Finite Group Databasebeta

← #115486

Subgroups — #115486

Order 512.

1482 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
1C22×#460C211yesmaximal, characteristic
2C2×#1982C211yesmaximal, characteristic
3C2×#1957C211yesmaximal, characteristic
4C22×Q8⋊C8C211yesmaximal, characteristic, complemented
5#13926C211yesmaximal, characteristic, complemented
6C2×#1296C211yesmaximal, characteristic, complemented
7C2×#1279C211yesmaximal, characteristic, complemented
8C23×SD16—21no—
9C22×#127C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#460—22no—
12C2×#460—22no—
13C2×#460—22no—
14C2×#460—22no—
15C2×#460—22no—
16C2×#460C414yescomplemented
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#102—21no—
20C22×#102—22no—
21C22×#102—22no—
22C22×C4×C8C2211yescharacteristic
23C2×#391—21no—
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1982—22no—
27#1982—22no—
28#1957—22no—
29#1957—22no—
30C2×Q8⋊C8—22no—
31C2×Q8⋊C8—22no—
32C2×Q8⋊C8—22no—
33#1441—22no—
34C4⋊#95—22no—
35C4⋊#95—21no—
36#1296—22no—
37#1279—22no—
38C23×Q8D411yescharacteristic
39C23×D4—21no—
40C23×D4—21no—
41C23×D4—21no—
42C22×SD16—24no—
43C22×SD16—24no—
44C22×SD16—24no—
45C22×SD16—42no—
46C22×SD16—42no—
47C22×SD16—42no—
48C22×SD16—42no—
49C23×C8—21no—
50C23×C8—21no—
51C23×C8—21no—
52C2×#127—21no—
53C2×#127—21no—
54C2×#127—21no—
55C2×#127—21no—
56C2×#127—21no—
57C2×#127—21no—
58C2×#127—21no—
59C2×#127—21no—
60C2×#127—21no—
61C2×#127C2×C412yes—
62C2×#127D412yes—
63C2×#127D412yes—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8C2×C412yes—
75C2×C4×Q8D412yes—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D412yes—
81C22×C4⋊C4D412yes—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#460—28no—
86#460—28no—
87#460—28no—
88#460—28no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2×C412yes—
98C2×C4⋊C8D412yes—
99C2×C4⋊C8D412yes—
100C2×C4⋊C8D412yes—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—28no—
105C2×#102—28no—
106C2×#102—44no—
107C2×#102—48no—
108C2×#102—44no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8D412yes—
115#391—42no—
116#391—42no—
117#384—21no—
118#384—21no—
119#384—21no—
120#384—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C2×#95—24no—
126C2×#95—21no—
127C2×#95—24no—
128Q8⋊C8—28no—
129C25—41no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8C22⋊C412yes—
142C22×Q8D812yes—
143C22×Q8SD1612yes—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—44no—
154C22×D4—44no—
155C22×D4—41no—
156C22×D4—41no—
157C22×D4—41no—
158C22×D4—41no—
159C22×D4—41no—
160C22×D4—41no—
161C22×D4—41no—
162C22×D4—41no—
163C22×D4—41no—
164C22×D4—41no—
165C22×D4—41no—
166C22×D4—42no—
167C22×D4—41no—
168C22×D4—41no—
169C22×D4—42no—
170C22×D4—41no—
171C22×D4—41no—
172C22×D4—41no—
173C22×D4—42no—
174C22×D4—41no—
175C22×D4—42no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C2×SD16—216no—
184C2×SD16—48no—
185C2×SD16—48no—
186C2×SD16—48no—
187C2×SD16—48no—
188C2×SD16—48no—
189C2×SD16—48no—
190C22×C8—24no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—42no—
197C22×C8—41no—
198C22×C8—41no—
199C22×C8—41no—
200C22×C8—41no—
201C22×C8—41no—
202C22×C8—41no—
203C22×C8—41no—
204C22×C8—41no—
205C22×C8—42no—
206C22×C8—42no—
207#127—22no—
208#127—22no—
209#127—22no—
210#127—22no—
211#127—22no—
212#127—22no—
213#127—22no—
214#127—22no—
215#127—22no—
216#127—41no—
217#127—41no—
218#127—41no—
219#127—41no—
220#127—41no—
221#127—41no—
222#127C22⋊C414yes—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—41no—
232C4×Q8—41no—
233C4×Q8—41no—
234C4×Q8—41no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×Q8—41no—
238C4×Q8—41no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4C22⋊C412yes—
269C2×C4⋊C4SD1614yescomplemented
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—41no—
273C2×C42—41no—
274C2×C42—41no—
275C2×C42—41no—
276C2×C42—41no—
277C2×C42—41no—
278C2×C42—41no—
279C2×C42—41no—
280C2×C42—41no—
