Finite Group Databasebeta

← #6805896

Subgroups — #6805896

Order 512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 775 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36595C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic, complemented
3#33230C211yesmaximal, characteristic, complemented
4#30905C214yesmaximal, complemented
5#30900C214yesmaximal, complemented
6#30219C211yesmaximal, characteristic, complemented
7#30218C211yesmaximal, characteristic
8#29193C211yesmaximal, characteristic, complemented
9#29192C211yesmaximal, characteristic, complemented
10#29139C211yesmaximal, characteristic, complemented
11#28444C211yesmaximal, characteristic, complemented
12#28444C211yesmaximal, characteristic, complemented
13#28441C211yesmaximal, characteristic
14#24739C214yesmaximal, complemented
15#24206C214yesmaximal, complemented
16#22270C211yesmaximal, characteristic, complemented
17#22268C211yesmaximal, characteristic, complemented
18#21692C211yesmaximal, characteristic, complemented
19#21340C211yesmaximal, characteristic, complemented
20C2×#559C2212yes—
21C2×#494C2211yescharacteristic
22C2×#492C2211yescharacteristic
23C2×#492C2211yescharacteristic
24C2×#489C2214yes—
25C2×#464C2214yes—
26C2×#464C2214yes—
27C2×#460C2214yes—
28C2×#440C2214yes—
29#2868C2211yescharacteristic
30#2867C2211yescharacteristic
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2858C2211yescharacteristic
34#2858C2211yescharacteristic
35#2582C2212yes—
36#2567—21no—
37#2552—21no—
38#2550—21no—
39#2550—21no—
40#2548—21no—
41#2548—21no—
42#2548C2212yes—
43#2544C2212yes—
44#2544C2212yes—
45#2507—21no—
46#2507—21no—
47#2451C2212yes—
48#2451C2212yes—
49#2445—21no—
50#2445—21no—
51#2445—21no—
52#2445—21no—
53#2369—21no—
54#2366—21no—
55#2366—21no—
56#2355—21no—
57Q8×C4⋊C4C2212yes—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×C4⋊Q8C2211yescharacteristic
62C4×#127C2211yescharacteristic
63C4×#127C2211yescharacteristic
64C42×Q8C2211yescharacteristic
65#2068C2214yes—
66#2067C2214yes—
67#2042C2214yes—
68#2042C2214yes—
69#2041C2214yes—
70#2041C2214yes—
71#2023C2214yes—
72#2015C2214yes—
73#2015C2214yes—
74#2014C2214yes—
75#1985C2212yes—
76#1973C2214yes—
77#1959C2214yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1957C2211yescharacteristic
82#1957C2211yescharacteristic
83#1957C2211yescharacteristic
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1956C2211yescharacteristic
87#1956C2211yescharacteristic
88#1940C2212yes—
89#1917C2212yes—
90#1917C2212yes—
91#1880C2214yes—
92#1878C2214yes—
93#1878C2214yes—
94#1877C2214yes—
95#1863C2214yes—
96#1862C2214yes—
97#1845C2211yescharacteristic
98#1834C2211yescharacteristic
99#1834C2211yescharacteristic
100#1802C2214yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×Q16C2314yes—
108C22×SD16C2314yes—
109C22×SD16C2314yes—
110#559—48no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2312yes—
116C2×#127—21no—
117C2×#127—21no—
118C2×#127—21no—
119C2×#127—21no—
120C2×#127C2311yescharacteristic
121C2×#127C2311yescharacteristic
122C2×#127C2312yes—
123C2×#127C2312yes—
124C2×#125—21no—
125C2×#125—21no—
126C2×#124—22no—
127C2×#124C2312yes—
