Finite Group Databasebeta

← #6480600

Subgroups — #6480600

Order 512.

1600 conjugacy classes of subgroups.

The subgroup lattice has 774 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35320C211yesmaximal, characteristic, complemented
2#35288C211yesmaximal, characteristic
3#33287C211yesmaximal, characteristic, complemented
4#33278C211yesmaximal, characteristic, complemented
5#33252C211yesmaximal, characteristic, complemented
6#33239C211yesmaximal, characteristic, complemented
7#30355C214yesmaximal, complemented
8#29568C214yesmaximal, complemented
9#28656C211yesmaximal, characteristic, complemented
10#28653C211yesmaximal, characteristic, complemented
11#28613C211yesmaximal, characteristic, complemented
12#28613C211yesmaximal, characteristic, complemented
13#25905C211yesmaximal, characteristic
14#25856C211yesmaximal, characteristic, complemented
15#25812C211yesmaximal, characteristic, complemented
16#25718C211yesmaximal, characteristic
17#24518C214yesmaximal, complemented
18#22058C214yesmaximal, complemented
19C4×#433C211yesmaximal, characteristic, complemented
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#498C2211yescharacteristic
24C2×#492C2211yescharacteristic
25C2×#492C2211yescharacteristic
26C2×#487C2211yescharacteristic
27C2×#487C2211yescharacteristic
28C2×#486C2211yescharacteristic
29C2×#486C2211yescharacteristic
30C2×#448C2214yes—
31C2×#433C2212yes—
32#2862C2211yescharacteristic
33#2862C2211yescharacteristic
34#2862C2211yescharacteristic
35#2862C2211yescharacteristic
36#2859C2211yescharacteristic
37#2858C2211yescharacteristic
38#2857C2211yescharacteristic
39#2856C2211yescharacteristic
40#2644C2212yes—
41#2642C2212yes—
42#2642C2212yes—
43#2624C2212yes—
44#2568—21no—
45#2568—21no—
46#2568—21no—
47#2568—21no—
48#2513—21no—
49#2503—21no—
50#2503—21no—
51#2503—21no—
52#2503—21no—
53#2490—21no—
54#2454—21no—
55#2448—21no—
56#2447—21no—
57#2427—21no—
58#2426—21no—
59#2420—21no—
60#2324C2212yes—
61#2319C2212yes—
62#2310C2212yes—
63#2310C2212yes—
64C4×C4⋊Q8C2211yescharacteristic
65C4×#127C2211yescharacteristic
66C4×#117C2211yescharacteristic
67#2068C2214yes—
68#2066C2214yes—
69#2059C2214yes—
70#2055C2214yes—
71#2055C2214yes—
72#2043C2214yes—
73#2025C2214yes—
74#2013C2214yes—
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1973C2214yes—
80#1973C2214yes—
81#1968C2214yes—
82#1968C2214yes—
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1956C2211yescharacteristic
87#1935C2214yes—
88#1935C2214yes—
89#1933C2214yes—
90#1933C2214yes—
91#1928C2214yes—
92#1927C2214yes—
93#1877C2214yes—
94#1877C2214yes—
95#1875C2214yes—
96#1854C2214yes—
97#1787C2214yes—
98#1768C2212yes—
99C4×C4⋊C8C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101#1750C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1746C2212yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C23×Q8C2312yes—
108C22×SD16C2314yes—
109C22×SD16C2314yes—
110#533—42no—
111#533—42no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—42no—
115C2×C4⋊Q8—42no—
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×#127C2311yescharacteristic
123C2×#125—21no—
124C2×#125C2312yes—
125C2×#124—21no—
126C2×#124—21no—
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8—21no—
129C2×C22⋊Q8—21no—
130C2×#121—21no—
131C2×#121—21no—
132C2×#121C2312yes—
133C2×C4×Q8—21no—
134C2×C4×Q8—21no—
135C2×C4×Q8—21no—
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4C2312yes—
