Finite Group Databasebeta

← #373710

Subgroups — #373710

Order 512.

1734 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2C2×#1854C212yesmaximal
3C2×#1817C211yesmaximal, characteristic
4#13756C211yesmaximal, characteristic
5#11324C211yesmaximal, characteristic, complemented
6#11323C211yesmaximal, characteristic, complemented
7#11267C211yesmaximal, characteristic, complemented
8#11267C211yesmaximal, characteristic, complemented
9#10908C212yesmaximal, complemented
10#10907C212yesmaximal, complemented
11#10731C212yesmaximal
12C22×#117—21no—
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×#107C2212yes—
17C22×#106C2212yes—
18C22×C22⋊C8C2212yes—
19#2285—28no—
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382—21no—
23C2×#380—21no—
24C2×#380C2212yes—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2212yes—
27#1854—28no—
28#1817C418yescomplemented
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C22⋊#95—21no—
43C22⋊#95—21no—
44C4×#95—21no—
45C4×#95—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#117—28no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—24no—
57C2×#117—24no—
58C2×#117—42no—
59C2×#117—42no—
60C2×#117—44no—
61C2×#117—44no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—22no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C42—24no—
76C22×C42—24no—
77C22×C42—22no—
78C22×C42—21no—
79C22×C42—21no—
80C22×C42C2×C414yes—
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42Q814yes—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#107—24no—
87C2×#107—24no—
88C2×#107—42no—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—42no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8C2×C418yes—
95#384—24no—
96#384—24no—
97#382—28no—
98#380—28no—
99#380—24no—
100#380D418yes—
101C4×C4⋊C4—28no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—24no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—42no—
119#340—42no—
120#336C2×C414yes—
121#336C2×C414yes—
122#336C2×C418yes—
123C25C2×D411yescharacteristic
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—41no—
147C23×C4—41no—
148C23×C4C4⋊C414yes—
149C23×C4C4⋊C414yes—
150C23×C4C22×C412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D414yes—
157C23×C4C2×Q812yes—
158C23×C4#4114yes—
159C23×C4C2411yescharacteristic
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8C22×C414yes—
173C22×C8C22×C414yes—
174#117—48no—
175#117—48no—
176#117—48no—
177#117—48no—
178#117—432no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4#4112yes—
216C2×C4⋊C4#4112yes—
217C2×C4⋊C4#4112yes—
218C2×C4⋊C4#4112yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C42—216no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—216no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—28no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—44no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—44no—
247C2×C42C4⋊C414yes—
248C2×C42C4⋊C414yes—
249C2×C42C22×C412yes—
250C2×C42C22×C412yes—
251C2×C42C2×Q812yes—
252C2×C42C2×Q812yes—
253#107—48no—
254#106—48no—
255C22⋊C8—216no—
256#95—28no—
257#95—28no—
258#95—42no—
259#95—42no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95—42no—
265#95—42no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24C2×C4⋊C412yes—
279C24#12012yes—
280C24#12312yes—
281C24C22×D411yescharacteristic
282C22×C4—28no—
283C22×C4—28no—
284C22×C4—28no—
285C22×C4—28no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—28no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4C2×C4⋊C412yes—
375C22×C4C2×C4⋊C412yes—
376C22×C4C2×C4⋊C412yes—
377C22×C4C2×C4⋊C412yes—
378C22×C4C2×C4⋊C412yes—
379C22×C4C4×D418yes—
380C22×C4#12012yes—
381C22×C4C22⋊Q818yes—
382C22×C4#12312yes—
383C22×C4#13512yes—
384C22×C4#13512yes—
385C22×C4AGL(1,ℤ/8)12yes—
386C22×C4AGL(1,ℤ/8)12yes—
387C22×C4#13712yes—
388C22×C4#13712yes—
389C22×C4C23×C411yescharacteristic