128C2×#124C2312yes—
129C2×#124C2312yes—
130C2×#124C2312yes—
131C2×#123—22no—
132C2×#123—22no—
133C2×C4×Q8D411yescharacteristic
134C2×C4×Q8D411yescharacteristic
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2311yescharacteristic
140C2×C4×D4—22no—
141C2×C4×D4—21no—
142C2×C4×D4—21no—
143C2×C4×D4—21no—
144C2×C4×D4—21no—
145C2×C4×D4C2312yes—
146C2×C4×D4C2312yes—
147#494—24no—
148#492—24no—
149#492—24no—
150#489—48no—
151#464—48no—
152#464—48no—
153#460—48no—
154#440—48no—
155C2×#107C2314yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2312yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×#102C2314yes—
169C2×#97C2312yes—
170C2×#97C2312yes—
171C2×C4×C8C2311yescharacteristic
172C2×C4×C8C2311yescharacteristic
173C2×C4×C8C2311yescharacteristic
174C2×C4×C8C2311yescharacteristic
175#398—22no—
176#398—22no—
177#398—22no—
178#398—22no—
179#395—22no—
180#395—22no—
181#394—22no—
182#394—22no—
183#394—22no—
184#394—22no—
185#394—22no—
186#394—22no—
187#391—22no—
188#391—22no—
189#391—22no—
190#391C2312yes—
191#391C2312yes—
192#391C2312yes—
193#391C2312yes—
194#390—22no—
195#390—22no—
196#390—21no—
197#390—21no—
198#390—21no—
199#390—21no—
200#390—21no—
201#390—21no—
202#390—21no—
203#390—21no—
204#390—21no—
205#390—21no—
206#390—21no—
207#390—21no—
208#390C2312yes—
209#390C2312yes—
210#389—22no—
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—21no—
220#389—21no—
221#389—21no—
222#389—21no—
223#389—22no—
224#389—22no—
225#389C2312yes—
226#389C2312yes—
227#389C2312yes—
228#389C2312yes—
229#387—22no—
230#387—22no—
231#386—22no—
232#386—22no—
233#386—22no—
234#386—22no—
235#384—22no—
236#384—21no—
237#384—21no—
238#384C2311yescharacteristic
239#384C2311yescharacteristic
240#384C2311yescharacteristic
241#384C2311yescharacteristic
242#384C2312yes—
243#382—21no—
244#382—21no—
245C4×C4⋊C4—22no—
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—21no—
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×Q8—22no—
276C22×Q8C2×D411yescharacteristic
277C22×Q8C2412yes—
278C22×Q8C2412yes—
279C22×D4—22no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4—21no—
283C22×D4—21no—
284C22×D4—22no—
285C22×D4—22no—
286C22×D4—22no—
287C22×D4C2412yes—
288C22×D4C2412yes—
289C23×C4—22no—
290C23×C4—22no—
291C2×Q16—28no—
292C2×Q16—44no—
293C2×Q16—44no—
294C2×SD16—28no—
295C2×SD16—28no—
296C2×SD16—28no—
297C2×SD16—44no—
298C2×SD16—44no—
299C2×SD16—44no—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C22×C8C2412yes—
303C22×C8C2412yes—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—44no—
308C4⋊Q8—44no—
309C4⋊Q8—82no—
310C4⋊Q8#4114yes—
311#127—22no—
312#127—22no—
313#127—22no—
314#127—22no—
315#127—44no—
316#127—44no—
317#127—44no—
318#127—44no—
319#127—82no—
320#127—82no—
321#125—82no—
322#125—82no—
323#124—28no—
324#124—44no—
325#124—44no—
326#124—44no—
327#124—48no—
328#123—48no—
329#123—48no—
330C4×Q8—24no—
331C4×Q8—24no—
332C4×Q8—28no—
333C4×Q8—42no—
334C4×Q8—44no—
335C4×Q8—44no—
336C4×Q8—44no—
337C4×D4—28no—
338C4×D4—44no—
339C4×D4—48no—
340C4×D4—82no—
341C4×D4—82no—
342C4×D4—82no—
343C4×D4—82no—
