Finite Group Databasebeta

← #7630535

Subgroups — #7630535

Order 512.

5112 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35612C218yesmaximal, complemented
2#35602C218yesmaximal, complemented
3#35581C218yesmaximal, complemented
4#35580C218yesmaximal, complemented
5#35174C212yesmaximal
6#35173C211yesmaximal, characteristic, complemented
7#35172C212yesmaximal
8#35167C211yesmaximal, characteristic, complemented
9#35159C212yesmaximal, complemented
10#35052C214yesmaximal, complemented
11#35033C214yesmaximal, complemented
12#35014C214yesmaximal, complemented
13#35012C214yesmaximal, complemented
14C4×#544C212yesmaximal, complemented
15#34074C212yesmaximal, complemented
16#34058C212yesmaximal
17C2×#2275C211yesmaximal, characteristic
18C2×#544C2212yes—
19C2×#533C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2857C2214yes—
24#2857C2214yes—
25#2851C2218yes—
26#2848C2218yes—
27#2848C2218yes—
28#2659C2214yes—
29#2659C2214yes—
30#2646C2218yes—
31#2626C2218yes—
32#2626C2218yes—
33#2608C22116yescomplemented
34#2607C22116yescomplemented
35#2589C22116yescomplemented
36#2587C22116yescomplemented
37#2584C2218yescomplemented
38#2580C2218yes—
39#2574C22116yescomplemented
40#2569C22116yescomplemented
41#2567C2214yescomplemented
42#2566C2218yes—
43#2563C2214yes—
44#2556C22116yescomplemented
45#2552C2214yescomplemented
46#2551C2214yescomplemented
47#2550C2218yescomplemented
48#2549C22116yescomplemented
49#2546C2214yescomplemented
50#2542C2214yes—
51#2538C2214yes—
52#2538C2214yes—
53#2522C22116yescomplemented
54#2509C2218yescomplemented
55#2508C2218yescomplemented
56#2507C2218yescomplemented
57#2506C2218yescomplemented
58#2497C2218yescomplemented
59#2492C2218yescomplemented
60#2465C22116yescomplemented
61#2454C2218yes—
62#2454C2218yes—
63#2452C2218yes—
64#2452C2218yes—
65#2441C2218yescomplemented
66#2441C2218yescomplemented
67#2439C2218yescomplemented
68#2439C2218yescomplemented
69#2438C22116yescomplemented
70#2438C22116yescomplemented
71#2429C2218yescomplemented
72#2423C2218yescomplemented
73#2331C2214yescomplemented
74#2331C2218yescomplemented
75#2331C2214yescomplemented
76#2330C2214yescomplemented
77#2330C2214yescomplemented
78#2330C2218yescomplemented
79#2329C2214yes—
80#2329C2214yes—
81#2329C2214yes—
82#2329C2214yes—
83#2329C2218yes—
84#2319C2218yes—
85#2319C2218yes—
86#2317C2214yes—
87#2316C2212yes—
88#2316C2212yes—
89#2316C2212yes—
90#2312C2218yes—
91#2309C2214yes—
92C4×C4⋊Q8C2218yescomplemented
93C4×#125C2218yescomplemented
94C4×C22⋊Q8C2218yes—
95#2296C2214yes—
96#2295C2218yescomplemented
97#2295C2218yescomplemented
98#2293C2218yes—
99#2293C2218yes—
100#2285C2212yes—
101#2280C2218yes—
102#2275C2218yes—
103#2275C2218yes—
104C4×#117C2214yes—
105C2×C4×C4⋊C4C2211yescharacteristic
106C2×C4×C4⋊C4C2211yescharacteristic
107C2×C4×C4⋊C4C2211yescharacteristic
108C2×C4×C22⋊C4C2212yes—
