Finite Group Databasebeta

← #6798758

Subgroups — #6798758

Order 512.

1485 conjugacy classes of subgroups.

The subgroup lattice has 749 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36209C211yesmaximal, characteristic
2#36129C211yesmaximal, characteristic
3#33290C211yesmaximal, characteristic, complemented
4#31749C214yesmaximal, complemented
5#31746C214yesmaximal, complemented
6#29845C211yesmaximal, characteristic, complemented
7#29844C211yesmaximal, characteristic, complemented
8#29207C211yesmaximal, characteristic, complemented
9#28719C211yesmaximal, characteristic, complemented
10#27878C211yesmaximal, characteristic, complemented
11#27875C211yesmaximal, characteristic, complemented
12#26029C214yesmaximal, complemented
13#25904C211yesmaximal, characteristic, complemented
14#25902C211yesmaximal, characteristic, complemented
15#24976C214yesmaximal
16#21835C211yesmaximal, characteristic, complemented
17#21833C211yesmaximal, characteristic, complemented
18C4×#487C211yesmaximal, characteristic, complemented
19#21264C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27#2862C2211yescharacteristic
28#2859C2211yescharacteristic
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2550C2212yes—
34#2536C2212yes—
35#2508—21no—
36#2508—21no—
37#2495—21no—
38#2495—21no—
39#2492—21no—
40#2489C2212yes—
41#2489C2212yes—
42#2485—21no—
43#2485—21no—
44#2451—21no—
45#2446—21no—
46#2442C2212yes—
47#2439C2212yes—
48#2429—21no—
49#2417—21no—
50#2409—21no—
51#2360—21no—
52#2356C2212yes—
53#2352C2212yes—
54#2346—21no—
55#2346—21no—
56Q8×C22⋊C4—21no—
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59#2285C2211yescharacteristic
60#2285C2211yescharacteristic
61C42×D4C2211yescharacteristic
62#2077C2214yes—
63#2076C2214yes—
64#2074C2214yes—
65#2068C2214yes—
66#2066C2214yes—
67#2059C2214yes—
68#2036C2214yes—
69#2035C2214yes—
70#2026C2214yes—
71#2024C2214yes—
72#1991C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1990C2211yescharacteristic
80#1990C2211yescharacteristic
81#1990C2211yescharacteristic
82#1989C2214yes—
83#1988C2214yes—
84#1956C2211yescharacteristic
85#1954C2212yes—
86#1949C2214yes—
87#1939C2212yes—
88#1933C2214yes—
89#1923C2212yes—
90#1923C2212yes—
91#1921C2214yes—
92#1920C2214yes—
93#1873C2214yes—
94#1872C2214yes—
95#1845C2211yescharacteristic
96#1844C2211yescharacteristic
97#1835C2214yes—
98#1834C2211yescharacteristic
99#1825C2214yes—
100#1823C2214yes—
101#1805C2214yes—
102C4×C4⋊C8C2211yescharacteristic
103C4×#103C2214yes—
104C4×C22⋊C8C2211yescharacteristic
105C4×C22⋊C8C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C23×Q8—21no—
108#533—24no—
109#533—42no—
110C2×C4⋊Q8—24no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—42no—
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2311yescharacteristic
119C2×#125C2312yes—
120C2×#125C2312yes—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8C2312yes—
126C2×C22⋊Q8C2312yes—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8C2312yes—
131C2×C4×D4D411yescharacteristic
132C2×C4×D4D411yescharacteristic
133C2×C4×D4C2311yescharacteristic
134C2×#117—21no—
135C2×#117—21no—
136C2×#117C2312yes—
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2312yes—
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#487—24no—
142#487—42no—
143#487—42no—
144#431—42no—
145#431—42no—
146C2×#107C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2312yes—
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2312yes—
165#400—22no—
166#400—22no—
167#400—22no—
168#398—22no—
169#398—22no—
170#398—22no—
171#397—22no—
172#397—22no—
173#396—22no—
174#395—22no—
175#394—22no—
176#393—22no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391—21no—
181#391—21no—
182#391—21no—
183#391—21no—
184#391—22no—
185#391—22no—
186#391C2312yes—
187#391C2312yes—
188#391C2312yes—
189#389—21no—
190#389—21no—
191#389—21no—
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—22no—
203#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#388—22no—
207#388—22no—
208#388—22no—
209#388—22no—
210#388—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#386—22no—
216#385—22no—
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#383—22no—
222#382—21no—
223#382—21no—
224#382—22no—
225#382—22no—
226#382C2312yes—
227#382C2312yes—
228#382C2312yes—
229#380—21no—
230#380—21no—
231#380—21no—
232#380—21no—
233#380—21no—
234#380—21no—
235#380—21no—
236#380—21no—
237#380—22no—
238#380C2312yes—
239#380C2312yes—
240#380C2312yes—
241#380C2312yes—
242C4×C4⋊C4—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⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2312yes—
252C4×C4⋊C4C2311yescharacteristic
253C4×C22⋊C4—22no—
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4D411yescharacteristic
256C4×C22⋊C4D411yescharacteristic
257C4×C22⋊C4D411yescharacteristic
258C4×C22⋊C4C2311yescharacteristic
259C4×C22⋊C4C2311yescharacteristic
260#376—22no—
261C43C2311yescharacteristic
262#340C2314yes—
263#340C2314yes—
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—21no—
278C22×Q8—22no—
279C22×Q8—44no—
280C22×Q8—44no—
281C22×Q8—82no—
282C22×Q8C2412yes—
