Finite Group Databasebeta

← #373784

Subgroups — #373784

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2C2×#1948C211yesmaximal, characteristic, complemented
3C2×#1802C211yesmaximal, characteristic
4C2×C4×#102C211yesmaximal, characteristic
5#14151C211yesmaximal, characteristic, complemented
6#11121C212yesmaximal
7#11066C212yesmaximal
8#10908C212yesmaximal, complemented
9#10885C212yesmaximal, complemented
10#10841C211yesmaximal, characteristic, complemented
11#10834C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117—21no—
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×C4×C8C2211yescharacteristic
20#2354—24no—
21#2354—24no—
22C2×#390C2212yes—
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#388—22no—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#380—21no—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1948—24no—
33#1802C418yescomplemented
34C4×#102—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1441—22no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4C2×C414yes—
55C2×C4×D4D414yes—
56C2×C4×D4Q814yes—
57C2×#117—28no—
58C2×#117—44no—
59C2×#117—44no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C42—22no—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79C2×#106—24no—
80C2×#106—24no—
81C2×#106—42no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8C2×C414yes—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—22no—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—42no—
92C2×#102—42no—
93C2×#102C2×C418yes—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96C2×C4×C8—22no—
97#390—24no—
98#390—24no—
99#389—24no—
100#389—42no—
101#389—44no—
102#388—216no—
103#384—22no—
104#384—22no—
105#384C2×C414yes—
106#384D414yes—
107#380—28no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95D412yes—
120#340—24no—
121#340—24no—
122#340—22no—
123#340—22no—
124#340—42no—
125#340C2×C418yes—
126#336—24no—
127#336—42no—
128C25—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4C22×C412yes—
141C22×D4C2×D412yes—
142C22×D4C2×Q812yes—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8C22×C414yes—
179C22×C8#4112yes—
180C22×C8#4112yes—
181C4×D4—28no—
182C4×D4—28no—
183C4×D4—28no—
184C4×D4C4⋊C4116yescomplemented
185#117—432no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—81no—
225C2×C4⋊C4—81no—
226C2×C4⋊C4—81no—
227C2×C4⋊C4—81no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4SD1614yes—
230C2×C4⋊C4C22×C414yes—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×Q812yes—
235C2×C4⋊C4#4112yes—
236C2×C4⋊C4#4112yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—82no—
245C2×C22⋊C4—82no—
246C2×C42—28no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—44no—
254C2×C42—44no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42C22×C412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×Q812yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264#106—48no—
265C4⋊C8—28no—
266#102—28no—
267#102—28no—
268#102—48no—
269C4×C8—44no—
270#95—28no—
271#95—24no—
272#95—44no—
273#95—44no—
274#95—42no—
275#95—82no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24—41no—
291C24C22×D411yescharacteristic
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4C2×C4⋊C414yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—216no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4C4×D412yes—
380C22×C4C4×D412yes—
381C22×C4#12012yes—
382C22×C4#12012yes—
383C22×C4C22⋊Q812yes—
384C22×C4C22⋊Q812yes—
385C22×C4C2×SD1614yes—
386C22×C4C2×SD1612yes—
387C22×C4#13512yes—
388C22×C4AGL(1,ℤ/8)12yes—
389C22×C4AGL(1,ℤ/8)12yes—
390C22×C4C23×C411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×Q811yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4H(5,𝔽2)11yescharacteristic
401C22×C4H(5,𝔽2)11yescharacteristic
402C22×C42−1+411yescharacteristic
403C22×C42−1+411yescharacteristic
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—28no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—42no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—42no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—48no—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—84no—
449C4⋊C4—84no—
450C4⋊C4—88no—
451C4⋊C4C2×C4⋊C414yes—
452C22⋊C4—216no—
453C22⋊C4—416no—
454C22⋊C4—88no—
455C42—416no—
456C42—42no—
457C42—42no—
458C42—42no—
459C42—42no—
460C42—84no—
461C42C2×C4⋊C414yes—
462C42C4×D414yes—
463C42C22⋊Q814yes—
464C42#12314yes—
465C23—24no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—81no—
490C23—81no—
491C23—81no—
