Finite Group Databasebeta

← #6469531

Subgroups — #6469531

Order 512.

1604 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35288C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#29527C214yesmaximal, complemented
4#29179C211yesmaximal, characteristic, complemented
5#29177C211yesmaximal, characteristic, complemented
6#28655C211yesmaximal, characteristic, complemented
7#28655C211yesmaximal, characteristic, complemented
8#27962C211yesmaximal, characteristic, complemented
9#27907C211yesmaximal, characteristic
10#25903C211yesmaximal, characteristic
11#25813C211yesmaximal, characteristic, complemented
12#24615C214yesmaximal, complemented
13#23787C214yesmaximal, complemented
14#21726C211yesmaximal, characteristic, complemented
15C4×#500C211yesmaximal, characteristic, complemented
16#21646C211yesmaximal, characteristic, complemented
17C4×#486C211yesmaximal, characteristic, complemented
18#21590C214yesmaximal, complemented
19C4×#433C211yesmaximal, characteristic, complemented
20C22×C4⋊Q8C2211yescharacteristic
21C2×#500C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#482C2214yes—
26C2×#433C2212yes—
27#2859C2211yescharacteristic
28#2858C2211yescharacteristic
29#2857C2211yescharacteristic
30#2856C2211yescharacteristic
31#2644C2212yes—
32#2624C2212yes—
33#2513—21no—
34#2490—21no—
35#2454—21no—
36#2448—21no—
37#2447—21no—
38#2427—21no—
39#2426—21no—
40#2420—21no—
41#2369—21no—
42#2369—21no—
43#2365—21no—
44#2365—21no—
45#2345—21no—
46#2345—21no—
47Q8×C22⋊C4—21no—
48Q8×C22⋊C4—21no—
49#2331C2212yes—
50#2331C2212yes—
51#2324C2212yes—
52#2319C2212yes—
53#2316C2212yes—
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×#127C2211yescharacteristic
60#2285C2211yescharacteristic
61#2274C2212yes—
62C4×#117C2211yescharacteristic
63#2101C2214yes—
64#2092C2214yes—
65#2088C2214yes—
66#2087C2214yes—
67#2067C2214yes—
68#2067C2214yes—
69#2057C2214yes—
70#2023C2214yes—
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1957C2211yescharacteristic
78#1957C2211yescharacteristic
79#1943C2214yes—
80#1942C2214yes—
81#1936C2214yes—
82#1934C2214yes—
83#1925C2214yes—
84#1925C2214yes—
85#1921C2214yes—
86#1919C2214yes—
87#1874C2214yes—
88#1852C2214yes—
89#1845C2211yescharacteristic
90#1844C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1825C2214yes—
94#1776C2214yes—
95C4×#107C2214yes—
96#1768C2212yes—
97C4×C4⋊C8C2211yescharacteristic
98C4×C4⋊C8C2211yescharacteristic
99#1762C2214yes—
100C4×#102C2214yes—
101#1750C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1746C2212yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C23×Q8—21no—
108C2×C4⋊Q8—24no—
109C2×C4⋊Q8—24no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×#127C2311yescharacteristic
117C2×#125—21no—
118C2×#124—21no—
119C2×#124—21no—
120C2×C22⋊Q8—21no—
121C2×#121—21no—
122C2×#121—21no—
123C2×#121C2312yes—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4C2312yes—
132C2×#117C2312yes—
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2312yes—
137C22×C42C2311yescharacteristic
138#500—24no—
139#500—24no—
140#486—24no—
141#486—24no—
142#482—216no—
143#433—28no—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×#97C2312yes—
157C2×#97C2312yes—
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2312yes—
163C2×C4×C8C2312yes—
164#399—22no—
165#397—22no—
166#397—22no—
167#397—22no—
168#397—22no—
169#396—22no—
170#394—22no—
171#394—22no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—22no—
179#391—22no—
180#391—22no—
181#391—22no—
182#390—22no—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390C2312yes—
190#390C2312yes—
191#390C2312yes—
192#390C2312yes—
193#389—22no—
194#389—22no—
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#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#388—22no—
212#388—22no—
213#388—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#386—22no—
219#385—22no—
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#382—21no—
225#382C2312yes—
226#381—22no—
227#380—21no—
228#380—21no—
229#380—21no—
230#380—22no—
231#380—22no—
232#380—22no—
233#380C2312yes—
234C4×C4⋊C4—22no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2312yes—
252C4×C22⋊C4—22no—
253C4×C22⋊C4—22no—
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4D411yescharacteristic
256C4×C22⋊C4D411yescharacteristic
257C4×C22⋊C4D411yescharacteristic
258#376C2311yescharacteristic
259#376C2311yescharacteristic
260C43C2311yescharacteristic
261C43C2311yescharacteristic
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274C22×Q8—28no—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×Q8—22no—
278C22×Q8—21no—
279C22×Q8—82no—
280C22×Q8—82no—
281C22×D4—22no—
282C22×D4—21no—
283C22×D4—21no—
284C22×D4C2412yes—
285C23×C4—22no—
286C23×C4—22no—
