Finite Group Databasebeta

← #364877

Subgroups — #364877

Order 512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 757 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×M16C211yesmaximal, characteristic, complemented
2#34952C211yesmaximal, characteristic
3C2×#1806C211yesmaximal, characteristic
4C2×#1759C211yesmaximal, characteristic
5#13850C211yesmaximal, characteristic, complemented
6#11209C212yesmaximal, complemented
7#11053C212yesmaximal
8#10933C212yesmaximal
9#10886C212yesmaximal, complemented
10#10851C211yesmaximal, characteristic, complemented
11#10840C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14Q8×M16—24no—
15Q8×M16—24no—
16C2×#446—21no—
17C2×#446—22no—
18C2×C8×Q8—21no—
19C2×#423C2212yes—
20C2×#423C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C22×#104C2212yes—
23C22×#104C2211yescharacteristic
24C2×C4×M16C2212yes—
25C2×C4×M16C2211yescharacteristic
26#2582C412yescomplemented
27#2581C411yescharacteristic, complemented
28#2581C411yescharacteristic, complemented
29#2580C411yescharacteristic, complemented
30#2580C411yescharacteristic, complemented
31#2579—22no—
32#2538C412yescomplemented
33#2487—21no—
34#2409—21no—
35#2396—21no—
36#2396—21no—
37C4×C22⋊Q8—22no—
38#2297C2212yes—
39#2296C2212yes—
40#2285C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×#376C2211yescharacteristic
43#1806—24no—
44#1759—24no—
45C2×#339C2212yes—
46C2×#339C2212yes—
47C2×#339C2211yescharacteristic
48C2×#339C2212yes—
49C2×#339C2211yescharacteristic
50C2×#339C2212yes—
51C2×#339C2212yes—
52#1446—21no—
53#1445—21no—
54#1441—22no—
55C4×#95C2212yes—
56C4×#95C2212yes—
57C22×#41C2311yescharacteristic
58C23×Q8D411yescharacteristic
59C22×M16C2312yes—
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C2×C22⋊Q8—22no—
63C4×#41—22no—
64C4×#41—22no—
65C4×#41—22no—
66C4×#41—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×Q8C2×C412yes—
76C2×C4×Q8D411yescharacteristic
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8D412yes—
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4C2×C411yescharacteristic
83C2×#117—21no—
84C2×#117C2311yescharacteristic
85C2×#117C2312yes—
86C22×C4⋊C4—21no—
87C22×C4⋊C4D412yes—
88C22×C4⋊C4D411yescharacteristic
89C22×C42—21no—
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2312yes—
94C22×C42C2311yescharacteristic
95#446—216no—
96#446—28no—
97C8×Q8—28no—
98#423—24no—
99#423—24no—
100#423—22no—
101#423D414yes—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—21no—
106C2×#104—22no—
107C2×#104—22no—
108C2×#104—22no—
109C2×#104—22no—
110C2×#104—22no—
111C2×#104—24no—
112C2×#104—24no—
113C2×#104—42no—
114C2×#104Q814yescomplemented
115C4×M16—24no—
116C4×M16—24no—
117C4×M16—22no—
118C4×M16D414yes—
119C2×#97—22no—
120C2×#97—21no—
121C2×C4×C8—22no—
122C2×C4×C8—21no—
123#400—21no—
124#400—21no—
125#399—21no—
126#399—21no—
127#398—22no—
128#398—22no—
129#396—22no—
130#396—22no—
131#395—21no—
132#395—21no—
133#393—21no—
134#393—21no—
135#391—22no—
136#391C2×C412yes—
137#391C2×C412yes—
138#390C2×C412yes—
139#390C2×C412yes—
140#389—22no—
141#389C2×C412yes—
142#389C2×C412yes—
143#389C2×C412yes—
144#389C2×C412yes—
145#387—22no—
146#387—22no—
147#386—22no—
148#386—22no—
149#385—22no—
150#385—22no—
151#385—22no—
152#385—22no—
153#384C2×C412yes—
154#382C2×C411yescharacteristic
155#382C2×C411yescharacteristic
156#380—21no—
157#380—21no—
158#380—22no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C22⋊C4—22no—
166C4×C22⋊C4—22no—
167#376—22no—
168#376—22no—
169C2×#95—22no—
170C2×#95—22no—
171C2×#95—22no—
172C2×#95—22no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176#339—24no—
177#339—22no—
178#339—22no—
179#339—24no—
180#339—24no—
181#339—42no—
182#339—42no—
183C2×#41—21no—
184C2×#41—21no—
185C2×#41—21no—
