Finite Group Databasebeta

← #400657

Subgroups — #400657

Order 512.

1650 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2C2×#2056C212yesmaximal
3C2×#2043C211yesmaximal, characteristic
4C2×#2016C211yesmaximal, characteristic
5C2×#1878C212yesmaximal
6#14703C211yesmaximal, characteristic
7#11207C211yesmaximal, characteristic
8#11077C212yesmaximal
9#11050C211yesmaximal, characteristic
10#10832C211yesmaximal, characteristic
11#10789C212yesmaximal
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C22×C4⋊C8C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C4×C4⋊Q8—28no—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#391C2212yes—
24C2×#391C2211yescharacteristic
25C2×#389—21no—
26C2×#389—21no—
27C2×#389C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4—21no—
31C2×C43C2211yescharacteristic
32#2056—28no—
33#2043—24no—
34#2016—24no—
35#1878—28no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1436—21no—
43#1436—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46#1434—21no—
47#1434—21no—
48C23×Q8C2311yescharacteristic
49C23×Q8C2312yes—
50C22×Q16—24no—
51C22×Q16—24no—
52C22×Q16—24no—
53C22×Q16—24no—
54C22×Q16—24no—
55C22×Q16—42no—
56C22×Q16—42no—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8D414yes—
63C2×C4×Q8—24no—
64C2×C4×Q8—24no—
65C2×C4×Q8—24no—
66C2×C4×Q8—24no—
67C2×C4×Q8—42no—
68C2×C4×Q8—42no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C42—22no—
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×C4⋊C8—24no—
88C2×C4⋊C8—24no—
89C2×C4⋊C8—42no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—42no—
100C2×#103—42no—
101C2×#103—42no—
102#391—24no—
103#391—22no—
104#391—24no—
105#391—22no—
106#391—22no—
107#391D418yes—
108#391D414yes—
109#389—28no—
110#389—28no—
111#389—22no—
112#389—22no—
113#384—24no—
114C4×C4⋊C4—28no—
115C4×C4⋊C4—28no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C43—24no—
130#340—24no—
131#340—24no—
132#340—42no—
133#336—24no—
134#336—22no—
135#336—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—21no—
140C22×Q8—22no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—21no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—21no—
156C22×Q8C2×D412yes—
157C22×Q8C2×D414yes—
158C22×Q8C2×D414yes—
159C22×Q8#4114yes—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C2×Q16—216no—
174C2×Q16—48no—
175C2×Q16—48no—
176C2×Q16—48no—
177C2×Q16—48no—
178C2×Q16—48no—
179C2×Q16—48no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8#4114yes—
194C4⋊Q8—216no—
195C4⋊Q8—216no—
196C4×Q8—48no—
197C4×Q8—48no—
198C4×Q8—48no—
199C4×Q8—48no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×D412yes—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
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—28no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—41no—
256C2×C42—41no—
257C2×C42—44no—
258C2×C42—44no—
259C2×C42Q1614yescomplemented
260C2×C42C2×D414yes—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42#4112yes—
264C2×C42#4112yes—
265C4⋊C8—48no—
266#103—48no—
267#103—48no—
268#103—48no—
269#103—48no—
270#95—28no—
271#95—28no—
272#95—24no—
273#95—24no—
274#95—42no—
275#95—42no—
276#95—42no—
277#95—42no—
278#95—42no—
279#95—42no—
280#95—42no—
281#95—42no—
282#95—42no—
283C24C22×D411yescharacteristic
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—22no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—22no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—41no—
315C2×Q8—42no—
316C2×Q8—41no—
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#12118yes—
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—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—21no—
346C22×C4—24no—
347C22×C4—24no—
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—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4#12012yes—
399C22×C4#12012yes—
400C22×C4#12112yes—
401C22×C4#12112yes—
402C22×C4#12112yes—
403C22×C4#12112yes—
404C22×C4C2×Q1612yes—
405C22×C4C2×Q1612yes—
406C22×C4C2×Q1612yes—
407C22×C4#13512yes—
408C22×C4#13712yes—
409C22×C4#13712yes—
