Finite Group Databasebeta

← #6489455

Subgroups — #6489455

Order 512.

1456 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35288C211yesmaximal, characteristic
2#35284C211yesmaximal, characteristic, complemented
3#33212C211yesmaximal, characteristic, complemented
4#33209C211yesmaximal, characteristic, complemented
5#33048C211yesmaximal, characteristic
6#33042C211yesmaximal, characteristic
7#30390C214yesmaximal, complemented
8#29220C211yesmaximal, characteristic, complemented
9#29179C211yesmaximal, characteristic, complemented
10#29177C211yesmaximal, characteristic, complemented
11#28716C211yesmaximal, characteristic, complemented
12#28689C211yesmaximal, characteristic, complemented
13#28655C211yesmaximal, characteristic, complemented
14#28655C211yesmaximal, characteristic, complemented
15#28514C211yesmaximal, characteristic, complemented
16#27874C214yesmaximal, complemented
17#24848C214yesmaximal, complemented
18#22126C214yesmaximal, complemented
19C4×#424C211yesmaximal, characteristic, complemented
20C2×#500C2211yescharacteristic
21C2×#500C2211yescharacteristic
22C2×#486C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#478C2214yes—
25C2×#424C2212yes—
26C2×C4×M16C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2868C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2858C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2856C2211yescharacteristic
36#2647C2212yes—
37#2644C2212yes—
38#2629C2212yes—
39#2624C2212yes—
40#2513—21no—
41#2507—21no—
42#2507—21no—
43#2497—21no—
44#2497—21no—
45#2490—21no—
46#2454—21no—
47#2448—21no—
48#2447—21no—
49#2446—21no—
50#2446—21no—
51#2429—21no—
52#2429—21no—
53#2427—21no—
54#2426—21no—
55#2420—21no—
56#2330C2212yes—
57#2324C2212yes—
58#2319C2212yes—
59#2314C2212yes—
60C4×C4⋊Q8C2211yescharacteristic
61C4×C4⋊Q8C2211yescharacteristic
62C4×C4⋊Q8C2211yescharacteristic
63C4×#127C2211yescharacteristic
64C4×#117C2211yescharacteristic
65#2101C2214yes—
66#2092C2214yes—
67#2088C2214yes—
68#2087C2214yes—
69#2083C2214yes—
70#2071C2214yes—
71#2067C2214yes—
72#2067C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1989C2214yes—
80#1957C2211yescharacteristic
81#1957C2211yescharacteristic
82#1947C2214yes—
83#1944C2214yes—
84#1942C2214yes—
85#1936C2214yes—
86#1934C2214yes—
87#1925C2214yes—
88#1925C2214yes—
89#1921C2214yes—
90#1919C2214yes—
91#1895C2214yes—
92#1893C2214yes—
93#1845C2211yescharacteristic
94#1844C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1809C2214yes—
98#1808C2214yes—
99#1802C2214yes—
100#1768C2212yes—
101#1768C2212yes—
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104C4×C4⋊C8C2211yescharacteristic
105C4×C4⋊C8C2211yescharacteristic
106#1729C2212yes—
107C22×M16C2312yes—
108C23×C8C2312yes—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×#127C2311yescharacteristic
115C2×#125—21no—
116C2×#125C2312yes—
117C2×#124—21no—
118C2×#124—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×#121—21no—
123C2×#121—21no—
124C2×#121C2312yes—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C2312yes—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#500—42no—
135#500—42no—
136#486—42no—
137#486—42no—
138#478—216no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106C2314yes—
143#424—28no—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C4×M16—42no—
153C2×#97C2312yes—
154C2×C4×C8—42no—
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2312yes—
160#400—22no—
161#400—22no—
162#399—22no—
163#398—22no—
164#398—22no—
165#398—22no—
166#398—22no—
167#397—22no—
168#397—22no—
169#397—22no—
170#397—22no—
171#396—22no—
172#396—22no—
173#396—22no—
174#394—22no—
175#394—22no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391—21no—
181#391—21no—
182#390—22no—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390C2312yes—
190#390C2312yes—
191#390C2312yes—
192#390C2312yes—
193#389—21no—
194#389—21no—
195#389—21no—
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—22no—
206#389—22no—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#389C2312yes—
213#389C2312yes—
214#389C2312yes—
215#388—22no—
216#388—22no—
217#388—22no—
218#388—22no—
219#388—22no—
220#387—22no—
221#387—22no—
222#387—22no—
223#387—22no—
224#386—22no—
225#385—22no—
226#384—21no—
227#384—21no—
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#384C2311yescharacteristic
231#384C2311yescharacteristic
232#384C2312yes—
233#384C2312yes—
234#382—21no—
235#382C2312yes—
236#381—22no—
237#380—21no—
238#380—21no—
239#380—21no—
240#380—22no—
241#380C2312yes—
242#380C2312yes—