492C23—81no—
493C23—81no—
494C23—81no—
495C23—81no—
496C23—81no—
497C23—81no—
498C23—81no—
499C23—81no—
500C23C2×C4×D411yescharacteristic
501C23C2×#12011yescharacteristic
502C23C2×C22⋊Q811yescharacteristic
503C23#54611yescharacteristic
504C23#54711yescharacteristic
505C23#54811yescharacteristic
506C23#55011yescharacteristic
507C23C22×SD1611yescharacteristic
508C23C2×#13511yescharacteristic
509C23C2×AGL(1,ℤ/8)11yescharacteristic
510C23C2×AGL(1,ℤ/8)11yescharacteristic
511C23#57611yescharacteristic
512C23#57711yescharacteristic
513C23#57711yescharacteristic
514C23#57811yescharacteristic
515D4—24no—
516D4—24no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—42no—
522D4—42no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—432no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—82no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—81no—
570C2×C4—82no—
571C2×C4—81no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—82no—
579C2×C4#38514yes—
580C2×C4#38514yes—
581C2×C4C4×SD1614yes—
582C2×C4#44214yes—
583C2×C4#44914yes—
584C2×C4#45014yes—
585C2×C4#47614yes—
586C2×C4#47814yes—
587C2×C4C22×C4⋊C411yescharacteristic
588C2×C4C2×C4×D411yescharacteristic
589C2×C4C4×#4111yescharacteristic
590C2×C4C4×#4111yescharacteristic
591C2×C4#51811yescharacteristic
592C2×C4#51911yescharacteristic
593C2×C4C2×C22⋊Q811yescharacteristic
594C2×C4C2×#12311yescharacteristic
595C2×C4#52511yescharacteristic
596C2×C4#52511yescharacteristic
597C2×C4#53311yescharacteristic
598C2×C4#53311yescharacteristic
599C2×C4#53711yescharacteristic
600C2×C4#53711yescharacteristic
601C2×C4#54411yescharacteristic
602C2×C4#54411yescharacteristic
603C2×C4#54611yescharacteristic
604C2×C4#54711yescharacteristic
605C2×C4#54811yescharacteristic
606C2×C4#55011yescharacteristic
607C2×C4#55211yescharacteristic
608C2×C4#55211yescharacteristic
609C2×C4#55311yescharacteristic
610C2×C4#55311yescharacteristic
611C2×C4#55511yescharacteristic
612C2×C4#55511yescharacteristic
613C2×C4#55611yescharacteristic
614C2×C4#55611yescharacteristic
615C8—44no—
616C8—42no—
617C8—42no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22C2×#38511yescharacteristic
634C22#236211yescharacteristic
635C22#236311yescharacteristic
636C22#240711yescharacteristic
637C22#240711yescharacteristic
638C22#243911yescharacteristic
639C22#244111yescharacteristic
640C22C2×C4×SD1611yescharacteristic
641C22C2×#44211yescharacteristic
642C22#294611yescharacteristic
643C22#294711yescharacteristic
644C22C2×#45011yescharacteristic
645C22C2×#44911yescharacteristic
646C22#301211yescharacteristic
647C22#301311yescharacteristic
648C22C2×#47611yescharacteristic
649C22C2×#47811yescharacteristic
650C22#308311yescharacteristic
651C22#308411yescharacteristic
652C22#329811yescharacteristic
653C22#330311yescharacteristic
654C22#332511yescharacteristic
655C22#333211yescharacteristic
656C22#333511yescharacteristic
657C22#334111yescharacteristic
658C22#335311yescharacteristic
659C22#335411yescharacteristic
660C22#336011yescharacteristic
661C22#336211yescharacteristic
662C22#336711yescharacteristic
663C22#336811yescharacteristic
664C22#339211yescharacteristic
665C22#339311yescharacteristic
666C22#339711yescharacteristic
667C22#339911yescharacteristic
668C4—24no—
669C4—44no—
670C4—44no—
671C4—82no—
672C4—82no—
673C4—84no—
674C4—82no—
675C4—84no—
676C4#186418yes—
677C4#229611yescharacteristic
678C4#231611yescharacteristic
679C4#236211yescharacteristic
680C4#236311yescharacteristic
681C4#243911yescharacteristic
682C4#244111yescharacteristic
683C4#248511yescharacteristic
684C4#248511yescharacteristic
685C2—84no—
686C2#1412911yescharacteristic
687C2C2×#186411yescharacteristic
688C2#2087811yescharacteristic
689C2#2306111yescharacteristic
690C2#2306611yescharacteristic
691C2#2309411yescharacteristic
692C2#2309911yescharacteristic
693C2#2411711yescharacteristic
694C2#2412011yescharacteristic
695C2#2418511yescharacteristic
696C2#2418811yescharacteristic
697C2#2602111yescharacteristic
698C2#2602411yescharacteristic
699C2#2608711yescharacteristic
700C2#2609011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ 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×#1948C2semidirect
5. #14151C2semidirect
8. #10908C2semidirect
9. #10885C2semidirect
10. #10841C2semidirect
11. #10834C2semidirect
33. #1802C4semidirect
184. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#19482
C2×#18022
C2×C4×#1022
#141512
#111212
#110662
#109082
#108852
#108412
#108342