Finite Group Databasebeta

← #360324

Subgroups — #360324

Order 512.

1770 conjugacy classes of subgroups.

The subgroup lattice has 783 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1817C211yesmaximal, characteristic
2C2×#1817C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic
4C2×C4×C22⋊C8C211yesmaximal, characteristic
5C2×C4×#95C211yesmaximal, characteristic
6#11298C212yesmaximal, complemented
7#11255C212yesmaximal, complemented
8#10876C212yesmaximal, complemented
9#10797C212yesmaximal, complemented
10#10712C211yesmaximal, characteristic
11#10712C211yesmaximal, characteristic
12C23×C42C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C4×C8C2212yes—
21C22×#95C411yescharacteristic
22C22×#95C411yescharacteristic
23C22×#95C411yescharacteristic
24C22×#95C411yescharacteristic
25#1817—24no—
26#1817—24no—
27#1797—24no—
28C4×C22⋊C8—24no—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C4×#95C411yescharacteristic
40C4×#95C411yescharacteristic
41C4×#95C411yescharacteristic
42C4×#95C411yescharacteristic
43C4×#95C2211yescharacteristic
44C4×#95C2211yescharacteristic
45C4×#95C2211yescharacteristic
46C4×#95C2211yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4C2311yescharacteristic
54C23×C8C2312yes—
55C23×C8C2312yes—
56C23×C8C2312yes—
57C23×C8C2312yes—
58C22×C42—22no—
59C22×C42—22no—
60C22×C42—22no—
61C22×C42—22no—
62C22×C42—22no—
63C22×C42—22no—
64C22×C42—22no—
65C22×C42—22no—
66C22×C42—22no—
67C22×C42—22no—
68C22×C42—24no—
69C22×C42—24no—
70C22×C42—42no—
71C22×C42—42no—
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×C4⋊C8—24no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—42no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8D414yes—
98C2×C22⋊C8D414yes—
99C2×C4×C8—24no—
100C2×C4×C8—24no—
101C2×C4×C8—42no—
102C2×#95—24no—
103C2×#95—24no—
104C2×#95—24no—
105C2×#95C818yescomplemented
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115C2×#95C2×C411yescharacteristic
116C2×#95C2×C411yescharacteristic
117C2×#95C2×C411yescharacteristic
118C2×#95C2×C411yescharacteristic
119C2×#95C2×C411yescharacteristic
120C2×#95C2×C411yescharacteristic
121C2×#95C2×C411yescharacteristic
122#336—24no—
123#336—24no—
124#336—24no—
125#336—24no—
126#336—24no—
127#336—24no—
128#336—24no—
129#336—24no—
130#336—42no—
131#336—42no—
132C25C22×C411yescharacteristic
133C23×C4—22no—
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—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
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—41no—
166C23×C4—41no—
167C23×C4—41no—
168C23×C4—41no—
169C23×C4—41no—
170C23×C4—41no—
171C23×C4—41no—
172C23×C4—41no—
173C23×C4—41no—
174C23×C4—41no—
175C23×C4C2×C814yes—
176C23×C4C2×C814yes—
177C23×C4C2×C814yes—
178C23×C4M1614yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C23×C4#4112yes—
195C23×C4#4112yes—
196C23×C4#4112yes—
197C23×C4#4112yes—
198C23×C4C2411yescharacteristic
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
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—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8C2×D414yes—
226C22×C8C2×D414yes—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—48no—
268C2×C42—48no—
269C2×C42—48no—
270C2×C42—48no—
271C2×C42—41no—
272C2×C42—41no—
273C2×C42—41no—
274C2×C42—41no—
275C2×C42—44no—
276C2×C42—44no—
277C2×C42—44no—
278C2×C42—44no—
279C2×C42#4114yes—
280C2×C42#4114yes—
281C4⋊C8—48no—
282C22⋊C8—28no—
283C22⋊C8—28no—
284C22⋊C8—44no—
285C22⋊C8—44no—
286C4×C8—48no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95C2×C814yes—
298#95C2×C814yes—
299#95C2×C814yes—
300#95C2×C814yes—
301#95M1614yes—
302#95M1614yes—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—21no—
311C24#11712yes—
312C24C4×D412yes—
313C24C4×D412yes—
314C24C22×C812yes—
315C24C2×M1612yes—
316C24#13112yes—
317C24#13112yes—
318C24C23×C411yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4#11712yes—
444C22×C4#11712yes—
445C22×C4#11712yes—
446C22×C4#11712yes—
447C22×C4#11712yes—
448C22×C4C4×D412yes—
449C22×C4C4×D412yes—
450C22×C4#12114yes—
451C22×C4#12314yes—
452C22×C4#12414yes—
453C22×C4#12614yes—
454C22×C4C22×C812yes—
455C22×C4C22×C812yes—
456C22×C4C22×C812yes—
457C22×C4C22×C812yes—
458C22×C4C22×C812yes—
459C22×C4C22×C812yes—
460C22×C4C2×M1612yes—
461C22×C4C2×M1612yes—
462C22×C4C2×M1612yes—
463C22×C4C2×M1612yes—
464C22×C4C2×M1612yes—
465C22×C4C2×M1612yes—
466C22×C4C2×M1612yes—
467C22×C4C2×M1612yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4111yescharacteristic
471C22×C4C2×#4111yescharacteristic
