Finite Group Databasebeta

← #7674639

Subgroups — #7674639

Order 512.

4864 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#44465C218yesmaximal, complemented
2#43515C218yesmaximal, complemented
3#39553C218yesmaximal, complemented
4#39286C218yesmaximal, complemented
5#39283C218yesmaximal, complemented
6#37897C214yesmaximal, complemented
7#37894C214yesmaximal, complemented
8#35672C212yesmaximal, complemented
9#35633C212yesmaximal
10#35339C211yesmaximal, characteristic, complemented
11#35316C211yesmaximal, characteristic, complemented
12#35313C212yesmaximal, complemented
13C2×#2621C211yesmaximal, characteristic, complemented
14#34276C212yesmaximal, complemented
15#34209C211yesmaximal, characteristic, complemented
16#34184C211yesmaximal, characteristic, complemented
17#34065C211yesmaximal, characteristic
18#34048C211yesmaximal, characteristic, complemented
19C2×#536C2212yes—
20C22×C4⋊Q8C2211yescharacteristic
21C2×#525C2211yescharacteristic
22C22×C22⋊Q8C2212yes—
23C2×#520C2212yes—
24C2×#519C2212yes—
25C2×C4×#41C2211yescharacteristic
26C22×#117C2211yescharacteristic
27#2864C2214yescomplemented
28#2863C2218yescomplemented
29#2857C2214yescomplemented
30#2802C22116yescomplemented
31#2796C22116yes—
32#2787C22116yescomplemented
33#2777C22116yes—
34#2772C22116yes—
35#2766C22116yescomplemented
36#2747C22116yescomplemented
37#2741C22116yes—
38#2647C2214yescomplemented
39#2645C2218yescomplemented
40#2621C2214yes—
41#2621C2218yes—
42#2621C2214yes—
43#2621C2214yescomplemented
44#2597C22116yes—
45#2596C22116yes—
46#2594C22116yes—
47#2590C22116yescomplemented
48#2585C22116yescomplemented
49#2577C22116yescomplemented
50#2570C2218yescomplemented
51#2569C2218yes—
52#2569C22116yescomplemented
53#2547C2214yescomplemented
54#2546C2214yescomplemented
55#2534C2218yes—
56#2514C2218yescomplemented
57#2511C2218yescomplemented
58#2510C22116yes—
59#2493C22116yes—
60#2486C2218yes—
61#2486C2218yes—
62#2484C2218yes—
63#2482C2218yes—
64#2429C2218yes—
65#2427C2218yes—
66#2423C2218yes—
67#2423C2218yes—
68#2421C2218yes—
69#2421C2218yes—
70#2417C2218yes—
71#2415C2218yes—
72C2×#398C2214yes—
73C2×#397C2214yes—
74#2365C2218yescomplemented
75#2362C2218yescomplemented
76#2360C2218yescomplemented
77#2359C22116yes—
78#2351C2214yescomplemented
79#2349C2218yes—
80#2349C2214yescomplemented
81#2344C2218yes—
82#2327C2212yescomplemented
83#2327C2212yescomplemented
84#2327C2214yescomplemented
85#2326C2214yescomplemented
86#2324C2214yes—
87#2323C2214yes—
88#2322C2214yes—
89#2321C2214yes—
90#2317C2212yes—
91#2317C2212yes—
92#2317C2214yescomplemented
93#2316C2212yes—
94#2314C2212yes—
95#2314C2212yes—
96#2314C2214yes—
97#2313C2214yes—
98#2312C2214yes—
99#2310C2212yes—
100#2310C2212yes—
101C4×C4⋊Q8C2214yescomplemented
102C4×C4⋊Q8C2214yescomplemented
103C4×#124C2214yescomplemented
104#2298C2218yes—
105#2297C2211yescharacteristic
106#2296C2211yescharacteristic
107C2×#391C2212yes—
108#2293C2214yes—
109#2290C2214yes—
110#2285C2211yescharacteristic
111#2285C2211yescharacteristic
112#2285C2212yescomplemented
113C2×#382C2211yescharacteristic
114C2×#382C2212yescomplemented
115#2280C2212yes—
116#2280C2211yescharacteristic
117C4×#117C2212yes—
118C4×#117C2211yescharacteristic
119C22×#41C2311yescharacteristic
120C23×Q8C2312yes—
121C24×C4C2311yescharacteristic
122#536—216no—
123C2×C4⋊Q8—28no—
124C2×C4⋊Q8C2312yes—
125C2×C4⋊Q8C2314yes—
126C2×C4⋊Q8C2312yes—
127C2×C4⋊Q8C2314yes—
128C2×#124C2314yes—
129#525—28no—
130C2×#123C2312yes—
131C2×#123C2318yes—
132C2×C22⋊Q8—48no—
