Finite Group Databasebeta

← #6495519

Subgroups — #6495519

Order 512.

1543 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic, complemented
2#35277C211yesmaximal, characteristic
3#33316C211yesmaximal, characteristic, complemented
4#33289C211yesmaximal, characteristic, complemented
5#33238C211yesmaximal, characteristic, complemented
6#33226C211yesmaximal, characteristic, complemented
7#29845C211yesmaximal, characteristic, complemented
8#29844C211yesmaximal, characteristic, complemented
9#29843C211yesmaximal, characteristic
10#29842C211yesmaximal, characteristic
11#29203C214yesmaximal, complemented
12#28668C214yesmaximal, complemented
13#27904C214yesmaximal, complemented
14#25684C214yesmaximal, complemented
15#22221C211yesmaximal, characteristic, complemented
16#22220C211yesmaximal, characteristic, complemented
17#22219C211yesmaximal, characteristic, complemented
18#22218C211yesmaximal, characteristic, complemented
19#20462C211yesmaximal, characteristic, complemented
20C2×#542C2212yes—
21C2×#501C2214yes—
22C2×#488C2214yes—
23C2×#487C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#486C2211yescharacteristic
26C2×#486C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29#2868C2211yescharacteristic
30#2868C2211yescharacteristic
31#2859C2211yescharacteristic
32#2858C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2857C2211yescharacteristic
36#2856C2211yescharacteristic
37#2647C2212yes—
38#2640C2212yes—
39#2629C2212yes—
40#2620C2212yes—
41#2512—21no—
42#2512—21no—
43#2507—21no—
44#2507—21no—
45#2497—21no—
46#2497—21no—
47#2491—21no—
48#2491—21no—
49#2452—21no—
50#2452—21no—
51#2446—21no—
52#2446—21no—
53#2429—21no—
54#2429—21no—
55#2422—21no—
56#2422—21no—
57#2330C2212yes—
58#2326C2212yes—
59#2314C2212yes—
60C4×C4⋊Q8C2211yescharacteristic
61C4×C4⋊Q8C2211yescharacteristic
62C4×C4⋊Q8C2211yescharacteristic
63C4×#127C2211yescharacteristic
64C4×#117C2211yescharacteristic
65#2068C2214yes—
66#2067C2214yes—
67#2066C2214yes—
68#2065C2214yes—
69#2036C2214yes—
70#2035C2214yes—
71#2034C2214yes—
72#2033C2214yes—
73#1990C2214yes—
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1959C2214yes—
79#1959C2214yes—
80#1958C2214yes—
81#1957C2211yescharacteristic
82#1957C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1928C2214yes—
86#1927C2214yes—
87#1926C2214yes—
88#1925C2214yes—
89#1865C2214yes—
90#1864C2214yes—
91#1863C2214yes—
92#1862C2214yes—
93#1847C2214yes—
94#1845C2211yescharacteristic
95#1845C2211yescharacteristic
96#1844C2211yescharacteristic
97#1844C2211yescharacteristic
98#1834C2214yes—
99#1768C2212yes—
100#1768C2212yes—
101#1750C2212yes—
102#1750C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
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⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2312yes—
116C2×#127C2311yescharacteristic
117C2×#126—22no—
118C2×#125C2312yes—
119C2×#124—22no—
120C2×#124C2312yes—
121C2×#124C2312yes—
122C2×#123—22no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—22no—
126C2×C22⋊Q8C2312yes—
127C2×#121—21no—
128C2×#121—21no—
129C2×#121C2312yes—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8C2312yes—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4C2312yes—
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#501—48no—
140#488—48no—
141#487—24no—
142#487—24no—
143#486—24no—
144#486—24no—
145#431—24no—
146#431—24no—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×#106C2314yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×#97C2312yes—
164C2×#97C2312yes—
165C2×C4×C8C2312yes—
166C2×C4×C8C2312yes—
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171#400—22no—
172#400—22no—
173#398—22no—
174#398—22no—
175#398—22no—
176#398—22no—
177#397—22no—
178#397—22no—
179#396—22no—
180#396—22no—
181#396—22no—
182#396—22no—
183#394—22no—
184#394—22no—
185#393—22no—
186#393—22no—
187#391—21no—
188#391—21no—
189#391—21no—
190#391—21no—
191#391C2312yes—
192#391C2312yes—
193#390—21no—
194#390—21no—
195#390—21no—
196#390—21no—
197#390—21no—
198#390—21no—
199#390—21no—
200#390—21no—
201#390C2312yes—
202#390C2312yes—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389—22no—
216#389—22no—
217#389C2312yes—
218#389C2312yes—
219#389C2312yes—
220#389C2312yes—
221#388—22no—
222#388—22no—
223#388—22no—
224#388—22no—
225#387—22no—
226#387—22no—
227#387—22no—
228#387—22no—
229#385—22no—
230#385—22no—
231#384—21no—
232#384—21no—
233#384—21no—
234#384—21no—
235#384C2311yescharacteristic
236#384C2311yescharacteristic
237#384C2311yescharacteristic
238#384C2311yescharacteristic
239#384C2312yes—
240#384C2312yes—
241#384C2312yes—
242#382C2312yes—
243#380—21no—
244#380—21no—
245#380—21no—
246#380—21no—
247#380C2312yes—
248#380C2312yes—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4D411yescharacteristic
256C4×C4⋊C4D411yescharacteristic
257C4×C4⋊C4C2312yes—
258C4×C4⋊C4C2312yes—
259C4×C22⋊C4D411yescharacteristic
260C4×C22⋊C4D411yescharacteristic
261C4×C22⋊C4D411yescharacteristic
262C4×C22⋊C4D411yescharacteristic
263#376C2311yescharacteristic
