Finite Group Databasebeta

← #7677545

Subgroups — #7677545

Order 512.

4738 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59435C214yesmaximal, complemented
2#59434C214yesmaximal, complemented
3#59427C214yesmaximal, complemented
4#59426C214yesmaximal, complemented
5#45166C218yesmaximal, complemented
6#44983C218yesmaximal, complemented
7#39511C214yesmaximal
8#37958C218yesmaximal, complemented
9#36177C214yesmaximal, complemented
10#35544C214yesmaximal, complemented
11#35173C211yesmaximal, characteristic, complemented
12#35151C212yesmaximal, complemented
13#34946C212yesmaximal
14#34214C212yesmaximal, complemented
15#34179C212yesmaximal, complemented
16C2×#2354C211yesmaximal, characteristic
17C4×#544C211yesmaximal, characteristic, complemented
18C2×#544C2211yescharacteristic
19C2×#533C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2212yes—
22#2857C2214yes—
23#2855C2218yes—
24#2808C2218yes—
25#2800C22116yes—
26#2783C22116yescomplemented
27#2777C22116yescomplemented
28#2775C22116yescomplemented
29#2759C22116yescomplemented
30#2727C2218yes—
31#2659C2214yes—
32#2626C2218yes—
33#2604C22116yescomplemented
34#2594C22116yescomplemented
35#2592C2218yes—
36#2584C2218yescomplemented
37#2571C2214yescomplemented
38#2568C2214yescomplemented
39#2567C2214yescomplemented
40#2563C2212yes—
41#2563C2212yes—
42#2562C2214yes—
43#2562C2214yes—
44#2561C2218yes—
45#2547C2212yes—
46#2547C2212yes—
47#2546C2214yescomplemented
48#2545C2218yes—
49#2533C22116yescomplemented
50#2532C22116yescomplemented
51#2514C2218yescomplemented
52#2514C2218yescomplemented
53#2512C2218yes—
54#2509C2218yescomplemented
55#2503C2214yes—
56#2497C2218yescomplemented
57#2497C2218yescomplemented
58#2496C2218yescomplemented
59#2496C2218yescomplemented
60#2495C2218yescomplemented
61#2495C2218yescomplemented
62#2494C2218yescomplemented
63#2494C2218yescomplemented
64#2493C22116yes—
65#2480C22116yes—
66#2469C22116yescomplemented
67#2448C2218yes—
68#2441C2218yescomplemented
69#2441C2218yescomplemented
70#2439C2218yescomplemented
71#2439C2218yescomplemented
72#2435C2214yes—
73#2398C2218yes—
74#2365C2214yescomplemented
75#2360C2218yes—
76#2360C2218yes—
77#2356C2218yescomplemented
78#2355C2214yescomplemented
79#2355C2214yescomplemented
80#2355C2214yescomplemented
81#2354C2214yes—
82#2354C2214yes—
83#2354C2218yes—
84Q8×C4⋊C4C2214yescomplemented
85Q8×C4⋊C4C2214yescomplemented
86#2347C2218yescomplemented
87#2347C2218yescomplemented
88#2346C2218yescomplemented
89#2346C2218yescomplemented
90#2331C2214yescomplemented
91#2330C2214yescomplemented
92#2330C2218yescomplemented
93#2329C2214yes—
94#2317C2212yes—
95#2317C2214yes—
96#2316C2211yescharacteristic
97#2316C2212yes—
98#2316C2212yes—
99#2312C2218yes—
100C4×C4⋊Q8C2214yescomplemented
101C4×C4⋊Q8C2214yescomplemented
102C4×C4⋊Q8C2214yescomplemented
103C4×#125C2214yescomplemented
104C4×#124C2214yes—
105C4×#124C2214yes—
106C4×C22⋊Q8C2214yes—
107C2×#390C2212yes—
108C2×#388C2214yes—
