Finite Group Databasebeta

← #6583860

Subgroups — #6583860

Order 512.

1920 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3372C211yesmaximal, characteristic, complemented
2C2×#3361C211yesmaximal, characteristic
3#36538C211yesmaximal, characteristic, complemented
4#36180C211yesmaximal, characteristic, complemented
5#33238C211yesmaximal, characteristic, complemented
6#33198C211yesmaximal, characteristic, complemented
7#33164C211yesmaximal, characteristic
8#28645C211yesmaximal, characteristic, complemented
9#28638C211yesmaximal, characteristic, complemented
10#27781C214yesmaximal, complemented
11#26477C214yesmaximal, complemented
12#25934C211yesmaximal, characteristic, complemented
13#25930C211yesmaximal, characteristic
14#25651C211yesmaximal, characteristic, complemented
15#25648C211yesmaximal, characteristic, complemented
16#24740C214yesmaximal, complemented
17#23011C214yesmaximal, complemented
18#21714C211yesmaximal, characteristic, complemented
19C4×#440C211yesmaximal, characteristic, complemented
20C2×#559C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3372—28no—
25#3361—28no—
26C2×#491C2212yes—
27C2×#488C2212yes—
28C2×#487C2211yescharacteristic
29C2×#487C2211yescharacteristic
30C2×#486C2211yescharacteristic
31C2×#486C2211yescharacteristic
32C2×#479C2212yes—
33C2×#477C2211yescharacteristic
34C2×#477C2211yescharacteristic
35C2×#478C2212yes—
36C2×#463C2212yes—
37C2×#463C2214yes—
38C2×#461C2211yescharacteristic
39C2×#461C2212yes—
40C2×#459C2211yescharacteristic
41C2×#446C2211yescharacteristic
42C2×C8×Q8C2211yescharacteristic
43C2×#440C2212yes—
44C2×#440C2212yes—
45C2×C4×SD16C2211yescharacteristic
46C2×C4×D8C2212yes—
47#2868C2211yescharacteristic
48#2861—21no—
49#2858C2211yescharacteristic
50#2857—21no—
51#2568C2211yescharacteristic
52#2567C2211yescharacteristic
53#2567C2211yescharacteristic
54#2566—21no—
55#2558—24no—
56#2552C2214yes—
57#2548C2211yescharacteristic
58#2545—21no—
59#2545C2214yes—
60#2502—21no—
61#2502—21no—
62#2440—24no—
63#2367C2214yes—
64D4×C4⋊C4C2214yes—
65C4×C4⋊Q8C2211yescharacteristic
66C4×#124—21no—
67C2×#384—21no—
68C42×Q8C2211yescharacteristic
69C42×D4C2211yescharacteristic
70#2068C2214yes—
71#2067C2214yes—
72#2043C2214yes—
73#2042C2214yes—
74#2026C2214yes—
75#2025C2214yes—
76#1987C2211yescharacteristic
77#1987C2211yescharacteristic
78#1984C2211yescharacteristic
79#1984C2211yescharacteristic
80#1983C2214yes—
81#1982C2211yescharacteristic
82#1981C2211yescharacteristic
83#1959C2214yes—
84#1959C2214yes—
85#1947C2214yes—
86#1940C2214yes—
87#1936C2214yes—
88#1933C2214yes—
89#1879C2214yes—
90#1877C2214yes—
91#1863C2214yes—
92#1862C2214yes—
93#1842C2211yescharacteristic
94#1803C2214yes—
95#1776C2212yes—
96C4×#107C2211yescharacteristic
97#1771C2214yes—
98#1763C2212yes—
99#1762C2212yes—
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102C4×#97C2211yescharacteristic
103C22×SD16C2312yes—
104C22×SD16C2311yescharacteristic
105C22×D8C2312yes—
106#559—24no—
107#559—42no—
108#557—24no—
109#557—42no—
110#554—24no—
111#554—24no—
112#554—42no—
113#554—42no—
114C2×C4⋊Q8—22no—
115C2×C4⋊Q8D411yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×#127C2311yescharacteristic
121C2×#125—21no—
122C2×#125—22no—
123C2×#125C2312yes—
124C2×#125C2312yes—
125C2×#124—21no—
126C2×#124—21no—
127C2×#124C2314yes—
128C2×#124C2312yes—
129C2×#123—22no—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—22no—
133C2×C22⋊Q8—22no—
134C2×C4×Q8—21no—
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2311yescharacteristic
138C2×C4×Q8C2311yescharacteristic
139C2×C4×Q8C2311yescharacteristic
140C2×C4×D4—21no—
141C2×C4×D4C2312yes—
142C2×C4×D4C2312yes—
143C2×C4×D4C2312yes—
144C2×C4×D4C2311yescharacteristic
145C2×C4×D4C2312yes—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—24no—
149C22×C42—21no—
150#491—28no—
151#488—28no—
152#487—24no—
153#487—42no—
154#486—24no—
155#486—42no—
156#479—28no—
157#478—28no—
158#477—24no—
159#477—24no—
160#463—28no—
161#463—216no—
162#461—28no—
