Finite Group Databasebeta

← #6630807

Subgroups — #6630807

Order 512.

1626 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35291C211yesmaximal, characteristic, complemented
2#35288C211yesmaximal, characteristic
3#33287C211yesmaximal, characteristic, complemented
4#33286C211yesmaximal, characteristic, complemented
5#33271C214yesmaximal, complemented
6#33256C211yesmaximal, characteristic, complemented
7#33239C211yesmaximal, characteristic, complemented
8#33207C211yesmaximal, characteristic, complemented
9#33170C211yesmaximal, characteristic
10#33137C214yesmaximal, complemented
11#33049C211yesmaximal, characteristic, complemented
12#33048C211yesmaximal, characteristic
13C4×#498C211yesmaximal, characteristic, complemented
14C4×#492C211yesmaximal, characteristic, complemented
15#21647C211yesmaximal, characteristic, complemented
16#21646C211yesmaximal, characteristic, complemented
17#21523C214yesmaximal, complemented
18#21280C214yesmaximal, complemented
19C4×#405C211yesmaximal, characteristic, complemented
20C2×#498C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#465C2214yes—
25C2×C4×SD16C2214yes—
26C2×#405C2212yes—
27C2×#405C2212yes—
28C2×C4×M16C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30#2862C2211yescharacteristic
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2858C2211yescharacteristic
34#2857C2211yescharacteristic
35#2857C2211yescharacteristic
36#2857C2211yescharacteristic
37#2856C2211yescharacteristic
38#2645C2212yes—
39#2644C2212yes—
40#2625C2212yes—
41#2624C2212yes—
42#2514—21no—
43#2513—21no—
44#2509—21no—
45#2508—21no—
46#2495—21no—
47#2494—21no—
48#2490—21no—
49#2489—21no—
50#2454—21no—
51#2453—21no—
52#2448—21no—
53#2447—21no—
54#2427—21no—
55#2426—21no—
56#2421—21no—
57#2420—21no—
58#2324C2212yes—
59#2323C2212yes—
60#2320C2212yes—
61#2319C2212yes—
62C4×C4⋊Q8C2211yescharacteristic
63C4×C4⋊Q8C2211yescharacteristic
64C4×C4⋊Q8C2211yescharacteristic
65C4×#127C2211yescharacteristic
66C4×#117C2211yescharacteristic
67#1990C2211yescharacteristic
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1968C2214yes—
72#1968C2214yes—
73#1968C2214yes—
74#1968C2214yes—
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1956C2211yescharacteristic
78#1956C2211yescharacteristic
79#1941C2214yes—
80#1941C2214yes—
81#1941C2214yes—
82#1941C2214yes—
83#1936C2214yes—
84#1936C2214yes—
85#1936C2214yes—
86#1936C2214yes—
87#1935C2214yes—
88#1935C2214yes—
89#1935C2214yes—
90#1935C2214yes—
91#1895C2214yes—
92#1894C2214yes—
93#1844C2211yescharacteristic
94#1844C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1807C2214yes—
98C4×#106C2214yes—
99#1763C2214yes—
100#1762C2214yes—
101#1726C2212yes—
102#1726C2212yes—
103C4×#97C2211yescharacteristic
104C4×#97C2211yescharacteristic
105C42×C8C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×SD16C2314yes—
108C22×M16C2312yes—
109C23×C8C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×#127C2311yescharacteristic
114C2×#125—21no—
115C2×#125—21no—
116C2×#124—21no—
117C2×#124—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8C2312yes—
121C2×#121—21no—
122C2×#121—21no—
123C2×#121C2312yes—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2312yes—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4C2312yes—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#498—24no—
