Finite Group Databasebeta

← #364643

Subgroups — #364643

Order 512.

2182 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×D4C211yesmaximal, characteristic
2C2×C4×#123C211yesmaximal, characteristic
3C2×#1869C211yesmaximal, characteristic
4C2×#1817C211yesmaximal, characteristic, complemented
5C22×#336C211yesmaximal, characteristic, complemented
6#11202C212yesmaximal, complemented
7#11147C212yesmaximal, complemented
8#10874C212yesmaximal, complemented
9#10714C211yesmaximal, characteristic, complemented
10#10714C212yesmaximal, complemented
11#10714C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C22×#123C411yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C23×C42C2211yescharacteristic
18C2×C8×D4—24no—
19C2×C8×D4—24no—
20C2×C8×D4—24no—
21C22×C4⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2212yes—
25C22×C22⋊C8C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C4×#123—24no—
28C4×#123—24no—
29C2×#388C412yes—
30C2×#387C412yes—
31C2×#386C411yescharacteristic
32C2×#385C411yescharacteristic
33C2×#381C411yescharacteristic
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1869—24no—
38#1817—24no—
39C2×#336—24no—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47C2×#336C2212yes—
48C2×#336C2212yes—
49C23×D4C2×C411yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4C2×C411yescharacteristic
52C24×C4C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—24no—
61C23×C8—24no—
62C23×C8—22no—
63C23×C8—22no—
64C23×C8—24no—
65C23×C8—42no—
66C23×C8—42no—
67C23×C8—42no—
68C23×C8D414yes—
69C23×C8C2312yes—
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2312yes—
74C23×C8C2311yescharacteristic
75C2×#123—24no—
76C2×#123—24no—
77C2×#123C818yescomplemented
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×C4×D4D414yes—
83C2×C4×D4D414yes—
84C22×C4⋊C4C2×C412yes—
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C4⋊C4C2×C412yes—
87C22×C22⋊C4C2×C411yescharacteristic
88C22×C22⋊C4C2×C412yes—
89C22×C22⋊C4C2×C411yescharacteristic
90C22×C22⋊C4C2×C411yescharacteristic
91C22×C22⋊C4C2×C411yescharacteristic
92C22×C22⋊C4C2×C412yes—
93C22×C42—22no—
94C22×C42—22no—
95C22×C42—22no—
96C22×C42—22no—
97C22×C42—22no—
98C22×C42—22no—
99C22×C42C2×C412yes—
100C22×C42C2312yes—
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103C8×D4—232no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—24no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8—24no—
117C2×C22⋊C8—42no—
118C2×C22⋊C8D414yes—
119C2×C4×C8—22no—
120C2×C4×C8—22no—
121C2×C4×C8—22no—
122#388—28no—
123#387—28no—
124#386—24no—
125#385—24no—
126#381—24no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—24no—
129C4×C22⋊C4—24no—
130C4×C22⋊C4—24no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C2×#95C2×C412yes—
134C2×#95C2×C412yes—
135C2×#95C2×C412yes—
136C2×#95C2×C412yes—
137C2×#95C2×C412yes—
138#336—24no—
139#336—24no—
140#336—24no—
141#336—24no—
142#336—24no—
143#336—24no—
144#336—24no—
145#336—42no—
146#336—42no—
147C25C22×C411yescharacteristic
148C25C22×C411yescharacteristic
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4C22⋊C414yes—
154C22×D4C2×C814yes—
155C22×D4M1614yes—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—42no—
179C23×C4C2×C814yescomplemented
180C23×C4C22×C412yes—
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C412yes—
184C23×C4C22×C412yes—
185C23×C4C22×C412yes—
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4C2×D412yes—
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C23×C4#4112yes—
195C23×C4#4112yes—
196C23×C4C2411yescharacteristic
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—22no—
200C22×C8—28no—
201C22×C8—28no—
202C22×C8—28no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—28no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—44no—
230C22×C8—48no—
231C22×C8—48no—
232C22×C8—48no—
233C22×C8—48no—
234C22×C8—48no—
235C22×C8—48no—
236C22×C8—44no—
237C22×C8—44no—
238C22×C8—44no—
239C22×C8—44no—
240C22×C8C2×D412yes—
241C22×C8C2×D412yes—
242C22×C8C2×D412yes—
243C22×C8C2×D412yes—
244C22×C8#4114yes—
245C22×C8#4114yes—
246#123—232no—
247C4×D4—28no—
248C4×D4—28no—
249C4×D4—28no—
250C4×D4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4C22⋊C414yes—
259C2×C4⋊C4C2×C818yescomplemented
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4C22⋊C414yes—
275C2×C22⋊C4C22⋊C414yes—
276C2×C22⋊C4C2×C814yescomplemented
277C2×C22⋊C4C2×C818yescomplemented
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C4⋊C8—28no—
304C22⋊C8—28no—
305C22⋊C8—28no—
306C22⋊C8—48no—
307C22⋊C8—44no—
308C4×C8—28no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—21no—
324C24—21no—
325C24—22no—
326C24C2×C22⋊C412yes—
327C24C2×C22⋊C412yes—
328C24C4×D412yes—
329C24C4×D412yes—