109C2×#384C2211yescharacteristic
110#2283C2218yes—
111#2280C2214yes—
112#2280C2214yes—
113C2×#380C2212yes—
114#2274C2214yes—
115C4×#117C2212yes—
116C2×C4×C4⋊C4C2211yescharacteristic
117C2×C4×C4⋊C4C2212yes—
118C23×D4C2311yescharacteristic
119C24×C4C2312yes—
120#544—28no—
121#533—216no—
122C2×C4⋊Q8C2312yes—
123C2×C4⋊Q8C2312yes—
124C2×C4⋊Q8C2314yes—
125C2×C4⋊Q8C2314yes—
126C2×#126C23116yes—
127C2×#125C2314yes—
128C2×#125C2314yes—
129C2×#124C2314yes—
130C2×#124C2314yes—
131C2×C22⋊Q8C2314yes—
132C2×C22⋊Q8C2314yes—
133C2×C22⋊Q8C2314yes—
134C2×C4×Q8C2314yes—
135C2×C4×Q8C2314yes—
136C2×C4×Q8C2314yes—
137C2×C4×Q8C2318yes—
138C2×C4×Q8C2318yes—
139C2×C4×D4—28no—
140C2×C4×D4C2314yes—
141C2×C4×D4C2312yes—
142C2×C4×D4C2312yes—
143C2×#117—216no—
144C2×#117C2314yes—
145C2×#117C2312yes—
146C2×#117C2312yes—
147C2×#117C2318yes—
148C2×#117C2318yes—
149C22×C4⋊C4C2311yescharacteristic
150C22×C4⋊C4C2312yes—
151C22×C4⋊C4C2311yescharacteristic
152C22×C4⋊C4C2312yes—
153C22×C4⋊C4C2312yes—
154C22×C4⋊C4C2311yescharacteristic
155C22×C22⋊C4C2312yes—
156C22×C22⋊C4C2314yes—
157C22×C22⋊C4C2314yes—
158C22×C42C2314yes—
159C22×C42C2312yes—
160C22×C42C2311yescharacteristic
161#401C23116yes—
162#400C2318yes—
163#400C2318yes—
164#400C23116yes—
165#400C23116yes—
166#400C23116yes—
167#399C23116yes—
168#398C23116yes—
169#397C2318yes—
170#397C2318yes—
171#397C23116yes—
172#397C23116yes—
173#397C23116yes—
174#396C23116yes—
175#396C23116yes—
176#396C23116yes—
177#395C23116yes—
178#394C2318yes—
179#394C2318yes—
180#393C2318yes—
181#393C2318yes—
182#393C23116yes—
183#391C2318yes—
184#391C2318yes—
185#391C2318yes—
186#391C2318yes—
187#391C2314yes—
188#391C2318yes—
189#391C2314yes—
190#391C2318yes—
191#391C2314yes—
192#390C2314yes—
193#390C2314yes—
194#390C2318yes—
195#390C2318yes—
196#389C2314yes—
197#389C2314yes—
198#389C2318yes—
199#389C2314yes—
200#389C2318yes—
201#389C2318yes—
202#389C2314yes—
203#389C2318yes—
204#389C2318yes—
205#389C2318yes—
206#389C2318yes—
207#389C2318yes—
208#389C2318yes—
209#389C2318yes—
210#389C2318yes—
211#389C2318yes—
212#389C2314yes—
213#389C23116yes—
214#389C23116yes—
215#389C2318yes—
216#388C2318yes—
217#388C2318yes—
218#388C2318yes—
219#388C2318yes—
220#388C23116yes—
221#388C23116yes—
222#388C23116yes—
223#388C23116yes—
224#388C23116yes—
225#388C23116yes—
226#388C23116yes—
227#388C23116yes—
228#388C23116yes—
229#387C2318yes—
230#387C2318yes—
231#387C2318yes—
232#387C23116yes—
233#387C23116yes—
234#387C23116yes—
235#387C23116yes—
236#387C23116yes—
237#387C23116yes—
238#387C23116yes—
239#385C2318yes—
240#385C23116yes—
241#384C2314yes—
242#384C2314yes—
243#384C2314yes—
244#384C2318yes—
245#384C2314yes—
246#383C23116yes—
247#383C23116yes—
248#382C2314yes—
249#382C2314yes—
250#382C2314yes—
251#382C2318yes—
252#382C2318yes—
253#382C2318yes—
254#381C2318yes—
255#381C2318yes—
256#380C2314yes—
257#380C2314yes—
258#380C2314yes—
259#380C2314yes—
260#380C2314yes—
261#380C2314yes—
262#380C2314yes—
263#380C2314yes—