109C2×C4×C22⋊C4C2212yes—
110C2×C4×C22⋊C4C2212yes—
111C2×#376C2211yescharacteristic
112C22×#95C2212yes—
113C23×D4C2311yescharacteristic
114C24×C4C2312yes—
115C24×C4C2312yes—
116C24×C4C2312yes—
117#544—216no—
118#533—216no—
119C2×C4⋊Q8C2318yes—
120C2×C4⋊Q8C2312yes—
121C2×C4⋊Q8C2312yes—
122C2×C4⋊Q8C2314yes—
123C2×C4⋊Q8C2314yes—
124C2×#125C2318yes—
125C2×#125C2314yes—
126C2×#125C2318yes—
127C2×C22⋊Q8C2314yes—
128C2×C22⋊Q8C2314yes—
129C2×C22⋊Q8C2318yes—
130C2×C22⋊Q8C2318yes—
131C2×C22⋊Q8C2318yes—
132C2×C22⋊Q8C2318yes—
133C2×C22⋊Q8C2318yes—
134C2×C4×Q8C2314yes—
135C2×C4×Q8C2314yes—
136C2×C4×Q8C2318yes—
137C2×C4×D4—28no—
138C2×C4×D4—28no—
139C2×C4×D4Q8116yescomplemented
140C2×C4×D4C2314yes—
141C2×C4×D4C2314yes—
142C2×C4×D4C2318yes—
143C2×C4×D4C2318yes—
144C2×#117C2314yes—
145C2×#117C2314yes—
146C2×#117C2318yes—
147C2×#117C2318yes—
148C22×C4⋊C4C2311yescharacteristic
149C22×C4⋊C4C2311yescharacteristic
150C22×C4⋊C4C2311yescharacteristic
151C22×C22⋊C4C2312yes—
152C22×C22⋊C4C2312yes—
153C22×C22⋊C4C2312yes—
154C22×C42C2311yescharacteristic
155C22×C42C2312yes—
156C22×C42C2312yes—
157C22×C42C2312yes—
158C22×C42C2311yescharacteristic
159C22×C42C2311yescharacteristic
160#400C23116yes—
161#400C23116yes—
162#399C23116yes—
163#399C23116yes—
164#398C23116yes—
165#398C23116yes—
166#398C23116yes—
167#398C23116yes—
168#396C23116yes—
169#396C23116yes—
170#396C23116yes—
171#396C23116yes—
172#395C23116yes—
173#395C23116yes—
174#393C23116yes—
175#393C23116yes—
176#391C23116yes—
177#391C2318yes—
178#391C2318yes—
179#391C2318yes—
180#391C2318yes—
181#391C2318yes—
182#391C2318yes—
183#391C2318yes—
184#390C2318yes—
185#390C2318yes—
186#390C2318yes—
187#390C2318yes—
188#389C2318yes—
189#389C2318yes—
190#389C2318yes—
191#389C2318yes—
192#389C23116yes—
193#389C2318yes—
194#389C2318yes—
195#389C2318yes—
196#389C2318yes—
197#389C2318yes—
198#389C2318yes—
199#389C2318yes—
200#389C2318yes—
201#389C2318yes—
202#389C2318yes—
203#389C2318yes—
204#389C2314yes—
205#389C2314yes—
206#389C2314yes—
207#389C2314yes—
208#389C2318yes—
209#389C2318yes—
210#389C2318yes—
211#389C2318yes—
212#388C23116yes—
213#388C23116yes—
214#388C23116yes—
215#388C23116yes—
216#387C23116yes—
217#387C23116yes—
218#387C2318yes—
219#387C2318yes—
220#387C2318yes—
221#387C2318yes—
222#387C23116yes—
223#387C2318yes—
224#387C2318yes—
225#387C2318yes—
226#387C2318yes—
227#387C23116yes—
228#387C23116yes—
229#385C23116yes—
230#385C23116yes—
231#385C23116yes—
232#385C23116yes—
233#385C2318yes—
234#385C2318yes—
235#385C23116yes—
236#384C2318yes—
237#384C2314yes—
238#384C2312yes—
239#384C2312yes—
240#384C2314yes—
241#382C2318yes—
242#382C2318yes—
243#382C2314yes—
244#381C23116yes—
245#380C2318yes—
246#380C2318yes—
247#380C2318yes—
248#380C2318yes—
249#380C2318yes—
250#380C2318yes—
251#380C2318yes—
252#380C2318yes—
253#380C2314yes—
254#380C2314yes—
255#380C2318yes—