330C24C22×C812yes—
331C24C22×C812yes—
332C24C2×M1612yes—
333C24C2×M1612yes—
334C24#13112yes—
335C24#13112yes—
336C24C23×C411yescharacteristic
337C2×D4—28no—
338C2×D4—28no—
339C2×D4—28no—
340C2×D4—28no—
341C2×D4—28no—
342C2×D4—28no—
343C2×D4C22⋊C8116yes—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—28no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4C2×C22⋊C412yes—
428C22×C4C2×C22⋊C412yes—
429C22×C4C2×C22⋊C412yes—
430C22×C4C2×C22⋊C412yes—
431C22×C4C4×D412yes—
432C22×C4C4×D412yes—
433C22×C4#12014yes—
434C22×C4#12114yes—
435C22×C4#12114yes—
436C22×C4#12314yes—
437C22×C4C22×C814yes—
438C22×C4C22×C814yes—
439C22×C4C22×C812yes—
440C22×C4C2×M1612yes—
441C22×C4#13114yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C42−1+412yes—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—416no—
467C2×C8—416no—
468C2×C8—416no—
469C2×C8—42no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—42no—
474C2×C8—41no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—42no—
487C2×C8—42no—
488C2×C8#12314yescomplemented
489C4⋊C4—28no—
490C4⋊C4—216no—
491C4⋊C4—48no—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—216no—
495C22⋊C4—28no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—42no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42—42no—
508C42—42no—
509C42—42no—
510C42—48no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—24no—
528C23—22no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—22no—
539C23—41no—
540C23—42no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—41no—
549C23#38614yes—
550C23C2×C22⋊C814yes—
551C23C2×C22⋊C814yes—
552C23#40814yes—
553C23C8×D414yes—
554C23C8×D414yes—
555C23#43514yes—
556C23#43514yes—
557C23C22×C22⋊C411yescharacteristic
558C23C2×C4×D411yescharacteristic
559C23C2×C4×D411yescharacteristic
560C23C4×#4112yes—
561C23#52012yes—
562C23C23×C811yescharacteristic
563C23C22×M1611yescharacteristic
564C23C2×#13111yescharacteristic
565C23C2×#13111yescharacteristic
566C23C2×#13112yes—
567C23#56812yes—
568D4—232no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—416no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—416no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4#38614yes—
632C2×C4C2×C22⋊C814yes—
633C2×C4C2×#12011yescharacteristic
634C2×C4C2×#12111yescharacteristic
635C2×C4C2×#12111yescharacteristic
636C2×C4C2×#12311yescharacteristic
637C2×C4C2×#12311yescharacteristic
638C2×C4#52511yescharacteristic
639C2×C4#52912yes—
640C2×C4#53411yescharacteristic
641C2×C4#53611yescharacteristic
642C2×C4#53912yes—
643C2×C4#54612yes—
644C2×C4#54612yes—
645C2×C4#54612yes—
646C2×C4#54712yes—
647C2×C4#54712yes—
648C2×C4#54712yes—
649C2×C4#55112yes—
650C2×C4#55212yes—
651C8—24no—
652C8—42no—
653C8—42no—
654C8—42no—
655C8—44no—
656C8—82no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—28no—
662C22—28no—
663C22—24no—
664C22—24no—
665C22—28no—
666C22—28no—
667C22—24no—
668C22—28no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—44no—
674C22—44no—
675C22—42no—
676C22—42no—
677C22—44no—
678C22#186918yes—
679C22C2×#38611yescharacteristic
680C22C4×#12311yescharacteristic
681C22#232211yescharacteristic
682C22#233912yes—
683C22#235912yes—
684C22C22×C22⋊C811yescharacteristic
685C22C2×#40811yescharacteristic
686C22#288012yes—
687C22C2×C8×D411yescharacteristic
688C22C2×C8×D411yescharacteristic
689C22C2×#43511yescharacteristic
690C22C2×#43511yescharacteristic
691C22#292911yescharacteristic
692C22#293011yescharacteristic
693C22#293312yes—
694C22C8×#4112yes—
695C22#296612yes—
696C22#296812yes—
697C22#297012yes—
698C22#298812yes—
699C22#299012yes—
700C22#299212yes—
701C22#299412yes—
702C4—24no—
703C4—24no—
704C4—42no—
705C4—44no—
706C4—42no—
707C4—44no—
708C4—44no—
709C4—82no—
710C4#240511yescharacteristic
711C4#240611yescharacteristic
712C4#241912yes—
713C4#242512yes—
714C4#248611yescharacteristic
715C4#253411yescharacteristic
716C2—28no—
717C2—44no—
718C2#1396811yescharacteristic
719C2C2×#186911yescharacteristic
720C2#2089911yescharacteristic
721C2C8×#12311yescharacteristic
722C2#2156111yescharacteristic
723C2#2206311yescharacteristic
724C2#2206511yescharacteristic
725C2#2246912yes—
726C2#2247012yes—
727C2#2290012yes—
728C2#2290812yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1817C2semidirect
5. C22×#336C2semidirect
6. #11202C2semidirect
7. #11147C2semidirect
8. #10874C2semidirect
9. #10714C2semidirect
10. #10714C2semidirect
11. #10714C2semidirect
77. C2×#123C8semidirect
179. C23×C4C2×C8semidirect
259. C2×C4⋊C4C2×C8semidirect
276. C2×C22⋊C4C2×C8semidirect
277. C2×C22⋊C4C2×C8semidirect
488. C2×C8#123semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D42
C2×C4×#1232
C2×#18692
C2×#18172
C22×#3362
#112022
#111472
#108742
#107142