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41C2×Q812yes—
194C2×#41#4112yes—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—44no—
198C22×Q8C22⋊C411yescharacteristic
199C22×Q8C22⋊C412yes—
200C22×Q8C22⋊C411yescharacteristic
201C22×Q8C22×C411yescharacteristic
202C22×D4—21no—
203C22×D4C22×C411yescharacteristic
204C23×C4—21no—
205C23×C4—22no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4C2×D412yes—
211C23×C4C2×D412yes—
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2411yescharacteristic
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—22no—
219C2×M16—21no—
220C2×M16—22no—
221C2×M16—21no—
222C2×M16—22no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—22no—
226C2×M16—21no—
227C2×M16—22no—
228C2×M16C2×D414yes—
229C2×M16C2×D412yes—
230C2×M16C2×D412yes—
231C2×M16C2×Q812yes—
232C2×M16#4112yes—
233C22×C8—21no—
234C22×C8—22no—
235C22×C8—21no—
236C22⋊Q8—48no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8—24no—
242C4×Q8—24no—
243C4×Q8—22no—
244C4×Q8—24no—
245C4×Q8—22no—
246C4×Q8—24no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—42no—
258C4×D4—42no—
259C4×D4—42no—
260C4×D4—42no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—22no—
266#117—22no—
267#117—22no—
268#117—22no—
269#117—42no—
270#117—42no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4C22⋊C412yes—
286C2×C4⋊C4C22⋊C412yes—
287C2×C4⋊C4C22⋊C412yes—
288C2×C4⋊C4C22⋊C412yes—
289C2×C4⋊C4C22⋊C412yes—
290C2×C4⋊C4C22⋊C411yescharacteristic
291C2×C4⋊C4C22⋊C411yescharacteristic
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C4⋊C4C22×C411yescharacteristic
294C2×C4⋊C4C22×C412yes—
295C2×C4⋊C4C22×C412yes—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42C22×C412yes—
331C2×C42C22×C412yes—
332C2×C42C22×C411yescharacteristic
333C2×C42C22×C411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×Q812yes—
341C2×C42#4112yes—
342C2×C42#4112yes—
343C2×C42#4112yes—
344C4⋊C8—28no—
345C4⋊C8—28no—
346C4⋊C8—24no—
347C4⋊C8—24no—
348#104—28no—
349#104—28no—
350#104—48no—
351#97—28no—
352#97—24no—
353C4×C8—28no—
354C4×C8—24no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—44no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95—42no—
372#95—42no—
373#95—42no—
374C24—21no—
375C24C22×D411yescharacteristic
376#41—22no—
377#41—22no—
378#41—22no—
379#41—22no—
380#41—22no—
381#41—22no—
382#41—22no—
383#41—22no—
384#41—41no—
385#41—41no—
386#41—41no—
387#41—41no—
388#41—41no—
389#41—41no—
390#41—41no—
391#41—41no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—48no—
405C2×Q8—48no—
406C2×Q8C4×Q812yes—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—41no—
418C2×D4C4×Q812yes—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—82no—
461C22×C4C2×C22⋊C412yes—
462C22×C4C2×C22⋊C412yes—
463C22×C4C2×C22⋊C412yes—
464C22×C4C2×C22⋊C411yescharacteristic
465C22×C4C2×C22⋊C411yescharacteristic
466C22×C4C2×C22⋊C411yescharacteristic
467C22×C4C2×C22⋊C411yescharacteristic
468C22×C4C2×C22⋊C412yes—
469C22×C4#12112yes—
470C22×C4#12112yes—
471C22×C4#12112yes—
472C22×C4#12112yes—
473C22×C4C23×C411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×Q811yescharacteristic
478C22×C4C2×#4111yescharacteristic
479C22×C4C2×#4111yescharacteristic
480C22×C4C2×#4111yescharacteristic
481C22×C4C2×#4111yescharacteristic
482C22×C4C2×#4111yescharacteristic
483C22×C4H(5,𝔽2)11yescharacteristic
484C22×C42−1+411yescharacteristic
485C22×C42−1+411yescharacteristic
486C22×C42−1+411yescharacteristic
487M16—22no—
488M16—22no—
489M16—22no—
490M16—22no—
491M16—22no—
492M16—24no—
493M16—24no—
494M16—42no—
495M16—42no—
496M16—41no—
497M16—41no—
498M16C22⋊Q814yescomplemented
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—22no—
504C2×C8—41no—
505C2×C8—42no—
506C2×C8—42no—
