Finite Group Databasebeta

← #6495007

Subgroups — #6495007

Order 512.

1472 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic
2#35275C211yesmaximal, characteristic
3#33316C211yesmaximal, characteristic, complemented
4#33311C211yesmaximal, characteristic, complemented
5#33297C211yesmaximal, characteristic, complemented
6#33289C211yesmaximal, characteristic, complemented
7#30307C211yesmaximal, characteristic, complemented
8#30307C211yesmaximal, characteristic, complemented
9#30305C211yesmaximal, characteristic, complemented
10#30304C211yesmaximal, characteristic, complemented
11#29217C214yesmaximal, complemented
12#28698C214yesmaximal, complemented
13#27959C214yesmaximal
14#25822C214yesmaximal
15#21984C211yesmaximal, characteristic
16#21984C211yesmaximal, characteristic
17#21982C211yesmaximal, characteristic
18#21981C211yesmaximal, characteristic
19#20462C211yesmaximal, characteristic
20C2×#544C2212yes—
21C2×#501C2214yes—
22C2×#499C2214yes—
23C2×#431C2211yescharacteristic
24C2×#431C2211yescharacteristic
25#2868C2211yescharacteristic
26#2868C2211yescharacteristic
27#2868C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2660C2212yes—
34#2660C2212yes—
35#2629C2212yes—
36#2628C2212yes—
37#2570—21no—
38#2570—21no—
39#2552—21no—
40#2552—21no—
41#2551—21no—
42#2551—21no—
43#2550—21no—
44#2550—21no—
45#2441—21no—
46#2441—21no—
47#2440—21no—
48#2440—21no—
49#2440—21no—
50#2440—21no—
51#2439—21no—
52#2439—21no—
53#2331C2212yes—
54#2330C2212yes—
55#2317C2212yes—
56#2285C2211yescharacteristic
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59C2×#384C2211yescharacteristic
60C4×#117C2211yescharacteristic
61#2085C2214yes—
62#2085C2214yes—
63#2084C2214yes—
64#2083C2214yes—
65#2059C2214yes—
66#2059C2214yes—
67#2058C2214yes—
68#2057C2214yes—
69#1993C2214yes—
70#1992C2214yes—
71#1990C2214yes—
72#1990C2214yes—
73#1949C2214yes—
74#1949C2214yes—
75#1948C2214yes—
76#1947C2214yes—
77#1854C2214yes—
78#1854C2214yes—
79#1853C2214yes—
80#1852C2214yes—
81#1847C2214yes—
82#1846C2214yes—
83#1845C2211yescharacteristic
84#1845C2211yescharacteristic
85#1845C2211yescharacteristic
86#1845C2211yescharacteristic
87#1844C2211yescharacteristic
88#1844C2211yescharacteristic
89#1844C2211yescharacteristic
90#1844C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1768C2212yes—
100#1768C2212yes—
101#1750C2212yes—
102#1750C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107#544—48no—
108C2×C4⋊Q8—22no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8C2312yes—
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2312yes—
118C2×#125—22no—
119C2×#125—21no—
120C2×#125—21no—
121C2×#125C2312yes—
122C2×#125C2312yes—
123C2×#125C2312yes—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8—22no—
126C2×#117—21no—
127C2×#117—21no—
128C2×#117C2312yes—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2312yes—
134C22×C42C2311yescharacteristic
135#501—48no—
136#499—48no—
137#431Q818yescomplemented
138#431Q818yescomplemented
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×#97C2312yes—
152C2×#97C2312yes—
153C2×C4×C8C2312yes—
154C2×C4×C8C2312yes—
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159#400—22no—
160#399—22no—
161#398—22no—
162#398—22no—
163#398—22no—
164#398—22no—
165#398—22no—
166#398—22no—
167#396—22no—
168#396—22no—
169#395—22no—
170#395—22no—
171#395—22no—
172#395—22no—
173#395—22no—
174#393—22no—
175#391—22no—
176#391—22no—
177#389—22no—
178#389—22no—
179#389—21no—
180#389—21no—
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—21no—
194#389—21no—
195#389—22no—
196#389—22no—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#385—22no—
206#385—22no—
207#384—21no—
208#384—21no—
209#384—21no—
210#384—21no—
211#384—21no—
212#384—21no—
213#384—21no—
214#384—21no—
215#384C2312yes—
216#384C2312yes—
217#384C2312yes—
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2312yes—
227#382—21no—
228#382—21no—
229#382—21no—
230#382—21no—
231#382C2312yes—
232#382C2312yes—
233#382C2312yes—
234#382C2312yes—
235#380—21no—
236#380—21no—
237#380—21no—
238#380—21no—
239#380C2312yes—
240#380C2312yes—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C22⋊C4D411yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255#376C2311yescharacteristic
256#376C2311yescharacteristic
257C43C2311yescharacteristic
258C43C2311yescharacteristic
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271C22×Q8—22no—
272C22×Q8—22no—
273C22×Q8—22no—
274C23×C4—22no—
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2411yescharacteristic