264#376C2311yescharacteristic
265C43C2311yescharacteristic
266C43C2311yescharacteristic
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×Q8—22no—
278C22×Q8C2412yes—
279C22×D4—22no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C2412yes—
283C23×C4—22no—
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2411yescharacteristic
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—28no—
298C4⋊Q8—82no—
299C4⋊Q8—82no—
300C4⋊Q8—82no—
301C4⋊Q8—82no—
302#127—22no—
303#127—22no—
304#126—48no—
305#125—28no—
306#124—28no—
307#124—28no—
308#124—48no—
309#123—48no—
310C22⋊Q8—44no—
311C22⋊Q8—48no—
312C22⋊Q8—82no—
313C22⋊Q8—82no—
314#121—44no—
315#121—82no—
316#121—82no—
317C4×Q8—44no—
318C4×Q8—82no—
319C4×Q8—82no—
320C4×D4—44no—
321C4×D4—82no—
322C4×D4—82no—
323#117—24no—
324#117—24no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
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—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
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⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C22⋊C4C2×D411yescharacteristic
378C2×C22⋊C4C2×D411yescharacteristic
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400#107—44no—
401#107—44no—
402#106—44no—
403#106—44no—
404C4⋊C8—24no—
405C4⋊C8—24no—
406#103—28no—
407#103—28no—
408#103—44no—
409#103—44no—
410#102—28no—
411#102—28no—
412#102—44no—
413#102—44no—
414C22⋊C8—24no—
415C22⋊C8—24no—
416#97—24no—
417#97—24no—
418C4×C8—22no—
419C4×C8—22no—
420C4×C8—24no—
421C4×C8—24no—
422C4×C8#4114yes—
423C4×C8#4114yes—
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#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446C24—22no—
447C24C22×D411yescharacteristic
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—24no—
451C2×Q8—24no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—44no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—82no—
460C2×Q8—82no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—24no—
464C2×D4—24no—
465C2×D4—44no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—82no—
473C2×D4—82no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—44no—
492C22×C4—82no—
493C22×C4—82no—
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C2511yescharacteristic
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8C2×#4114yes—
519C2×C8C2×#4114yes—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C4⋊C4—82no—
556C4⋊C4—82no—
557C4⋊C4—82no—
558C4⋊C4—82no—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4—82no—
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—22no—
585C42—22no—
586C42—21no—
587C42—21no—
588C42—22no—
589C42—22no—
590C42—44no—
591C42—82no—
592C42—82no—
593C42C2×#4112yes—
594C42C2×#4112yes—
595C42H(5,𝔽2)14yes—
596C422−1+414yes—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23C23×D411yescharacteristic
612Q8—44no—
613Q8—44no—
614Q8—84no—
615Q8—84no—
616Q8—164no—
617D4—44no—
618D4—44no—
619D4—84no—
620D4—84no—
621D4—164no—
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—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—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—44no—
652C2×C4—44no—
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—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—164no—
677C2×C4#53212yes—
678C2×C4#53212yes—
679C2×C4#53512yes—
680C2×C4#53512yes—
681C2×C4#53612yes—
682C2×C4#53612yes—
683C2×C4#57612yes—
684C2×C4#57712yes—
685C2×C4#57712yes—
686C2×C4#57812yes—
687C2×C4C22×#4111yescharacteristic
688C2×C4C22×#4111yescharacteristic
689C2×C4C2×H(5,𝔽2)12yes—
690C2×C4C2×2−1+412yes—
691C8—24no—
692C8—24no—
693C8—42no—
694C8—42no—
695C22—41no—
696C22—41no—
697C22—41no—
698C22—41no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22C2×#53211yescharacteristic
710C22C2×#53511yescharacteristic
711C22C2×#53611yescharacteristic
712C22C2×#57611yescharacteristic
713C22C2×#57711yescharacteristic
714C22C2×#57711yescharacteristic
715C22C2×#57811yescharacteristic
716C4—22no—
717C4—22no—
718C4—22no—
719C4—22no—
720C4—41no—
721C4—41no—
722C4—41no—
723C4—41no—
724C4—41no—
725C4—41no—
726C4—41no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4#317814yes—
735C4#317914yes—
736C4#351112yes—
737C4#351112yes—
738C2—81no—
739C2—162no—
740C2#3529811yescharacteristic
741C2C2×#317811yescharacteristic
742C2C2×#317911yescharacteristic
743C2#6082211yescharacteristic
744C2#6082411yescharacteristic
745C2#6088011yescharacteristic
746C2#6088111yescharacteristic
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. #35284C2semidirect
3. #33316C2semidirect
4. #33289C2semidirect
5. #33238C2semidirect
6. #33226C2semidirect
7. #29845C2semidirect
8. #29844C2semidirect
11. #29203C2semidirect
12. #28668C2semidirect
13. #27904C2semidirect
14. #25684C2semidirect
15. #22221C2semidirect
16. #22220C2semidirect
17. #22219C2semidirect
18. #22218C2semidirect
19. #20462C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#352772
#333162
#332892
#332382
#332262
#298452
#298442
#298432
#298422
#292032
#286682
#279042
#256842
#222212
#222202
#222192
#222182
#204622