Finite Group Databasebeta

← #6495598

Subgroups — #6495598

Order 512.

1539 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic, complemented
2#35277C211yesmaximal, characteristic
3#33316C211yesmaximal, characteristic, complemented
4#33289C211yesmaximal, characteristic, complemented
5#33230C211yesmaximal, characteristic, complemented
6#33226C211yesmaximal, characteristic, complemented
7#29857C214yesmaximal, complemented
8#29193C211yesmaximal, characteristic, complemented
9#29192C211yesmaximal, characteristic, complemented
10#29191C211yesmaximal, characteristic, complemented
11#29190C211yesmaximal, characteristic, complemented
12#28668C214yesmaximal, complemented
13#27892C211yesmaximal, characteristic, complemented
14#27891C211yesmaximal, characteristic
15#27890C211yesmaximal, characteristic
16#27889C211yesmaximal, characteristic, complemented
17#25684C214yesmaximal, complemented
18#22233C214yesmaximal, complemented
19C4×#433C211yesmaximal, characteristic, complemented
20C2×#542C2212yes—
21C2×#501C2214yes—
22C2×#489C2214yes—
23C2×#488C2214yes—
24C2×#433C2212yes—
25#2868C2211yescharacteristic
26#2868C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2857C2211yescharacteristic
32#2856C2211yescharacteristic
33#2640C2212yes—
34#2629C2212yes—
35#2628C2212yes—
36#2620C2212yes—
37#2570—21no—
38#2570—21no—
39#2550—21no—
40#2550—21no—
41#2512—21no—
42#2512—21no—
43#2491—21no—
44#2491—21no—
45#2452—21no—
46#2452—21no—
47#2441—21no—
48#2441—21no—
49#2439—21no—
50#2439—21no—
51#2422—21no—
52#2422—21no—
53#2331C2212yes—
54#2330C2212yes—
55#2326C2212yes—
56C4×C4⋊Q8C2211yescharacteristic
57C4×#127C2211yescharacteristic
58#2285C2211yescharacteristic
59#2285C2211yescharacteristic
60C4×#117C2211yescharacteristic
61#2068C2214yes—
62#2067C2214yes—
63#2066C2214yes—
64#2065C2214yes—
65#2036C2214yes—
66#2035C2214yes—
67#2034C2214yes—
68#2033C2214yes—
69#1990C2214yes—
70#1990C2214yes—
71#1959C2214yes—
72#1958C2214yes—
73#1957C2211yescharacteristic
74#1957C2211yescharacteristic
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1956C2211yescharacteristic
78#1956C2211yescharacteristic
79#1956C2211yescharacteristic
80#1956C2211yescharacteristic
81#1928C2214yes—
82#1927C2214yes—
83#1926C2214yes—
84#1925C2214yes—
85#1865C2214yes—
86#1864C2214yes—
87#1863C2214yes—
88#1862C2214yes—
89#1847C2214yes—
90#1845C2211yescharacteristic
91#1845C2211yescharacteristic
92#1844C2211yescharacteristic
93#1844C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1768C2212yes—
99C4×C4⋊C8C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101#1750C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1746C2212yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107#542—48no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×#127C2311yescharacteristic
116C2×#126—22no—
117C2×#125—21no—
118C2×#125—21no—
119C2×#125C2312yes—
120C2×#124—22no—
121C2×#124C2312yes—
122C2×#124C2312yes—
123C2×#123—22no—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8C2312yes—
126C2×#121—21no—
127C2×#121—21no—
128C2×#121C2312yes—
129C2×C4×Q8C2312yes—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4C2312yes—
133C2×#117—21no—
134C2×#117—21no—
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#501—48no—
139#489—216no—
140#488—48no—
141#433—28no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158C2×#97C2312yes—
159C2×#97C2312yes—
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2312yes—
165C2×C4×C8C2312yes—
166#400—22no—
167#399—22no—
168#398—22no—
169#398—22no—
170#397—22no—
171#397—22no—
172#396—22no—
173#396—22no—
174#396—22no—
175#396—22no—
176#395—22no—
177#394—22no—
178#394—22no—
179#393—22no—
180#393—22no—
181#393—22no—
182#391—22no—
183#391—22no—
184#391C2312yes—
185#391C2312yes—
186#390—21no—
187#390—21no—
188#390—21no—
189#390—21no—
190#390—21no—
191#390—21no—
192#390—21no—
193#390—21no—
194#390C2312yes—
195#390C2312yes—
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—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—22no—
209#389—22no—
210#389C2312yes—
211#389C2312yes—
212#389C2312yes—
213#389C2312yes—
214#388—22no—
215#388—22no—
216#387—22no—
217#387—22no—
218#387—22no—
219#387—22no—
220#385—22no—
221#385—22no—
222#384—21no—
223#384—21no—
224#384—21no—
225#384—21no—
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2312yes—
229#384C2312yes—
230#384C2311yescharacteristic
231#384C2311yescharacteristic
232#384C2311yescharacteristic
233#384C2311yescharacteristic
234#382—21no—
235#382—21no—
236#382—21no—
237#382—21no—
238#382C2312yes—
239#380—21no—
240#380—21no—
241#380—21no—
242#380—21no—
243#380C2312yes—
244#380C2312yes—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4D411yescharacteristic
253C4×C4⋊C4C2312yes—
254C4×C4⋊C4C2312yes—
255C4×C22⋊C4D411yescharacteristic