283C22×D4C2×D411yescharacteristic
284C23×C4—22no—
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2411yescharacteristic
290C23×C4C2411yescharacteristic
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C4⋊Q8—24no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—42no—
304C4⋊Q8—44no—
305C4⋊Q8—44no—
306C4⋊Q8#4114yes—
307#125—24no—
308#125—42no—
309#125—44no—
310C22⋊Q8—44no—
311C22⋊Q8—44no—
312C22⋊Q8—44no—
313C22⋊Q8—44no—
314C22⋊Q8—44no—
315C22⋊Q8—82no—
316C4×Q8—28no—
317C4×Q8—44no—
318C4×Q8—44no—
319C4×Q8—82no—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—42no—
323#117—24no—
324#117—42no—
325#117—44no—
326#117—44no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
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—42no—
354C2×C4⋊C4—44no—
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C22⋊C4C2×D411yescharacteristic
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—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×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400#107—44no—
401#106—44no—
402C4⋊C8—24no—
403C4⋊C8—42no—
404C4⋊C8—42no—
405#103—28no—
406#103—44no—
407#103—44no—
408#103—44no—
409C22⋊C8—42no—
410C22⋊C8—42no—
411C22⋊C8—42no—
412C22⋊C8—42no—
413C4×C8—22no—
414C4×C8—22no—
415C4×C8—22no—
416C4×C8—22no—
417C4×C8—24no—
418C4×C8—42no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—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#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447C24C22×D411yescharacteristic
448C24C22×D411yescharacteristic
449C2×Q8—24no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—24no—
453C2×Q8—24no—
454C2×Q8—44no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—44no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—42no—
464C2×Q8—82no—
465C2×Q8—816no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—41no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—82no—
499C22×C4#13514yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
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×C4C2511yescharacteristic
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C4⋊C4—22no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—41no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
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—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—82no—
557C4⋊C4—82no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4C2×#4114yes—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—41no—
568C22⋊C4—41no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—82no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—21no—
585C42—21no—
586C42—21no—
587C42—21no—
588C42—22no—
589C42—44no—
590C42—44no—
591C42—44no—
592C42—44no—
593C42—82no—
594C42C2×#4112yes—
595C42H(5,𝔽2)12yes—
596C422−1+414yes—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—21no—
610C23C2×#13512yes—
611C23C23×D411yescharacteristic
612Q8—48no—
613Q8—44no—
614Q8—44no—
615Q8—88no—
616Q8—84no—
617Q8—84no—
618D4—44no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—28no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—24no—
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—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
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—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—88no—
679C2×C4#53612yes—
680C2×C4#53612yes—
681C2×C4#54612yes—
682C2×C4#54612yes—
683C2×C4C2×#13512yes—
684C2×C4C2×#13512yes—
685C2×C4#57512yes—
686C2×C4#57712yes—
687C2×C4#57812yes—
688C2×C4C22×#4111yescharacteristic
689C2×C4C2×H(5,𝔽2)11yescharacteristic
690C2×C4C2×2−1+412yes—
691C2×C4#58512yes—
692C8—42no—
693C8—42no—
694C8—42no—
695C8—42no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22C2×#53611yescharacteristic
711C22C2×#54611yescharacteristic
712C22#350011yescharacteristic
713C22C22×#13511yescharacteristic
714C22C2×#57511yescharacteristic
715C22C2×#57711yescharacteristic
716C22C2×#57811yescharacteristic
717C4—22no—
718C4—22no—
719C4—22no—
720C4—41no—
721C4—41no—
722C4—41no—
723C4—41no—
724C4—41no—
725C4—44no—
726C4—41no—
727C4—41no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4#329614yes—
736C4#332714yes—
737C4#351812yes—
738C4#354712yes—
739C2—42no—
740C2—81no—
741C2#3895811yescharacteristic
742C2#6084611yescharacteristic
743C2#6087811yescharacteristic
744C2C2×#329611yescharacteristic
745C2C2×#332711yescharacteristic
746C2#6252811yescharacteristic
747C2#6254311yescharacteristic
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. #33290C2semidirect
4. #31749C2semidirect
5. #31746C2semidirect
6. #29845C2semidirect
7. #29844C2semidirect
8. #29207C2semidirect
9. #28719C2semidirect
10. #27878C2semidirect
11. #27875C2semidirect
12. #26029C2semidirect
13. #25904C2semidirect
14. #25902C2semidirect
16. #21835C2semidirect
17. #21833C2semidirect
18. C4×#487C2semidirect

Maximal subgroups

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

subgroupindex
#362092
#361292
#332902
#317492
#317462
#298452
#298442
#292072
#287192
#278782
#278752
#260292
#259042
#259022
#249762
#218352
#218332
C4×#4872
#212642