Finite Group Databasebeta

← #6805791

Subgroups — #6805791

Order 512.

1614 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36595C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic, complemented
3#33231C211yesmaximal, characteristic
4#30835C214yesmaximal, complemented
5#30833C214yesmaximal, complemented
6#30204C211yesmaximal, characteristic, complemented
7#30203C211yesmaximal, characteristic, complemented
8#29207C211yesmaximal, characteristic, complemented
9#28749C211yesmaximal, characteristic
10#28655C211yesmaximal, characteristic, complemented
11#28655C211yesmaximal, characteristic, complemented
12#28514C211yesmaximal, characteristic, complemented
13#28509C211yesmaximal, characteristic, complemented
14#26472C214yesmaximal, complemented
15#26334C214yesmaximal, complemented
16#22220C211yesmaximal, characteristic, complemented
17#22219C211yesmaximal, characteristic, complemented
18#21726C211yesmaximal, characteristic, complemented
19#21340C211yesmaximal, characteristic, complemented
20C2×#559—21no—
21C2×#500C2211yescharacteristic
22C2×#490C2214yes—
23C2×#486C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#478C2214yes—
26C2×#476C2214yes—
27#2868C2211yescharacteristic
28#2867C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2858C2211yescharacteristic
32#2858C2211yescharacteristic
33#2582—21no—
34#2567—21no—
35#2552C2212yes—
36#2550C2212yes—
37#2550C2212yes—
38#2548—21no—
39#2548—21no—
40#2548C2212yes—
41#2544—21no—
42#2544—21no—
43#2507—21no—
44#2507—21no—
45#2451—21no—
46#2451—21no—
47#2445—21no—
48#2445—21no—
49#2445C2212yes—
50#2445C2212yes—
51#2369—21no—
52#2366—21no—
53#2366C2212yes—
54#2355C2212yes—
55Q8×C4⋊C4—21no—
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×#127C2211yescharacteristic
61C4×#127C2211yescharacteristic
62C42×Q8C2211yescharacteristic
63#2094C2214yes—
64#2092C2214yes—
65#2073C2214yes—
66#2071C2214yes—
67#2067C2214yes—
68#2067C2214yes—
69#2059C2214yes—
70#2041C2214yes—
71#2025C2214yes—
72#2023C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1985C2212yes—
78#1982C2214yes—
79#1961C2214yes—
80#1957C2211yescharacteristic
81#1957C2211yescharacteristic
82#1957C2211yescharacteristic
83#1957C2211yescharacteristic
84#1957C2211yescharacteristic
85#1957C2211yescharacteristic
86#1949C2214yes—
87#1943C2214yes—
88#1940C2212yes—
89#1926C2214yes—
90#1925C2214yes—
91#1925C2214yes—
92#1925C2214yes—
93#1917C2212yes—
94#1917C2212yes—
95#1864C2214yes—
96#1862C2214yes—
97#1845C2211yescharacteristic
98#1844C2211yescharacteristic
99#1844C2211yescharacteristic
100#1776C2214yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107#559—48no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2312yes—
113C2×#127—21no—
114C2×#127—21no—
115C2×#127—21no—
116C2×#127—21no—
117C2×#127C2311yescharacteristic
118C2×#127C2311yescharacteristic
119C2×#127C2312yes—
120C2×#127C2312yes—
121C2×#125C2312yes—
122C2×#125C2312yes—
123C2×#124—22no—
124C2×#124—21no—
125C2×#124—21no—
126C2×#124—21no—
127C2×#124—21no—
128C2×#123—22no—
129C2×#123—22no—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×Q8D411yescharacteristic
135C2×C4×Q8D411yescharacteristic
136C2×C4×Q8C2311yescharacteristic
137C2×C4×D4—22no—
138C2×C4×D4—21no—
139C2×C4×D4—21no—
140C2×C4×D4—21no—
141C2×C4×D4—21no—
142C2×C4×D4C2312yes—
143C2×C4×D4C2312yes—
144#500—24no—
145#490—48no—
146#486—24no—
147#486—24no—
148#478—48no—
149#476—48no—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×#106C2314yes—
153C2×#106C2314yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#97C2312yes—
166C2×#97C2312yes—
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171#398—22no—
172#398—22no—
173#398—22no—
174#398—22no—
175#395—22no—
176#395—22no—
177#394—22no—
178#394—22no—
179#394—22no—
180#394—22no—
181#394—22no—
182#394—22no—
183#391—22no—
184#391—21no—
185#391—21no—
186#391—21no—
187#391—21no—
188#391—22no—
189#391—22no—
190#390—22no—
191#390—22no—
192#390—21no—
193#390—21no—
194#390—21no—
195#390—21no—
196#390—21no—
197#390—21no—
198#390—21no—
199#390—21no—
200#390C2312yes—
201#390C2312yes—
202#390C2312yes—
203#390C2312yes—
204#390C2312yes—
205#390C2312yes—
206#389—22no—
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#389—21no—
218#389—21no—
219#389—22no—
220#389—22no—
221#389C2312yes—
222#389C2312yes—
223#389C2312yes—
224#389C2312yes—
225#387—22no—
226#387—22no—
227#386—22no—
228#386—22no—
229#386—22no—
230#386—22no—
231#384—22no—
232#384—21no—
233#384—21no—
234#384C2311yescharacteristic
235#384C2311yescharacteristic
236#384C2311yescharacteristic
237#384C2311yescharacteristic
238#384C2312yes—
239#382C2312yes—
240#382C2312yes—
241C4×C4⋊C4—22no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4C2312yes—
253C4×C4⋊C4C2312yes—
254C4×C4⋊C4C2312yes—
255C4×C4⋊C4C2312yes—
256C4×C4⋊C4C2311yescharacteristic
257C4×C4⋊C4C2311yescharacteristic
258C4×C4⋊C4C2311yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261C43C2311yescharacteristic
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—22no—
276C22×Q8C2×D411yescharacteristic
277C22×D4—21no—
278C22×D4—21no—
279C22×D4—22no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4—22no—
283C22×D4—22no—
284C22×D4—22no—