139C2×#117C2312yes—
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C22×C4⋊C4—21no—
143C22×C4⋊C4—21no—
144C22×C42C2311yescharacteristic
145#500—42no—
146#498—42no—
147#492—42no—
148#492—42no—
149#487—42no—
150#487—42no—
151#486—42no—
152#486—42no—
153#448—216no—
154#433—28no—
155C2×#107C2314yes—
156C2×#106C2314yes—
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2312yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×C22⋊C8C2312yes—
168C2×C22⋊C8C2312yes—
169C2×#97C2312yes—
170C2×#97C2312yes—
171C2×C4×C8C2311yescharacteristic
172C2×C4×C8C2311yescharacteristic
173C2×C4×C8C2311yescharacteristic
174C2×C4×C8C2311yescharacteristic
175C2×C4×C8C2312yes—
176C2×C4×C8C2312yes—
177#400—22no—
178#400—22no—
179#400—22no—
180#400—22no—
181#400—22no—
182#400—22no—
183#400—22no—
184#400—22no—
185#399—22no—
186#397—22no—
187#397—22no—
188#397—22no—
189#397—22no—
190#396—22no—
191#394—22no—
192#394—22no—
193#391—22no—
194#391—22no—
195#391—22no—
196#391—22no—
197#391—21no—
198#391—21no—
199#391—21no—
200#391—21no—
201#391—21no—
202#391—21no—
203#391—21no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391C2312yes—
208#391C2312yes—
209#391C2312yes—
210#391C2312yes—
211#391C2312yes—
212#391C2312yes—
213#391C2312yes—
214#391C2312yes—
215#390—22no—
216#390—21no—
217#390—21no—
218#390—21no—
219#390—21no—
220#390—21no—
221#390—21no—
222#390C2312yes—
223#390C2312yes—
224#390C2312yes—
225#390C2312yes—
226#389—21no—
227#389—21no—
228#389—21no—
229#389—21no—
230#389C2312yes—
231#389C2312yes—
232#389C2312yes—
233#389C2312yes—
234#388—22no—
235#388—22no—
236#388—22no—
237#386—22no—
238#385—22no—
239#384C2311yescharacteristic
240#384C2311yescharacteristic
241#382—21no—
242#382—21no—
243#382—21no—
244#382—21no—
245#382—21no—
246#382—21no—
247#382—21no—
248#382—21no—
249#382—21no—
250#381—22no—
251#381—22no—
252#381—22no—
253#381—22no—
254#381—22no—
255#380—21no—
256#380—22no—
257#380C2312yes—
258C4×C4⋊C4—21no—
259C4×C4⋊C4D411yescharacteristic
260C4×C4⋊C4D411yescharacteristic
261C4×C4⋊C4D411yescharacteristic
262C4×C4⋊C4D411yescharacteristic
263C4×C4⋊C4C2312yes—
264C4×C22⋊C4D411yescharacteristic
265C4×C22⋊C4D411yescharacteristic
266C4×C22⋊C4D411yescharacteristic
267C4×C22⋊C4D411yescharacteristic
268#376C2311yescharacteristic
269#376C2311yescharacteristic
270C43C2311yescharacteristic
271C43C2311yescharacteristic
272#340C2314yes—
273#340C2314yes—
274#340C2314yes—
275#340C2314yes—
276#336C2312yes—
277#336C2312yes—
278#336C2312yes—
279#336C2312yes—
280C22×Q8—21no—
281C22×Q8—21no—
282C22×Q8—21no—
283C22×Q8—22no—
284C22×Q8—28no—
285C22×Q8—42no—
286C22×Q8—42no—
287C22×Q8—44no—
288C22×Q8C2412yes—
289C22×Q8C2412yes—
290C22×D4—22no—
291C22×D4—21no—
292C22×D4—21no—
293C22×D4C2412yes—
294C23×C4—22no—
295C23×C4—22no—
296C23×C4C2×D411yescharacteristic
297C23×C4C2×D411yescharacteristic
298C23×C4C2411yescharacteristic
299C2×SD16—28no—
300C2×SD16—28no—
301C2×SD16—44no—
302C2×SD16—44no—
303C2×SD16—44no—
304C2×SD16—44no—
305C22×C8C2412yes—
306C22×C8C2412yes—
307C22×C8C2412yes—
308C22×C8C2412yes—
309C4⋊Q8—22no—
310C4⋊Q8—22no—
311C4⋊Q8—22no—
312C4⋊Q8—22no—
313C4⋊Q8—22no—
314C4⋊Q8—22no—
315C4⋊Q8—22no—
316C4⋊Q8—22no—
317C4⋊Q8—22no—
318C4⋊Q8—22no—
319C4⋊Q8—42no—
320C4⋊Q8—42no—
321C4⋊Q8—44no—
322C4⋊Q8—44no—
323#127—22no—
324#127—22no—
325#125—42no—
326#125—42no—
327#125—82no—
328#124—82no—
329#124—82no—
330C22⋊Q8—44no—
331C22⋊Q8—44no—
332C22⋊Q8—82no—