281C2×C42—41no—
282C2×C42C22×C411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42#4111yescharacteristic
291C2×C42#4111yescharacteristic
292C2×C42#4111yescharacteristic
293C2×C42#4111yescharacteristic
294C4⋊C8—22no—
295C4⋊C8—22no—
296C4⋊C8—22no—
297C4⋊C8—22no—
298C4⋊C8—22no—
299C4⋊C8—41no—
300C4⋊C8—41no—
301C4⋊C8C22⋊C414yes—
302#102—216no—
303#102—416no—
304#102—416no—
305C4×C8—22no—
306C4×C8—22no—
307C4×C8—41no—
308C4×C8—41no—
309#95—48no—
310#95—41no—
311#95—41no—
312#95—84no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—81no—
317C24—81no—
318C24—82no—
319C24—81no—
320C24—81no—
321C24—82no—
322C24—82no—
323C24—82no—
324C24#12011yescharacteristic
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8#10214yescomplemented
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—44no—
357C2×D4—48no—
358C2×D4—44no—
359C2×D4—48no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—44no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—44no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—84no—
388C2×D4—82no—
389C2×D4—82no—
390C2×D4—84no—
391C2×D4—82no—
392C2×D4—82no—
393C2×D4—82no—
394C2×D4—81no—
395C2×D4—81no—
396C2×D4—81no—
397C2×D4—81no—
398C2×D4—82no—
399C2×D4—81no—
400C2×D4—81no—
401C2×D4—81no—
402C2×D4—81no—
403C2×D4—81no—
404C2×D4—81no—
405C2×D4—81no—
406C2×D4—81no—
407C2×D4—81no—
408C2×D4—81no—
409C2×D4—81no—
410C2×D4—81no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C4×D411yescharacteristic
439C22×C4C4×D411yescharacteristic
440C22×C4#12011yescharacteristic
441C22×C4#12011yescharacteristic
442C22×C4#12011yescharacteristic
443C22×C4#12011yescharacteristic
444C22×C4#12111yescharacteristic
445C22×C4#12111yescharacteristic
446C22×C4#12111yescharacteristic
447C22×C4#12111yescharacteristic
448C22×C4#12111yescharacteristic
449C22×C4#12111yescharacteristic
450C22×C4#12111yescharacteristic
451C22×C4#12111yescharacteristic
452C22×C4#12311yescharacteristic
453C22×C4#12711yescharacteristic
454C22×C4#12711yescharacteristic
455C22×C4C2×D811yescharacteristic
456C22×C4C2×SD1611yescharacteristic
457C22×C4C2×SD1612yes—
458C22×C4#13511yescharacteristic
459C22×C4#13511yescharacteristic
460C22×C4#13512yes—
461C22×C4AGL(1,ℤ/8)11yescharacteristic
462C22×C4AGL(1,ℤ/8)11yescharacteristic
463C22×C4AGL(1,ℤ/8)11yescharacteristic
464C22×C4AGL(1,ℤ/8)12yes—
465C22×C4#13711yescharacteristic
466C22×C4#13712yes—
467SD16—432no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—44no—
471C2×C8—42no—
472C2×C8—44no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—44no—
482C2×C8—82no—
483C2×C8—81no—
484C2×C8—81no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—82no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4—82no—
502C4⋊C4—81no—
503C4⋊C4—81no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C42—42no—
507C42—42no—
508C42—42no—
509C42—42no—
510C42—81no—
511C42—81no—
512C42—81no—
513C42—81no—
514C42C2×C22⋊C411yescharacteristic
515C42C2×C22⋊C411yescharacteristic
516C42C4×D411yescharacteristic
517C42C4×D411yescharacteristic
518C42#12011yescharacteristic
519C42#12011yescharacteristic
520C42#12011yescharacteristic
521C42#12111yescharacteristic
522C42#12111yescharacteristic
523C42#12111yescharacteristic
524C42#12111yescharacteristic
525C42#12311yescharacteristic
526C42#12311yescharacteristic
527C42#12311yescharacteristic
528C42#12411yescharacteristic
529C42#12411yescharacteristic
530C23—44no—
531C23—82no—
532C23—82no—
533C23—84no—
534C23—84no—
535C23—82no—
536C23—82no—
537C23—84no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—84no—
543C23—82no—
544C23—82no—
545C23—162no—
546C23—162no—
547C23—162no—
548C23—162no—
549C23—162no—
550C23—162no—
551C23—162no—
552C23—162no—
553C23#38611yescharacteristic
554C23#39211yescharacteristic
555C23#39211yescharacteristic
556C23#44711yescharacteristic
557C23#44811yescharacteristic
558C23#44812yes—
559C23#44911yescharacteristic