256C4×C22⋊C4D411yescharacteristic
257C4×C22⋊C4D411yescharacteristic
258C4×C22⋊C4D411yescharacteristic
259#376C2311yescharacteristic
260#376C2311yescharacteristic
261C43C2311yescharacteristic
262C43C2311yescharacteristic
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271C22×Q8—22no—
272C22×Q8—22no—
273C22×Q8C2412yes—
274C22×D4—22no—
275C22×D4—21no—
276C22×D4—21no—
277C22×D4C2412yes—
278C23×C4—22no—
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2411yescharacteristic
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C4⋊Q8—28no—
287C4⋊Q8—28no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—82no—
291C4⋊Q8—82no—
292C4⋊Q8—82no—
293C4⋊Q8—82no—
294#127—22no—
295#127—22no—
296#126—48no—
297#125—28no—
298#125—82no—
299#125—82no—
300#124—28no—
301#124—28no—
302#124—48no—
303#123—48no—
304C22⋊Q8—44no—
305C22⋊Q8—48no—
306#121—44no—
307#121—82no—
308#121—82no—
309C4×Q8—44no—
310C4×D4—44no—
311C4×D4—82no—
312C4×D4—82no—
313#117—24no—
314#117—24no—
315#117—82no—
316#117—82no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
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—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
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⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4C2×D411yescharacteristic
369C2×C22⋊C4C2×D411yescharacteristic
370C2×C22⋊C4C2×D411yescharacteristic
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393#107—44no—
394#107—44no—
395#106—44no—
396#106—44no—
397C4⋊C8—24no—
398C4⋊C8—24no—
399#103—28no—
400#103—28no—
401#103—44no—
402#103—44no—
403#102—28no—
404#102—28no—
405#102—44no—
406#102—44no—
407C22⋊C8—24no—
408C22⋊C8—24no—
409#97—24no—
410#97#4118yes—
411C4×C8—24no—
412C4×C8—24no—
413C4×C8—22no—
414C4×C8—22no—
415C4×C8—22no—
416C4×C8—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#95—22no—
430#95—22no—
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439C24—22no—
440C24C22×D411yescharacteristic
441C2×Q8—24no—
442C2×Q8—24no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8—82no—
448C2×Q8—82no—
449C2×Q8—82no—
450C2×Q8—82no—
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—82no—
463C2×D4—82no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—44no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C2511yescharacteristic
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8C2×#4114yes—
509C2×C8C2×#4114yes—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—21no—
578C42—21no—
579C42—44no—
580C42—82no—
581C42—82no—
582C42C2×#4114yes—
583C42H(5,𝔽2)14yes—
584C422−1+412yes—
585C422−1+412yes—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23C23×D411yescharacteristic
601Q8—44no—
602Q8—44no—
603Q8—164no—
604Q8—164no—
605D4—44no—
606D4—44no—
607D4—84no—
608D4—84no—
609D4—164no—
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—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
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—21no—
638C2×C4—21no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—164no—
665C2×C4#53212yes—
666C2×C4#53212yes—
667C2×C4#53512yes—
668C2×C4#53512yes—
669C2×C4#53612yes—
670C2×C4#53612yes—
671C2×C4#57612yes—
672C2×C4#57712yes—
673C2×C4#57712yes—
674C2×C4#57812yes—
675C2×C4C22×#4112yes—
676C2×C4C2×H(5,𝔽2)12yes—
677C2×C4C2×2−1+411yescharacteristic
678C2×C4C2×2−1+411yescharacteristic
679C8—24no—
680C8—24no—
681C8—42no—
682C8—42no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22—41no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22C2×#53211yescharacteristic
698C22C2×#53511yescharacteristic
699C22C2×#53611yescharacteristic
700C22C2×#57611yescharacteristic
701C22C2×#57711yescharacteristic
702C22C2×#57711yescharacteristic
703C22C2×#57811yescharacteristic
704C4—22no—
705C4—22no—
706C4—22no—
707C4—22no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4#324314yes—
723C4#324414yes—
724C4#351112yes—
725C4#351112yes—
726C2—81no—
727C2—162no—
728C2#3529811yescharacteristic
729C2#6072211yescharacteristic
730C2#6072311yescharacteristic
731C2#6082211yescharacteristic
732C2#6082411yescharacteristic
733C2C2×#324311yescharacteristic
734C2C2×#324411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
1. #35294C2semidirect
3. #33316C2semidirect
4. #33289C2semidirect
5. #33230C2semidirect
6. #33226C2semidirect
7. #29857C2semidirect
8. #29193C2semidirect
9. #29192C2semidirect
10. #29191C2semidirect
11. #29190C2semidirect
12. #28668C2semidirect
13. #27892C2semidirect
16. #27889C2semidirect
17. #25684C2semidirect
18. #22233C2semidirect
19. C4×#433C2semidirect

Maximal subgroups

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

subgroupindex
#352942
#352772
#333162
#332892
#332302
#332262
#298572
#291932
#291922
#291912
#291902
#286682
#278922
#278912
#278902
#278892
#256842
#222332
C4×#4332