163#461—24no—
164#459—24no—
165#446—24no—
166C8×Q8—24no—
167#440—28no—
168#440—28no—
169C4×SD16D418yes—
170C4×D8—28no—
171C2×#107C2312yes—
172C2×#107C2312yes—
173C2×#107C2312yes—
174C2×#106C2311yescharacteristic
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2311yescharacteristic
178C2×C4⋊C8C2312yes—
179C2×C4⋊C8C2314yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2314yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#103C2314yes—
188C2×#103C2311yescharacteristic
189C2×#102C2312yes—
190C2×#102C2312yes—
191C2×#102C2314yes—
192C2×#102C2312yes—
193C2×#102C2312yes—
194C2×#102C2314yes—
195C2×#102C2311yescharacteristic
196C2×#97C2312yes—
197C2×#97C2312yes—
198C2×#97C2312yes—
199C2×C4×C8C2311yescharacteristic
200C2×C4×C8C2312yes—
201C2×C4×C8C2311yescharacteristic
202#400—22no—
203#398—24no—
204#398—24no—
205#396—24no—
206#396—24no—
207#395—22no—
208#395—22no—
209#394—22no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391D411yescharacteristic
214#391D411yescharacteristic
215#391C2314yes—
216#390—21no—
217#390—21no—
218#390—21no—
219#390—22no—
220#390C2314yes—
221#389—24no—
222#389D414yes—
223#389C2314yes—
224#389C2314yes—
225#388—24no—
226#388—24no—
227#388—24no—
228#388—24no—
229#385—24no—
230#385—24no—
231#385—24no—
232#384—21no—
233#384—21no—
234#384—24no—
235#384—21no—
236#384—21no—
237#384D411yescharacteristic
238#384C2314yes—
239#382—21no—
240#380—24no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4C2311yescharacteristic
244C4×C4⋊C4C2311yescharacteristic
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2314yes—
248C4×C22⋊C4—22no—
249C4×C22⋊C4—21no—
250C4×C22⋊C4—24no—
251C43—21no—
252C43C2311yescharacteristic
253#340C2314yes—
254#340C2314yes—
255#340C2312yes—
256#340C2312yes—
257#340C2312yes—
258#336C2314yes—
259#336C2314yes—
260C22×Q8—21no—
261C22×Q8—21no—
262C22×Q8C2×D411yescharacteristic
263C22×Q8C2×D411yescharacteristic
264C22×Q8C2412yes—
265C22×Q8C2411yescharacteristic
266C22×D4—21no—
267C22×D4—21no—
268C22×D4C2412yes—
269C22×D4C2411yescharacteristic
270C23×C4—21no—
271C23×C4—22no—
272C23×C4—22no—
273C23×C4—22no—
274C23×C4—22no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—28no—
278C2×SD16C2×D414yes—
279C2×D8—24no—
280C2×D8—28no—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2411yescharacteristic
285C22×C8C2411yescharacteristic
286C4⋊Q8—24no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—42no—
294C4⋊Q8—42no—
295C4⋊Q8—84no—
296#127—22no—
297#127—22no—
298#125—24no—
299#125—24no—
300#125—42no—
301#125—42no—
302#125—44no—
303#125—48no—
304#124—24no—
305#124—28no—
306#124—28no—
307#124—42no—
308#124—44no—
309#124—44no—
310#123—48no—
311C22⋊Q8—48no—
312C22⋊Q8—48no—
313C22⋊Q8—44no—
314C22⋊Q8—44no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—24no—
321C4×Q8—24no—
322C4×Q8—24no—
323C4×Q8—42no—
324C4×Q8—44no—
325C4×Q8C2×D414yes—
326C4×D4—24no—
327C4×D4—24no—
328C4×D4—24no—
329C4×D4—22no—
330C4×D4—24no—
331C4×D4—42no—
332C4×D4—42no—
333C4×D4—42no—
334C4×D4—44no—
335C4×D4C2×D414yes—
336C4×D4#4118yes—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—44no—
355C2×C4⋊C4—44no—
356C2×C4⋊C4—88no—
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D412yes—
361C2×C4⋊C4C2×D414yes—
362C2×C4⋊C4C2×D414yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2411yescharacteristic
367C2×C4⋊C4C2411yescharacteristic
368C2×C4⋊C4C2412yes—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—24no—
380C2×C42—21no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—21no—
384C2×C42—22no—
385C2×C42—21no—
386C2×C42—22no—
387C2×C42—44no—
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D412yes—
391C2×C42C2412yes—
392C2×C42C2412yes—
393C2×C42C2411yescharacteristic
394C2×C42C2412yes—
395#107—24no—
396#107—24no—
397#107—24no—
398#106C2×D414yes—
399C4⋊C8—24no—
400C4⋊C8—24no—
401C4⋊C8—22no—
402C4⋊C8—22no—
403C4⋊C8—28no—
404#103—24no—
405#103—24no—
406#103—24no—
407#103—24no—
408#103—24no—
409#103—24no—
410#103—28no—
411#103—44no—
