Finite Group Databasebeta

← #6805860

Subgroups — #6805860

Order 512.

1614 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic
2#36574C211yesmaximal, characteristic, complemented
3#33226C211yesmaximal, characteristic
4#30888C214yesmaximal, complemented
5#30885C214yesmaximal, complemented
6#30220C211yesmaximal, characteristic, complemented
7#30219C211yesmaximal, characteristic
8#29193C211yesmaximal, characteristic, complemented
9#29192C211yesmaximal, characteristic
10#29144C211yesmaximal, characteristic, complemented
11#28444C211yesmaximal, characteristic
12#28444C211yesmaximal, characteristic
13#28440C211yesmaximal, characteristic, complemented
14#24736C214yesmaximal, complemented
15#24195C214yesmaximal, complemented
16#22270C211yesmaximal, characteristic, complemented
17#22268C211yesmaximal, characteristic
18C4×#494C211yesmaximal, characteristic, complemented
19#21339C211yesmaximal, characteristic, complemented
20C2×#556C2212yes—
21C2×#494C2211yescharacteristic
22C2×#494C2211yescharacteristic
23C2×#492C2211yescharacteristic
24C2×#488C2214yes—
25C2×#464C2214yes—
26C2×#464C2214yes—
27C2×#462C2214yes—
28C2×C4×Q16C2214yes—
29#2868C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2858C2211yescharacteristic
34#2858C2211yescharacteristic
35#2584—21no—
36#2584—21no—
37#2581C2212yes—
38#2547C2212yes—
39#2545C2212yes—
40#2545C2212yes—
41#2538—21no—
42#2538—21no—
43#2512—21no—
44#2512—21no—
45#2512—21no—
46#2512—21no—
47#2509—21no—
48#2509—21no—
49#2508—21no—
50#2508—21no—
51#2453C2212yes—
52#2453C2212yes—
53#2370—21no—
54#2370—21no—
55#2354—21no—
56#2354—21no—
57#2351C2212yes—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×C4⋊Q8C2211yescharacteristic
62C4×C4⋊Q8C2211yescharacteristic
63C4×#127C2211yescharacteristic
64C42×Q8C2211yescharacteristic
65#2068C2214yes—
66#2067C2214yes—
67#2043C2214yes—
68#2043C2214yes—
69#2042C2214yes—
70#2042C2214yes—
71#2026C2214yes—
72#2016C2214yes—
73#2015C2214yes—
74#2015C2214yes—
75#1984C2212yes—
76#1969C2214yes—
77#1958C2214yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1956C2211yescharacteristic
87#1956C2211yescharacteristic
88#1939C2212yes—
89#1916C2212yes—
90#1916C2212yes—
91#1880C2214yes—
92#1878C2214yes—
93#1878C2214yes—
94#1877C2214yes—
95#1863C2214yes—
96#1862C2214yes—
97#1845C2211yescharacteristic
98#1834C2211yescharacteristic
99#1834C2211yescharacteristic
100#1805C2214yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×Q16C2314yes—
108C22×Q16C2314yes—
109C22×SD16C2314yes—
110#556—48no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2312yes—
116C2×#127C2311yescharacteristic
117C2×#126—22no—
118C2×#126—22no—
119C2×#125—21no—
120C2×#125—21no—
121C2×#125C2312yes—
122C2×#124C2312yes—
123C2×#124C2312yes—
124C2×#124C2312yes—
125C2×#124C2312yes—
126C2×C22⋊Q8—22no—
127C2×C4×Q8D411yescharacteristic
128C2×C4×Q8D411yescharacteristic
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2311yescharacteristic
135C2×C4×D4—21no—
136C2×C4×D4—21no—
137C2×C4×D4C2312yes—
138C2×#117—22no—
139#494—24no—
140#494—24no—
141#492—24no—
142#488—48no—
143#464—48no—
144#464—48no—
145#462—48no—
146C4×Q16—48no—
147C2×#107C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2312yes—
165C2×C4×C8C2312yes—
166C2×C4×C8C2311yescharacteristic
167#400—22no—
168#400—22no—
169#400—22no—
170#398—22no—
171#398—22no—
172#397—22no—
173#397—22no—
174#396—22no—
175#396—22no—
176#396—22no—
177#396—22no—
178#395—22no—
179#391C2312yes—
180#391C2312yes—
181#391C2312yes—
182#391C2312yes—
183#391C2312yes—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390—21no—
190#390—21no—
191#390—21no—
192#390—21no—
193#390—21no—
194#390C2312yes—
195#389—22no—
196#389—22no—
197#389—22no—
198#389—22no—
199#389—21no—
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#389—21no—
213#389—21no—
214#389—21no—
215#389—21no—
216#389—21no—
217#389C2312yes—
218#389C2312yes—
219#389C2312yes—
220#389C2312yes—
221#388—22no—
222#388—22no—
223#388—22no—
224#388—22no—
225#387—22no—
226#387—22no—
227#387—22no—
228#387—22no—
229#384—21no—
230#384—21no—
231#384—21no—
232#384—21no—
233#384C2311yescharacteristic
234#384C2311yescharacteristic
235#384C2311yescharacteristic
236#384C2311yescharacteristic
237#382—21no—
238#382—21no—
239#382—22no—
240#382C2312yes—
241#380—22no—
242#380—22no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4—22no—
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4D411yescharacteristic
256C4×C4⋊C4D411yescharacteristic
257C4×C4⋊C4D411yescharacteristic
258C4×C4⋊C4D411yescharacteristic
259C4×C4⋊C4D411yescharacteristic