410C22×C4#13712yes—
411C22×C4#13712yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C2×#4112yes—
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4112yes—
418C22×C4C2×#4111yescharacteristic
419C22×C4H(5,𝔽2)11yescharacteristic
420C22×C4H(5,𝔽2)12yes—
421C22×C42−1+411yescharacteristic
422Q16—432no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C4⋊C4—216no—
438C4⋊C4—216no—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—416no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—84no—
455C42—28no—
456C42—24no—
457C42—24no—
458C42—24no—
459C42—24no—
460C42—24no—
461C42—24no—
462C42—28no—
463C42—44no—
464C42—44no—
465C42—416no—
466C23C2×#12011yescharacteristic
467C23C2×#12111yescharacteristic
468C23C2×#12111yescharacteristic
469C23#52511yescharacteristic
470C23#53211yescharacteristic
471C23#53511yescharacteristic
472C23#53711yescharacteristic
473C23C22×Q1611yescharacteristic
474C23C2×#13511yescharacteristic
475C23C2×#13711yescharacteristic
476C23C2×#13711yescharacteristic
477C23C2×#13711yescharacteristic
478C23C2×#13711yescharacteristic
479C23#57511yescharacteristic
480C23#57511yescharacteristic
481Q8—28no—
482Q8—28no—
483Q8—44no—
484Q8—44no—
485Q8—44no—
486Q8—44no—
487Q8—44no—
488Q8—44no—
489Q8—44no—
490Q8—44no—
491Q8—82no—
492Q8—82no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
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—28no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—82no—
556C2×C4—81no—
557C2×C4—82no—
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#45114yes—
574C2×C4#46214yes—
575C2×C4#46214yes—
576C2×C4#46414yes—
577C2×C4C2×#12112yes—
578C2×C4#52512yes—
579C2×C4#52512yes—
580C2×C4#52912yes—
581C2×C4#53812yes—
582C2×C4#53912yes—
583C2×C4D4211yescharacteristic
584C2×C4D4212yes—
585C2×C4#54612yes—
586C2×C4#54612yes—
587C2×C4#54612yes—
588C2×C4#54711yescharacteristic
589C2×C4#54712yes—
590C2×C4#54711yescharacteristic
591C2×C4#55511yescharacteristic
592C2×C4#55912yes—
593C8—44no—
594C8—82no—
595C22#239211yescharacteristic
596C22#240811yescharacteristic
597C22#243011yescharacteristic
598C22#253711yescharacteristic
599C22#254411yescharacteristic
600C22#263611yescharacteristic
601C22#284711yescharacteristic
602C22C2×#45111yescharacteristic
603C22#300211yescharacteristic
604C22#300411yescharacteristic
605C22#300711yescharacteristic
606C22C2×#46211yescharacteristic
607C22C2×#46211yescharacteristic
608C22C2×#46411yescharacteristic
609C22#303511yescharacteristic
610C22#303711yescharacteristic
611C22#303711yescharacteristic
612C22#303711yescharacteristic
613C22#303911yescharacteristic
614C22#310411yescharacteristic
615C22#310811yescharacteristic
616C22#311311yescharacteristic
617C22#311711yescharacteristic
618C22#317011yescharacteristic
619C22#317211yescharacteristic
620C22#318111yescharacteristic
621C22#318511yescharacteristic
622C22#320911yescharacteristic
623C22#321011yescharacteristic
624C22#321811yescharacteristic
625C22#321911yescharacteristic
626C22#324811yescharacteristic
627C22#324911yescharacteristic
628C22#325211yescharacteristic
629C22#325511yescharacteristic
630C4—24no—
631C4—24no—
632C4—24no—
633C4—24no—
634C4—42no—
635C4—42no—
636C4—48no—
637C4—82no—
638C4—84no—
639C4—82no—
640C4—161no—
641C4—161no—
642C4#242412yes—
643C4#242512yes—
644C4#245512yes—
645C4#245912yes—
646C2#1480611yescharacteristic
647C2#2367911yescharacteristic
648C2#2368511yescharacteristic
649C2#2421211yescharacteristic
650C2#2421611yescharacteristic
651C2#2549811yescharacteristic
652C2#2550411yescharacteristic
653C2#2825411yescharacteristic
654C2#2825511yescharacteristic
655C2#2861611yescharacteristic
656C2#2861911yescharacteristic
657C2#3013111yescharacteristic
658C2#3013211yescharacteristic
659C2#3290611yescharacteristic
660C2#3290911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
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
259. C2×C42Q16semidirect

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×#20562
C2×#20432
C2×#20162
C2×#18782
#147032
#112072
#110772
#110502
#108322
#107892