Finite Group Databasebeta

← #366936

Subgroups — #366936

Order 512.

1374 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34947C211yesmaximal, characteristic
2C2×#1995C211yesmaximal, characteristic, complemented
3C2×#1986C211yesmaximal, characteristic, complemented
4C2×#1806C211yesmaximal, characteristic
5C4⋊C4×M16C211yesmaximal, characteristic, complemented
6C2×#1759C211yesmaximal, characteristic
7#13848C211yesmaximal, characteristic, complemented
8#11359C212yesmaximal, complemented
9#11075C212yesmaximal
10#10933C212yesmaximal
11#10851C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C22×#104C2211yescharacteristic
16C22×#104C2212yes—
17C22×#104C2211yescharacteristic
18C2×C4×M16C2212yes—
19C2×C4×M16C2211yescharacteristic
20#2568C411yescharacteristic, complemented
21#2567C411yescharacteristic, complemented
22#2566C412yescomplemented
23#2565C412yescomplemented
24#2564—21no—
25#2564—21no—
26#2563C411yescharacteristic, complemented
27#2542C411yescharacteristic, complemented
28#2417—22no—
29#2396—21no—
30#2396—21no—
31C4×C22⋊Q8—22no—
32#2297C2212yes—
33#2285C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×#376C2211yescharacteristic
38#1995—24no—
39#1986—24no—
40#1806—24no—
41#1771—22no—
42#1770—21no—
43C8×C4⋊C4—21no—
44#1759—24no—
45C2×#339C2212yes—
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2212yes—
49C2×#339C2211yescharacteristic
50C2×#339C2212yes—
51C2×#339C2211yescharacteristic
52#1728C2212yes—
53#1444—21no—
54#1442—21no—
55#1441—22no—
56C4×#95C2212yes—
57C22×#41C2311yescharacteristic
58C22×M16C2312yes—
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61#533—22no—
62#533—22no—
63#533—22no—
64#533—22no—
65C2×C4⋊Q8C2×C411yescharacteristic
66C2×C4⋊Q8C2×C411yescharacteristic
67C2×#125C2×C412yes—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—22no—
71C4×#41—22no—
72C4×#41—22no—
73C4×#41—22no—
74C4×#41—22no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8C2×C411yescharacteristic
78C2×C4×Q8C2×C411yescharacteristic
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—22no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4D412yes—
89C22×C42—21no—
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2312yes—
94C22×C42C2311yescharacteristic
95#423—22no—
96#423D414yes—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×#104—22no—
100C2×#104—22no—
101C2×#104—22no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—24no—
105C2×#104—24no—
106C2×#104—42no—
107C2×#104Q814yescomplemented
108C4×M16—22no—
109C4×M16—24no—
110C4×M16—42no—
111C4×M16D414yes—
112C2×#97—21no—
113C2×#97—22no—
114C2×C4×C8—21no—
115C2×C4×C8—22no—
116#400—21no—
117#400—21no—
118#399—21no—
119#399—21no—
120#398—22no—
121#398—22no—
122#396—22no—
123#396—22no—
124#395—21no—
125#395—21no—
126#393—21no—
127#393—21no—
128#391—22no—
129#391C2×C412yes—
130#391C2×C412yes—
131#390C2×C412yes—
132#390C2×C412yes—
133#389—22no—
134#387—22no—
135#387—22no—
136#386—22no—
137#386—22no—
138#385—22no—
139#385—22no—
140#384C2×C412yes—
141#384C2×C412yes—
142#382C2×C412yes—
143#382C2×C412yes—
144#380—22no—
145#380—22no—
146#380—22no—
147C4×C4⋊C4—22no—
148C4×C4⋊C4—22no—
149C4×C4⋊C4—22no—
150C4×C4⋊C4—22no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—42no—
153C4×C4⋊C4C2×C411yescharacteristic
154C4×C4⋊C4C2×C412yes—
155C4×C4⋊C4C2×C411yescharacteristic
156C4×C4⋊C4D411yescharacteristic
157C4×C4⋊C4D411yescharacteristic
158C4×C4⋊C4D412yes—
159C4×C22⋊C4—22no—
160C4×C22⋊C4—22no—
161#376—22no—
162#376—22no—
163C2×#95—22no—
164C2×#95—22no—
165C2×#95—21no—
166C2×#95—21no—
167C2×#95—22no—
168C2×#95—21no—
169C2×#95D411yescharacteristic
170C2×#95D411yescharacteristic
171#339—24no—
172#339—24no—
173#339—24no—
174#339—24no—
175#339—24no—
176#339—22no—
177#339—22no—
178#336—22no—
179#336—22no—
180C2×#41—21no—
181C2×#41—21no—
182C2×#41—21no—
183C2×#41—21no—
