Finite Group Databasebeta

← #6849170

Subgroups — #6849170

Order 512.

1685 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic
2#34929C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#32052C214yesmaximal, complemented
5#31840C214yesmaximal, complemented
6#29664C212yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#28401C212yesmaximal, complemented
9#28255C211yesmaximal, characteristic, complemented
10#27815C212yesmaximal, complemented
11#27681C212yesmaximal, complemented
12#25573C212yesmaximal
13#24652C212yesmaximal, complemented
14#24122C212yesmaximal
15#22795C211yesmaximal, characteristic
16#21860C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×#554C2212yes—
19C2×#533C2212yes—
20C2×#487C2212yes—
21C2×#477C2212yes—
22C2×#462C2211yescharacteristic
23C2×#462C2212yes—
24C2×#462C2212yes—
25C2×C4×Q16C2212yes—
26C2×#431C2212yes—
27#2868C2211yescharacteristic
28#2857C2212yes—
29#2567C2212yes—
30#2556—22no—
31#2552—21no—
32#2552—21no—
33#2551C2211yescharacteristic
34#2546C2211yescharacteristic
35#2542C2211yescharacteristic
36#2538C2211yescharacteristic
37#2506—21no—
38#2502C2214yes—
39#2492C2212yes—
40#2452—21no—
41#2440—22no—
42#2440—22no—
43#2429C2212yes—
44#2396—22no—
45D4×C4⋊C4—21no—
46D4×C4⋊C4—21no—
47C4×C4⋊Q8—21no—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×C22⋊Q8C2212yes—
51#2293—21no—
52C2×#384C2212yes—
53C42×Q8C2211yescharacteristic
54C42×D4C2211yescharacteristic
55#2100C2214yes—
56#2095C2214yes—
57#2072C2214yes—
58#2056C2214yes—
59#2036C2214yes—
60#2024C2214yes—
61#2016C2214yes—
62#2001C2214yes—
63#2001C2214yes—
64#2001C2214yes—
65#1990C2212yes—
66#1990C2214yes—
67#1987C2212yes—
68#1981C2211yescharacteristic
69#1969C2214yes—
70#1956C2214yes—
71#1954C2212yes—
72#1947C2211yescharacteristic
73#1947C2212yes—
74#1947C2212yes—
75#1933C2214yes—
76#1928C2212yes—
77#1923C2214yes—
78#1878C2212yes—
79#1872C2214yes—
80#1872C2214yes—
81#1872C2214yes—
82#1852C2214yes—
83#1842C2211yescharacteristic
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1803C2211yescharacteristic
90#1803C2212yes—
91#1797C2212yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×Q16C2312yes—
96C22×Q16C2314yes—
97#554—44no—
98#554—44no—
99#533—28no—
100#533—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×#125—22no—
110C2×#125—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2314yes—
117C2×C4×Q8—22no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4—22no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4C2314yes—
131C22×C42C2312yes—
132#487—44no—
133#477—44no—
134#462—24no—
135#462—44no—
136#462—44no—
137C4×Q16—44no—
138#431—44no—
139C2×#107C2314yes—
140C2×#107C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C4×C8C2312yes—
158C2×C4×C8C2314yes—
159#400—22no—
160#399—22no—
161#398—22no—
162#398—22no—
163#398—24no—
164#398—24no—
165#396—22no—
166#396—22no—
167#395—24no—
168#395—22no—
169#393—22no—
170#391—22no—
171#391C2312yes—
172#391C2312yes—
173#391C2314yes—
174#391C2312yes—
175#390—21no—
176#390—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—22no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389C2312yes—
191#388—22no—
192#388—22no—
193#387—22no—
194#385—22no—
195#385—24no—
196#385—24no—
197#384—21no—
198#384—21no—
199#384—21no—
200#384—22no—
201#384—22no—
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#384C2314yes—
205#384C2312yes—
206#384C2312yes—
207#380—22no—
208#380—22no—
209#380C2312yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—22no—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4C2312yes—
228C43—21no—
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2314yes—
236#340C2312yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8C2412yes—
242C22×Q8C2412yes—
243C22×Q8C2411yescharacteristic
244C22×D4C2×D411yescharacteristic
245C23×C4—24no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2412yes—
249C2×Q16—24no—
250C2×Q16—42no—
251C2×Q16—42no—
252C2×Q16—44no—
253C2×Q16—44no—
254C2×Q16—44no—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—42no—
267C4⋊Q8—42no—
268C4⋊Q8—44no—
269C4⋊Q8—82no—
270C4⋊Q8—82no—
271#125—48no—
272#125—48no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276C22⋊Q8—416no—
277C22⋊Q8—42no—
278C22⋊Q8—42no—
