Finite Group Databasebeta

← #6495194

Subgroups — #6495194

Order 512.

1630 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35277C211yesmaximal, characteristic
2#35275C211yesmaximal, characteristic, complemented
3#33297C211yesmaximal, characteristic, complemented
4#33289C211yesmaximal, characteristic, complemented
5#33230C211yesmaximal, characteristic, complemented
6#33226C211yesmaximal, characteristic, complemented
7#30211C214yesmaximal, complemented
8#29144C211yesmaximal, characteristic, complemented
9#29143C211yesmaximal, characteristic, complemented
10#29141C211yesmaximal, characteristic, complemented
11#29139C211yesmaximal, characteristic, complemented
12#28730C214yesmaximal, complemented
13#27975C211yesmaximal, characteristic, complemented
14#27974C211yesmaximal, characteristic
15#27973C211yesmaximal, characteristic
16#27972C211yesmaximal, characteristic, complemented
17#25916C214yesmaximal, complemented
18#21871C214yesmaximal, complemented
19C4×#433C211yesmaximal, characteristic, complemented
20C2×#544C2212yes—
21C2×#542C2212yes—
22C2×#489C2214yes—
23C2×#488C2214yes—
24C2×#433C2212yes—
25#2868C2211yescharacteristic
26#2868C2211yescharacteristic
27#2868C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2857C2211yescharacteristic
32#2856C2211yescharacteristic
33#2660C2212yes—
34#2660C2212yes—
35#2640C2212yes—
36#2620C2212yes—
37#2552—21no—
38#2552—21no—
39#2551—21no—
40#2551—21no—
41#2512—21no—
42#2512—21no—
43#2491—21no—
44#2491—21no—
45#2452—21no—
46#2452—21no—
47#2440—21no—
48#2440—21no—
49#2440—21no—
50#2440—21no—
51#2422—21no—
52#2422—21no—
53#2326C2212yes—
54#2317C2212yes—
55C4×C4⋊Q8C2211yescharacteristic
56C4×#127C2211yescharacteristic
57#2285C2211yescharacteristic
58C2×#384C2211yescharacteristic
59C4×#117C2211yescharacteristic
60#2074C2214yes—
61#2073C2214yes—
62#2072C2214yes—
63#2071C2214yes—
64#2026C2214yes—
65#2025C2214yes—
66#2024C2214yes—
67#2023C2214yes—
68#1993C2214yes—
69#1992C2214yes—
70#1990C2214yes—
71#1990C2214yes—
72#1959C2214yes—
73#1958C2214yes—
74#1957C2211yescharacteristic
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1957C2211yescharacteristic
78#1956C2211yescharacteristic
79#1956C2211yescharacteristic
80#1956C2211yescharacteristic
81#1956C2211yescharacteristic
82#1875C2214yes—
83#1874C2214yes—
84#1873C2214yes—
85#1872C2214yes—
86#1834C2211yescharacteristic
87#1834C2211yescharacteristic
88#1834C2211yescharacteristic
89#1834C2211yescharacteristic
90#1834C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1834C2211yescharacteristic
94#1805C2214yes—
95#1804C2214yes—
96#1803C2214yes—
97#1802C2214yes—
98#1768C2212yes—
99C4×C4⋊C8C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101#1750C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1746C2212yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107#544—48no—
108#542—216no—
109C2×C4⋊Q8—22no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2312yes—
120C2×#127C2311yescharacteristic
121C2×#126—22no—
122C2×#125—22no—
123C2×#125C2312yes—
124C2×#125C2312yes—
125C2×#124—22no—
126C2×#124C2312yes—
127C2×#124C2312yes—
128C2×#123—22no—
129C2×C22⋊Q8—22no—
130C2×C22⋊Q8—22no—
131C2×C22⋊Q8—22no—
132C2×C22⋊Q8C2312yes—
133C2×#121—21no—
134C2×#121—21no—
135C2×#121C2312yes—
136C2×C4×Q8C2312yes—
137C2×C4×D4—21no—
138C2×C4×D4—21no—
139C2×C4×D4C2312yes—
140C2×#117C2312yes—
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C22×C4⋊C4—21no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4C2312yes—