472C22×C4C2×#4111yescharacteristic
473C22×C4C2×#4111yescharacteristic
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C42−1+412yes—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—42no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—42no—
490C2×C8—42no—
491C2×C8—42no—
492C2×C8—42no—
493C2×C8—42no—
494C2×C8—42no—
495C2×C8—42no—
496C2×C8—42no—
497C2×C8—42no—
498C2×C8—42no—
499C2×C8—42no—
500C2×C8—42no—
501C2×C8—42no—
502C2×C8—42no—
503C2×C8—42no—
504C2×C8—42no—
505C2×C8—42no—
506C42—24no—
507C42—24no—
508C42—24no—
509C42—24no—
510C42—44no—
511C42—44no—
512C42—44no—
513C42—44no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—44no—
527C42—44no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23#38814yes—
557C23#43114yes—
558C23#43314yes—
559C23C8×D414yes—
560C23C8×D414yes—
561C23#43514yes—
562C23#43614yes—
563C23C2×#11711yescharacteristic
564C23C2×#11711yescharacteristic
565C23C2×#11711yescharacteristic
566C23C2×C4×D411yescharacteristic
567C23C2×C4×D411yescharacteristic
568C23#51811yescharacteristic
569C23#51911yescharacteristic
570C23C23×C811yescharacteristic
571C23C22×M1611yescharacteristic
572C23C22×M1611yescharacteristic
573C23C22×M1611yescharacteristic
574C23C2×#13111yescharacteristic
575C23C2×#13111yescharacteristic
576C23#56811yescharacteristic
577C23#56811yescharacteristic
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—416no—
634C2×C4—416no—
635C2×C4—416no—
636C2×C4—416no—
637C2×C4—416no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—416no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4#38814yes—
676C2×C4#43114yes—
677C2×C4#43114yes—
678C2×C4#43214yes—
679C2×C4C2×#12111yescharacteristic
680C2×C4C2×#12311yescharacteristic
681C2×C4C2×#12411yescharacteristic
682C2×C4C2×#12611yescharacteristic
683C2×C4#52912yes—
684C2×C4#52912yes—
685C2×C4#53912yes—
686C2×C4#53912yes—
687C2×C4#54612yes—
688C2×C4#54612yes—
689C2×C4#54712yes—
690C2×C4#54712yes—
691C2×C4#55112yes—
692C2×C4#55112yes—
693C2×C4#55212yes—
694C2×C4#55212yes—
695C8—44no—
696C8—44no—
697C8—82no—
698C8—82no—
699C22—24no—
700C22—24no—
701C22—24no—
702C22—24no—
703C22—24no—
704C22—24no—
705C22—24no—
706C22—24no—
707C22—24no—
708C22—24no—
709C22—24no—
710C22—24no—
711C22—24no—
712C22—24no—
713C22#192318yes—
714C22#228311yescharacteristic
715C22C2×#38811yescharacteristic
716C22#231311yescharacteristic
717C22#234211yescharacteristic
718C22#234211yescharacteristic
719C22#234411yescharacteristic
720C22#234411yescharacteristic
721C22C2×#43111yescharacteristic
722C22C2×#43111yescharacteristic
723C22C2×#43111yescharacteristic
724C22C2×#43211yescharacteristic
725C22C2×#43311yescharacteristic
726C22#292011yescharacteristic
727C22#292011yescharacteristic
728C22#292111yescharacteristic
729C22#292411yescharacteristic
730C22#292411yescharacteristic
731C22#292411yescharacteristic
732C22#292411yescharacteristic
733C22C2×C8×D411yescharacteristic
734C22C2×C8×D411yescharacteristic
735C22C2×#43511yescharacteristic
736C22C2×#43611yescharacteristic
737C22D4×M1611yescharacteristic
738C22D4×M1611yescharacteristic
739C22D4×M1611yescharacteristic
740C22D4×M1611yescharacteristic
741C22#297211yescharacteristic
742C22#297311yescharacteristic
743C22#297411yescharacteristic
744C22#297411yescharacteristic
745C22#298211yescharacteristic
746C22#298311yescharacteristic
747C22#298411yescharacteristic
748C22#298411yescharacteristic
749C4—24no—
750C4—24no—
751C4—24no—
752C4—24no—
753C4—42no—
754C4—42no—
755C4—42no—
756C4—42no—
757C4—42no—
758C4—42no—
759C4—42no—
760C4—42no—
761C4—42no—
762C4#241912yes—
763C4#242512yes—
764C4#248812yes—
765C4#249312yes—
766C2—28no—
767C2#1387411yescharacteristic
768C2#2077711yescharacteristic
769C2#2077911yescharacteristic
770C2C2×#192311yescharacteristic
771C2#2104911yescharacteristic
772C2#2191211yescharacteristic
773C2#2191411yescharacteristic
774C2#2255211yescharacteristic
775C2#2255211yescharacteristic
776C2#2255411yescharacteristic
777C2#2255511yescharacteristic
778C2#2259911yescharacteristic
779C2#2259911yescharacteristic
780C2#2260111yescharacteristic
781C2#2260311yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC25C22×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
6. #11298C2semidirect
7. #11255C2semidirect
8. #10876C2semidirect
9. #10797C2semidirect
105. C2×#95C8semidirect

Maximal subgroups

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

subgroupindex
C2×#18172
C2×#17972
C2×C4×C22⋊C82
C2×C4×#952
#112982
#112552
#108762
#107972
#107122