Finite Group Databasebeta

← #6469537

Subgroups — #6469537

Order 512.

1494 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35291C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#29531C214yesmaximal, complemented
4#29179C211yesmaximal, characteristic, complemented
5#29178C211yesmaximal, characteristic, complemented
6#28656C211yesmaximal, characteristic, complemented
7#28653C211yesmaximal, characteristic, complemented
8#27960C211yesmaximal, characteristic
9#27910C211yesmaximal, characteristic
10#25904C211yesmaximal, characteristic
11#25812C211yesmaximal, characteristic
12#24614C214yesmaximal
13#23792C214yesmaximal, complemented
14#21727C211yesmaximal, characteristic, complemented
15C4×#500C211yesmaximal, characteristic, complemented
16#21647C211yesmaximal, characteristic, complemented
17C4×#487C211yesmaximal, characteristic, complemented
18#21585C214yesmaximal
19C4×#433C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#500C2211yescharacteristic
22C2×#498C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#484C2214yes—
26C2×#433C2212yes—
27#2862C2211yescharacteristic
28#2859C2211yescharacteristic
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2645C2212yes—
32#2625C2212yes—
33#2514—21no—
34#2509—21no—
35#2508—21no—
36#2495—21no—
37#2494—21no—
38#2489—21no—
39#2453—21no—
40#2421—21no—
41#2360—21no—
42#2360—21no—
43#2360—21no—
44#2360—21no—
45#2355—21no—
46#2355—21no—
47Q8×C4⋊C4—21no—
48Q8×C4⋊C4—21no—
49#2330C2212yes—
50#2330C2212yes—
51#2323C2212yes—
52#2320C2212yes—
53#2317C2212yes—
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C2×#384C2211yescharacteristic
61#2274C2212yes—
62C4×#117C2211yescharacteristic
63#2099C2214yes—
64#2095C2214yes—
65#2087C2214yes—
66#2086C2214yes—
67#2068C2214yes—
68#2066C2214yes—
69#2059C2214yes—
70#2026C2214yes—
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1956C2211yescharacteristic
78#1956C2211yescharacteristic
79#1944C2214yes—
80#1942C2214yes—
81#1935C2214yes—
82#1933C2214yes—
83#1928C2214yes—
84#1927C2214yes—
85#1921C2214yes—
86#1918C2214yes—
87#1873C2214yes—
88#1854C2214yes—
89#1845C2211yescharacteristic
90#1845C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1823C2214yes—
94#1776C2214yes—
95C4×#107C2214yes—
96#1768C2212yes—
97C4×C4⋊C8C2211yescharacteristic
98C4×C4⋊C8C2211yescharacteristic
99#1763C2214yes—
100C4×#103C2214yes—
101#1750C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1746C2212yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107#533—24no—
108#533—24no—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×#125—21no—
114C2×#125C2312yes—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2312yes—
127C2×#117C2312yes—
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2312yes—
133C22×C42C2311yescharacteristic
134#500—24no—
135#498—24no—
136#487—24no—
137#487—24no—
138#484—216no—
139#433—28no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×#97C2312yes—
153C2×#97C2312yes—
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2312yes—
159C2×C4×C8C2312yes—
160#401—22no—
161#400—22no—
162#400—22no—
163#400—22no—
164#400—22no—
165#398—22no—
166#397—22no—
167#397—22no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391C2312yes—
177#391C2312yes—
178#391C2312yes—
179#391C2312yes—
180#389—22no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—22no—
194#389—22no—
195#389—22no—
196#389—22no—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#388—22no—
206#388—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#385—22no—
215#385—22no—
216#384—22no—
217#384—21no—
218#384—21no—
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#384C2312yes—
224#383—22no—
225#382—22no—
226#382—21no—
227#382—21no—
228#382—21no—
229#382—22no—
230#380—21no—
231#380—21no—
232#380—21no—
233#380C2312yes—
234C4×C4⋊C4—22no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2312yes—
252C4×C22⋊C4—22no—
253C4×C22⋊C4—22no—
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4D411yescharacteristic
256C4×C22⋊C4D411yescharacteristic
257C4×C22⋊C4D411yescharacteristic
