Finite Group Databasebeta

← #364651

Subgroups — #364651

Order 512.

2192 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#480C211yesmaximal, characteristic
2C2×#2361C211yesmaximal, characteristic
3C2×#1893C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic, complemented
5#13904C211yesmaximal, characteristic, complemented
6#11219C212yesmaximal, complemented
7#11189C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10788C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic, complemented
11#10714C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C22×C4×D4—21no—
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#480—24no—
19C2×#480—24no—
20C2×#480C418yescomplemented
21C22×#107C2212yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25#2361—24no—
26#2361—24no—
27C2×#390C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#388—21no—
30C2×#386—21no—
31C2×#385—21no—
32C2×#381—21no—
33C2×#381—21no—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2211yescharacteristic
37#1893—24no—
38C2×#340—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1445—21no—
47#1445—21no—
48C4⋊#95C2212yes—
49C4⋊#95C2212yes—
50C23×D4—21no—
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C24×C4—21no—
54C24×C4—21no—
55C24×C4C2311yescharacteristic
56C23×C8—22no—
57C23×C8—24no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—42no—
61C23×C8C2312yes—
62C23×C8C2312yes—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121C2×C414yes—
67C2×#121D414yes—
68C2×#121D414yes—
69C2×C4×D4—28no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—24no—
73C2×C4×D4—42no—
74C2×C4×D4—44no—
75C2×C4×D4—44no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4C2×C414yescomplemented
88C22×C4⋊C4D414yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4D411yescharacteristic
100C22×C22⋊C4D411yescharacteristic
101C22×C42—21no—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104#480—232no—
105C2×#107—24no—
106C2×#107—24no—
107C2×#107C2×C418yescomplemented
108C2×#102—24no—
109C2×#102—24no—
110C2×#102—24no—
111C2×#102—24no—
112C2×#102—42no—
113C2×#102C2×C418yescomplemented
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8C2×C414yescomplemented
117#390—22no—
118#390—22no—
119#389—22no—
120#389—22no—
121#388—28no—
122#386—28no—
123#385—28no—
124#381—44no—
125#381—44no—
126#380—22no—
127#380—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—21no—
139C2×#95D411yescharacteristic
140C2×#95D411yescharacteristic
141#340—24no—
142#340—24no—
143#340—24no—
144#340—24no—
145#340—42no—
146#336—24no—
147#336—42no—
148C25—21no—
149C25C2×D411yescharacteristic
150C22×D4—24no—
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4—42no—
164C22×D4C22×C412yes—
165C22×D4C2×D412yes—
166C22×D4C2×D412yes—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—24no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—82no—
192C23×C4—82no—
193C23×C4C22×C412yes—
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D412yes—
199C23×C4C2×D412yes—
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D412yes—
202C23×C4#4112yes—
203C23×C4#4112yes—
204C23×C4C2411yescharacteristic
205C22×C8—28no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—48no—
219C22×C8—48no—
220C22×C8—44no—
221C22×C8—44no—
222C22×C8C22×C414yes—
223C22×C8#4114yes—
224#121—28no—
225#121—28no—
226#121—28no—
227#121C22⋊C4116yes—
228C4×D4—432no—
229C4×D4—48no—
230C4×D4—44no—
231C4×D4—44no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—48no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4#4114yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—82no—
291C2×C22⋊C4D814yes—
292C2×C42—24no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—42no—
302C2×C42—42no—
303#107—216no—
304#102—216no—
305#102—48no—
306C22⋊C8—28no—
307#95—24no—
308#95—24no—
309#95—22no—
310#95—22no—
311#95—44no—
312#95—44no—
313#95—42no—
314#95—44no—
315#95—44no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—42no—
324C24—42no—
325C24—41no—
326C24—41no—
327C24—41no—
328C24—41no—
329C24—42no—
330C24—42no—
331C24C4×D412yes—
332C24#12012yes—
333C24#12112yes—
334C24C2×D812yes—
335C24#13512yes—
336C24AGL(1,ℤ/8)12yes—
337C24AGL(1,ℤ/8)12yes—
338C24C22×D411yescharacteristic
339C2×D4—216no—