256#379C23116yes—
257C4×C4⋊C4C2314yes—
258C4×C4⋊C4C2318yes—
259C4×C4⋊C4C23116yes—
260C4×C4⋊C4C2318yes—
261C4×C4⋊C4C2314yes—
262C4×C4⋊C4C2314yes—
263C4×C4⋊C4C2314yes—
264C4×C4⋊C4C2314yes—
265C4×C4⋊C4C2314yes—
266C4×C4⋊C4C2314yes—
267C4×C4⋊C4C2314yes—
268C4×C4⋊C4C2318yes—
269C4×C4⋊C4C2318yes—
270C4×C4⋊C4C2318yes—
271C4×C22⋊C4C2318yes—
272C4×C22⋊C4C23116yes—
273C4×C22⋊C4C23116yes—
274C4×C22⋊C4C2318yes—
275C4×C22⋊C4C2318yes—
276C4×C22⋊C4C2318yes—
277C4×C22⋊C4C2318yes—
278C4×C22⋊C4C2318yes—
279#376C2318yes—
280#376C2314yes—
281#376C2314yes—
282C2×#95C23116yes—
283C2×#95C2311yescharacteristic
284C2×#95C2312yes—
285C2×#95C2311yescharacteristic
286C2×#95C2311yescharacteristic
287C2×#95C2311yescharacteristic
288C2×#95C2312yes—
289C2×#95C2312yes—
290C43C2318yes—
291C25C2412yes—
292C22×Q8C2414yes—
293C22×Q8C2418yes—
294C22×Q8C2414yes—
295C22×D4—24no—
296C22×D4—24no—
297C22×D4C2×Q818yes—
298C22×D4C2414yes—
299C23×C4—28no—
300C23×C4—28no—
301C23×C4—82no—
302C23×C4—82no—
303C23×C4—82no—
304C23×C4—82no—
305C23×C4—82no—
306C23×C4—82no—
307C23×C4C2×Q8116yes—
308C23×C4C2411yescharacteristic
309C23×C4C2412yes—
310C23×C4C2418yes—
311C23×C4C2418yes—
312C23×C4C2411yescharacteristic
313C23×C4C2411yescharacteristic
314C23×C4C2412yes—
315C23×C4C2418yes—
316C23×C4C2411yescharacteristic
317C23×C4C2411yescharacteristic
318C23×C4C2412yes—
319C23×C4C2411yescharacteristic
320C23×C4C2411yescharacteristic
321C4⋊Q8—232no—
322C4⋊Q8—28no—
323C4⋊Q8—28no—
324C4⋊Q8—84no—
325C4⋊Q8—84no—
326#125—232no—
327#125—216no—
328#125—88no—
329C22⋊Q8—216no—
330C22⋊Q8—216no—
331C22⋊Q8—232no—
332C22⋊Q8—88no—
333C22⋊Q8—88no—
334C22⋊Q8—88no—
335C22⋊Q8—88no—
336C4×Q8—216no—
337C4×Q8—216no—
338C4×Q8—88no—
339C4×D4—232no—
340C4×D4—216no—
341C4×D4—216no—
342C4×D4C2×Q8164yes—
343#117—216no—
344#117—216no—
345#117—88no—
346#117—88no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4C2×Q818yes—
349C2×C4⋊C4#4118yes—
350C2×C4⋊C4C2414yes—
351C2×C4⋊C4C2414yes—
352C2×C4⋊C4C2414yes—
353C2×C4⋊C4C2414yes—
354C2×C4⋊C4C2414yes—
355C2×C4⋊C4C2418yes—
356C2×C4⋊C4C2418yes—
357C2×C4⋊C4C2414yes—
358C2×C4⋊C4C2414yes—
359C2×C4⋊C4C2418yes—
360C2×C4⋊C4C24116yes—
361C2×C4⋊C4C2418yes—
362C2×C4⋊C4C2418yes—
363C2×C4⋊C4C2418yes—
364C2×C4⋊C4C2418yes—
365C2×C4⋊C4C2418yes—
366C2×C4⋊C4C24116yes—
367C2×C4⋊C4C2414yes—
368C2×C4⋊C4C2414yes—
369C2×C4⋊C4C2418yes—
370C2×C4⋊C4C2418yes—
371C2×C4⋊C4C24116yes—
372C2×C4⋊C4C2418yes—
373C2×C4⋊C4C2418yes—
374C2×C4⋊C4C2414yes—
375C2×C4⋊C4C2414yes—
376C2×C4⋊C4C2418yes—
377C2×C4⋊C4C2414yes—
378C2×C4⋊C4C2414yes—
379C2×C22⋊C4—28no—
380C2×C22⋊C4—28no—
381C2×C22⋊C4C2×Q8116yes—
382C2×C22⋊C4C2414yes—
383C2×C22⋊C4C2418yes—
384C2×C22⋊C4C2414yes—
385C2×C22⋊C4C2418yes—
386C2×C22⋊C4C2418yes—
387C2×C22⋊C4C24116yes—
388C2×C22⋊C4C24116yes—
389C2×C22⋊C4C24116yes—
390C2×C22⋊C4C24116yes—
391C2×C22⋊C4C2418yes—
392C2×C42—24no—
393C2×C42—82no—