146C22×C42C2311yescharacteristic
147#489—48no—
148#488—48no—
149#433—28no—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×#97C2312yes—
163C2×#97C2312yes—
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2312yes—
169C2×C4×C8C2312yes—
170#398—22no—
171#398—22no—
172#398—22no—
173#398—22no—
174#397—22no—
175#397—22no—
176#396—22no—
177#396—22no—
178#395—22no—
179#395—22no—
180#395—22no—
181#395—22no—
182#394—22no—
183#394—22no—
184#393—22no—
185#393—22no—
186#391C2312yes—
187#391C2312yes—
188#390—21no—
189#390—21no—
190#390—21no—
191#390—21no—
192#390—21no—
193#390—21no—
194#390—21no—
195#390—21no—
196#390C2312yes—
197#390C2312yes—
198#389—22no—
199#389—22no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
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#388—22no—
213#388—22no—
214#385—22no—
215#385—22no—
216#385—22no—
217#385—22no—
218#384—21no—
219#384—21no—
220#384—21no—
221#384—21no—
222#384—21no—
223#384—21no—
224#384—21no—
225#384—21no—
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2312yes—
229#384C2312yes—
230#384C2312yes—
231#384C2312yes—
232#384C2311yescharacteristic
233#384C2311yescharacteristic
234#384C2311yescharacteristic
235#384C2311yescharacteristic
236#382C2312yes—
237#382C2312yes—
238#382C2312yes—
239#380—21no—
240#380—21no—
241#380—21no—
242#380—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4D411yescharacteristic
251C4×C22⋊C4D411yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255#376C2311yescharacteristic
256#376C2311yescharacteristic
257C43C2311yescharacteristic
258C43C2311yescharacteristic
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271C22×Q8—22no—
272C22×Q8—22no—
273C22×Q8—22no—
274C22×Q8C2412yes—
275C22×D4—22no—
276C22×D4—21no—
277C22×D4—21no—
278C22×D4C2412yes—
279C23×C4—22no—
280C23×C4—22no—
281C23×C4C2×D411yescharacteristic
282C23×C4C2×D411yescharacteristic
283C23×C4C2411yescharacteristic
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C4⋊Q8—28no—
289C4⋊Q8—28no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—44no—
293C4⋊Q8—48no—
294C4⋊Q8—82no—
295C4⋊Q8—82no—
296C4⋊Q8—82no—
297C4⋊Q8—82no—
298C4⋊Q8—82no—
299C4⋊Q8—82no—
300#127—22no—
301#127—22no—
302#126—48no—
303#125—28no—
304#125—44no—
305#125—48no—
306#124—28no—
307#124—28no—
308#124—48no—
309#123—48no—
310C22⋊Q8—28no—
311C22⋊Q8—48no—
312C22⋊Q8—48no—
313C22⋊Q8—48no—
314#121—28no—
315#121—82no—
316#121—82no—
317C4×Q8—28no—
318C4×D4—28no—
319C4×D4—82no—
320C4×D4—82no—
321#117—24no—
322#117—44no—
323#117#4118yes—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
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—44no—
357C2×C4⋊C4—82no—
358C2×C4⋊C4—82no—
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
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×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4C2×D411yescharacteristic
382C2×C22⋊C4C2×D411yescharacteristic
383C2×C22⋊C4C2×D411yescharacteristic
384C2×C22⋊C4C2×D411yescharacteristic
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C4⋊C8—24no—
406C4⋊C8—24no—
407#103—44no—
408#103—44no—
409#103—44no—
410#103—44no—
411#102—44no—
412#102—44no—
413#102—44no—
414#102—44no—
415C22⋊C8—24no—
416C22⋊C8—24no—
417#97—24no—
418#97—24no—
419C4×C8—24no—
420C4×C8—24no—
421C4×C8—22no—
422C4×C8—22no—
423C4×C8—22no—
424C4×C8—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—24no—
