Finite Group Databasebeta

← #365592

Subgroups — #365592

Order 512.

1816 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2352C211yesmaximal, characteristic
2C2×#1955C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic, complemented
4C2×#1854C212yesmaximal, complemented
5C2×#1852C212yesmaximal, complemented
6C2×#1847C211yesmaximal, characteristic, complemented
7#13906C211yesmaximal, characteristic
8#11096C212yesmaximal, complemented
9#11095C212yesmaximal, complemented
10#10850C211yesmaximal, characteristic, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#107C2212yes—
17C22×#106C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21C22×#97C2211yescharacteristic
22#2352—24no—
23#2352—24no—
24C2×#390—21no—
25C2×#388—21no—
26C2×#387—21no—
27C2×#386—21no—
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4—21no—
37#1955C418yescomplemented
38#1947—24no—
39#1854—28no—
40#1852—28no—
41#1847—24no—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47#1439—21no—
48#1439—21no—
49C4⋊#95—21no—
50C23×D4D411yescharacteristic
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4C2×C414yes—
61C2×C4×D4Q814yes—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—24no—
65C2×#117—42no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4D411yescharacteristic
86C22×C22⋊C4D411yescharacteristic
87C22×C42—21no—
88C22×C42—21no—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91C2×#107—24no—
92C2×#107—24no—
93C2×#107—24no—
94C2×#107—24no—
95C2×#107—24no—
96C2×#107—42no—
97C2×#106—24no—
98C2×#106—24no—
99C2×#106—42no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8C2×C414yescomplemented
103C2×C22⋊C8—24no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—24no—
107C2×C22⋊C8C2×C418yes—
108C2×C22⋊C8C2×C418yescomplemented
109C2×#97—22no—
110C2×#97—22no—
111C2×#97C2×C414yescomplemented
112#390—44no—
113#388—28no—
114#387—28no—
115#386—28no—
116#384—22no—
117#384—22no—
118#382—22no—
119#382D414yes—
120#380—22no—
121#380—22no—
122#380—22no—
123#380—22no—
124#380—24no—
125#380—24no—
126#380—42no—
127#380—42no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4D414yes—
132C4×C22⋊C4—28no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—21no—
141#340—24no—
142#340—24no—
143#340—24no—
144#340—24no—
145#340D418yescomplemented
146C25C2×D411yescharacteristic
147C25C2×D411yescharacteristic
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—41no—
153C22×D4—41no—
154C22×D4C4⋊C414yes—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—44no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—82no—
182C23×C4C22×C412yes—
183C23×C4C22×C412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×Q812yes—
189C23×C4C2×Q812yes—
190C23×C4C2411yescharacteristic
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8C22×C414yes—
202C22×C8C22×C414yes—
203C22×C8C2×D414yes—
204C22×C8#4114yes—
205C4×D4—28no—
206C4×D4—28no—
207C4×D4—28no—
208C4×D4—28no—
209#117—48no—
210#117—44no—
211#117—44no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
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—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—48no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4C4⋊C414yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D412yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—82no—
268C2×C22⋊C4—82no—
269C2×C22⋊C4—82no—
270C2×C22⋊C4C4⋊C414yes—
271C2×C22⋊C4C4⋊C414yes—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—44no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—82no—
279C2×C42C22×C412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×Q812yes—
283C2×C42#4112yes—
284C2×C42#4112yes—
285#107—48no—
286#107—48no—
287#106—48no—
288C4⋊C8—28no—
289C22⋊C8—216no—
290C22⋊C8—216no—
291#97—28no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—44no—
298#95—44no—
299#95—42no—
300C24—21no—
301C24—21no—
302C24—22no—
303C24—22no—
304C24—21no—
305C24—21no—
306C24—22no—
307C24—22no—
308C24—21no—
309C24—21no—
310C24—22no—
311C24—22no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—22no—
317C24—22no—
318C24C2×C4⋊C412yes—
319C24C2×C4⋊C412yes—
320C24AGL(1,ℤ/8)12yes—
321C24#13712yes—
322C24C22×D411yescharacteristic
323C2×D4—28no—