264#380C2314yes—
265#380C2318yes—
266#380C2318yes—
267#380C2318yes—
268#380C2318yes—
269#379C2318yes—
270#379C2318yes—
271C4×C4⋊C4C2312yes—
272C4×C4⋊C4C2314yes—
273C4×C4⋊C4C2314yes—
274C4×C4⋊C4C2318yes—
275C4×C4⋊C4C2312yes—
276C4×C4⋊C4C2312yes—
277C4×C4⋊C4C2312yes—
278C4×C4⋊C4C2318yes—
279C4×C4⋊C4C2318yes—
280C4×C4⋊C4C2318yes—
281C4×C4⋊C4C2318yes—
282C4×C4⋊C4C2314yes—
283C4×C4⋊C4C2314yes—
284C4×C4⋊C4C2318yes—
285C4×C4⋊C4C2318yes—
286C4×C4⋊C4C2318yes—
287C4×C4⋊C4C2318yes—
288C4×C4⋊C4C2314yes—
289C4×C4⋊C4C2318yes—
290C4×C22⋊C4C2314yes—
291C4×C22⋊C4C23116yes—
292C4×C22⋊C4C2318yes—
293C4×C22⋊C4C2318yes—
294C4×C22⋊C4C2314yes—
295C4×C22⋊C4C2318yes—
296C4×C22⋊C4C2318yes—
297#376C2314yes—
298#376C23116yes—
299C2×#95C2312yes—
300C2×#95C2314yes—
301C43C2314yes—
302C25C2412yes—
303C22×Q8C2414yes—
304C22×Q8C2414yes—
305C22×Q8C2414yes—
306C22×D4—24no—
307C22×D4—42no—
308C22×D4—42no—
309C22×D4C2412yes—
310C22×D4C2412yes—
311C23×C4—28no—
312C23×C4—28no—
313C23×C4—28no—
314C23×C4C2412yes—
315C23×C4C2414yes—
316C23×C4C2414yes—
317C23×C4C2411yescharacteristic
318C23×C4C2414yes—
319C23×C4C2412yes—
320C23×C4C2411yescharacteristic
321C23×C4C2412yes—
322C23×C4C2411yescharacteristic
323C4⋊Q8—216no—
324C4⋊Q8—28no—
325C4⋊Q8—28no—
326C4⋊Q8—48no—
327#126—816no—
328#125—216no—
329#125—216no—
330#124—48no—
331#124—48no—
332C22⋊Q8—216no—
333C22⋊Q8—216no—
334C22⋊Q8—216no—
335C4×Q8—216no—
336C4×Q8—216no—
337C4×Q8—48no—
338C4×Q8—416no—
339C4×Q8—88no—
340C4×D4—216no—
341C4×D4—28no—
342C4×D4—28no—
343#117—216no—
344#117—28no—
345#117—28no—
346#117—232no—
347#117—232no—
348C2×C4⋊C4—28no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—44no—
352C2×C4⋊C4—82no—
353C2×C4⋊C4#4118yes—
354C2×C4⋊C4C2418yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2414yes—
359C2×C4⋊C4C2414yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2414yes—
362C2×C4⋊C4C2414yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2418yes—
367C2×C4⋊C4C2418yes—
368C2×C4⋊C4C2414yes—
369C2×C4⋊C4C2414yes—
370C2×C4⋊C4C2414yes—
371C2×C4⋊C4C2418yes—
372C2×C4⋊C4C2418yes—
373C2×C4⋊C4C2414yes—
374C2×C4⋊C4C24116yes—
375C2×C4⋊C4C2414yes—
376C2×C4⋊C4C2418yes—
377C2×C4⋊C4C24116yes—
378C2×C4⋊C4C2418yes—
379C2×C4⋊C4C2418yes—
380C2×C4⋊C4C2418yes—
381C2×C4⋊C4C2418yes—
382C2×C4⋊C4C2414yes—
383C2×C4⋊C4C2418yes—
384C2×C4⋊C4C2414yes—
385C2×C22⋊C4—216no—
386C2×C22⋊C4—28no—
387C2×C22⋊C4—84no—
388C2×C22⋊C4C24116yes—
389C2×C22⋊C4C2414yes—
390C2×C22⋊C4C2414yes—
391C2×C22⋊C4C2418yes—
392C2×C22⋊C4C2414yes—
393C2×C22⋊C4C2414yes—
394C2×C22⋊C4C24116yes—
395C2×C22⋊C4C2418yes—
396C2×C22⋊C4C2418yes—
397C2×C22⋊C4C2418yes—
398C2×C22⋊C4C2418yes—
399C2×C22⋊C4C2418yes—
400C2×C22⋊C4C2418yes—
401C2×C22⋊C4C2418yes—
402C2×C22⋊C4C2418yes—
403C2×C42—216no—
404C2×C42—44no—
405C2×C42#4118yes—
406C2×C42C2418yes—
407C2×C42C2418yes—
408C2×C42C2418yes—
409C2×C42C2414yes—
410C2×C42C2414yes—
411C2×C42C2414yes—
412C2×C42C2414yes—
413C2×C42C2414yes—
414C2×C42C2414yes—
415C2×C42C24116yes—