184C2×#41—21no—
185C2×#41—21no—
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41C2×Q812yes—
191C2×#41#4112yes—
192C22×Q8—21no—
193C22×Q8—21no—
194C22×Q8C22×C411yescharacteristic
195C22×D4—21no—
196C22×D4C22×C411yescharacteristic
197C23×C4—21no—
198C23×C4—22no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4C2×D412yes—
204C23×C4C2×D412yes—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2411yescharacteristic
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—22no—
213C2×M16—22no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—22no—
217C2×M16—22no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—22no—
221C2×M16C2×D412yes—
222C2×M16C2×D414yes—
223C2×M16C2×D412yes—
224C2×M16C2×Q812yes—
225C2×M16#4112yes—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233#125—24no—
234#125—24no—
235C22⋊Q8—48no—
236C22⋊Q8—42no—
237C22⋊Q8—42no—
238C22⋊Q8—42no—
239C22⋊Q8—42no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—42no—
245C4×Q8—42no—
246C4×Q8—42no—
247C4×Q8—42no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—42no—
253C4×D4—42no—
254C4×D4—42no—
255C4×D4—42no—
256#117—21no—
257#117—21no—
258#117—22no—
259#117—22no—
260#117—21no—
261#117—21no—
262#117—22no—
263#117—22no—
264#117—42no—
265#117—42no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4C22⋊C412yes—
283C2×C4⋊C4C22⋊C412yes—
284C2×C4⋊C4C22⋊C412yes—
285C2×C4⋊C4C22⋊C412yes—
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C412yes—
289C2×C4⋊C4C22×C412yes—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—22no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—42no—
323C2×C42—42no—
324C2×C42C22×C412yes—
325C2×C42C22×C412yes—
326C2×C42C22×C411yescharacteristic
327C2×C42C22×C411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×Q812yes—
335C2×C42#4112yes—
336C2×C42#4112yes—
337C2×C42#4112yes—
338C4⋊C8—24no—
339C4⋊C8—24no—
340#104—28no—
341#104—28no—
342#104—48no—
343#97—24no—
344#97—44no—
345C4×C8—24no—
346C4×C8—44no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—42no—
359#95—42no—
360#95—42no—
361#95C22⋊C412yes—
362#95C22⋊C412yes—
363#95C22⋊C411yescharacteristic
364#95C22⋊C411yescharacteristic
365#95C22⋊C411yescharacteristic
366#95C22⋊C411yescharacteristic
367C24—21no—
368C24C22×D411yescharacteristic
369#41—22no—
370#41—22no—
371#41—22no—
372#41—22no—
373#41—22no—
374#41—22no—
375#41—22no—
376#41—22no—
377#41—41no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382#41—41no—
383#41—41no—
384#41—41no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—82no—
394C2×Q8C4×Q812yes—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4C4×Q812yes—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—82no—
449C22×C4C2×C22⋊C412yes—
450C22×C4C2×C22⋊C412yes—
451C22×C4C2×C22⋊C412yes—
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C411yescharacteristic
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C412yes—
457C22×C4#12412yes—
458C22×C4#12412yes—
459C22×C4#12712yes—
460C22×C4#12712yes—
461C22×C4C23×C411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×Q811yescharacteristic
466C22×C4C2×#4111yescharacteristic
467C22×C4C2×#4111yescharacteristic
468C22×C4C2×#4111yescharacteristic
469C22×C4C2×#4111yescharacteristic
470C22×C4C2×#4111yescharacteristic
471C22×C4H(5,𝔽2)11yescharacteristic
472C22×C42−1+411yescharacteristic
473C22×C42−1+411yescharacteristic
474C22×C42−1+411yescharacteristic
475M16—22no—
476M16—22no—
477M16—22no—
478M16—22no—
479M16—22no—
480M16—24no—
481M16—24no—
482M16—42no—
483M16—42no—
484M16—41no—
485M16—41no—
486M16C22⋊Q814yescomplemented
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—42no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—42no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—416no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C22⋊C4—44no—
521C22⋊C4—41no—
522C22⋊C4—42no—
