Finite Group Databasebeta

← #6583854

Subgroups — #6583854

Order 512.

1918 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3367C211yesmaximal, characteristic
2C2×#3360C211yesmaximal, characteristic
3#36276C211yesmaximal, characteristic, complemented
4#36188C211yesmaximal, characteristic, complemented
5#33236C211yesmaximal, characteristic, complemented
6#33201C211yesmaximal, characteristic, complemented
7#33162C211yesmaximal, characteristic, complemented
8#28647C211yesmaximal, characteristic, complemented
9#28639C211yesmaximal, characteristic, complemented
10#27784C214yesmaximal, complemented
11#26474C214yesmaximal, complemented
12#25934C211yesmaximal, characteristic, complemented
13#25929C211yesmaximal, characteristic, complemented
14#25650C211yesmaximal, characteristic, complemented
15#25646C211yesmaximal, characteristic, complemented
16#24745C214yesmaximal
17#23012C214yesmaximal, complemented
18#21714C211yesmaximal, characteristic, complemented
19C4×#442C211yesmaximal, characteristic
20C2×#556C2211yescharacteristic
21C2×#556C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×D4×Q8C2211yescharacteristic
24#3367—28no—
25#3360—28no—
26C2×#489C2212yes—
27C2×#488C2212yes—
28C2×#487C2211yescharacteristic
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#487C2211yescharacteristic
32C2×#479C2212yes—
33C2×#477C2212yes—
34C2×#476C2211yescharacteristic
35C2×#474C2211yescharacteristic
36C2×#463C2212yes—
37C2×#463C2212yes—
38C2×#461C2211yescharacteristic
39C2×#461C2211yescharacteristic
40C2×#461C2214yes—
41C2×#446C2211yescharacteristic
42C2×C8×Q8C2211yescharacteristic
43C2×#442C2212yes—
44C2×#440C2212yes—
45C2×C4×SD16C2212yes—
46C2×C4×D8C2211yescharacteristic
47#2868C2211yescharacteristic
48#2862C2211yescharacteristic
49#2857—21no—
50#2857—21no—
51#2568C2211yescharacteristic
52#2568C2211yescharacteristic
53#2567C2211yescharacteristic
54#2550C2214yes—
55#2549—24no—
56#2547C2214yes—
57#2546C2211yescharacteristic
58#2502—21no—
59#2499—21no—
60#2499—21no—
61#2496—24no—
62#2434—21no—
63#2365C2214yes—
64#2349C2214yes—
65C4×C4⋊Q8C2211yescharacteristic
66C2×#391—21no—
67#2285—21no—
68C42×D4C2211yescharacteristic
69C42×D4C2211yescharacteristic
70#2066C2214yes—
71#2066C2214yes—
72#2042C2214yes—
73#2042C2214yes—
74#2026C2214yes—
75#2024C2214yes—
76#1988C2211yescharacteristic
77#1987C2211yescharacteristic
78#1984C2211yescharacteristic
79#1984C2211yescharacteristic
80#1981C2211yescharacteristic
81#1981C2211yescharacteristic
82#1981C2214yes—
83#1960C2214yes—
84#1960C2214yes—
85#1949C2214yes—
86#1940C2214yes—
87#1935C2214yes—
88#1933C2214yes—
89#1877C2214yes—
90#1877C2214yes—
91#1865C2214yes—
92#1863C2214yes—
93#1842C2211yescharacteristic
94#1804C2214yes—
95#1776C2212yes—
96C4×#106C2211yescharacteristic
97#1771C2214yes—
98#1762C2212yes—
99#1762C2212yes—
100C4×#102C2211yescharacteristic
101C4×#102C2211yescharacteristic
102C4×#97C2211yescharacteristic
103C23×Q8—21no—
104C22×SD16C2312yes—
105C22×SD16C2312yes—
106C22×D8C2311yescharacteristic
107#556—24no—
108#556—24no—
109#556—42no—
110#556—42no—
111#554—24no—
112#554—42no—
113D4×Q8—24no—
114D4×Q8—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8D411yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2314yes—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2312yes—
123C2×#126—22no—
124C2×#126—22no—
125C2×#125C2312yes—
126C2×#124C2312yes—
127C2×#124C2312yes—
128C2×C22⋊Q8—21no—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—22no—
133C2×C22⋊Q8—22no—
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2311yescharacteristic
137C2×C4×Q8C2311yescharacteristic
138C2×C4×D4C2312yes—
139C2×C4×D4C2312yes—
140C2×C4×D4C2312yes—
141C2×C4×D4C2311yescharacteristic
142C2×C4×D4C2311yescharacteristic
143C2×C4×D4C2312yes—
144C2×#117—21no—
145C2×#117—21no—
146C22×C4⋊C4—21no—
147C22×C42—21no—
148C22×C42—21no—
149#489—28no—
150#488—28no—
151#487—24no—
152#487—24no—
153#487—42no—
154#487—42no—
155#479—28no—
156#477—28no—