394C2×C42—82no—
395C2×C42—82no—
396C2×C42C2×Q818yes—
397C2×C42#4118yes—
398C2×C42C2418yes—
399C2×C42C2418yes—
400C2×C42C2414yes—
401C2×C42C2418yes—
402C2×C42C2418yes—
403C2×C42C2418yes—
404C2×C42C2418yes—
405C2×C42C2418yes—
406C2×C42C2414yes—
407C2×C42C2414yes—
408C2×C42C2418yes—
409C2×C42C2414yes—
410C2×C42C2412yes—
411C2×C42C2412yes—
412#95C2414yes—
413#95C2418yes—
414#95C2418yes—
415#95C2418yes—
416#95C2412yes—
417#95C2412yes—
418#95C2414yes—
419#95C2412yes—
420#95C2412yes—
421#95C2418yes—
422#95C2418yes—
423#95C2418yes—
424#95C2418yes—
425#95C24116yes—
426#95C24116yes—
427#95C2418yes—
428#95C2418yes—
429#95C24116yes—
430#95C2418yes—
431#95C24116yes—
432#95C2418yes—
433#95C24116yes—
434#95C2418yes—
435#95C2418yes—
436C24—24no—
437C24—24no—
438C24—42no—
439C24—42no—
440C24—42no—
441C24—42no—
442C24C22×Q818yes—
443C24C2514yes—
444C24C2511yescharacteristic
445C2×Q8—28no—
446C2×Q8—28no—
447C2×Q8—216no—
448C2×Q8—48no—
449C2×Q8—44no—
450C2×Q8—44no—
451C2×Q8—48no—
452C2×Q8—44no—
453C2×Q8—44no—
454C2×D4—232no—
455C2×D4—28no—
456C2×D4—28no—
457C2×D4C22×Q8116yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—216no—
462C22×C4—216no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—816no—
488C22×C4—816no—
489C22×C4—816no—
490C22×C4C22×Q814yes—
491C22×C4C22×Q814yes—
492C22×C4C22×Q818yes—
493C22×C4C22×Q8132yes—
494C22×C4C22×Q814yes—
495C22×C4C2×#4114yes—
496C22×C4C2×#4114yes—
497C22×C4C2×#4118yes—
498C22×C4C2×#4114yes—
499C22×C4C2×#4114yes—
500C22×C4C2512yes—
501C22×C4C2512yes—
502C22×C4C2514yes—
503C22×C4C2518yes—
504C22×C4C2518yes—
505C22×C4C2518yes—
506C22×C4C2518yes—
507C22×C4C2512yes—
508C22×C4C2512yes—
509C22×C4C2514yes—
510C22×C4C2511yescharacteristic
511C22×C4C2512yes—
512C22×C4C2514yes—
513C22×C4C2511yescharacteristic
514C22×C4C2512yes—
515C4⋊C4—28no—
516C4⋊C4—28no—
517C4⋊C4—28no—
518C4⋊C4—28no—
519C4⋊C4—28no—
520C4⋊C4—216no—
521C4⋊C4—216no—
522C4⋊C4—28no—
523C4⋊C4—28no—
524C4⋊C4—216no—
525C4⋊C4—232no—
526C4⋊C4—44no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—416no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—48no—
537C4⋊C4—48no—
538C4⋊C4—416no—
539C4⋊C4—48no—
540C4⋊C4—48no—
541C4⋊C4—44no—
542C4⋊C4C22×Q8116yes—
543C4⋊C4C2×#41116yes—
544C22⋊C4—216no—
545C22⋊C4—216no—
546C22⋊C4—216no—
547C22⋊C4—28no—
548C22⋊C4—28no—
549C22⋊C4—416no—
550C22⋊C4—416no—
551C22⋊C4—416no—
552C22⋊C4—416no—
553C22⋊C4C22×Q8132yes—
554C42—28no—
555C42—216no—
556C42—216no—
557C42—48no—
558C42—48no—
559C42—44no—
560C42—44no—
561C42—48no—
562C42—48no—
563C42—48no—
564C42—48no—
565C42C22×Q8116yes—
566C42C2×#4118yes—
567C42C2×#4118yes—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—216no—
579C23—48no—
580C23—48no—
581C23—48no—
582C23—48no—
583C23—48no—
584C23—48no—
585C23C23×Q818yes—
586C23C23×Q812yes—
587C23C22×#4112yes—
588C23C22×#4112yes—
589C23C2611yescharacteristic
