Finite Group Databasebeta

← #6797225

Subgroups — #6797225

Order 512.

1497 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36210C211yesmaximal, characteristic
2#36129C211yesmaximal, characteristic
3#33287C211yesmaximal, characteristic, complemented
4#31829C214yesmaximal, complemented
5#31731C214yesmaximal
6#29845C211yesmaximal, characteristic, complemented
7#29808C211yesmaximal, characteristic, complemented
8#29176C211yesmaximal, characteristic, complemented
9#28616C211yesmaximal, characteristic, complemented
10#27962C211yesmaximal, characteristic, complemented
11#27889C211yesmaximal, characteristic, complemented
12#26750C214yesmaximal, complemented
13#26678C214yesmaximal, complemented
14#25902C211yesmaximal, characteristic, complemented
15#25600C211yesmaximal, characteristic, complemented
16#21864C211yesmaximal, characteristic, complemented
17#21834C211yesmaximal, characteristic, complemented
18#21647C211yesmaximal, characteristic, complemented
19#21265C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#494C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27#2862C2211yescharacteristic
28#2859C2211yescharacteristic
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2562—21no—
34#2550—21no—
35#2514C2212yes—
36#2509—21no—
37#2495—21no—
38#2494—21no—
39#2494—21no—
40#2492C2212yes—
41#2485—21no—
42#2451C2212yes—
43#2446—21no—
44#2442—21no—
45#2439—21no—
46#2429—21no—
47#2421C2212yes—
48#2417C2212yes—
49#2415C2212yes—
50#2409—21no—
51#2370—21no—
52#2362C2212yes—
53#2360C2212yes—
54#2352—21no—
55#2347—21no—
56Q8×C22⋊C4—21no—
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59#2285C2211yescharacteristic
60#2285C2211yescharacteristic
61C42×D4C2211yescharacteristic
62#2101C2214yes—
63#2088C2214yes—
64#2066C2214yes—
65#2056C2214yes—
66#2056C2214yes—
67#2035C2214yes—
68#2035C2214yes—
69#2024C2214yes—
70#2001C2214yes—
71#2001C2214yes—
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1987C2214yes—
79#1956C2211yescharacteristic
80#1956C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1955C2212yes—
84#1940C2212yes—
85#1935C2214yes—
86#1935C2214yes—
87#1935C2214yes—
88#1927C2214yes—
89#1924C2212yes—
90#1924C2212yes—
91#1920C2214yes—
92#1878C2214yes—
93#1878C2214yes—
94#1873C2214yes—
95#1872C2214yes—
96#1844C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1825C2214yes—
100#1824C2214yes—
101#1805C2214yes—
102C4×C4⋊C8C2211yescharacteristic
103#1763C2214yes—
104C4×C22⋊C8C2211yescharacteristic
105C4×C22⋊C8C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C23×Q8—21no—
108C22×Q16C2314yes—
109#533—24no—
110#533—42no—
111C2×C4⋊Q8—24no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—42no—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2311yescharacteristic
120C2×#125—21no—
121C2×#125—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8C2312yes—
125C2×C22⋊Q8C2312yes—
126C2×C22⋊Q8C2312yes—
127C2×C22⋊Q8C2312yes—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2312yes—
132C2×C4×D4D411yescharacteristic
133C2×C4×D4D411yescharacteristic
134C2×C4×D4C2311yescharacteristic
135C2×#117—21no—
136C2×#117—21no—
137C2×#117C2312yes—
138C22×C4⋊C4—21no—
139C22×C4⋊C4C2312yes—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#494—42no—
143#487—24no—
144#487—42no—
145#431—42no—
146#431—42no—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×#97C2312yes—
162C2×#97C2312yes—
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167#401—22no—
168#400—22no—
169#400—22no—
170#400—22no—
171#398—22no—
172#398—22no—
173#397—22no—
174#397—22no—
175#396—22no—
176#395—22no—
177#394—22no—
178#393—22no—
179#391—21no—
180#391—21no—
181#391—21no—
182#391—21no—
183#391—22no—
184#391—22no—
185#391C2312yes—
186#391C2312yes—
187#391C2312yes—
188#391C2312yes—
189#391C2312yes—
190#391C2312yes—
191#389—22no—
192#389—21no—
193#389—21no—
194#389—21no—
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#389—21no—
204#389—22no—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#388—22no—
210#388—22no—
211#388—22no—
212#388—22no—
213#388—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#387—22no—
219#386—22no—
220#385—22no—
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#383—22no—
226#382—21no—
227#382—21no—
228#382—21no—
229#382—21no—
230#382C2312yes—
231#380—21no—
232#380—21no—
233#380—21no—
234#380—21no—
235#380—21no—
236#380—21no—
237#380—21no—
238#380—21no—
239#380—21no—
240#380—21no—
241#380—22no—
242#380C2312yes—
243#380C2312yes—
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—22no—
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4D411yescharacteristic
253C4×C4⋊C4C2312yes—
254C4×C4⋊C4C2312yes—
255C4×C4⋊C4C2311yescharacteristic
256C4×C22⋊C4—22no—
257C4×C22⋊C4D411yescharacteristic
258C4×C22⋊C4D411yescharacteristic
259C4×C22⋊C4D411yescharacteristic
260C4×C22⋊C4D411yescharacteristic
261C4×C22⋊C4C2311yescharacteristic
262C4×C22⋊C4C2311yescharacteristic
263C43C2311yescharacteristic
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#340C2314yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×Q8—22no—
278C22×Q8—44no—
279C22×Q8—44no—
280C22×Q8—82no—
281C22×Q8C2412yes—
282C22×Q8C2412yes—
283C22×D4C2×D411yescharacteristic
284C23×C4—22no—