157#476—24no—
158#474—24no—
159#463—28no—
160#463—28no—
161#461—24no—
162#461—24no—
163#461—216no—
164#446—24no—
165C8×Q8—24no—
166#442—28no—
167#440—28no—
168C4×SD16—28no—
169C4×D8D418yes—
170C2×#107C2312yes—
171C2×#107C2311yescharacteristic
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2314yes—
177C2×C4⋊C8C2311yescharacteristic
178C2×C4⋊C8C2311yescharacteristic
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#103C2314yes—
188C2×#103C2314yes—
189C2×#102C2312yes—
190C2×#102C2312yes—
191C2×#102C2312yes—
192C2×#102C2312yes—
193C2×#102C2311yescharacteristic
194C2×#102C2311yescharacteristic
195C2×#97C2312yes—
196C2×#97C2312yes—
197C2×#97C2312yes—
198C2×C4×C8C2311yescharacteristic
199C2×C4×C8C2312yes—
200C2×C4×C8C2311yescharacteristic
201#400—22no—
202#400—22no—
203#398—24no—
204#398—22no—
205#397—24no—
206#397—24no—
207#395—22no—
208#395—24no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—21no—
214#391—21no—
215#391—22no—
216#391—24no—
217#391D411yescharacteristic
218#391D411yescharacteristic
219#391C2314yes—
220#391C2314yes—
221#389—24no—
222#389—24no—
223#389D414yes—
224#389C2314yes—
225#389C2314yes—
226#387—24no—
227#387—24no—
228#385—24no—
229#385—24no—
230#384—21no—
231#384—21no—
232#384—22no—
233#384—21no—
234#384D411yescharacteristic
235#382—21no—
236#382—21no—
237#382C2314yes—
238#380—24no—
239#380—24no—
240#380—24no—
241C4×C4⋊C4—24no—
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2311yescharacteristic
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2314yes—
247C4×C22⋊C4—21no—
248C4×C22⋊C4—22no—
249C4×C22⋊C4—21no—
250C4×C22⋊C4—22no—
251C2×#95—24no—
252C43C2311yescharacteristic
253#340C2314yes—
254#340C2314yes—
255#340C2312yes—
256#340C2312yes—
257#340C2312yes—
258#336C2314yes—
259#336C2314yes—
260C22×Q8—21no—
261C22×Q8—22no—
262C22×Q8—21no—
263C22×Q8—44no—
264C22×Q8—44no—
265C22×Q8—82no—
266C22×Q8C2×D411yescharacteristic
267C22×Q8C2×D411yescharacteristic
268C22×Q8C2412yes—
269C22×Q8C2412yes—
270C22×D4C2411yescharacteristic
271C22×D4C2411yescharacteristic
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—22no—
275C23×C4—22no—
276C23×C4—22no—
277C23×C4—22no—
278C2×SD16—24no—
279C2×SD16—24no—
280C2×SD16—28no—
281C2×SD16—28no—
282C2×D8—24no—
283C2×D8C2×D414yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2411yescharacteristic
288C22×C8C2411yescharacteristic
289C4⋊Q8—24no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—28no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—28no—
300C4⋊Q8—42no—
301C4⋊Q8—42no—
302C4⋊Q8—82no—
303#126—48no—
304#126—48no—
305#125—24no—
306#125—42no—
307#124—24no—
308#124—24no—
309#124—42no—
310#124—42no—
311C22⋊Q8—48no—
312C22⋊Q8—48no—
313C22⋊Q8—44no—
314C22⋊Q8—44no—
315C22⋊Q8—44no—
316C22⋊Q8—44no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8—24no—
322C4×Q8—24no—
323C4×Q8—42no—
324C4×Q8—42no—
325C4×D4—24no—
326C4×D4—24no—
327C4×D4—24no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—24no—
331C4×D4—24no—
332C4×D4—42no—
333C4×D4—42no—
334C4×D4C2×D414yes—
335C4×D4C2×D414yes—
336C4×D4#4118yes—
337#117—44no—
338#117—44no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—44no—
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2×D412yes—
359C2×C4⋊C4C2×D414yes—
360C2×C4⋊C4C2×D414yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2411yescharacteristic
365C2×C4⋊C4C2411yescharacteristic
366C2×C4⋊C4C2412yes—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—22no—
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—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42—22no—
383C2×C42—44no—
384C2×C42—82no—
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D412yes—
388C2×C42C2412yes—
389C2×C42C2412yes—
390C2×C42C2411yescharacteristic
391C2×C42C2412yes—
392#107—24no—
393#107C2×D414yes—
394#106—24no—
395#106—24no—
396C4⋊C8—24no—
397C4⋊C8—24no—
398C4⋊C8—22no—
399C4⋊C8—22no—
400C4⋊C8—28no—
401#103—24no—
402#103—24no—
403#103—24no—
404#103—24no—
405#103—24no—