590Q8—816no—
591Q8—88no—
592Q8—88no—
593D4—232no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—22no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—432no—
632C2×C4—48no—
633C2×C4—48no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—48no—
638C2×C4—44no—
639C2×C4—816no—
640C2×C4—816no—
641C2×C4#533116yes—
642C2×C4C23×Q814yes—
643C2×C4C23×Q814yes—
644C2×C4C23×Q818yes—
645C2×C4C23×Q814yes—
646C2×C4C22×#4112yes—
647C2×C4C22×#4112yes—
648C2×C4C22×#4114yes—
649C2×C4C22×#4118yes—
650C2×C4C22×#4114yes—
651C2×C4C22×#4114yes—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—28no—
659C22—28no—
660C22—28no—
661C22—432no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22C2×#53314yes—
667C22C24×Q811yescharacteristic
668C22C23×#4111yescharacteristic
669C22C23×#4111yescharacteristic
670C22H(7,𝔽2)11yescharacteristic
671C22H(7,𝔽2)11yescharacteristic
672C22H(7,𝔽2)11yescharacteristic
673C222−1+611yescharacteristic
674C4—24no—
675C4—28no—
676C4—28no—
677C4—24no—
678C4—24no—
679C4—22no—
680C4—22no—
681C4—42no—
682C4—48no—
683C4—48no—
684C4—42no—
685C4—44no—
686C4—48no—
687C4—48no—
688C4C2×#53318yes—
689C2—24no—
690C2—48no—
691C2C22×#53311yescharacteristic
692C2#6284311yescharacteristic
693C2#6284611yescharacteristic
694C2#6285111yescharacteristic
695C2#6285411yescharacteristic
696C2#6285511yescharacteristic
697C2#6285611yescharacteristic
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. #35612C2semidirect
2. #35602C2semidirect
3. #35581C2semidirect
4. #35580C2semidirect
6. #35173C2semidirect
8. #35167C2semidirect
9. #35159C2semidirect
10. #35052C2semidirect
11. #35033C2semidirect
12. #35014C2semidirect
13. #35012C2semidirect
14. C4×#544C2semidirect
15. #34074C2semidirect
33. #2608C22semidirect
34. #2607C22semidirect
35. #2589C22semidirect
36. #2587C22semidirect
37. #2584C22semidirect
39. #2574C22semidirect
40. #2569C22semidirect
41. #2567C22semidirect
44. #2556C22semidirect
45. #2552C22semidirect
46. #2551C22semidirect
47. #2550C22semidirect
48. #2549C22semidirect
49. #2546C22semidirect
53. #2522C22semidirect
54. #2509C22semidirect
55. #2508C22semidirect
56. #2507C22semidirect
57. #2506C22semidirect
58. #2497C22semidirect
59. #2492C22semidirect
60. #2465C22semidirect
65. #2441C22semidirect
66. #2441C22semidirect
67. #2439C22semidirect
68. #2439C22semidirect
69. #2438C22semidirect
70. #2438C22semidirect
71. #2429C22semidirect
72. #2423C22semidirect
73. #2331C22semidirect
74. #2331C22semidirect
75. #2331C22semidirect
76. #2330C22semidirect
77. #2330C22semidirect
78. #2330C22semidirect
92. C4×C4⋊Q8C22semidirect
93. C4×#125C22semidirect
96. #2295C22semidirect
97. #2295C22semidirect
139. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#356122
#356022
#355812
#355802
#351742
#351732
#351722
#351672
#351592
#350522
#350332
#350142
#350122
C4×#5442
#340742
#340582
C2×#22752