446C2×Q8—24no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—44no—
450C2×Q8—44no—
451C2×Q8—44no—
452C2×Q8—82no—
453C2×Q8—82no—
454C2×Q8—82no—
455C2×Q8—82no—
456C2×Q8—82no—
457C2×Q8—82no—
458C2×D4—24no—
459C2×D4—24no—
460C2×D4—22no—
461C2×D4—22no—
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—22no—
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—44no—
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×C8—22no—
519C2×C8—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
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—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
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#12414yes—
561C4⋊C4#12414yes—
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—82no—
573C22⋊C4—82no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4#12414yes—
577C22⋊C4#12414yes—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—21no—
590C42—21no—
591C42—44no—
592C42—44no—
593C42C2×#4114yes—
594C42C2×#4114yes—
595C42H(5,𝔽2)14yes—
596C422−1+412yes—
597C422−1+412yes—
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—44no—
614Q8—44no—
615Q8—164no—
616Q8—164no—
617Q8—164no—
618D4—44no—
619D4—44no—
620D4—84no—
621D4—84no—
622D4—164no—
623C2×C4—21no—
624C2×C4—21no—
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—24no—
638C2×C4—24no—
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—44no—
652C2×C4—44no—
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—42no—
673C2×C4—42no—
674C2×C4—164no—
675C2×C4—164no—
676C2×C4C2×#12412yes—
677C2×C4C2×#12412yes—
678C2×C4C2×#12412yes—
679C2×C4C2×#12412yes—
680C2×C4C2×#12412yes—
681C2×C4#53512yes—
682C2×C4#53512yes—
683C2×C4#53612yes—
684C2×C4#53612yes—
685C2×C4#57612yes—
686C2×C4#57712yes—
687C2×C4#57712yes—
688C2×C4#57812yes—
689C2×C4C22×#4112yes—
690C2×C4C2×H(5,𝔽2)12yes—
691C2×C4C2×2−1+411yescharacteristic
692C2×C4C2×2−1+411yescharacteristic
693C8—42no—
694C8—42no—
695C8—42no—
696C8—42no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—41no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22C22×#12411yescharacteristic
712C22C2×#53511yescharacteristic
713C22C2×#53611yescharacteristic
714C22C2×#57611yescharacteristic
715C22C2×#57711yescharacteristic
716C22C2×#57711yescharacteristic
717C22C2×#57811yescharacteristic
718C4—22no—
719C4—22no—
720C4—22no—
721C4—22no—
722C4—22no—
723C4—22no—
724C4—22no—
725C4—41no—
726C4—41no—
727C4—41no—
728C4—41no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4#324314yes—
737C4#324414yes—
738C4#351112yes—
739C4#351112yes—
740C2—42no—
741C2—162no—
742C2#3528911yescharacteristic
743C2#6062211yescharacteristic
744C2#6062311yescharacteristic
745C2#6082211yescharacteristic
746C2#6082411yescharacteristic
747C2C2×#324311yescharacteristic
748C2C2×#324411yescharacteristic
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. #35275C2semidirect
3. #33297C2semidirect
4. #33289C2semidirect
5. #33230C2semidirect
6. #33226C2semidirect
7. #30211C2semidirect
8. #29144C2semidirect
9. #29143C2semidirect
10. #29141C2semidirect
11. #29139C2semidirect
12. #28730C2semidirect
13. #27975C2semidirect
16. #27972C2semidirect
17. #25916C2semidirect
18. #21871C2semidirect
19. C4×#433C2semidirect

Maximal subgroups

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

subgroupindex
#352772
#352752
#332972
#332892
#332302
#332262
#302112
#291442
#291432
#291412
#291392
#287302
#279752
#279742
#279732
#279722
#259162
#218712
C4×#4332