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C4⋊Q8—28no—
283C4⋊Q8—28no—
284C4⋊Q8—44no—
285C4⋊Q8—48no—
286C4⋊Q8—82no—
287C4⋊Q8—82no—
288C4⋊Q8—82no—
289C4⋊Q8—82no—
290C4⋊Q8—82no—
291C4⋊Q8—82no—
292#125—28no—
293#125—28no—
294#125—44no—
295#125—48no—
296#125—82no—
297#125—82no—
298C22⋊Q8—48no—
299C22⋊Q8—48no—
300#117—24no—
301#117—24no—
302#117—44no—
303#117—82no—
304#117—82no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
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—22no—
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—22no—
343C2×C4⋊C4—44no—
344C2×C4⋊C4—82no—
345C2×C4⋊C4—82no—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4C2×D411yescharacteristic
362C2×C22⋊C4C2×D411yescharacteristic
363C2×C22⋊C4C2×D411yescharacteristic
364C2×C22⋊C4C2×D411yescharacteristic
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×Q814yes—
379C2×C42C2×Q814yes—
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386#107—44no—
387#107—44no—
388#107—44no—
389#107—44no—
390#106—44no—
391#106—44no—
392#106—44no—
393#106—44no—
394C4⋊C8C2×Q818yes—
395C4⋊C8C2×Q818yes—
396C22⋊C8C2×Q818yes—
397C22⋊C8C2×Q818yes—
398#97—24no—
399#97—24no—
400C4×C8—24no—
401C4×C8—24no—
402C4×C8C2×Q814yes—
403C4×C8C2×Q814yes—
404C4×C8C2×Q814yes—
405C4×C8C2×Q814yes—
406#95—22no—
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#95C2×D411yescharacteristic
423#95C2×D411yescharacteristic
424#95C2×D411yescharacteristic
425#95C2×D411yescharacteristic
426#95C2×D411yescharacteristic
427#95C2×D411yescharacteristic
428#95C2×D411yescharacteristic
429#95C2×D411yescharacteristic
430C24C22×D411yescharacteristic
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—82no—
435C2×Q8—82no—
436C2×Q8—82no—
437C2×Q8—82no—
438C2×Q8—82no—
439C2×Q8—82no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—44no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4C4⋊Q814yes—
458C22×C4C4⋊Q814yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×Q812yes—
473C22×C4C22×Q812yes—
474C22×C4C2511yescharacteristic
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8C22×Q814yes—
484C2×C8C22×Q814yes—
485C2×C8C22×Q814yes—
486C2×C8C22×Q814yes—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—82no—
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—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—44no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—44no—
555C42—82no—
556C42—82no—
557C42C4⋊Q814yes—
558C42C4⋊Q814yes—
559C42C22×Q812yes—
560C42C22×Q812yes—
561C42C22×Q812yes—
562C42C22×Q812yes—
563C42H(5,𝔽2)14yes—
564C422−1+414yes—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23C2×C4⋊Q812yes—
572C23C23×D411yescharacteristic
573Q8—164no—
574Q8—164no—
575Q8—164no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
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—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—164no—
631C2×C4C2×C4⋊Q812yes—
632C2×C4C2×C4⋊Q812yes—
633C2×C4C2×C4⋊Q812yes—
634C2×C4C2×C4⋊Q812yes—
635C2×C4C2×C4⋊Q812yes—
636C2×C4#53512yes—
637C2×C4#53512yes—
638C2×C4#53612yes—
639C2×C4#53612yes—
640C2×C4#57612yes—
641C2×C4#57712yes—
642C2×C4#57712yes—
643C2×C4#57812yes—
644C2×C4C23×Q811yescharacteristic
645C2×C4C23×Q811yescharacteristic
646C2×C4C2×H(5,𝔽2)12yes—
647C2×C4C2×2−1+412yes—
648C8—24no—
649C8—24no—
650C8—24no—
651C8—24no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22C22×C4⋊Q811yescharacteristic
660C22C2×#53511yescharacteristic
661C22C2×#53611yescharacteristic
662C22C2×#57611yescharacteristic
663C22C2×#57711yescharacteristic
664C22C2×#57711yescharacteristic
665C22C2×#57811yescharacteristic
666C4—22no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#316314yes—
686C4#316414yes—
687C4#350312yes—
688C4#350312yes—
689C2—42no—
690C2#3528111yescharacteristic
691C2C2×#316311yescharacteristic
692C2C2×#316411yescharacteristic
693C2#6082211yescharacteristic
694C2#6082411yescharacteristic
695C2#6087711yescharacteristic
696C2#6087811yescharacteristic
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. #33316C2semidirect
4. #33311C2semidirect
5. #33297C2semidirect
6. #33289C2semidirect
7. #30307C2semidirect
8. #30307C2semidirect
9. #30305C2semidirect
10. #30304C2semidirect
11. #29217C2semidirect
12. #28698C2semidirect
137. #431Q8semidirect
138. #431Q8semidirect

Maximal subgroups

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

subgroupindex
#352942
#352752
#333162
#333112
#332972
#332892
#303072
#303052
#303042
#292172
#286982
#279592
#258222
#219842
#219822
#219812
#204622