243#380C2312yes—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4C2312yes—
252C4×C4⋊C4C2312yes—
253C4×C4⋊C4C2312yes—
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×D4—22no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4C2412yes—
281C23×C4—22no—
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2411yescharacteristic
285C2×M16—24no—
286C2×M16—24no—
287C2×M16—42no—
288C22×C8—24no—
289C22×C8—24no—
290C22×C8—42no—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—82no—
302C4⋊Q8—82no—
303#127—22no—
304#127—22no—
305#125—28no—
306#125—82no—
307#124—82no—
308#124—82no—
309C22⋊Q8—82no—
310C22⋊Q8—82no—
311C22⋊Q8—82no—
312#121—28no—
313#121—82no—
314#121—82no—
315C4×Q8—82no—
316C4×Q8—82no—
317C4×Q8—82no—
318C4×D4—28no—
319C4×D4—82no—
320C4×D4—82no—
321#117—24no—
322#117—24no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
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—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
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⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397#107—28no—
398#107—44no—
399#107—44no—
400#106—28no—
401C4⋊C8—24no—
402C4⋊C8—24no—
403C4⋊C8—24no—
404C4⋊C8#4118yes—
405#102—28no—
406#102—28no—
407#102—44no—
408#102—44no—
409#97—42no—
410C4×C8—22no—
411C4×C8—22no—
412C4×C8—22no—
413C4×C8—22no—
414C4×C8—42no—
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#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—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×D4—24no—
451C2×D4—22no—
452C2×D4—24no—
453C2×D4—22no—
454C2×D4—44no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—82no—
462C2×D4—82no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—44no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
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×C42−1+414yes—
496C22×C4C2511yescharacteristic
497M16—48no—
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×C8—48no—
509C2×C8C2×#4114yes—
510C2×C8C2×#4114yes—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—44no—
526C4⋊C4—44no—
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—82no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—21no—
579C42—21no—
580C42—22no—
581C42—22no—
582C42—82no—
583C42—82no—
584C42—82no—
585C42C2×#4114yes—
586C42H(5,𝔽2)12yes—
587C422−1+412yes—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23#53612yes—
602C23C23×D411yescharacteristic
603Q8—84no—
604Q8—84no—
605Q8—84no—
606D4—44no—
607D4—44no—
608D4—84no—
609D4—84no—
610D4—164no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—24no—
618C2×C4—24no—
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—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—44no—
644C2×C4—44no—
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—42no—
665C2×C4—42no—
666C2×C4—164no—
667C2×C4#53212yes—
668C2×C4#53212yes—
669C2×C4#53612yes—
670C2×C4#53712yes—
671C2×C4#53712yes—
672C2×C4#57612yes—
673C2×C4#57612yes—
674C2×C4#57712yes—
675C2×C4#57812yes—
676C2×C4C22×#4112yes—
677C2×C4C2×H(5,𝔽2)11yescharacteristic
678C2×C4C2×2−1+412yes—
679C2×C4C2×2−1+411yescharacteristic
680C8—24no—
681C8—24no—
682C8—42no—
683C8—42no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—41no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22C2×#53211yescharacteristic
699C22C2×#53611yescharacteristic
700C22C2×#53711yescharacteristic
701C22C2×#57611yescharacteristic
702C22C2×#57611yescharacteristic
703C22C2×#57711yescharacteristic
704C22C2×#57811yescharacteristic
705C4—22no—
706C4—22no—
707C4—22no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—41no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4#325814yes—
724C4#325914yes—
725C4#352012yes—
726C4#352312yes—
727C2—42no—
728C2—162no—
729C2#3524111yescharacteristic
730C2#6071811yescharacteristic
731C2#6072211yescharacteristic
732C2#6087711yescharacteristic
733C2#6088011yescharacteristic
734C2C2×#325811yescharacteristic
735C2C2×#325911yescharacteristic
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
2. #35284C2semidirect
3. #33212C2semidirect
4. #33209C2semidirect
7. #30390C2semidirect
8. #29220C2semidirect
9. #29179C2semidirect
10. #29177C2semidirect
11. #28716C2semidirect
12. #28689C2semidirect
13. #28655C2semidirect
14. #28655C2semidirect
15. #28514C2semidirect
16. #27874C2semidirect
17. #24848C2semidirect
18. #22126C2semidirect
19. C4×#424C2semidirect

Maximal subgroups

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

subgroupindex
#352882
#352842
#332122
#332092
#330482
#330422
#303902
#292202
#291792
#291772
#287162
#286892
#286552
#285142
#278742
#248482
#221262
C4×#4242