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2411yescharacteristic
290C23×C4C2411yescharacteristic
291C2×Q16—44no—
292C2×Q16—44no—
293C2×Q16—44no—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C4⋊Q8—24no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—42no—
307C4⋊Q8—44no—
308C4⋊Q8—44no—
309C4⋊Q8#4114yes—
310#125—44no—
311#125—44no—
312C22⋊Q8—24no—
313C22⋊Q8—44no—
314C22⋊Q8—44no—
315C22⋊Q8—42no—
316C22⋊Q8—44no—
317C22⋊Q8—44no—
318C22⋊Q8—82no—
319C4×Q8—24no—
320C4×Q8—42no—
321C4×Q8—44no—
322C4×Q8—44no—
323C4×Q8—82no—
324C4×D4—24no—
325C4×D4—42no—
326C4×D4—42no—
327#117—44no—
328#117—44no—
329#117—44no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
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—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—42no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C22⋊C4—22no—
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⋊C4C2×D411yescharacteristic
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C22⋊C4C2×D411yescharacteristic
378C2×C22⋊C4C2×D411yescharacteristic
379C2×C22⋊C4C2×D411yescharacteristic
380C2×C22⋊C4C2×D411yescharacteristic
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
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
403#106—44no—
404C4⋊C8—42no—
405C4⋊C8—42no—
406C4⋊C8—42no—
407#103—28no—
408#103—44no—
409#103—44no—
410#103—44no—
411#103—44no—
412#103—44no—
413C22⋊C8—42no—
414C22⋊C8—42no—
415C22⋊C8—42no—
416C22⋊C8—42no—
417#97—42no—
418#97—42no—
419C4×C8—22no—
420C4×C8—22no—
421C4×C8—22no—
422C4×C8—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#95—22no—
436#95—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#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451C24C22×D411yescharacteristic
452C24C22×D411yescharacteristic
453C2×Q8—24no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—24no—
459C2×Q8—24no—
460C2×Q8—24no—
461C2×Q8—24no—
462C2×Q8—44no—
463C2×Q8—44no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8—82no—
470C2×Q8—816no—
471C2×D4—22no—
472C2×D4—41no—
473C2×D4—41no—
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—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—82no—
504C22×C4#13714yes—
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
518Q16—88no—
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×C8—22no—
530C2×C8—22no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—41no—
541C4⋊C4—41no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C4⋊C4C2×#4114yes—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—41no—
571C22⋊C4—41no—
572C22⋊C4—41no—
573C22⋊C4—41no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—82no—
579C22⋊C4—82no—
580C22⋊C4—82no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—21no—
590C42—21no—
591C42—21no—
592C42—21no—
593C42—22no—
594C42—22no—
595C42—44no—
596C42—44no—
597C42—82no—
598C42—82no—
599C42—82no—
600C42C2×#4112yes—
601C42H(5,𝔽2)12yes—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—21no—
612C23—21no—
613C23—21no—
614C23—21no—
615C23C2×#13712yes—
616C23C23×D411yescharacteristic
617Q8—48no—
618Q8—44no—
619Q8—44no—
620Q8—44no—
621Q8—44no—
622Q8—88no—
623Q8—84no—
624D4—44no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—28no—
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—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—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
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—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—88no—
687C2×C4#54612yes—
688C2×C4#54612yes—
689C2×C4C2×#13712yes—
690C2×C4C2×#13712yes—
691C2×C4#57512yes—
692C2×C4#57712yes—
693C2×C4#57812yes—
694C2×C4C22×#4111yescharacteristic
695C2×C4C2×H(5,𝔽2)11yescharacteristic
696C2×C4#58512yes—
697C2×C4#58512yes—
698C8—42no—
699C8—42no—
700C8—42no—
701C8—42no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22—41no—
715C22—41no—
716C22#344911yescharacteristic
717C22C2×#54611yescharacteristic
718C22#350011yescharacteristic
719C22C22×#13711yescharacteristic
720C22C2×#57511yescharacteristic
721C22C2×#57711yescharacteristic
722C22C2×#57811yescharacteristic
723C4—22no—
724C4—41no—
725C4—41no—
726C4—41no—
727C4—41no—
728C4—41no—
729C4—41no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—44no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4#331114yes—
742C4#331214yes—
743C4#352412yes—
744C4#355112yes—
745C2—42no—
746C2—81no—
747C2#3894811yescharacteristic
748C2#6096611yescharacteristic
749C2#6100811yescharacteristic
750C2C2×#331211yescharacteristic
751C2C2×#331111yescharacteristic
752C2#6254211yescharacteristic
753C2#6254511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33287C2semidirect
4. #31829C2semidirect
6. #29845C2semidirect
7. #29808C2semidirect
8. #29176C2semidirect
9. #28616C2semidirect
10. #27962C2semidirect
11. #27889C2semidirect
12. #26750C2semidirect
13. #26678C2semidirect
14. #25902C2semidirect
15. #25600C2semidirect
16. #21864C2semidirect
17. #21834C2semidirect
18. #21647C2semidirect

Maximal subgroups

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

subgroupindex
#362102
#361292
#332872
#318292
#317312
#298452
#298082
#291762
#286162
#279622
#278892
#267502
#266782
#259022
#256002
#218642
#218342
#216472
#212652