Finite Group Databasebeta

← #6497358

Subgroups — #6497358

Order 512.

1514 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35298C211yesmaximal, characteristic
2#35286C211yesmaximal, characteristic, complemented
3#33294C211yesmaximal, characteristic, complemented
4#33290C211yesmaximal, characteristic, complemented
5#33286C211yesmaximal, characteristic, complemented
6#33277C211yesmaximal, characteristic, complemented
7#30229C211yesmaximal, characteristic, complemented
8#30228C211yesmaximal, characteristic, complemented
9#30204C211yesmaximal, characteristic, complemented
10#30204C211yesmaximal, characteristic, complemented
11#27829C214yesmaximal, complemented
12#27743C214yesmaximal, complemented
13#27726C214yesmaximal, complemented
14#27710C214yesmaximal, complemented
15#21859C211yesmaximal, characteristic
16#21835C211yesmaximal, characteristic, complemented
17#21410C211yesmaximal, characteristic
18#21322C211yesmaximal, characteristic, complemented
19#20472C211yesmaximal, characteristic, complemented
20C2×#533C2211yescharacteristic
21C2×#532C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24#2868C2211yescharacteristic
25#2862C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2858C2211yescharacteristic
32#2647—21no—
33#2647—21no—
34#2620—21no—
35#2620—21no—
36#2571—21no—
37#2571C2212yes—
38#2566C2212yes—
39#2566C2212yes—
40#2552—21no—
41#2552C2212yes—
42#2545—21no—
43#2545—21no—
44#2500C2212yes—
45#2500C2212yes—
46#2442—21no—
47#2442—21no—
48#2442C2212yes—
49#2442C2212yes—
50#2432—21no—
51#2432—21no—
52#2326—21no—
53#2326—21no—
54#2317—21no—
55#2316—21no—
56#2285C2211yescharacteristic
57C2×#384C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2101C2214yes—
60#2101C2214yes—
61#2099C2214yes—
62#2099C2214yes—
63#2073C2214yes—
64#2073C2214yes—
65#2025C2214yes—
66#2025C2214yes—
67#1992C2214yes—
68#1992C2214yes—
69#1991C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1990C2211yescharacteristic
80#1990C2211yescharacteristic
81#1990C2211yescharacteristic
82#1989C2214yes—
83#1957C2211yescharacteristic
84#1957C2211yescharacteristic
85#1957C2211yescharacteristic
86#1957C2211yescharacteristic
87#1936C2214yes—
88#1936C2214yes—
89#1934C2214yes—
90#1934C2214yes—
91#1895C2214yes—
92#1874C2214yes—
93#1836C2214yes—
94#1835C2214yes—
95#1825C2214yes—
96#1802C2214yes—
97#1771C2212yes—
98#1771C2212yes—
99#1762C2214yes—
100#1753C2212yes—
101#1753C2212yes—
102#1744C2211yescharacteristic
103#1744C2211yescharacteristic
104#1736C2214yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×#41—21no—
108C22×M16C2312yes—
109#533—24no—
110#533—24no—
111#532—24no—
112#532—24no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×#127C2311yescharacteristic
119C2×#125—21no—
120C2×#125—21no—
121C2×#125C2312yes—
122C2×#125C2312yes—
123C2×#125C2312yes—
124C2×#124—21no—
125C2×#124—21no—
126C2×#124—21no—
127C2×#124—21no—
128C2×#124—21no—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—21no—
131C2×#121—21no—
132C2×#121C2312yes—
133C2×C4×Q8—21no—
134C2×C4×Q8—21no—
135C2×C4×D4—21no—
136C2×C4×D4C2312yes—
137C2×#117—21no—
138C2×#117D411yescharacteristic
139C2×#117D411yescharacteristic
140C2×#117C2312yes—
141C22×C4⋊C4—21no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4C2312yes—
144C22×C42C2311yescharacteristic
145#431—24no—
146C2×C4⋊C8C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×C22⋊C8C2312yes—
152C4×M16—24no—
153C2×#97C2312yes—
154C2×#97C2312yes—
155C2×#97C2312yes—
156C2×C4×C8C2312yes—
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2312yes—
162#400—22no—
163#400—22no—
164#400—22no—
165#400—22no—
166#398—22no—
167#398—22no—
168#398—22no—
169#398—22no—
170#396—22no—
171#396—22no—
172#396—22no—
173#396—22no—
174#394—22no—
175#394—22no—
176#394—22no—
177#394—22no—
178#391—21no—
179#391—21no—
180#391—21no—
181#391—21no—
182#390—21no—
183#390—21no—
184#390—21no—
185#390—21no—
186#390C2312yes—
187#390C2312yes—
188#390C2312yes—
189#390C2312yes—
190#389—22no—
191#389—22no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#388—22no—
209#388—22no—
210#388—22no—
211#388—22no—
212#385—22no—
213#385—22no—
214#385—22no—
215#385—22no—
216#385—22no—
217#385—22no—
218#384—22no—
219#384—21no—
220#384—21no—
221#384—21no—
222#384—21no—
223#384C2312yes—
224#384C2312yes—
225#384C2312yes—
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#382—22no—
231#382—21no—
232#382—21no—
233#382—21no—
234#382C2312yes—
235#382C2312yes—
236#382C2312yes—
237#382C2312yes—
238#380—21no—
239#380—22no—
240#380—22no—
241#380C2312yes—
242C4×C4⋊C4—21no—
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2311yescharacteristic
247C4×C4⋊C4C2311yescharacteristic
248C4×C22⋊C4D411yescharacteristic
249C4×C22⋊C4D411yescharacteristic
250C4×C22⋊C4C2311yescharacteristic
251C4×C22⋊C4C2311yescharacteristic
252#376D411yescharacteristic
253#376D411yescharacteristic
254C43C2311yescharacteristic
255C43C2311yescharacteristic
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C2×#41—44no—
273C2×#41—82no—
274C2×#41—82no—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×Q8—21no—
278C22×D4—22no—