285C22×D4C2412yes—
286C22×D4C2412yes—
287C23×C4—22no—
288C23×C4—22no—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—44no—
297C4⋊Q8—44no—
298C4⋊Q8—82no—
299C4⋊Q8#4114yes—
300#127—22no—
301#127—22no—
302#127—28no—
303#127—22no—
304#127—22no—
305#127—44no—
306#127—44no—
307#127—44no—
308#127—82no—
309#127—82no—
310#125—44no—
311#125—44no—
312#124—44no—
313#124—48no—
314#124—44no—
315#124—82no—
316#124—82no—
317#123—48no—
318#123—48no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—42no—
322C4×Q8—44no—
323C4×Q8—44no—
324C4×Q8—82no—
325C4×Q8—82no—
326C4×D4—44no—
327C4×D4—44no—
328C4×D4—44no—
329C4×D4—48no—
330C4×D4—44no—
331C4×D4—82no—
332C4×D4—82no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
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—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⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
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⋊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⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
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×C42C2×D411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406#107—28no—
407#107—44no—
408#106—28no—
409#106—44no—
410C4⋊C8—24no—
411C4⋊C8—24no—
412C4⋊C8—24no—
413#102—28no—
414#102—28no—
415#102—28no—
416#102—28no—
417#102—44no—
418#102—44no—
419#102—44no—
420#102—44no—
421#97—24no—
422#97—24no—
423C4×C8—22no—
424C4×C8—22no—
425C4×C8—22no—
426C4×C8—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#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445C24—22no—
446C24—22no—
447C2×Q8—22no—
448C2×Q8—24no—
449C2×Q8—22no—
450C2×Q8—41no—
451C2×Q8—44no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—82no—
458C2×Q8—82no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—24no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—44no—
468C2×D4—44no—
469C2×D4—44no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4—82no—
486C2×D4—82no—
487C2×D4—82no—
488C2×D4—82no—
489C2×D4—82no—
490C2×D4—82no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—82no—
500C22×C4—82no—
501C22×C4—82no—
502C22×C4—82no—
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C2511yescharacteristic
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8H(5,𝔽2)14yes—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—22no—
544C4⋊C4—41no—
545C4⋊C4—41no—
546C4⋊C4—41no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C4⋊C4C2×#4114yes—
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—
578C22⋊C4—82no—
579C22⋊C4—82no—
580C22⋊C4—82no—
581C22⋊C4—82no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—22no—
597C42—22no—
598C42—44no—
599C42—44no—
600C42—82no—
601C42—82no—
602C42AGL(1,ℤ/8)14yes—
603C42C2×#4112yes—
604C422−1+414yes—
605C422−1+412yes—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23C23×D411yescharacteristic
621Q8—44no—
622Q8—48no—
623Q8—84no—
624Q8—164no—
625D4—44no—
626D4—44no—
627D4—44no—
628D4—44no—
629D4—88no—
630D4—84no—
631D4—84no—
632D4—84no—
633D4—84no—
634D4—164no—
635D4—164no—
636D4—164no—
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—24no—
649C2×C4—21no—
650C2×C4—21no—
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—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—164no—
688C2×C4—164no—
689C2×C4#53612yes—
690C2×C4#53612yes—
691C2×C4#54812yes—
692C2×C4#54812yes—
693C2×C4C2×AGL(1,ℤ/8)12yes—
694C2×C4C2×AGL(1,ℤ/8)12yes—
695C2×C4C2×AGL(1,ℤ/8)12yes—
696C2×C4#57512yes—
697C2×C4#57612yes—
698C2×C4#57712yes—
699C2×C4C22×#4111yescharacteristic
700C2×C4C2×H(5,𝔽2)12yes—
701C2×C4C2×2−1+412yes—
702C2×C4C2×2−1+411yescharacteristic
703C8—24no—
704C8—42no—
705C8—42no—
706C8—42no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22C2×#53611yescharacteristic
722C22C2×#54811yescharacteristic
723C22#349811yescharacteristic
724C22C22×AGL(1,ℤ/8)11yescharacteristic
725C22C2×#57511yescharacteristic
726C22C2×#57611yescharacteristic
727C22C2×#57711yescharacteristic
728C4—22no—
729C4—22no—
730C4—22no—
731C4—22no—
732C4—44no—
733C4—41no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—41no—
739C4—41no—
740C4—41no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4#336614yes—
747C4#337114yes—
748C4#352212yes—
749C4#355012yes—
750C2—162no—
751C2—162no—
752C2#3916811yescharacteristic
753C2#6086411yescharacteristic
754C2#6087711yescharacteristic
755C2C2×#336611yescharacteristic
756C2C2×#337111yescharacteristic
757C2#6242111yescharacteristic
758C2#6242611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2867C22
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. #36563C2semidirect
4. #30835C2semidirect
5. #30833C2semidirect
6. #30204C2semidirect
7. #30203C2semidirect
8. #29207C2semidirect
10. #28655C2semidirect
11. #28655C2semidirect
12. #28514C2semidirect
13. #28509C2semidirect
14. #26472C2semidirect
15. #26334C2semidirect
16. #22220C2semidirect
17. #22219C2semidirect
18. #21726C2semidirect
19. #21340C2semidirect

Maximal subgroups

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

subgroupindex
#365952
#365632
#332312
#308352
#308332
#302042
#302032
#292072
#287492
#286552
#285142
#285092
#264722
#263342
#222202
#222192
#217262
#213402