333#121—28no—
334#121—82no—
335#121—82no—
336C4×Q8—44no—
337C4×Q8—44no—
338C4×Q8—82no—
339C4×D4—44no—
340C4×D4—82no—
341C4×D4—82no—
342#117—24no—
343#117—42no—
344#117—42no—
345#117—42no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—44no—
366C2×C4⋊C4—44no—
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2×D411yescharacteristic
369C2×C4⋊C4C2×D411yescharacteristic
370C2×C4⋊C4C2×D411yescharacteristic
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2412yes—
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⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4C2×D411yescharacteristic
389C2×C22⋊C4C2×D411yescharacteristic
390C2×C22⋊C4C2×D411yescharacteristic
391C2×C22⋊C4C2×D411yescharacteristic
392C2×C42—22no—
393C2×C42—22no—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2×D411yescharacteristic
404C2×C42C2×D411yescharacteristic
405C2×C42C2×D411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409C2×C42C2411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2411yescharacteristic
412#107—44no—
413#106—44no—
414C4⋊C8—24no—
415C4⋊C8—24no—
416#103—28no—
417#103—28no—
418#103—44no—
419#103—44no—
420#103—44no—
421#103—44no—
422#102—44no—
423#102—44no—
424C22⋊C8—24no—
425C22⋊C8#4118yes—
426#97—24no—
427#97—42no—
428C4×C8—24no—
429C4×C8—22no—
430C4×C8—22no—
431C4×C8—22no—
432C4×C8—22no—
433C4×C8—42no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95—22no—
446#95—22no—
447#95—22no—
448#95—22no—
449#95—22no—
450#95—22no—
451#95—22no—
452#95—22no—
453#95—22no—
454#95—22no—
455#95—22no—
456#95C2×D411yescharacteristic
457#95C2×D411yescharacteristic
458#95C2×D411yescharacteristic
459#95C2×D411yescharacteristic
460#95C2×D411yescharacteristic
461#95C2×D411yescharacteristic
462#95C2×D411yescharacteristic
463#95C2×D411yescharacteristic
464C24—22no—
465C24C22×D411yescharacteristic
466C2×Q8—24no—
467C2×Q8—24no—
468C2×Q8—22no—
469C2×Q8—22no—
470C2×Q8—22no—
471C2×Q8—22no—
472C2×Q8—24no—
473C2×Q8—24no—
474C2×Q8—44no—
475C2×Q8—44no—
476C2×Q8—44no—
477C2×Q8—42no—
478C2×Q8—42no—
479C2×Q8—42no—
480C2×Q8—42no—
481C2×Q8—42no—
482C2×Q8—42no—
483C2×Q8—42no—
484C2×Q8—42no—
485C2×Q8—42no—
486C2×Q8—88no—
487C2×Q8—88no—
488C2×D4—24no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—24no—
492C2×D4—44no—
493C2×D4—42no—
494C2×D4—42no—
495C2×D4—42no—
496C2×D4—42no—
497C2×D4—42no—
498C2×D4—42no—
499C2×D4—82no—
500C2×D4—82no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—44no—
518C22×C4—44no—
519C22×C4—44no—
520C22×C4—44no—
521C22×C4—82no—
522C22×C4—82no—
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D411yescharacteristic
526C22×C4C22×D411yescharacteristic
527C22×C4C22×D411yescharacteristic
528C22×C4C22×D411yescharacteristic
529C22×C4C22×D411yescharacteristic
530C22×C4C22×D411yescharacteristic
531C22×C4C22×D411yescharacteristic
532C22×C4C22×D411yescharacteristic
533C22×C4C22×D411yescharacteristic
534C22×C4C22×D411yescharacteristic
535C22×C4C22×D411yescharacteristic
536C22×C4C2×#4114yes—
537C22×C4C2511yescharacteristic
538SD16—88no—
539SD16—88no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—22no—
549C2×C8—22no—
550C2×C8C2×#4114yes—
551C2×C8C2×#4114yes—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—24no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—24no—
562C4⋊C4—41no—
563C4⋊C4—41no—
564C4⋊C4—44no—
565C4⋊C4—44no—
566C4⋊C4—44no—
567C4⋊C4—44no—
568C4⋊C4—42no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—42no—