134#492—24no—
135#487—24no—
136#486—24no—
137#465—216no—
138C4×SD16—216no—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148#405—28no—
149#405—28no—
150C4×M16—24no—
151C2×#97C2312yes—
152C2×#97C2312yes—
153C2×#97C2312yes—
154C2×#97C2312yes—
155C2×#97C2312yes—
156C2×C4×C8—24no—
157C2×C4×C8C2312yes—
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2312yes—
161C2×C4×C8C2312yes—
162C2×C4×C8C2312yes—
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2312yes—
166#401—22no—
167#400—22no—
168#400—22no—
169#400—22no—
170#400—22no—
171#399—22no—
172#398—22no—
173#397—22no—
174#397—22no—
175#397—22no—
176#397—22no—
177#397—22no—
178#397—22no—
179#396—22no—
180#394—22no—
181#394—22no—
182#391—21no—
183#391—21no—
184#391—21no—
185#391—21no—
186#391—21no—
187#391—21no—
188#391C2312yes—
189#391C2312yes—
190#391C2312yes—
191#391C2312yes—
192#390—22no—
193#390—21no—
194#390—21no—
195#390—21no—
196#390—21no—
197#390—21no—
198#390—21no—
199#390C2312yes—
200#390C2312yes—
201#390C2312yes—
202#390C2312yes—
203#389—22no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389—21no—
216#389C2312yes—
217#389C2312yes—
218#389C2312yes—
219#389C2312yes—
220#389C2312yes—
221#389C2312yes—
222#389C2312yes—
223#389C2312yes—
224#388—22no—
225#388—22no—
226#388—22no—
227#388—22no—
228#388—22no—
229#387—22no—
230#387—22no—
231#387—22no—
232#386—22no—
233#385—22no—
234#384C2311yescharacteristic
235#384C2311yescharacteristic
236#384C2311yescharacteristic
237#384C2311yescharacteristic
238#383—22no—
239#382—21no—
240#382—21no—
241#382—21no—
242#382—21no—
243#382—22no—
244#381—22no—
245#380—21no—
246#380—21no—
247#380—21no—
248#380—21no—
249#380—22no—
250#380C2312yes—
251#380C2312yes—
252C4×C4⋊C4—21no—
253C4×C4⋊C4—21no—
254C4×C4⋊C4—21no—
255C4×C4⋊C4—21no—
256C4×C4⋊C4D411yescharacteristic
257C4×C4⋊C4D411yescharacteristic
258C4×C4⋊C4D411yescharacteristic
259C4×C4⋊C4D411yescharacteristic
260C4×C4⋊C4C2312yes—
261C4×C4⋊C4C2312yes—
262C4×C22⋊C4D411yescharacteristic
263C4×C22⋊C4D411yescharacteristic
264C4×C22⋊C4D411yescharacteristic
265C4×C22⋊C4D411yescharacteristic
266#376C2311yescharacteristic
267#376C2311yescharacteristic
268C43C2311yescharacteristic
269C43C2311yescharacteristic
270#340C2314yes—
271#340C2314yes—
272#340C2314yes—
273#340C2314yes—
274#340C2314yes—
275#340C2314yes—
276C22×Q8—21no—
277C22×Q8—21no—
278C22×Q8C2412yes—
279C22×D4—22no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C2412yes—
283C23×C4—22no—
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2411yescharacteristic
287C2×SD16—28no—
288C2×SD16—28no—
289C2×SD16—28no—
290C2×M16—24no—
291C2×M16—24no—
292C2×M16—24no—
293C22×C8—24no—
294C22×C8—24no—
295C22×C8#4118yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306#127—22no—
307#127—22no—
308#125—82no—
309#125—82no—
310#124—82no—
311#124—82no—
312C22⋊Q8—28no—
313C22⋊Q8—82no—
314C22⋊Q8—82no—
315#121—28no—
316#121—82no—
317#121—82no—
318C4×Q8—28no—
319C4×Q8—82no—
320C4×Q8—82no—
321C4×D4—28no—
322C4×D4—82no—
323C4×D4—82no—
324#117—24no—
325#117—24no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396#106—28no—
397#106—28no—
398#106—28no—
399#103—28no—
400#103—28no—
401#103—28no—
402#102—28no—
403#102—28no—