390C22×C4C22×D412yes—
391C22×C4C22×Q811yescharacteristic
392C22×C4C2×#4112yes—
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4H(5,𝔽2)12yes—
398C22×C42−1+412yes—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—416no—
436C42—24no—
437C42—24no—
438C42—24no—
439C42—24no—
440C42—28no—
441C42—28no—
442C42—48no—
443C42—48no—
444C42—416no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23#38514yes—
480C23C22×C4⋊C411yescharacteristic
481C23C2×#12011yescharacteristic
482C23C2×#12311yescharacteristic
483C23#52511yescharacteristic
484C23#52511yescharacteristic
485C23#53511yescharacteristic
486C23#53611yescharacteristic
487C23C2×#13511yescharacteristic
488C23C2×#13511yescharacteristic
489C23C2×AGL(1,ℤ/8)11yescharacteristic
490C23C2×AGL(1,ℤ/8)11yescharacteristic
491C23C2×#13711yescharacteristic
492C23C2×#13711yescharacteristic
493C23#57511yescharacteristic
494C23#57511yescharacteristic
495C2×C4—232no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—416no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
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—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—81no—
559C2×C4—82no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4#38514yes—
575C2×C4#42714yes—
576C2×C4#42814yes—
577C2×C4#42814yes—
578C2×C4C2×C4×D412yes—
579C2×C4C4×#4111yescharacteristic
580C2×C4C4×#4111yescharacteristic
581C2×C4#52012yes—
582C2×C4C2×C22⋊Q812yes—
583C2×C4#53311yescharacteristic
584C2×C4#53311yescharacteristic
585C2×C4#54412yes—
586C2×C4#54612yes—
587C2×C4#54612yes—
588C2×C4#54712yes—
589C2×C4#54712yes—
590C2×C4#55112yes—
591C2×C4#55112yes—
592C2×C4#55212yes—
593C2×C4#55212yes—
594C8—44no—
595C8—44no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22C2×#38511yescharacteristic
612C22#229611yescharacteristic
613C22#231711yescharacteristic
614C22#240411yescharacteristic
615C22#240611yescharacteristic
616C22#248211yescharacteristic
617C22#248411yescharacteristic
618C22C2×#42711yescharacteristic
619C22C2×#42811yescharacteristic
620C22C2×#42811yescharacteristic
621C22#291111yescharacteristic
622C22#300211yescharacteristic
623C22#300211yescharacteristic
624C22#300311yescharacteristic
625C22#300411yescharacteristic
626C22#309111yescharacteristic
627C22#309111yescharacteristic
628C22#309211yescharacteristic
629C22#309211yescharacteristic
630C22#310611yescharacteristic
631C22#310611yescharacteristic
632C22#310711yescharacteristic
633C22#310711yescharacteristic
634C22#311511yescharacteristic
635C22#311511yescharacteristic
636C22#311611yescharacteristic
637C22#311611yescharacteristic
638C22#320711yescharacteristic
639C22#321011yescharacteristic
640C22#321411yescharacteristic
641C22#321611yescharacteristic
642C22#322811yescharacteristic
643C22#323111yescharacteristic
644C22#323511yescharacteristic
645C22#323711yescharacteristic
646C4—24no—
647C4—24no—
648C4—24no—
649C4—24no—
650C4—42no—
651C4—82no—
652C4—84no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—84no—
657C4#235912yes—
658C4#235912yes—
659C4#243812yes—
660C4#243812yes—
661C2—44no—
662C2#1412811yescharacteristic
663C2#2084311yescharacteristic
664C2#2084411yescharacteristic
665C2#2136211yescharacteristic
666C2#2136411yescharacteristic
667C2#2198411yescharacteristic
668C2#2198811yescharacteristic
669C2#2393211yescharacteristic
670C2#2393511yescharacteristic
671C2#2406211yescharacteristic
672C2#2406511yescharacteristic
673C2#2758411yescharacteristic
674C2#2758711yescharacteristic
675C2#2761211yescharacteristic
676C2#2761511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11324C2semidirect
6. #11323C2semidirect
7. #11267C2semidirect
8. #11267C2semidirect
9. #10908C2semidirect
10. #10907C2semidirect
28. #1817C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
C2×#18542
C2×#18172
#137562
#113242
#113232
#112672
#109082
#109072
#107312