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—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2×D411yescharacteristic
369C2×C4⋊C4C2×D411yescharacteristic
370C2×C4⋊C4C2×D411yescharacteristic
371C2×C4⋊C4C2×D411yescharacteristic
372C2×C4⋊C4C2×D411yescharacteristic
373C2×C4⋊C4C2×D411yescharacteristic
374C2×C4⋊C4C2×D411yescharacteristic
375C2×C4⋊C4C2×D411yescharacteristic
376C2×C4⋊C4C2412yes—
377C2×C4⋊C4C2412yes—
378C2×C4⋊C4C2412yes—
379C2×C4⋊C4C2412yes—
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⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—22no—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—22no—
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×C42C2×D411yescharacteristic
407C2×C42C2×D411yescharacteristic
408C2×C42C2×D411yescharacteristic
409C2×C42C2×D411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2411yescharacteristic
412C2×C42C2411yescharacteristic
413C2×C42C2411yescharacteristic
414C2×C42C2411yescharacteristic
415C2×C42C2411yescharacteristic
416C2×C42C2411yescharacteristic
417#107—44no—
418C4⋊C8—24no—
419C4⋊C8—24no—
420C4⋊C8—24no—
421#103—28no—
422#103—44no—
423#103—44no—
424#103—44no—
425#102—28no—
426#102—44no—
427#102—44no—
428#102—44no—
429#102—44no—
430#102—44no—
431#97—24no—
432#97—24no—
433C4×C8—22no—
434C4×C8—22no—
435C4×C8—22no—
436C4×C8—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#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451#95C2×D411yescharacteristic
452#95C2×D411yescharacteristic
453#95C2×D411yescharacteristic
454#95C2×D411yescharacteristic
455C24—22no—
456C24—22no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—24no—
462C2×Q8—24no—
463C2×Q8—24no—
464C2×Q8—24no—
465C2×Q8—22no—
466C2×Q8—41no—
467C2×Q8—44no—
468C2×Q8—82no—
469C2×Q8—82no—
470C2×Q8C2×#4114yes—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—24no—
474C2×D4—24no—
475C2×D4—24no—
476C2×D4—24no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—44no—
480C2×D4—44no—
481C2×D4—44no—
482C2×D4—44no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—44no—
488C2×D4—42no—
489C2×D4—42no—
490C2×D4—44no—
491C2×D4—42no—
492C2×D4—42no—
493C2×D4—42no—
494C2×D4—42no—
495C2×D4—42no—
496C2×D4—42no—
497C2×D4—82no—
498C2×D4—82no—
499C2×D4—82no—
500C2×D4—82no—
501C2×D4—82no—
502C2×D4—82no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—44no—
510C22×C4—44no—
511C22×C4—82no—
512C22×C4—82no—
513C22×C4—82no—
514C22×C4—82no—
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D411yescharacteristic
526C22×C4C22×D411yescharacteristic
527C22×C4C22×D411yescharacteristic
528C22×C4C22×D411yescharacteristic
529C22×C4C2511yescharacteristic
530Q16—88no—
531SD16—416no—
532SD16—88no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8H(5,𝔽2)14yes—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—24no—
549C4⋊C4—24no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—24no—
554C4⋊C4—24no—
555C4⋊C4—24no—
556C4⋊C4—24no—
557C4⋊C4—41no—
558C4⋊C4—41no—
559C4⋊C4—41no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—44no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—42no—
566C4⋊C4—42no—
567C4⋊C4—42no—
568C4⋊C4—42no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—42no—
572C4⋊C4—42no—