572C4⋊C4—42no—
573C4⋊C4—42no—
574C4⋊C4—42no—
575C4⋊C4—42no—
576C4⋊C4—42no—
577C4⋊C4—42no—
578C4⋊C4—42no—
579C4⋊C4—82no—
580C4⋊C4—82no—
581C4⋊C4—82no—
582C4⋊C4—82no—
583C22⋊C4—22no—
584C22⋊C4—22no—
585C22⋊C4—41no—
586C22⋊C4—41no—
587C22⋊C4—44no—
588C22⋊C4—44no—
589C22⋊C4—44no—
590C22⋊C4—82no—
591C22⋊C4—82no—
592C22⋊C4—82no—
593C22⋊C4—82no—
594C22⋊C4—82no—
595C22⋊C4—82no—
596C22⋊C4—82no—
597C22⋊C4—82no—
598C22⋊C4—82no—
599C42—22no—
600C42—22no—
601C42—22no—
602C42—22no—
603C42—22no—
604C42—22no—
605C42—22no—
606C42—22no—
607C42—22no—
608C42—22no—
609C42—22no—
610C42—22no—
611C42—21no—
612C42—21no—
613C42—22no—
614C42—22no—
615C42—44no—
616C42—82no—
617C42H(5,𝔽2)12yes—
618C42H(5,𝔽2)12yes—
619C23—21no—
620C23—21no—
621C23—21no—
622C23—21no—
623C23—21no—
624C23—21no—
625C23—42no—
626C23—42no—
627C23—42no—
628C23—42no—
629C23—42no—
630C23—42no—
631C23—42no—
632C23#53212yes—
633C23C23×D411yescharacteristic
634Q8—44no—
635Q8—44no—
636Q8—44no—
637Q8—44no—
638Q8—88no—
639Q8—84no—
640Q8—84no—
641Q8—84no—
642D4—44no—
643D4—44no—
644D4—84no—
645D4—84no—
646D4—164no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—21no—
668C2×C4—21no—
669C2×C4—21no—
670C2×C4—21no—
671C2×C4—21no—
672C2×C4—21no—
673C2×C4—21no—
674C2×C4—21no—
675C2×C4—21no—
676C2×C4—21no—
677C2×C4—21no—
678C2×C4—21no—
679C2×C4—21no—
680C2×C4—21no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—42no—
701C2×C4—42no—
702C2×C4—42no—
703C2×C4—42no—
704C2×C4—88no—
705C2×C4—164no—
706C2×C4#53212yes—
707C2×C4#53512yes—
708C2×C4#53512yes—
709C2×C4#57712yes—
710C2×C4#57712yes—
711C2×C4#57712yes—
712C2×C4#57812yes—
713C2×C4C22×#4112yes—
714C2×C4C2×H(5,𝔽2)11yescharacteristic
715C2×C4C2×H(5,𝔽2)11yescharacteristic
716C2×C4#58512yes—
717C8—24no—
718C8—24no—
719C8—42no—
720C8—42no—
721C22—22no—
722C22—22no—
723C22—22no—
724C22—41no—
725C22—41no—
726C22—41no—
727C22—41no—
728C22—82no—
729C22—82no—
730C22—82no—
731C22—82no—
732C22—82no—
733C22—82no—
734C22—82no—
735C22C2×#53211yescharacteristic
736C22C2×#53511yescharacteristic
737C22#344911yescharacteristic
738C22C2×#57711yescharacteristic
739C22C2×#57711yescharacteristic
740C22C2×#57711yescharacteristic
741C22C2×#57811yescharacteristic
742C4—22no—
743C4—41no—
744C4—41no—
745C4—41no—
746C4—41no—
747C4—41no—
748C4—41no—
749C4—41no—
750C4—41no—
751C4—41no—
752C4—41no—
753C4—82no—
754C4—82no—
755C4—82no—
756C4—82no—
757C4—82no—
758C4—82no—
759C4—82no—
760C4#322114yes—
761C4#322514yes—
762C4#351412yes—
763C4#351412yes—
764C2—42no—
765C2—162no—
766C2#3513111yescharacteristic
767C2#6071211yescharacteristic
768C2#6072311yescharacteristic
769C2C2×#322111yescharacteristic
770C2C2×#322511yescharacteristic
771C2#6100811yescharacteristic
772C2#6101211yescharacteristic
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. #35320C2semidirect
3. #33287C2semidirect
4. #33278C2semidirect
5. #33252C2semidirect
6. #33239C2semidirect
7. #30355C2semidirect
8. #29568C2semidirect
9. #28656C2semidirect
10. #28653C2semidirect
11. #28613C2semidirect
12. #28613C2semidirect
14. #25856C2semidirect
15. #25812C2semidirect
17. #24518C2semidirect
18. #22058C2semidirect
19. C4×#433C2semidirect

Maximal subgroups

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

subgroupindex
#353202
#352882
#332872
#332782
#332522
#332392
#303552
#295682
#286562
#286532
#286132
#259052
#258562
#258122
#257182
#245182
#220582
C4×#4332