
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—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4#12012yes—
402C22×C4#12012yes—
403C22×C4#12112yes—
404C22×C4#12112yes—
405C22×C4#12312yes—
406C22×C4#12312yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4112yes—
414C22×C4C2×#4112yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C4H(5,𝔽2)12yes—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—42no—
433C2×C8—42no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—48no—
441C4⋊C4—416no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—84no—
457C22⋊C4—416no—
458C22⋊C4—48no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C42—48no—
463C42—84no—
464C42#12114yes—
465C42#12114yes—
466C42#12314yes—
467C42#12414yes—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
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—
503C23C2×#12011yescharacteristic
504C23C2×#12111yescharacteristic
505C23C2×#12311yescharacteristic
506C23#54612yes—
507C23#54612yes—
508C23C2×#13712yes—
509C23#57512yes—
510C23#57611yescharacteristic
511C23#57711yescharacteristic
512C23#57711yescharacteristic
513C23#57811yescharacteristic
514Q8—24no—
515Q8—24no—
516Q8—42no—
517Q8—44no—
518Q8—44no—
519Q8—42no—
520D4—416no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
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—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4#39414yes—
583C2×C4#39414yes—
584C2×C4C2×#12111yescharacteristic
585C2×C4C2×#12111yescharacteristic
586C2×C4C2×#12311yescharacteristic
587C2×C4#52512yes—
588C2×C4C2×#12411yescharacteristic
589C2×C4C2×#12411yescharacteristic
590C2×C4#52912yes—
591C2×C4#53211yescharacteristic
592C2×C4#53512yes—
593C2×C4#53511yescharacteristic
594C2×C4#53511yescharacteristic
595C2×C4#54012yes—
596C2×C4#54612yes—
597C2×C4#55012yes—
598C2×C4#55312yes—
599C2×C4#55312yes—
600C2×C4#55312yes—
601C2×C4#55512yes—
602C8—44no—
603C8—81no—
604C8—81no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22C2×#39411yescharacteristic
621C22#242012yes—
622C22#243111yescharacteristic
623C22#243511yescharacteristic
624C22#253512yes—
625C22#300411yescharacteristic
626C22#300511yescharacteristic
627C22#301211yescharacteristic
628C22#301311yescharacteristic
629C22#305612yes—
630C22#306211yescharacteristic
631C22#306311yescharacteristic
632C22#309512yes—
633C22#309811yescharacteristic
634C22#309911yescharacteristic
635C22#330912yes—
636C22#331112yes—
637C22#331212yes—
638C22#331312yes—
639C22#331412yes—
640C22#331612yes—
641C22#331712yes—
642C22#332012yes—
643C4—24no—
644C4—24no—
645C4—44no—
646C4—84no—
647C4—82no—
648C4—82no—
649C4—84no—
650C4—161no—
651C4—161no—
652C4#241412yes—
653C4#249012yes—
654C4#250011yescharacteristic
655C4#250411yescharacteristic
656C4#261911yescharacteristic
657C4#285011yescharacteristic
658C2—84no—
659C2#1521911yescharacteristic
660C2#2348111yescharacteristic
661C2#2348511yescharacteristic
662C2#2498312yes—
663C2#2498412yes—
664C2#2557911yescharacteristic
665C2#2558011yescharacteristic
666C2#2589311yescharacteristic
667C2#2589411yescharacteristic
668C2#2815312yes—
669C2#2815512yes—
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
3. C2×#1836C2semidirect
4. C2×#1805C2semidirect
5. C2×#1763C2semidirect
7. #11200C2semidirect
8. #11199C2semidirect
9. #10970C2semidirect
10. #10970C2semidirect
11. #10849C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#19552
C2×#18362
C2×#18052
C2×#17632
#139052
#112002
#111992
#109702
#108492