260C4×C4⋊C4C2311yescharacteristic
261C4×C4⋊C4C2311yescharacteristic
262C4×C4⋊C4C2311yescharacteristic
263C43C2311yescharacteristic
264C43C2311yescharacteristic
265C43C2311yescharacteristic
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C22×Q8C2×D411yescharacteristic
276C22×Q8C2412yes—
277C22×Q8C2412yes—
278C22×Q8C2412yes—
279C22×D4—21no—
280C22×D4—21no—
281C22×D4C2412yes—
282C23×C4—22no—
283C2×Q16—28no—
284C2×Q16—28no—
285C2×Q16—28no—
286C2×Q16—44no—
287C2×Q16—44no—
288C2×Q16—44no—
289C2×SD16—28no—
290C2×SD16—44no—
291C2×SD16—44no—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—44no—
302C4⋊Q8—44no—
303C4⋊Q8#4114yes—
304#127—22no—
305#127—22no—
306#126—48no—
307#126—48no—
308#125—44no—
309#125—44no—
310#125—44no—
311#124—28no—
312#124—44no—
313#124—44no—
314#124—44no—
315C22⋊Q8—48no—
316C4×Q8—24no—
317C4×Q8—24no—
318C4×Q8—28no—
319C4×Q8—28no—
320C4×Q8—42no—
321C4×Q8—44no—
322C4×Q8—44no—
323C4×Q8—44no—
324C4×D4—44no—
325C4×D4—44no—
326C4×D4—44no—
327#117—48no—
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—22no—
335C2×C4⋊C4—22no—
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—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
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⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
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×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405#107—44no—
406C4⋊C8—24no—
407C4⋊C8—24no—
408C4⋊C8—24no—
409#103—28no—
410#103—44no—
411#103—44no—
412#103—44no—
413#103—44no—
414#103—44no—
415#103—44no—
416#102—28no—
417#102—44no—
418#102—44no—
419C4×C8—24no—
420C4×C8—22no—
421C4×C8—22no—
422C4×C8—22no—
423C4×C8—24no—
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#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447C24—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—24no—
454C2×Q8—24no—
455C2×Q8—24no—
456C2×Q8—24no—
457C2×Q8—24no—
458C2×Q8—24no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—41no—
462C2×Q8C2×#4114yes—
463C2×D4—22no—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—22no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C2511yescharacteristic
498Q16—416no—
499Q16—88no—
500SD16—88no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8H(5,𝔽2)14yes—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—44no—
583C42—44no—
584C42—44no—
585C42—82no—
586C42—82no—
587C42#13514yes—
588C42C2×#4112yes—
589C42H(5,𝔽2)14yes—
590C422−1+412yes—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23C23×D411yescharacteristic
599Q8—28no—
600Q8—44no—
601Q8—44no—
602Q8—44no—
603Q8—44no—
604Q8—44no—
605Q8—44no—
606D4—44no—
607D4—44no—
608D4—84no—
609D4—84no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—24no—
622C2×C4—21no—
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—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
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—88no—
664C2×C4#53512yes—
665C2×C4#53512yes—
666C2×C4#54812yes—
667C2×C4#54812yes—
668C2×C4C2×#13512yes—
669C2×C4C2×#13512yes—
670C2×C4C2×#13512yes—
671C2×C4#57512yes—
672C2×C4#57812yes—
673C2×C4#57812yes—
674C2×C4C22×#4111yescharacteristic
675C2×C4C2×H(5,𝔽2)12yes—
676C2×C4C2×H(5,𝔽2)12yes—
677C2×C4C2×2−1+411yescharacteristic
678C8—24no—
679C8—42no—
680C8—42no—
681C8—42no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22C2×#53511yescharacteristic
690C22C2×#54811yescharacteristic
691C22#349811yescharacteristic
692C22C22×#13511yescharacteristic
693C22C2×#57511yescharacteristic
694C22C2×#57811yescharacteristic
695C22C2×#57811yescharacteristic
696C4—22no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—44no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—41no—
707C4—41no—
708C4—41no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4#336414yes—
716C4#337714yes—
717C4#351912yes—
718C4#354812yes—
719C2—162no—
720C2#3916911yescharacteristic
721C2#6078211yescharacteristic
722C2#6082111yescharacteristic
723C2C2×#336411yescharacteristic
724C2C2×#337711yescharacteristic
725C2#6241811yescharacteristic
726C2#6242511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36574C2semidirect
4. #30888C2semidirect
5. #30885C2semidirect
6. #30220C2semidirect
8. #29193C2semidirect
10. #29144C2semidirect
13. #28440C2semidirect
14. #24736C2semidirect
15. #24195C2semidirect
16. #22270C2semidirect
18. C4×#494C2semidirect
19. #21339C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#365742
#332262
#308882
#308852
#302202
#302192
#291932
#291922
#291442
#284442
#284402
#247362
#241952
#222702
#222682
C4×#4942
#213392