324C2×D4—28no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—48no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
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—48no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—84no—
417C22×C4—88no—
418C22×C4—88no—
419C22×C4C2×C4⋊C412yes—
420C22×C4C2×C4⋊C412yes—
421C22×C4C2×C4⋊C412yes—
422C22×C4C2×C4⋊C412yes—
423C22×C4#12012yes—
424C22×C4#12012yes—
425C22×C4#12312yes—
426C22×C4#12312yes—
427C22×C4C23×C411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×Q811yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4112yes—
435C22×C4H(5,𝔽2)12yes—
436C22×C42−1+412yes—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8C4×D418yes—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—28no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—48no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C4⋊C4#12714yes—
475C22⋊C4—28no—
476C22⋊C4—28no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—84no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C42—48no—
483C42—84no—
484C42—84no—
485C42C4×D414yes—
486C42C4×D414yes—
487C42C22⋊Q814yes—
488C42C22⋊Q814yes—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—22no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23#42814yes—
535C23C22×C4⋊C411yescharacteristic
536C23C2×#12011yescharacteristic
537C23C2×#12311yescharacteristic
538C23D4212yes—
539C23#54712yes—
540C23C2×AGL(1,ℤ/8)11yescharacteristic
541C23C2×#13711yescharacteristic
542C23#57511yescharacteristic
543C23#57511yescharacteristic
544C23#57612yes—
545C23#57812yes—
546D4—416no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—816no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4#38514yes—
615C2×C4#38514yes—
616C2×C4C2×C4×D411yescharacteristic
617C2×C4C2×C4×D412yes—
618C2×C4C2×C4×D411yescharacteristic
619C2×C4#52012yes—
620C2×C4C2×C22⋊Q811yescharacteristic
621C2×C4C2×C22⋊Q811yescharacteristic
622C2×C4C2×#12711yescharacteristic
623C2×C4#53211yescharacteristic
624C2×C4#53212yes—
625C2×C4#53511yescharacteristic
626C2×C4#53611yescharacteristic
627C2×C4#54212yes—
628C2×C4D4212yes—
629C2×C4#54712yes—
630C2×C4#54812yes—
631C2×C4D4×Q812yes—
632C2×C4#55312yes—
633C2×C4#55412yes—
634C8—28no—
635C8—44no—
636C22—24no—
637C22—28no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—44no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—44no—
647C22—42no—
648C22—42no—
649C22—44no—
650C22—44no—
651C22—42no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—82no—
657C22—82no—
658C22C2×#38511yescharacteristic
659C22D4×C4⋊C412yes—
660C22#242212yes—
661C22#243211yescharacteristic
662C22#243311yescharacteristic
663C22C2×#42811yescharacteristic
664C22#291111yescharacteristic
665C22#291312yes—
666C22#300611yescharacteristic
667C22#300711yescharacteristic
668C22#300811yescharacteristic
669C22#301111yescharacteristic
670C22#309411yescharacteristic
671C22#309511yescharacteristic
672C22#309812yes—
673C22#326412yes—
674C22#326812yes—
675C22#326912yes—
676C22#327012yes—
677C22#334312yes—
678C22#334612yes—
679C22#334712yes—
680C22#334912yes—
681C4—24no—
682C4—24no—
683C4—42no—
684C4—42no—
685C4—44no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—84no—
690C4—162no—
691C4C4×#12711yescharacteristic
692C4#232611yescharacteristic
693C4#236112yes—
694C4#245212yes—
695C4#254211yescharacteristic
696C4#256611yescharacteristic
697C2—44no—
698C2—82no—
699C2#1397711yescharacteristic
700C2#2084611yescharacteristic
701C2#2084711yescharacteristic
702C2#2266012yes—
703C2#2266112yes—
704C2#2502712yes—
705C2#2502812yes—
706C2#2571311yescharacteristic
707C2#2571611yescharacteristic
708C2#2576711yescharacteristic
709C2#2577011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×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×#1947C2semidirect
4. C2×#1854C2semidirect
5. C2×#1852C2semidirect
6. C2×#1847C2semidirect
8. #11096C2semidirect
9. #11095C2semidirect
10. #10850C2semidirect
11. #10703C2semidirect
37. #1955C4semidirect
102. C2×C4⋊C8C2×C4semidirect
108. C2×C22⋊C8C2×C4semidirect
111. C2×#97C2×C4semidirect
145. #340D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23522
C2×#19552
C2×#19472
C2×#18542
C2×#18522
C2×#18472
#139062
#110962
#110952
#108502
#107032