258#376C2311yescharacteristic
259#376C2311yescharacteristic
260C43C2311yescharacteristic
261C43C2311yescharacteristic
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×Q8C2412yes—
278C23×C4—22no—
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2411yescharacteristic
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8#4114yes—
293C4⋊Q8#4114yes—
294#125—24no—
295#125—24no—
296#125—82no—
297C22⋊Q8—28no—
298C22⋊Q8—28no—
299C22⋊Q8—28no—
300C22⋊Q8—82no—
301C4×Q8—28no—
302C4×Q8—28no—
303C4×Q8—28no—
304C4×Q8—82no—
305C4×Q8—82no—
306C4×Q8—82no—
307C4×Q8—82no—
308C4×Q8—82no—
309#117—24no—
310#117—24no—
311#117—24no—
312#117—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—82no—
343C2×C4⋊C4—82no—
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4C2×D411yescharacteristic
359C2×C22⋊C4C2×D411yescharacteristic
360C2×C22⋊C4C2×D411yescharacteristic
361C2×C22⋊C4C2×D411yescharacteristic
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#107—28no—
388#107—28no—
389#107—28no—
390#106—28no—
391C4⋊C8—24no—
392C4⋊C8—24no—
393#103—28no—
394#103—28no—
395#103—28no—
396#103—28no—
397C22⋊C8—24no—
398C22⋊C8#4118yes—
399#97—24no—
400#97—24no—
401C4×C8—24no—
402C4×C8—22no—
403C4×C8—22no—
404C4×C8—22no—
405C4×C8—22no—
406C4×C8—24no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
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#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437C24C22×D411yescharacteristic
438C2×Q8—24no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—24no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—44no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C2×#4114yes—
485C22×C4C2511yescharacteristic
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8C2×#4114yes—
497C2×C8C2×#4114yes—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4C2×#4114yes—
539C4⋊C4C2×#4114yes—
540C22⋊C4—28no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—44no—
546C22⋊C4—82no—
547C42—28no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—21no—
561C42—21no—
562C42—22no—
563C42—44no—
564C42—82no—
565C42—82no—
566C42—82no—
567C42—82no—
568C42—82no—
569C42C2×#4112yes—
570C42C2×#4112yes—
571C422−1+414yes—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23#53212yes—
579C23C23×D411yescharacteristic
580Q8—48no—
581Q8—48no—
582Q8—44no—
583Q8—44no—
584Q8—84no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—164no—
644C2×C4#52512yes—
645C2×C4#52512yes—
646C2×C4#53212yes—
647C2×C4#53612yes—
648C2×C4#53612yes—
649C2×C4#57612yes—
650C2×C4#57712yes—
651C2×C4#57812yes—
652C2×C4#57812yes—
653C2×C4C22×#4111yescharacteristic
654C2×C4C22×#4111yescharacteristic
655C2×C4C22×#4112yes—
656C2×C4C2×2−1+412yes—
657C8—24no—
658C8—24no—
659C8—42no—
660C8—42no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22C2×#52511yescharacteristic
669C22C2×#53211yescharacteristic
670C22C2×#53611yescharacteristic
671C22C2×#57611yescharacteristic
672C22C2×#57711yescharacteristic
673C22C2×#57811yescharacteristic
674C22C2×#57811yescharacteristic
675C4—22no—
676C4—22no—
677C4—22no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—44no—
684C4—41no—
685C4—44no—
686C4—41no—
687C4—41no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4#312014yes—
695C4#312414yes—
696C4#348212yes—
697C4#348212yes—
698C2—42no—
699C2#3497211yescharacteristic
700C2C2×#312011yescharacteristic
701C2C2×#312411yescharacteristic
702C2#6071511yescharacteristic
703C2#6072211yescharacteristic
704C2#6087811yescharacteristic
705C2#6088011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29531C2semidirect
4. #29179C2semidirect
5. #29178C2semidirect
6. #28656C2semidirect
7. #28653C2semidirect
13. #23792C2semidirect
14. #21727C2semidirect
15. C4×#500C2semidirect
16. #21647C2semidirect
17. C4×#487C2semidirect

Maximal subgroups

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

subgroupindex
#352912
#342142
#295312
#291792
#291782
#286562
#286532
#279602
#279102
#259042
#258122
#246142
#237922
#217272
C4×#5002
#216472
C4×#4872
#215852
C4×#4332