Finite Group Databasebeta

← #6409218

Subgroups — #6409218

Order 512.

2014 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#23329C214yesmaximal, complemented
3#22987C214yesmaximal, complemented
4#22628C212yesmaximal, complemented
5#22403C212yesmaximal, complemented
6#22398C214yesmaximal, complemented
7#22176C212yesmaximal, complemented
8#22168C212yesmaximal, complemented
9#22143C214yesmaximal, complemented
10#21937C212yesmaximal, complemented
11#21911C212yesmaximal, complemented
12#20715C211yesmaximal, characteristic, complemented
13C2×#1797C211yesmaximal, characteristic, complemented
14C2×#532C2211yescharacteristic
15C2×#525C2212yes—
16C23×C42C2211yescharacteristic
17C2×#435C2214yes—
18C2×#435C2214yescomplemented
19C2×C8×D4C2214yes—
20C2×#432C2212yes—
21C22×C4⋊C8C2212yes—
22C2×#408C2214yes—
23C22×C22⋊C8C2212yes—
24#2857C411yescharacteristic
25#2857C411yescharacteristic
26#2856C411yescharacteristic
27#2856C411yescharacteristic
28#2854C412yes—
29#2854C412yes—
30#2850C412yes—
31#2850C412yes—
32#2847C412yes—
33#2847C412yes—
34C4×#121C2212yes—
35C4×#121C2212yes—
36C4×#121C2212yes—
37C4×#121C2212yes—
38#2297C2212yes—
39#2285C2211yescharacteristic
40#1955C2214yes—
41#1955C2214yescomplemented
42#1954C2214yes—
43#1940C2214yescomplemented
44#1924C2214yescomplemented
45#1924C2214yescomplemented
46#1923C2214yescomplemented
47#1923C2214yescomplemented
48#1917C2214yescomplemented
49#1916C2214yescomplemented
50#1870C2214yes—
51#1870C2214yescomplemented
52#1869C2214yes—
53#1869C2214yes—
54#1869C2214yes—
55#1869C2214yescomplemented
56#1851C2214yescomplemented
57#1851C2214yescomplemented
58#1850C2214yescomplemented
59#1850C2214yescomplemented
60#1817C2212yes—
61#1798C2214yes—
62#1797C2212yes—
63#1797C2212yes—
64#1797C2212yes—
65#1797C2212yes—
66#1797C2211yescharacteristic
67#1797C2212yes—
68#1797C2211yescharacteristic
69#1780C2214yes—
70#1779C2212yes—
71#1771C2214yescomplemented
72#1750C2214yes—
73#1749C2212yes—
74#1749C2212yes—
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C22×#41C2312yes—
78C24×C4C2×C411yescharacteristic
79C24×C4C2×C411yescharacteristic
80C24×C4C2311yescharacteristic
81C22×M16C2314yes—
82C23×C8C2312yes—
83C23×C8C2312yes—
84C23×C8C2314yes—
85#532—44no—
86C2×C4⋊Q8C2×C411yescharacteristic
87C2×#127C2×C411yescharacteristic
88C2×#124C2×C412yes—
89#525—48no—
90C2×#123C2×C414yes—
91C2×C22⋊Q8C2×C412yes—
92C2×C22⋊Q8C2×C412yes—
93C2×#121C2×C412yes—
94C2×#121C2×C412yes—
95C2×#121C2×C412yes—
96C2×#121C2×C412yes—
97C2×#121C2×C412yes—
98C2×#121C2×C412yes—
99C2×#121C2×C412yes—
100C2×#121C2×C412yes—
101C2×#120C2×C414yes—
102C2×C4×D4C2314yes—
103C2×C4×D4C2312yes—
104C2×C4×D4C2314yes—
105C2×C4×D4C2312yes—
106C2×#117C2312yes—
107C22×C42—44no—
108C22×C42C2×C411yescharacteristic
109C22×C42C2×C411yescharacteristic
110C22×C42C2×C411yescharacteristic
111C22×C42C2×C411yescharacteristic
112C22×C42C2×C411yescharacteristic
113C22×C42C2×C411yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#435—48no—
121#435—48no—
122C8×D4—48no—
123#432—44no—
124C2×C4⋊C8—44no—
125C2×C4⋊C8C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2314yes—
131#408—48no—
132C2×C22⋊C8—44no—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×#97C2314yes—
146C2×#97C2314yes—
147C2×C4×C8C2314yes—
148#391C2×C412yes—
149#391C2×C412yes—
150#390C2×C412yes—
151#390C2×C412yes—
152#390C2×C412yes—
153#390C2×C412yes—
154#390C2×C412yes—
155#390C2×C412yes—
156#389C2×C412yes—
157#389C2×C412yes—
158#389C2×C412yes—
159#389C2×C412yes—
160#388C2×C414yes—
161#388C2×C414yes—
162#388C2×C414yes—
163#388C2×C414yes—
164#386C2×C414yes—
165#386C2×C414yes—
166#386C2×C414yes—
167#386C2×C414yes—
168#386C2×C414yes—
169#386C2×C414yes—