416C2×C42C24116yes—
417C2×C42C2414yes—
418C2×C42C2414yes—
419C2×C42C2412yes—
420C2×C42C2412yes—
421#95C2418yes—
422#95C2418yes—
423#95C2418yes—
424#95C2418yes—
425#95C2418yes—
426#95C2418yes—
427#95C2414yes—
428#95C2414yes—
429#95C2414yes—
430#95C2414yes—
431#95C24116yes—
432#95C2418yes—
433#95C2418yes—
434#95C24116yes—
435#95C24116yes—
436#95C2418yes—
437#95C2418yes—
438#95C2418yes—
439#95C2418yes—
440#95C2418yes—
441#95C24116yes—
442#95C2418yes—
443#95C24116yes—
444#95C24116yes—
445C24—24no—
446C24—24no—
447C24—24no—
448C24—42no—
449C24—42no—
450C24—42no—
451C24—42no—
452C24C2514yes—
453C24C2511yescharacteristic
454C2×Q8—28no—
455C2×Q8—28no—
456C2×Q8—28no—
457C2×Q8—28no—
458C2×Q8—28no—
459C2×Q8—44no—
460C2×Q8—44no—
461C2×Q8—44no—
462C2×Q8—44no—
463C2×D4—24no—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—24no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—416no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—28no—
476C22×C4—24no—
477C22×C4—28no—
478C22×C4—28no—
479C22×C4—28no—
480C22×C4—28no—
481C22×C4—28no—
482C22×C4—28no—
483C22×C4—42no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—432no—
492C22×C4C2×#4114yes—
493C22×C4C2×#4118yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4114yes—
496C22×C4C2×#4114yes—
497C22×C42−1+414yes—
498C22×C4C2512yes—
499C22×C4C2512yes—
500C22×C4C2514yes—
501C22×C4C2514yes—
502C22×C4C2514yes—
503C22×C4C2514yes—
504C22×C4C2518yes—
505C22×C4C2518yes—
506C22×C4C2518yes—
507C22×C4C2514yes—
508C22×C4C2514yes—
509C22×C4C2512yes—
510C22×C4C2511yescharacteristic
511C22×C4C2512yes—
512C22×C4C2511yescharacteristic
513C4⋊C4—216no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—28no—
519C4⋊C4—28no—
520C4⋊C4—24no—
521C4⋊C4—216no—
522C4⋊C4—216no—
523C4⋊C4—28no—
524C4⋊C4—28no—
525C4⋊C4—28no—
526C4⋊C4—216no—
527C4⋊C4—28no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—28no—
531C4⋊C4—44no—
532C4⋊C4—48no—
533C4⋊C4—44no—
534C4⋊C4—48no—
535C4⋊C4—44no—
536C4⋊C4—48no—
537C4⋊C4—416no—
538C4⋊C4—416no—
539C4⋊C4—48no—
540C4⋊C4—48no—
541C4⋊C4—48no—
542C4⋊C4—48no—
543C4⋊C4C2×#41116yes—
544C22⋊C4—28no—
545C22⋊C4—216no—
546C22⋊C4—28no—
547C22⋊C4—232no—
548C22⋊C4—28no—
549C22⋊C4—28no—
550C22⋊C4—416no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—48no—
554C22⋊C4—48no—
555C22⋊C4—48no—
556C22⋊C4—48no—
557C22⋊C4—48no—
558C22⋊C4—48no—
559C42—216no—
560C42—216no—
561C42—216no—
562C42—28no—
563C42—28no—
564C42—28no—
565C42—28no—
566C42—28no—
567C42—44no—
568C42—44no—
569C42—416no—
570C42—44no—
571C42—416no—
572C42C2×#4118yes—
573C42C2×#4118yes—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—24no—
579C23—24no—
580C23—24no—
581C23—24no—
582C23—24no—
583C23—24no—
584C23—24no—
585C23—48no—
586C23—48no—
587C23—48no—
588C23—84no—
589C23—84no—
590C23—84no—
591C23—84no—
592C23C22×#4112yes—
593C23C22×#4112yes—
594C23C2×2−1+412yes—
595C23C2611yescharacteristic
596Q8—416no—
597Q8—416no—
598Q8—88no—
599D4—48no—
600D4—48no—