412#103C2×D414yes—
413#102—24no—
414#102—24no—
415#102—24no—
416#102—24no—
417#102—28no—
418#102—44no—
419#102C2×D414yes—
420#97—24no—
421#97—24no—
422#97—24no—
423C4×C8—24no—
424C4×C8—22no—
425C4×C8C2×D414yes—
426#95—21no—
427#95—22no—
428#95—24no—
429#95—22no—
430#95—21no—
431#95—24no—
432#95—24no—
433#95—24no—
434#95C2×D414yes—
435#95C2×D414yes—
436C24—21no—
437C24—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—21no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—24no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—24no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—82no—
454C2×Q8—82no—
455C2×Q8C22×D412yes—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—21no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—82no—
467C2×D4C22×D412yes—
468C2×D4C2×#4114yes—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—24no—
474C22×C4—28no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—84no—
487C22×C4—84no—
488C22×C4—84no—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C2511yescharacteristic
498SD16—28no—
499SD16—48no—
500D8—48no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—24no—
507C2×C8—24no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8C22×D412yes—
511C2×C8C22×D412yes—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—48no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—84no—
542C4⋊C4—84no—
543C4⋊C4—84no—
544C4⋊C4C22×D412yes—
545C4⋊C4C22×D412yes—
546C4⋊C4C2×#4114yes—
547C22⋊C4—24no—
548C22⋊C4—28no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—48no—
555C22⋊C4—48no—
556C22⋊C4—48no—
557C22⋊C4—84no—
558C42—24no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—42no—
566C42—44no—
567C42—44no—
568C42—44no—
569C42—42no—
570C42—42no—
571C42—82no—
572C42C22×D412yes—
573C42C2×#4114yes—
574C42H(5,𝔽2)12yes—
575C422−1+414yes—
576C23—22no—
577C23—24no—
578C23—42no—
579C23—41no—
580C23—41no—
581C23—44no—
582C23—42no—
583C23—44no—
584C23—42no—
585C23—42no—
586C23C23×D411yescharacteristic
587Q8—22no—
588Q8—22no—
589Q8—42no—
590Q8—44no—
591Q8—44no—
592Q8—42no—
593Q8—42no—
594Q8—44no—
595Q8—162no—
596Q8—162no—
597D4—22no—
598D4—24no—
599D4—42no—
600D4—42no—
601D4—44no—
602D4—84no—
603D4—162no—
604D4#53214yes—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—416no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—48no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—164no—
642C2×C4—164no—
643C2×C4—164no—
644C2×C4D4212yes—
645C2×C4D4212yes—
646C2×C4#54814yes—
647C2×C4#54814yes—
648C2×C4#57512yes—
649C2×C4#57812yes—
650C2×C4C23×D411yescharacteristic
651C2×C4C22×#4112yes—
652C2×C4C2×H(5,𝔽2)11yescharacteristic
653C2×C4C2×2−1+412yes—
654C8—22no—
655C8—22no—
656C8—42no—
657C8—42no—
658C8—42no—
659C22—24no—
660C22—42no—
661C22—44no—
662C22—42no—
663C22—44no—
664C22—82no—
665C22—82no—
666C22—84no—
667C22—82no—
668C22—84no—
669C22C2×D4211yescharacteristic
670C22C2×#54812yes—
671C22C2×#57511yescharacteristic
672C22C2×#57811yescharacteristic
673C22#358512yes—
674C4—24no—
675C4—22no—
676C4—24no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—44no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4#326914yes—
690C4C2×#53211yescharacteristic
691C4#349911yescharacteristic
692C4#351112yes—
693C2—44no—
694C2—162no—
695C2#3612211yescharacteristic
696C2C2×#326911yescharacteristic
697C2#6108211yescharacteristic
698C2#6131812yes—
699C2#6134812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3372C2semidirect
3. #36538C2semidirect
4. #36180C2semidirect
5. #33238C2semidirect
6. #33198C2semidirect
8. #28645C2semidirect
9. #28638C2semidirect
10. #27781C2semidirect
11. #26477C2semidirect
12. #25934C2semidirect
14. #25651C2semidirect
15. #25648C2semidirect
16. #24740C2semidirect
17. #23011C2semidirect
18. #21714C2semidirect
19. C4×#440C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33722
C2×#33612
#365382
#361802
#332382
#331982
#331642
#286452
#286382
#277812
#264772
#259342
#259302
#256512
#256482
#247402
#230112
#217142
C4×#4402