170#384C2×C411yescharacteristic
171#384C2×C411yescharacteristic
172#382C2×C412yes—
173#380C2×C412yes—
174#380C2×C412yes—
175#380C2×C412yes—
176#380C2×C412yes—
177#380C2×C412yes—
178#380C2×C412yes—
179C4×C4⋊C4C2312yes—
180C4×C4⋊C4C2312yes—
181C4×C22⋊C4C2314yes—
182C4×C22⋊C4C2314yes—
183#336C2312yes—
184#336C2312yes—
185#336C2312yes—
186#336C2312yes—
187#336C2312yes—
188#336C2314yes—
189#336C2312yes—
190#336C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25C22×C411yescharacteristic
195C2×#41—24no—
196C2×#41—24no—
197C2×#41—44no—
198C2×#41—44no—
199C22×Q8C22×C412yes—
200C22×D4C22×C414yes—
201C22×D4C22×C414yes—
202C22×D4C22×C412yes—
203C22×D4C22×C412yes—
204C22×D4C22×C414yes—
205C22×D4C22×C412yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4C22×C411yescharacteristic
217C23×C4C22×C411yescharacteristic
218C23×C4C22×C411yescharacteristic
219C23×C4C22×C411yescharacteristic
220C23×C4C22×C411yescharacteristic
221C23×C4C22×C411yescharacteristic
222C23×C4C22×C411yescharacteristic
223C23×C4C22×C411yescharacteristic
224C23×C4C22×C411yescharacteristic
225C23×C4C22×C411yescharacteristic
226C23×C4C22×C411yescharacteristic
227C23×C4C22×C411yescharacteristic
228C23×C4C22×C411yescharacteristic
229C23×C4C22×C411yescharacteristic
230C23×C4C22×C411yescharacteristic
231C23×C4C2414yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C2×M16—44no—
236C2×M16—44no—
237C2×M16—44no—
238C22×C8—24no—
239C22×C8—24no—
240C22×C8—42no—
241C22×C8—42no—
242C22×C8—44no—
243C22×C8—42no—
244C22×C8—42no—
245C22×C8—44no—
246C22×C8—44no—
247C22×C8C2414yes—
248C22×C8C2414yes—
249C22×C8C2414yes—
250C22×C8C2414yes—
251C4⋊Q8—42no—
252#127—42no—
253#124—44no—
254#123—48no—
255C22⋊Q8—44no—
256C22⋊Q8—44no—
257#121—28no—
258#121—44no—
259#121—44no—
260#121—44no—
261#121—44no—
262#121—44no—
263#121—44no—
264#121—44no—
265#120—48no—
266C4×D4—28no—
267C4×D4—28no—
268C4×D4—28no—
269C4×D4—44no—
270C4×D4—44no—
271C4×D4—42no—
272C4×D4—42no—
273#117—44no—
274C2×C4⋊C4C22×C412yes—
275C2×C4⋊C4C22×C414yes—
276C2×C4⋊C4C22×C412yes—
277C2×C4⋊C4C22×C412yes—
278C2×C4⋊C4C22×C412yes—
279C2×C4⋊C4C22×C412yes—
280C2×C4⋊C4C22×C414yes—
281C2×C4⋊C4C22×C412yes—
282C2×C4⋊C4C22×C412yes—
283C2×C4⋊C4C22×C412yes—
284C2×C22⋊C4C22×C414yes—
285C2×C22⋊C4C22×C414yes—
286C2×C22⋊C4C22×C414yes—
287C2×C22⋊C4C22×C414yes—
288C2×C22⋊C4C22×C414yes—
289C2×C22⋊C4C22×C414yes—
290C2×C22⋊C4C22×C414yes—
291C2×C22⋊C4C22×C414yes—
292C2×C22⋊C4C22×C414yes—
293C2×C22⋊C4C22×C414yes—
294C2×C22⋊C4C22×C414yes—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42C22×C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C22×C411yescharacteristic
311C2×C42C22×C411yescharacteristic
312C2×C42C22×C411yescharacteristic
313C2×C42C22×C411yescharacteristic
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C22×C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C22×C411yescharacteristic
319C2×C42C2414yes—
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324C4⋊C8—28no—
325C4⋊C8—44no—
326C4⋊C8—42no—
327C4⋊C8—42no—
328C4⋊C8—44no—
329C4⋊C8—44no—
330C22⋊C8—28no—
331C22⋊C8—24no—
332C22⋊C8—28no—
333C22⋊C8—24no—
334C22⋊C8—44no—
335C22⋊C8—44no—
336C22⋊C8—44no—
337C22⋊C8—44no—
338C22⋊C8—44no—
339C22⋊C8—44no—
340C22⋊C8—44no—
341C22⋊C8—44no—
342#97—28no—
343#97—44no—
344C4×C8—44no—
345#95C22×C414yes—
346#95C22×C414yes—
347#95C22×C414yes—
348#95C22×C414yes—
349#95C22×C414yes—
350#95C22×C414yes—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—41no—
357C24—41no—
358C24C23×C414yes—
359C24C23×C411yescharacteristic
360C24C23×C411yescharacteristic
361C24C23×C411yescharacteristic
362#41—88no—
363#41—88no—
364C2×Q8—24no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—28no—