601C2×C4—24no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—28no—
628C2×C4—28no—
629C2×C4—28no—
630C2×C4—24no—
631C2×C4—28no—
632C2×C4—28no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—28no—
638C2×C4—48no—
639C2×C4—44no—
640C2×C4—48no—
641C2×C4—416no—
642C2×C4—416no—
643C2×C4—88no—
644C2×C4—84no—
645C2×C4#556116yes—
646C2×C4C22×#4118yes—
647C2×C4C22×#4114yes—
648C2×C4C22×#4112yes—
649C2×C4C22×#4114yes—
650C2×C4C22×#4114yes—
651C2×C4C22×#4112yes—
652C2×C4C2×2−1+414yes—
653C2×C4#58512yes—
654C2×C4#58512yes—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—816no—
669C22C2×#55614yes—
670C22C23×#4111yescharacteristic
671C22C23×#4111yescharacteristic
672C22C22×2−1+411yescharacteristic
673C22C2×#58511yescharacteristic
674C22C2×#58511yescharacteristic
675C22H(7,𝔽2)11yescharacteristic
676C222−1+611yescharacteristic
677C4—28no—
678C4—22no—
679C4—24no—
680C4—44no—
681C4—44no—
682C4—48no—
683C4—48no—
684C4—48no—
685C4—44no—
686C4—44no—
687C4—44no—
688C4—42no—
689C4—42no—
690C4C2×#55618yes—
691C4#349314yes—
692C2—24no—
693C2—84no—
694C2C22×#55611yescharacteristic
695C2C2×#349311yescharacteristic
696C2#6284511yescharacteristic
697C2#6292911yescharacteristic
698C2#6295911yescharacteristic
699C2#6299111yescharacteristic
700C2#6300611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59435C2semidirect
2. #59434C2semidirect
3. #59427C2semidirect
4. #59426C2semidirect
5. #45166C2semidirect
6. #44983C2semidirect
8. #37958C2semidirect
9. #36177C2semidirect
10. #35544C2semidirect
11. #35173C2semidirect
12. #35151C2semidirect
14. #34214C2semidirect
15. #34179C2semidirect
17. C4×#544C2semidirect
26. #2783C22semidirect
27. #2777C22semidirect
28. #2775C22semidirect
29. #2759C22semidirect
33. #2604C22semidirect
34. #2594C22semidirect
36. #2584C22semidirect
37. #2571C22semidirect
38. #2568C22semidirect
39. #2567C22semidirect
47. #2546C22semidirect
49. #2533C22semidirect
50. #2532C22semidirect
51. #2514C22semidirect
52. #2514C22semidirect
54. #2509C22semidirect
56. #2497C22semidirect
57. #2497C22semidirect
58. #2496C22semidirect
59. #2496C22semidirect
60. #2495C22semidirect
61. #2495C22semidirect
62. #2494C22semidirect
63. #2494C22semidirect
66. #2469C22semidirect
68. #2441C22semidirect
69. #2441C22semidirect
70. #2439C22semidirect
71. #2439C22semidirect
74. #2365C22semidirect
77. #2356C22semidirect
78. #2355C22semidirect
79. #2355C22semidirect
80. #2355C22semidirect
84. Q8×C4⋊C4C22semidirect
85. Q8×C4⋊C4C22semidirect
86. #2347C22semidirect
87. #2347C22semidirect
88. #2346C22semidirect
89. #2346C22semidirect
90. #2331C22semidirect
91. #2330C22semidirect
92. #2330C22semidirect
100. C4×C4⋊Q8C22semidirect
101. C4×C4⋊Q8C22semidirect
102. C4×C4⋊Q8C22semidirect
103. C4×#125C22semidirect

Maximal subgroups

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

subgroupindex
#594352
#594342
#594272
#594262
#451662
#449832
#395112
#379582
#361772
#355442
#351732
#351512
#349462
#342142
#341792
C2×#23542
C4×#5442