560C23#44911yescharacteristic
561C23#44912yes—
562C23#45312yes—
563C23#45412yes—
564Q8—24no—
565Q8—24no—
566Q8—42no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—42no—
572D4—48no—
573D4—48no—
574D4—44no—
575D4—44no—
576D4—84no—
577D4—84no—
578D4—84no—
579D4—84no—
580D4—82no—
581D4—82no—
582D4—82no—
583D4—82no—
584D4—82no—
585D4—82no—
586D4—82no—
587D4—82no—
588D4—82no—
589D4—82no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4#37911yescharacteristic
623C2×C4#38611yescharacteristic
624C2×C4#38611yescharacteristic
625C2×C4#38611yescharacteristic
626C2×C4#39011yescharacteristic
627C2×C4#39011yescharacteristic
628C2×C4#39211yescharacteristic
629C2×C4#39211yescharacteristic
630C2×C4#39411yescharacteristic
631C2×C4#39411yescharacteristic
632C2×C4#39411yescharacteristic
633C2×C4#39411yescharacteristic
634C2×C4C2×#10211yescharacteristic
635C2×C4#41611yescharacteristic
636C2×C4#41711yescharacteristic
637C2×C4#41811yescharacteristic
638C2×C4C4×SD1612yes—
639C2×C4#44012yes—
640C2×C4#44711yescharacteristic
641C2×C4#44711yescharacteristic
642C2×C4#44911yescharacteristic
643C2×C4#44911yescharacteristic
644C2×C4#45012yes—
645C2×C4#45011yescharacteristic
646C2×C4#45011yescharacteristic
647C2×C4#45212yes—
648C2×C4#45211yescharacteristic
649C2×C4#45211yescharacteristic
650C2×C4#46012yes—
651C2×C4#46112yes—
652C2×C4#46312yes—
653C2×C4#46412yes—
654C2×C4#46512yes—
655C2×C4#46511yescharacteristic
656C2×C4#46511yescharacteristic
657C2×C4#46611yescharacteristic
658C2×C4#46611yescharacteristic
659C2×C4#46812yes—
660C2×C4#46811yescharacteristic
661C2×C4#46811yescharacteristic
662C2×C4#46911yescharacteristic
663C2×C4#46911yescharacteristic
664C2×C4#48011yescharacteristic
665C2×C4#48111yescharacteristic
666C2×C4#48211yescharacteristic
667C2×C4#48211yescharacteristic
668C2×C4#49211yescharacteristic
669C2×C4#49511yescharacteristic
670C2×C4#49611yescharacteristic
671C2×C4#49711yescharacteristic
672C8—44no—
673C8—44no—
674C8—82no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—164no—
679C22—164no—
680C22—162no—
681C22—162no—
682C22—162no—
683C22—162no—
684C22—164no—
685C22—164no—
686C22#143811yescharacteristic
687C22#162012yes—
688C22#162112yes—
689C22#162212yes—
690C22#162312yes—
691C22#162612yes—
692C22#162612yes—
693C22#162712yes—
694C22#162812yes—
695C22#187411yescharacteristic
696C22#187511yescharacteristic
697C22#188012yes—
698C22#188312yes—
699C22#199911yescharacteristic
700C22#199911yescharacteristic
701C22#200012yes—
702C22#200011yescharacteristic
703C22#200011yescharacteristic
704C22#200211yescharacteristic
705C22#200311yescharacteristic
706C22#200612yes—
707C22#201211yescharacteristic
708C22#201311yescharacteristic
709C4—21no—
710C4—21no—
711C4—21no—
712C4—21no—
713C4—44no—
714C4—82no—
715C4—84no—
716C4—162no—
717C4#178611yescharacteristic
718C4#178811yescharacteristic
719C4#187712yes—
720C4#189311yescharacteristic
721C4#189511yescharacteristic
722C4#196811yescharacteristic
723C4#197311yescharacteristic
724C4#201411yescharacteristic
725C4#201511yescharacteristic
726C4#203311yescharacteristic
727C4#203311yescharacteristic
728C4#203411yescharacteristic
729C4#203411yescharacteristic
730C4#205512yes—
731C2—164no—
732C2#897812yes—
733C2#897912yes—
734C2#1004711yescharacteristic
735C2#1004911yescharacteristic
736C2#1005111yescharacteristic
737C2#1005311yescharacteristic
738C2#1022912yes—
739C2#1023012yes—
740C2#1096911yescharacteristic
741C2#1101611yescharacteristic
742C2#1102311yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC22×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C22×Q8⋊C8C2semidirect
5. #13926C2semidirect
6. C2×#1296C2semidirect
7. C2×#1279C2semidirect
16. C2×#460C4semidirect
269. C2×C4⋊C4SD16semidirect
352. C2×Q8#102semidirect

Maximal subgroups

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

subgroupindex
C22×#4602
C2×#19822
C2×#19572
C22×Q8⋊C82
#139262
C2×#12962
C2×#12792