406#103—24no—
407#103—28no—
408#103—28no—
409#103—44no—
410#103—44no—
411#102—24no—
412#102—24no—
413#102—24no—
414#102—24no—
415#102C2×D414yes—
416#102C2×D414yes—
417#97—24no—
418#97—24no—
419#97—24no—
420C4×C8—24no—
421C4×C8—22no—
422C4×C8C2×D414yes—
423#95—24no—
424#95—24no—
425#95—24no—
426#95—22no—
427#95—21no—
428#95—22no—
429#95—21no—
430#95—24no—
431#95—24no—
432#95—24no—
433#95—24no—
434#95—24no—
435#95C2×D414yes—
436#95C2×D414yes—
437C24—21no—
438C24—21no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—24no—
448C2×Q8—24no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—24no—
452C2×Q8—24no—
453C2×Q8—44no—
454C2×Q8—44no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—82no—
460C2×Q8—82no—
461C2×Q8—816no—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4C22×D412yes—
468C2×D4C22×D412yes—
469C2×D4C2×#4114yes—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—28no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—44no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4—82no—
494C22×C4—84no—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C2511yescharacteristic
504SD16—48no—
505SD16—48no—
506D8—28no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8C22×D412yes—
517C2×C8C22×D412yes—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—48no—
545C4⋊C4—48no—
546C4⋊C4—84no—
547C4⋊C4—84no—
548C4⋊C4C22×D412yes—
549C4⋊C4C22×D412yes—
550C4⋊C4C2×#4114yes—
551C22⋊C4—24no—
552C22⋊C4—24no—
553C22⋊C4—28no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—42no—
568C42—44no—
569C42—44no—
570C42—44no—
571C42—42no—
572C42—42no—
573C42—82no—
574C42C22×D412yes—
575C42C2×#4114yes—
576C42H(5,𝔽2)12yes—
577C422−1+414yes—
578C23—22no—
579C23—24no—
580C23—22no—
581C23—42no—
582C23—41no—
583C23—41no—
584C23—42no—
585C23—42no—
586C23—41no—
587C23—41no—
588C23C23×D411yescharacteristic
589Q8—44no—
590Q8—44no—
591Q8—42no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—44no—
596Q8—44no—
597Q8—88no—
598Q8—84no—
599Q8—162no—
600D4—22no—
601D4—22no—
602D4—22no—
603D4—24no—
604D4—42no—
605D4#53214yes—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
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—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—48no—
631C2×C4—48no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—88no—
645C2×C4—88no—
646C2×C4—88no—
647C2×C4—164no—
648C2×C4D4212yes—
649C2×C4D4212yes—
650C2×C4#54814yes—
651C2×C4#54814yes—
652C2×C4#57512yes—
653C2×C4#57712yes—
654C2×C4C23×D411yescharacteristic
655C2×C4C22×#4112yes—
656C2×C4C2×H(5,𝔽2)11yescharacteristic
657C2×C4C2×2−1+412yes—
658C8—22no—
659C8—22no—
660C8—42no—
661C8—42no—
662C8—42no—
663C22—24no—
664C22—42no—
665C22—44no—
666C22—42no—
667C22—44no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—82no—
672C22—82no—
673C22C2×D4211yescharacteristic
674C22C2×#54812yes—
675C22C2×#57511yescharacteristic
676C22C2×#57711yescharacteristic
677C22#358612yes—
678C4—24no—
679C4—22no—
680C4—24no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—44no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4#326714yes—
694C4C2×#53211yescharacteristic
695C4#349911yescharacteristic
696C4#351112yes—
697C2—44no—
698C2—82no—
699C2#3612211yescharacteristic
700C2C2×#326711yescharacteristic
701C2#6108311yescharacteristic
702C2#6132312yes—
703C2#6134012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D8C23
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
3. #36276C2semidirect
4. #36188C2semidirect
5. #33236C2semidirect
6. #33201C2semidirect
7. #33162C2semidirect
8. #28647C2semidirect
9. #28639C2semidirect
10. #27784C2semidirect
11. #26474C2semidirect
12. #25934C2semidirect
13. #25929C2semidirect
14. #25650C2semidirect
15. #25646C2semidirect
17. #23012C2semidirect
18. #21714C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33672
C2×#33602
#362762
#361882
#332362
#332012
#331622
#286472
#286392
#277842
#264742
#259342
#259292
#256502
#256462
#247452
#230122
#217142
C4×#4422