279C22⋊Q8—42no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C22⋊Q8—48no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—48no—
292C4×Q8—41no—
293C4×Q8#4114yes—
294C4×D4—24no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—48no—
298C4×D4Q1618yescomplemented
299#117—48no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—84no—
332C2×C4⋊C4—84no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C42—21no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D412yes—
362C2×C42C2×D412yes—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370#107—42no—
371#107—44no—
372C4⋊C8—22no—
373C4⋊C8—22no—
374C4⋊C8—24no—
375C4⋊C8—24no—
376C4⋊C8—42no—
377#103—24no—
378#103—24no—
379#103—42no—
380#103—42no—
381#103—44no—
382#103—44no—
383#103—44no—
384#103—44no—
385#103—44no—
386C22⋊C8—44no—
387C22⋊C8—44no—
388C4×C8—24no—
389C4×C8—44no—
390#95—22no—
391#95—22no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24C22×D412yes—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—82no—
428C2×Q8—82no—
429C2×Q8—82no—
430C2×Q8C2×#4112yes—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4C2×Q1614yes—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—42no—
454C22×C4—48no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4C2×Q1618yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2511yescharacteristic
468Q16—48no—
469Q16—88no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—81no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4C2×Q1614yes—
520C4⋊C4#13714yes—
521C4⋊C4C2×#4112yes—
522C4⋊C42−1+412yes—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—48no—
529C22⋊C4—82no—
530C22⋊C4C2×Q1618yes—
531C42—24no—
532C42—24no—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—21no—
539C42—21no—
540C42—41no—
541C42—42no—
542C42—44no—
543C42—42no—
544C42—44no—
545C42—42no—
546C42—42no—
547C42—81no—
548C42C2×Q1614yes—
549C42C2×#4112yes—
550C42H(5,𝔽2)12yes—
551C42H(5,𝔽2)12yes—
552C42H(5,𝔽2)12yes—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23C22×Q1614yes—
560C23C23×D411yescharacteristic
561Q8—22no—
562Q8—22no—
563Q8—41no—
564Q8—41no—
565Q8—42no—
566Q8—42no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—42no—
572Q8—42no—
573Q8—88no—
574Q8—164no—
575D4—28no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—216no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—48no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—168no—
626C2×C4#53512yes—
627C2×C4#53512yes—
628C2×C4#54612yes—
629C2×C4#54612yes—
630C2×C4C22×Q1612yes—
631C2×C4C22×Q1614yes—
632C2×C4C22×Q1612yes—
633C2×C4C22×Q1612yes—
634C2×C4C2×#13714yes—
635C2×C4C2×#13712yes—
636C2×C4#57712yes—
637C2×C4C22×#4111yescharacteristic
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×2−1+411yescharacteristic
642C2×C4#58511yescharacteristic
643C8—42no—
644C8—44no—
645C8—42no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22C2×#53511yescharacteristic
654C22C2×#54611yescharacteristic
655C22#348911yescharacteristic
656C22C23×Q1611yescharacteristic
657C22C22×#13711yescharacteristic
658C22C2×#57711yescharacteristic
659C22#358511yescharacteristic
660C4—22no—
661C4—22no—
662C4—24no—
663C4—22no—
664C4—24no—
665C4—42no—
666C4—42no—
667C4—41no—
668C4—42no—
669C4—42no—
670C4—82no—
671C4—82no—
672C4—81no—
673C4—82no—
674C4—84no—
675C4—82no—
676C4#320914yes—
677C4#329914yes—
678C4#350611yescharacteristic
679C4#351111yescharacteristic
680C4#354611yescharacteristic
681C4#354811yescharacteristic
682C2—44no—
683C2#3947111yescharacteristic
684C2C2×#320911yescharacteristic
685C2#6083311yescharacteristic
686C2C2×#329911yescharacteristic
687C2#6116211yescharacteristic
688C2#6210011yescharacteristic
689C2#6214711yescharacteristic
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. #33198C2semidirect
4. #32052C2semidirect
5. #31840C2semidirect
6. #29664C2semidirect
7. #28717C2semidirect
8. #28401C2semidirect
9. #28255C2semidirect
10. #27815C2semidirect
11. #27681C2semidirect
13. #24652C2semidirect
16. #21860C2semidirect
17. C4×#462C2semidirect
298. C4×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#361802
#349292
#331982
#320522
#318402
#296642
#287172
#284012
#282552
#278152
#276812
#255732
#246522
#241222
#227952
#218602
C4×#4622