340C2×D4—22no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—42no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4C2×C22⋊C414yes—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—88no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4C2×C22⋊C414yes—
447C22×C4C4×D414yes—
448C22×C4C4×D412yes—
449C22×C4#12012yes—
450C22×C4#12114yes—
451C22×C4#12112yes—
452C22×C4#12314yes—
453C22×C4C2×D812yes—
454C22×C4C2×D812yes—
455C22×C4C23×C411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4111yescharacteristic
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4111yescharacteristic
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+412yes—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—416no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—48no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4C2×C22⋊C414yes—
500C4⋊C4#12314yes—
501C22⋊C4—28no—
502C22⋊C4—216no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C42—48no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—42no—
514C42—42no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—42no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—82no—
547C23—82no—
548C23—82no—
549C23—82no—
550C23—82no—
551C23—82no—
552C23—82no—
553C23—82no—
554C23#38614yes—
555C23C4×D814yes—
556C23#44214yes—
557C23#44714yes—
558C23#44914yes—
559C23#45914yes—
560C23#46314yes—
561C23C2×C4×D411yescharacteristic
562C23C2×#12011yescharacteristic
563C23C2×#12111yescharacteristic
564C23#54612yes—
565C23#54712yes—
566C23C22×D811yescharacteristic
567C23C2×#13511yescharacteristic
568C23C2×AGL(1,ℤ/8)11yescharacteristic
569C23C2×AGL(1,ℤ/8)11yescharacteristic
570C23#57712yes—
571C23#57812yes—
572D4—28no—
573D4—432no—
574D4—42no—
575D4—44no—
576D4—42no—
577D4—42no—
578D4—42no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—816no—
628C2×C4—816no—
629C2×C4—816no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4#38614yes—
644C2×C4C22×C22⋊C411yescharacteristic
645C2×C4C2×C4×D411yescharacteristic
646C2×C4C4×#4112yes—
647C2×C4#52012yes—
648C2×C4C2×#12111yescharacteristic
649C2×C4C2×#12311yescharacteristic
650C2×C4C2×#12311yescharacteristic
651C2×C4#52511yescharacteristic
652C2×C4#52912yes—
653C2×C4#53411yescharacteristic
654C2×C4#53611yescharacteristic
655C2×C4#53912yes—
656C2×C4#54612yes—
657C2×C4#54612yes—
658C2×C4#54712yes—
659C2×C4#54712yes—
660C2×C4#55112yes—
661C2×C4#55212yes—
662C8—44no—
663C8—44no—
664C22—24no—
665C22—24no—
666C22—28no—
667C22—28no—
668C22—28no—
669C22—24no—
670C22—28no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—82no—
678C22—82no—
679C22—84no—
680C22—84no—
681C22—84no—
682C22—84no—
683C22—82no—
684C22—82no—
685C22#187918yes—
686C22C2×#38611yescharacteristic
687C22#235912yes—
688C22#240511yescharacteristic
689C22#240611yescharacteristic
690C22#242512yes—
691C22C2×C4×D811yescharacteristic
692C22C2×#44211yescharacteristic
693C22#294712yes—
694C22C2×#44711yescharacteristic
695C22C2×#44911yescharacteristic
696C22#300911yescharacteristic
697C22#301111yescharacteristic
698C22C2×#45911yescharacteristic
699C22C2×#46311yescharacteristic
700C22#304612yes—
701C22#329412yes—
702C22#330012yes—
703C22#330812yes—
704C22#331012yes—
705C22#333412yes—
706C22#333712yes—
707C22#334312yes—
708C22#334512yes—
709C4—24no—
710C4—42no—
711C4—44no—
712C4—42no—
713C4—44no—
714C4—81no—
715C4—84no—
716C4—82no—
717C4—81no—
718C4—162no—
719C4C4×#12311yescharacteristic
720C4#232211yescharacteristic
721C4#233912yes—
722C4#241912yes—
723C4#248611yescharacteristic
724C4#253411yescharacteristic
725C2—28no—
726C2—84no—
727C2#1396811yescharacteristic
728C2C2×#187911yescharacteristic
729C2#2093111yescharacteristic
730C2#2291612yes—
731C2#2293812yes—
732C2#2403211yescharacteristic
733C2#2403511yescharacteristic
734C2#2408811yescharacteristic
735C2#2409111yescharacteristic
736C2#2520012yes—
737C2#2521212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C22×#340C2semidirect
5. #13904C2semidirect
6. #11219C2semidirect
7. #11189C2semidirect
8. #10907C2semidirect
9. #10788C2semidirect
10. #10730C2semidirect
11. #10714C2semidirect
20. C2×#480C4semidirect
87. C22×C4⋊C4C2×C4semidirect
107. C2×#107C2×C4semidirect
113. C2×#102C2×C4semidirect
116. C2×C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4802
C2×#23612
C2×#18932
C22×#3402
#139042
#112192
#111892
#109072
#107882
#107302
#107142