279C22×D4—21no—
280C22×D4—21no—
281C22×D4C2412yes—
282C23×C4—22no—
283C23×C4—22no—
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2411yescharacteristic
287C2×M16—24no—
288C2×M16—24no—
289C2×M16—42no—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—82no—
301C4⋊Q8—82no—
302#127—22no—
303#127—22no—
304#125—24no—
305#125—24no—
306#125—44no—
307#125—44no—
308#125—82no—
309#125—82no—
310#124—44no—
311#124—44no—
312#124—44no—
313#124—82no—
314#124—82no—
315C22⋊Q8—44no—
316C22⋊Q8—44no—
317#121—24no—
318#121—24no—
319#121—44no—
320C4×Q8—44no—
321C4×Q8—44no—
322C4×D4—24no—
323C4×D4—24no—
324C4×D4—44no—
325#117—24no—
326#117—24no—
327#117—42no—
328#117—42no—
329#117—44no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
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—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4—44no—
360C2×C4⋊C4—82no—
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4C2×D411yescharacteristic
385C2×C22⋊C4C2×D411yescharacteristic
386C2×C22⋊C4C2×D411yescharacteristic
387C2×C22⋊C4C2×D411yescharacteristic
388C2×C42—22no—
389C2×C42—22no—
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42#4114yes—
401C2×C42#4114yes—
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408C4⋊C8—24no—
409#102—28no—
410#102—44no—
411#102—44no—
412#102—44no—
413C22⋊C8—24no—
414#97—24no—
415#97—24no—
416#97—42no—
417C4×C8—24no—
418C4×C8—22no—
419C4×C8—22no—
420C4×C8—22no—
421C4×C8—22no—
422C4×C8—42no—
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#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445C24—22no—
446C24C22×D411yescharacteristic
447#41—816no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—24no—
460C2×D4—24no—
461C2×D4—44no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—82no—
469C2×D4—82no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
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—44no—
484C22×C4—44no—
485C22×C4—82no—
486C22×C4—82no—
487C22×C4—82no—
488C22×C4—82no—
489C22×C4#12414yes—
490C22×C4#12414yes—
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×#4112yes—
505C22×C4C2×#4112yes—
506C22×C4C2511yescharacteristic
507M16—48no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C82−1+414yes—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—44no—
535C4⋊C4—42no—
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—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C4⋊C4—82no—
556C4⋊C4—82no—
557C4⋊C4—82no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C22⋊C4—41no—
563C22⋊C4—41no—
564C22⋊C4—41no—
565C22⋊C4—41no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—44no—
591C42—44no—
592C42#12414yes—
593C42#12414yes—
594C42C2×#4112yes—
595C42C2×#4112yes—
596C42C2×#4112yes—
597C42C2×#4112yes—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23C2×#12412yes—
612C23C23×D411yescharacteristic
613Q8—84no—
614Q8—84no—
615Q8—84no—
616D4—44no—
617D4—44no—
618D4—88no—
619D4—84no—
620D4—84no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—28no—
628C2×C4—28no—
629C2×C4—24no—
630C2×C4—24no—
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—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
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—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—88no—
677C2×C4—164no—
678C2×C4C2×#12412yes—
679C2×C4C2×#12412yes—
680C2×C4C2×#12412yes—
681C2×C4C2×#12412yes—
682C2×C4C2×#12412yes—
683C2×C4#57512yes—
684C2×C4#57512yes—
685C2×C4#57612yes—
686C2×C4#57712yes—
687C2×C4C22×#4111yescharacteristic
688C2×C4C22×#4111yescharacteristic
689C2×C4C2×2−1+412yes—
690C2×C4#58512yes—
691C8—24no—
692C8—42no—
693C8—42no—
694C8—42no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—41no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22C22×#12411yescharacteristic
710C22#349911yescharacteristic
711C22#350011yescharacteristic
712C22C2×#57511yescharacteristic
713C22C2×#57511yescharacteristic
714C22C2×#57611yescharacteristic
715C22C2×#57711yescharacteristic
716C4—22no—
717C4—22no—
718C4—22no—
719C4—41no—
720C4—41no—
721C4—41no—
722C4—41no—
723C4—41no—
724C4—41no—
725C4—41no—
726C4—41no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4#313614yes—
735C4#314214yes—
736C4#346112yes—
737C4#349512yes—
738C2—42no—
739C2—162no—
740C2#3531211yescharacteristic
741C2C2×#313611yescharacteristic
742C2C2×#314211yescharacteristic
743C2#6248111yescharacteristic
744C2#6248311yescharacteristic
745C2#6254311yescharacteristic
746C2#6254611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
2. #35286C2semidirect
3. #33294C2semidirect
4. #33290C2semidirect
5. #33286C2semidirect
6. #33277C2semidirect
7. #30229C2semidirect
8. #30228C2semidirect
9. #30204C2semidirect
10. #30204C2semidirect
11. #27829C2semidirect
12. #27743C2semidirect
13. #27726C2semidirect
14. #27710C2semidirect
16. #21835C2semidirect
18. #21322C2semidirect
19. #20472C2semidirect

Maximal subgroups

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

subgroupindex
#352982
#352862
#332942
#332902
#332862
#332772
#302292
#302282
#302042
#278292
#277432
#277262
#277102
#218592
#218352
#214102
#213222
#204722