573C4⋊C4—82no—
574C4⋊C4—82no—
575C4⋊C4—82no—
576C4⋊C4—82no—
577C4⋊C4—82no—
578C4⋊C4—82no—
579C4⋊C4—82no—
580C4⋊C4—82no—
581C22⋊C4—44no—
582C22⋊C4—44no—
583C22⋊C4—44no—
584C22⋊C4—44no—
585C22⋊C4—44no—
586C22⋊C4—44no—
587C22⋊C4—82no—
588C22⋊C4—82no—
589C22⋊C4—82no—
590C22⋊C4—82no—
591C22⋊C4—82no—
592C22⋊C4—82no—
593C22⋊C4—82no—
594C22⋊C4—82no—
595C42—22no—
596C42—22no—
597C42—22no—
598C42—22no—
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—44no—
612C42—44no—
613C42—82no—
614C42—82no—
615C42#13714yes—
616C42C2×#4112yes—
617C42H(5,𝔽2)14yes—
618C422−1+412yes—
619C23—42no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—42no—
624C23—42no—
625C23—42no—
626C23—42no—
627C23—42no—
628C23—42no—
629C23—42no—
630C23—42no—
631C23—42no—
632C23—42no—
633C23C23×D411yescharacteristic
634Q8—28no—
635Q8—44no—
636Q8—44no—
637Q8—44no—
638Q8—44no—
639Q8—164no—
640D4—44no—
641D4—44no—
642D4—44no—
643D4—44no—
644D4—88no—
645D4—84no—
646D4—84no—
647D4—84no—
648D4—84no—
649D4—164no—
650D4—164no—
651D4—164no—
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—24no—
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—21no—
682C2×C4—21no—
683C2×C4—21no—
684C2×C4—21no—
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—164no—
703C2×C4—164no—
704C2×C4#53512yes—
705C2×C4#53512yes—
706C2×C4#54812yes—
707C2×C4#54812yes—
708C2×C4C2×#13712yes—
709C2×C4C2×#13712yes—
710C2×C4C2×#13712yes—
711C2×C4#57512yes—
712C2×C4#57612yes—
713C2×C4#57712yes—
714C2×C4C22×#4111yescharacteristic
715C2×C4C2×H(5,𝔽2)12yes—
716C2×C4C2×H(5,𝔽2)12yes—
717C2×C4C2×2−1+411yescharacteristic
718C8—24no—
719C8—42no—
720C8—42no—
721C8—42no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22—82no—
731C22—82no—
732C22—82no—
733C22—82no—
734C22—82no—
735C22—82no—
736C22C2×#53511yescharacteristic
737C22C2×#54811yescharacteristic
738C22#349811yescharacteristic
739C22C22×#13711yescharacteristic
740C22C2×#57511yescharacteristic
741C22C2×#57611yescharacteristic
742C22C2×#57711yescharacteristic
743C4—22no—
744C4—22no—
745C4—22no—
746C4—22no—
747C4—44no—
748C4—41no—
749C4—41no—
750C4—41no—
751C4—41no—
752C4—41no—
753C4—41no—
754C4—41no—
755C4—41no—
756C4—82no—
757C4—82no—
758C4—82no—
759C4—82no—
760C4—82no—
761C4#337114yes—
762C4#337514yes—
763C4#351912yes—
764C4#354812yes—
765C2—162no—
766C2—162no—
767C2#3916911yescharacteristic
768C2#6080411yescharacteristic
769C2#6082211yescharacteristic
770C2C2×#337511yescharacteristic
771C2C2×#337111yescharacteristic
772C2#6242211yescharacteristic
773C2#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
3. #33230C2semidirect
4. #30905C2semidirect
5. #30900C2semidirect
6. #30219C2semidirect
8. #29193C2semidirect
9. #29192C2semidirect
10. #29139C2semidirect
11. #28444C2semidirect
12. #28444C2semidirect
14. #24739C2semidirect
15. #24206C2semidirect
16. #22270C2semidirect
17. #22268C2semidirect
18. #21692C2semidirect
19. #21340C2semidirect

Maximal subgroups

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

subgroupindex
#365952
#365632
#332302
#309052
#309002
#302192
#302182
#291932
#291922
#291392
#284442
#284412
#247392
#242062
#222702
#222682
#216922
#213402