523C22⋊C4—41no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—44no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—41no—
543C42—41no—
544C42—42no—
545C42—42no—
546C42—41no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—42no—
551C42—42no—
552C42—42no—
553C42—42no—
554C42—42no—
555C42—42no—
556C42C4×D412yes—
557C42C4×D412yes—
558C42C4×D412yes—
559C42C4×D412yes—
560C42#12112yes—
561C42#12112yes—
562C42#12112yes—
563C42#12112yes—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—41no—
568C23—42no—
569C23—42no—
570C23—41no—
571C23C22×C22⋊C411yescharacteristic
572C23C2×#12411yescharacteristic
573C23C2×#12711yescharacteristic
574C23#54611yescharacteristic
575C23#54711yescharacteristic
576C23#54811yescharacteristic
577C23D4×Q811yescharacteristic
578Q8—22no—
579Q8—22no—
580Q8—41no—
581Q8—41no—
582Q8—84no—
583Q8—84no—
584D4—22no—
585D4—22no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—41no—
591D4—41no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—81no—
644C2×C4—82no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—82no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4#39012yes—
652C2×C4#39012yes—
653C2×C4#39012yes—
654C2×C4#39012yes—
655C2×C4#42114yes—
656C2×C4C2×C4×D411yescharacteristic
657C2×C4C2×C4×D411yescharacteristic
658C2×C4C2×C4×Q811yescharacteristic
659C2×C4C4×#4111yescharacteristic
660C2×C4#51911yescharacteristic
661C2×C4#52011yescharacteristic
662C2×C4C2×#12111yescharacteristic
663C2×C4C2×#12111yescharacteristic
664C2×C4C2×C22⋊Q811yescharacteristic
665C2×C4#52511yescharacteristic
666C2×C4#52911yescharacteristic
667C2×C4#53311yescharacteristic
668C2×C4#53611yescharacteristic
669C2×C4#53711yescharacteristic
670C2×C4#54111yescharacteristic
671C2×C4#54211yescharacteristic
672C2×C4#54611yescharacteristic
673C2×C4#54611yescharacteristic
674C2×C4#54711yescharacteristic
675C2×C4#54711yescharacteristic
676C2×C4#54811yescharacteristic
677C2×C4#54811yescharacteristic
678C2×C4D4×Q811yescharacteristic
679C2×C4D4×Q811yescharacteristic
680C2×C4#55211yescharacteristic
681C2×C4#55211yescharacteristic
682C2×C4#55411yescharacteristic
683C2×C4#55611yescharacteristic
684C8—24no—
685C8—42no—
686C8—44no—
687C22—22no—
688C22—22no—
689C22—21no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—22no—
694C22—22no—
695C22—44no—
696C22—42no—
697C22—81no—
698C22—81no—
699C22—82no—
700C22C2×#39011yescharacteristic
701C22#233911yescharacteristic
702C22Q8×C22⋊C411yescharacteristic
703C22#248911yescharacteristic
704C22#249211yescharacteristic
705C22#254011yescharacteristic
706C22#254211yescharacteristic
707C22C2×#42112yes—
708C22#290112yes—
709C4—22no—
710C4—22no—
711C4—22no—
712C4—22no—
713C4—24no—
714C4—42no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—84no—
719C4—82no—
720C4—82no—
721C4—81no—
722C4—82no—
723C4—81no—
724C4—82no—
725C4C4×C22⋊Q811yescharacteristic
726C4#232311yescharacteristic
727C4#236011yescharacteristic
728C4#236211yescharacteristic
729C4#241511yescharacteristic
730C4#241711yescharacteristic
731C4#242711yescharacteristic
732C4#242911yescharacteristic
733C2—21no—
734C2—21no—
735C2—21no—
736C2—21no—
737C2—82no—
738C2#1399711yescharacteristic
739C2#2125211yescharacteristic
740C2#2125311yescharacteristic
741C2#2250612yes—
742C2#2252912yes—
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 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1995C2semidirect
3. C2×#1986C2semidirect
5. C4⋊C4×M16C2semidirect
7. #13848C2semidirect
8. #11359C2semidirect
11. #10851C2semidirect
20. #2568C4semidirect
21. #2567C4semidirect
22. #2566C4semidirect
23. #2565C4semidirect
26. #2563C4semidirect
27. #2542C4semidirect
107. C2×#104Q8semidirect
486. M16C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#349472
C2×#19952
C2×#19862
C2×#18062
C4⋊C4×M162
C2×#17592
#138482
#113592
#110752
#109332
#108512