404#102—28no—
405#97—24no—
406#97—24no—
407#97—24no—
408#97—24no—
409#97—24no—
410C4×C8—24no—
411C4×C8—24no—
412C4×C8—24no—
413C4×C8—24no—
414C4×C8—22no—
415C4×C8—22no—
416C4×C8#4114yes—
417C4×C8#4114yes—
418C4×C8#4118yes—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447C24—22no—
448C24C22×D411yescharacteristic
449C2×Q8—22no—
450C2×Q8—24no—
451C2×Q8—24no—
452C2×Q8—22no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×D4—22no—
460C2×D4—24no—
461C2×D4—24no—
462C2×D4—22no—
463C2×D4—44no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—82no—
471C2×D4—82no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—44no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C2×#4114yes—
505C22×C4C2511yescharacteristic
506SD16—416no—
507M16—216no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—216no—
515C2×C8C2×#4114yes—
516C2×C8C2×#4114yes—
517C2×C8C2×#4114yes—
518C2×C8C2×#4114yes—
519C2×C8C2×#4114yes—
520C2×C8C2×#4114yes—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—21no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—21no—
587C42—44no—
588C42—82no—
589C42—82no—
590C42C2×#4112yes—
591C42C2×#4112yes—
592C42C2×#4114yes—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23#53212yes—
607C23C23×D411yescharacteristic
608Q8—44no—
609Q8—44no—
610Q8—84no—
611Q8—84no—
612D4—44no—
613D4—44no—
614D4—84no—
615D4—84no—
616D4—164no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—164no—
673C2×C4#53212yes—
674C2×C4#53212yes—
675C2×C4#53212yes—
676C2×C4#53212yes—
677C2×C4#53212yes—
678C2×C4#57712yes—
679C2×C4#57712yes—
680C2×C4#57712yes—
681C2×C4#57712yes—
682C2×C4C22×#4111yescharacteristic
683C2×C4C22×#4112yes—
684C2×C4C22×#4112yes—
685C2×C4C22×#4111yescharacteristic
686C8—24no—
687C8—24no—
688C8—24no—
689C8#52918yes—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22—41no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22C2×#53211yescharacteristic
705C22C2×#53211yescharacteristic
706C22C2×#53211yescharacteristic
707C22C2×#57711yescharacteristic
708C22C2×#57711yescharacteristic
709C22C2×#57711yescharacteristic
710C22C2×#57711yescharacteristic
711C4—22no—
712C4—22no—
713C4—22no—
714C4—41no—
715C4—41no—
716C4—41no—
717C4—41no—
718C4—41no—
719C4—41no—
720C4—41no—
721C4—41no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4#317314yes—
730C4#317414yes—
731C4C2×#52912yes—
732C4#344412yes—
733C2—42no—
734C2—162no—
735C2#3663111yescharacteristic
736C2C2×#317311yescharacteristic
737C2C2×#317411yescharacteristic
738C2#6071211yescharacteristic
739C2#6071211yescharacteristic
740C2#6071611yescharacteristic
741C2#6071611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35291C2semidirect
3. #33287C2semidirect
4. #33286C2semidirect
5. #33271C2semidirect
6. #33256C2semidirect
7. #33239C2semidirect
8. #33207C2semidirect
10. #33137C2semidirect
11. #33049C2semidirect
13. C4×#498C2semidirect
14. C4×#492C2semidirect
15. #21647C2semidirect
16. #21646C2semidirect
17. #21523C2semidirect
18. #21280C2semidirect
19. C4×#405C2semidirect

Maximal subgroups

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

subgroupindex
#352912
#352882
#332872
#332862
#332712
#332562
#332392
#332072
#331702
#331372
#330492
#330482
C4×#4982
C4×#4922
#216472
#216462
#215232
#212802
C4×#4052