371C2×D4—28no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—44no—
376C2×D4—42no—
377C2×D4—44no—
378C2×D4—42no—
379C2×D4—44no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—44no—
384C2×D4—42no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—416no—
447C22×C4—416no—
448C22×C4#13114yes—
449C22×C4C23×C411yescharacteristic
450C22×C4C23×C411yescharacteristic
451C22×C4C23×C411yescharacteristic
452C22×C4C23×C414yes—
453C22×C4C23×C414yes—
454C22×C4C23×C414yes—
455C22×C4C23×C411yescharacteristic
456C22×C4C23×C411yescharacteristic
457C22×C4C23×C411yescharacteristic
458C22×C4C23×C411yescharacteristic
459C22×C4C23×C411yescharacteristic
460C22×C4C23×C411yescharacteristic
461C22×C4C23×C411yescharacteristic
462C22×C4C23×C411yescharacteristic
463C22×C4H(5,𝔽2)14yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+412yes—
466C22×C4C2511yescharacteristic
467M16—88no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—84no—
481C2×C8—84no—
482C2×C8—88no—
483C4⋊C4—24no—
484C4⋊C4—28no—
485C4⋊C4—24no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C22⋊C4—28no—
494C22⋊C4—28no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—28no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—24no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42H(5,𝔽2)14yes—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—28no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23#51812yes—
553C23#52012yes—
554C23C2×#13112yes—
555C23#56812yes—
556C23#56812yes—
557C23C24×C411yescharacteristic
558Q8—84no—
559D4—416no—
560D4—84no—
561D4—84no—
562D4—84no—
563D4—88no—
564D4—88no—
565C2×C4—28no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4#51812yes—
626C2×C4#51812yes—
627C2×C4#51812yes—
628C2×C4#52012yes—
629C2×C4C2×#13112yes—
630C2×C4C2×#13112yes—
631C2×C4#56812yes—
632C2×C4C2×H(5,𝔽2)12yes—
633C2×C4C2×H(5,𝔽2)12yes—
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×2−1+411yescharacteristic
636C8—44no—
637C8—44no—
638C8—44no—
639C8—44no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—416no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22#298114yes—
669C22C2×#51811yescharacteristic
670C22C2×#51811yescharacteristic
671C22C2×#52011yescharacteristic
672C22C22×#13111yescharacteristic
673C22C2×#56811yescharacteristic
674C22C2×#56811yescharacteristic
675C22C2×#56811yescharacteristic
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—44no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—44no—
689C4—44no—
690C4#353612yes—
691C4#353712yes—
692C2—24no—
693C2—42no—
694C2—42no—
695C2—84no—
696C2#3437911yescharacteristic
697C2C2×#298111yescharacteristic
698C2#6023111yescharacteristic
699C2#6023311yescharacteristic
700C2#6023711yescharacteristic
701C2#6026511yescharacteristic
702C2#6026911yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radical#36624C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23329C2semidirect
3. #22987C2semidirect
4. #22628C2semidirect
5. #22403C2semidirect
6. #22398C2semidirect
7. #22176C2semidirect
8. #22168C2semidirect
9. #22143C2semidirect
10. #21937C2semidirect
11. #21911C2semidirect
12. #20715C2semidirect
13. C2×#1797C2semidirect
18. C2×#435C22semidirect
41. #1955C22semidirect
43. #1940C22semidirect
44. #1924C22semidirect
45. #1924C22semidirect
46. #1923C22semidirect
47. #1923C22semidirect
48. #1917C22semidirect
49. #1916C22semidirect
51. #1870C22semidirect
55. #1869C22semidirect
56. #1851C22semidirect
57. #1851C22semidirect
58. #1850C22semidirect
59. #1850C22semidirect
71. #1771C22semidirect

Maximal subgroups

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

subgroupindex
#366242
#233292
#229872
#226282
#224032
#223982
#221762
#221682
#221432
#219372
#219112
#207152
C2×#17972