507C2×C8—41no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C22⋊C4—44no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—44no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—41no—
558C42—42no—
559C42—42no—
560C42—41no—
561C42—41no—
562C42—42no—
563C42—42no—
564C42—42no—
565C42—42no—
566C42—42no—
567C42—42no—
568C42C4×D412yes—
569C42C4×D412yes—
570C42C4×D412yes—
571C42C4×D412yes—
572C42#12012yes—
573C42#12012yes—
574C42#12312yes—
575C42#12312yes—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—41no—
580C23—42no—
581C23—42no—
582C23—41no—
583C23C22×C22⋊C411yescharacteristic
584C23C2×#12111yescharacteristic
585C23C2×#12111yescharacteristic
586C23#54811yescharacteristic
587C23#54811yescharacteristic
588C23D4×Q811yescharacteristic
589C23#55011yescharacteristic
590Q8—28no—
591Q8—22no—
592Q8—22no—
593Q8—48no—
594Q8—44no—
595Q8—41no—
596Q8—41no—
597D4—22no—
598D4—22no—
599D4—42no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—41no—
604D4—41no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—82no—
653C2×C4—82no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4—82no—
657C2×C4—82no—
658C2×C4—81no—
659C2×C4—81no—
660C2×C4—82no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—82no—
664C2×C4#38612yes—
665C2×C4#38612yes—
666C2×C4#38612yes—
667C2×C4#38612yes—
668C2×C4#42114yes—
669C2×C4C2×C4×D411yescharacteristic
670C2×C4C2×C4×D411yescharacteristic
671C2×C4C2×C4×Q811yescharacteristic
672C2×C4C4×#4111yescharacteristic
673C2×C4#51911yescharacteristic
674C2×C4#52011yescharacteristic
675C2×C4C2×#12011yescharacteristic
676C2×C4C2×C22⋊Q811yescharacteristic
677C2×C4C2×#12311yescharacteristic
678C2×C4#52511yescharacteristic
679C2×C4#52911yescharacteristic
680C2×C4#53311yescharacteristic
681C2×C4#53611yescharacteristic
682C2×C4#53711yescharacteristic
683C2×C4#54111yescharacteristic
684C2×C4#54211yescharacteristic
685C2×C4#54611yescharacteristic
686C2×C4#54711yescharacteristic
687C2×C4#54811yescharacteristic
688C2×C4#54811yescharacteristic
689C2×C4#54811yescharacteristic
690C2×C4D4×Q811yescharacteristic
691C2×C4D4×Q811yescharacteristic
692C2×C4#55011yescharacteristic
693C2×C4#55211yescharacteristic
694C2×C4#55211yescharacteristic
695C2×C4#55411yescharacteristic
696C2×C4#55611yescharacteristic
697C8—24no—
698C8—42no—
699C8—44no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—44no—
709C22—42no—
710C22—82no—
711C22—81no—
712C22—81no—
713C22C2×#38611yescharacteristic
714C22Q8×C22⋊C411yescharacteristic
715C22#234111yescharacteristic
716C22#242611yescharacteristic
717C22#242911yescharacteristic
718C22#253611yescharacteristic
719C22#253811yescharacteristic
720C22C2×#42112yes—
721C22#290112yes—
722C4—22no—
723C4—22no—
724C4—24no—
725C4—22no—
726C4—22no—
727C4—42no—
728C4—42no—
729C4—42no—
730C4—42no—
731C4—84no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—81no—
737C4—81no—
738C4C4×C22⋊Q811yescharacteristic
739C4#232311yescharacteristic
740C4#236011yescharacteristic
741C4#236211yescharacteristic
742C4#240711yescharacteristic
743C4#240911yescharacteristic
744C4#242111yescharacteristic
745C4#242311yescharacteristic
746C2—21no—
747C2—21no—
748C2—21no—
749C2—21no—
750C2—82no—
751C2#1396911yescharacteristic
752C2#2090211yescharacteristic
753C2#2090311yescharacteristic
754C2#2252912yes—
755C2#2255112yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×Q8×M16C2semidirect
5. #13850C2semidirect
6. #11209C2semidirect
9. #10886C2semidirect
10. #10851C2semidirect
11. #10840C2semidirect
26. #2582C4semidirect
27. #2581C4semidirect
28. #2581C4semidirect
29. #2580C4semidirect
30. #2580C4semidirect
32. #2538C4semidirect
114. C2×#104Q8semidirect
498. M16C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×M162
#349522
C2×#18062
C2×#17592
#138502
#112092
#110532
#109332
#108862
#108512
#108402