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2411yescharacteristic
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C4⋊Q8—28no—
295C4⋊Q8—28no—
296C4⋊Q8—24no—
297C4⋊Q8—24no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8#4114yes—
303C4⋊Q8#4114yes—
304#127—22no—
305#127—22no—
306#125—82no—
307#124—82no—
308#124—82no—
309C22⋊Q8—82no—
310#121—28no—
311#121—82no—
312#121—82no—
313C4×Q8—82no—
314C4×Q8—82no—
315C4×Q8—82no—
316C4×Q8—82no—
317C4×Q8—82no—
318C4×D4—28no—
319C4×D4—82no—
320C4×D4—82no—
321#117—24no—
322#117—24no—
323#117—28no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
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×C4⋊C4C2412yes—
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×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399#107—28no—
400#107—28no—
401#107—28no—
402#106—28no—
403C4⋊C8—24no—
404C4⋊C8—24no—
405#102—28no—
406#102—28no—
407#102—28no—
408#102—28no—
409C22⋊C8—24no—
410C22⋊C8#4118yes—
411#97—24no—
412#97—24no—
413C4×C8—24no—
414C4×C8—22no—
415C4×C8—22no—
416C4×C8—22no—
417C4×C8—22no—
418C4×C8—24no—
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#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443C24—22no—
444C24C22×D411yescharacteristic
445C2×Q8—28no—
446C2×Q8—24no—
447C2×Q8—24no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—82no—
456C2×Q8—82no—
457C2×Q8—816no—
458C2×D4—24no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—44no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—82no—
470C2×D4—82no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—28no—
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×C4—82no—
492C22×C4—82no—
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×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C2×#4114yes—
507C22×C4C2511yescharacteristic
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×C8C2×#4114yes—
519C2×C8C2×#4114yes—
520C4⋊C4—28no—
521C4⋊C4—28no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—24no—
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—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4C2×#4114yes—
554C4⋊C4C2×#4114yes—
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—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—21no—
584C42—21no—
585C42—22no—
586C42—44no—
587C42—82no—
588C42—82no—
589C42—82no—
590C42—82no—
591C42—82no—
592C42C2×#4112yes—
593C42C2×#4112yes—
594C422−1+414yes—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23#53212yes—
609C23C23×D411yescharacteristic
610Q8—48no—
611Q8—48no—
612Q8—88no—
613Q8—84no—
614D4—44no—
615D4—44no—
616D4—84no—
617D4—84no—
618D4—164no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—21no—
628C2×C4—21no—
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—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
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—88no—
675C2×C4—164no—
676C2×C4#52512yes—
677C2×C4#52512yes—
678C2×C4#53212yes—
679C2×C4#53612yes—
680C2×C4#53612yes—
681C2×C4#57612yes—
682C2×C4#57612yes—
683C2×C4#57712yes—
684C2×C4#57812yes—
685C2×C4C22×#4111yescharacteristic
686C2×C4C22×#4111yescharacteristic
687C2×C4C22×#4112yes—
688C2×C4C2×2−1+412yes—
689C8—24no—
690C8—24no—
691C8—42no—
692C8—42no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—41no—
697C22—41no—
698C22—41no—
699C22—41no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22C2×#52511yescharacteristic
708C22C2×#53211yescharacteristic
709C22C2×#53611yescharacteristic
710C22C2×#57611yescharacteristic
711C22C2×#57611yescharacteristic
712C22C2×#57711yescharacteristic
713C22C2×#57811yescharacteristic
714C4—22no—
715C4—22no—
716C4—22no—
717C4—41no—
718C4—41no—
719C4—41no—
720C4—41no—
721C4—41no—
722C4—41no—
723C4—44no—
724C4—44no—
725C4—41no—
726C4—41no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4#312114yes—
733C4#312214yes—
734C4#348212yes—
735C4#348212yes—
736C2—42no—
737C2—162no—
738C2#3497211yescharacteristic
739C2C2×#312111yescharacteristic
740C2C2×#312211yescharacteristic
741C2#6071811yescharacteristic
742C2#6072211yescharacteristic
743C2#6087711yescharacteristic
744C2#6088011yescharacteristic
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
2. #34216C2semidirect
3. #29527C2semidirect
4. #29179C2semidirect
5. #29177C2semidirect
6. #28655C2semidirect
7. #28655C2semidirect
8. #27962C2semidirect
11. #25813C2semidirect
12. #24615C2semidirect
13. #23787C2semidirect
14. #21726C2semidirect
15. C4×#500C2semidirect
16. #21646C2semidirect
17. C4×#486C2semidirect
18. #21590C2semidirect
19. C4×#433C2semidirect

Maximal subgroups

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

subgroupindex
#352882
#342162
#295272
#291792
#291772
#286552
#279622
#279072
#259032
#258132
#246152
#237872
#217262
C4×#5002
#216462
C4×#4862
#215902
C4×#4332