133C2×C22⋊Q8C2312yes—
134C2×C22⋊Q8C2318yes—
135C2×C22⋊Q8C2318yes—
136C2×C22⋊Q8C2318yes—
137C2×#121C2312yes—
138C2×#120C2312yes—
139#520—48no—
140#519—48no—
141C4×#41—28no—
142C2×C4×Q8C2312yes—
143C2×C4×Q8C2314yes—
144C2×C4×Q8C2314yes—
145C2×C4×Q8C2314yes—
146C2×C4×Q8C2314yes—
147C2×C4×Q8C2314yes—
148C2×C4×D4C2314yes—
149C2×C4×D4C2312yes—
150C2×C4×D4C2314yes—
151C2×C4×D4C2314yes—
152C2×C4×D4C2318yes—
153C2×#117—28no—
154C2×#117C2311yescharacteristic
155C2×#117C2311yescharacteristic
156C2×#117C2312yes—
157C2×#117C2312yes—
158C2×#117C2312yes—
159C2×#117C2312yes—
160C2×#117C2311yescharacteristic
161C2×#117C2312yes—
162C2×#117C2314yes—
163C2×#117C2314yes—
164C2×#117C2318yes—
165C22×C4⋊C4C2312yes—
166C22×C4⋊C4C2312yes—
167C22×C4⋊C4C2314yes—
168C22×C4⋊C4C2314yes—
169C22×C4⋊C4C2314yes—
170C22×C22⋊C4C2312yes—
171C22×C22⋊C4C2314yes—
172C22×C42C2312yes—
173C22×C42C2311yescharacteristic
174C22×C42C2311yescharacteristic
175#401C23116yes—
176#400C23116yes—
177#400C23116yes—
178#400C23116yes—
179#400C23116yes—
180#400C23116yes—
181#399C23116yes—
182#398C23116yes—
183#398C23116yes—
184#398C23116yes—
185#398C23116yes—
186#398C2318yes—
187#398C23116yes—
188#398C2318yes—
189#397C2318yes—
190#397C23116yes—
191#397C23116yes—
192#397C23116yes—
193#397C2318yes—
194#397C23116yes—
195#396C23116yes—
196#394C23116yes—
197#391C2312yes—
198#391C2314yes—
199#391C2314yes—
200#391C2314yes—
201#391C2318yes—
202#391C2318yes—
203#391C2318yes—
204#391C2314yes—
205#391C2318yes—
206#390C2318yes—
207#390C2312yes—
208#389C23116yes—
209#389C23116yes—
210#389C2318yes—
211#389C2318yes—
212#389C2318yes—
213#389C2318yes—
214#389C23116yes—
215#389C2318yes—
216#389C2314yes—
217#389C23116yes—
218#389C23116yes—
219#388C2318yes—
220#388C2318yes—
221#388C2318yes—
222#388C2318yes—
223#388C2318yes—
224#388C2318yes—
225#388C2318yes—
226#388C23116yes—
227#388C23116yes—
228#388C23116yes—
229#388C23116yes—
230#388C23116yes—
231#387C2318yes—
232#387C2318yes—
233#387C2318yes—
234#387C2318yes—
235#387C23116yes—
236#387C23116yes—
237#387C23116yes—
238#387C23116yes—
239#387C23116yes—
240#387C23116yes—
241#387C2318yes—
242#387C23116yes—
243#387C23116yes—
244#387C23116yes—
245#387C23116yes—
246#387C23116yes—
247#386C2314yes—
248#386C2318yes—
249#386C23116yes—
250#385C2318yes—
251#385C2318yes—
252#385C23116yes—
253#385C2318yes—
254#385C23116yes—
255#385C2314yes—
256#384C2312yes—
257#384C2314yes—
258#384C2312yes—
259#384C2314yes—
260#383C23116yes—
261#382C2311yescharacteristic
262#382C2311yescharacteristic
263#382C2312yes—
264#382C2314yes—
265#382C2314yes—
266#382C2314yes—
267#382C2312yes—
268#382C2318yes—
269#382C2312yes—
270#381C2318yes—
271#381C2318yes—
272#381C2314yes—
273#381C2312yes—
274#380C2314yes—
275#380C2314yes—
276#380C2318yes—
277#380C23116yes—
278#380C23116yes—
279#380C2314yes—
280#380C2314yes—
281#380C2314yes—
282#380C2318yes—
283#379C2314yes—
284#379C2312yes—
285C4×C4⋊C4C23116yes—
286C4×C4⋊C4C23116yes—
287C4×C4⋊C4C2314yes—
288C4×C4⋊C4C2314yes—
289C4×C4⋊C4C2314yes—
290C4×C4⋊C4C2314yes—
291C4×C4⋊C4C2318yes—
292C4×C4⋊C4C2314yes—
293C4×C4⋊C4C2314yes—
294C4×C22⋊C4C2318yes—
295C4×C22⋊C4C2314yes—
296C4×C22⋊C4C23116yes—
297C4×C22⋊C4C2314yes—
298C4×C22⋊C4C2314yes—
299C4×C22⋊C4C2318yes—
300#376C2314yes—
301#376C2312yes—
302C2×#95C2314yes—
303C2×#95C2318yes—
304C2×#95C2318yes—
305C43C2312yes—
306C43C2314yes—
307C25C2411yescharacteristic
308C2×#41—24no—
309C2×#41—24no—
310C2×#41#4118yes—
311C22×Q8—28no—
312C22×Q8—28no—
313C22×Q8—28no—
314C22×Q8C2414yes—
315C22×Q8C2414yes—
316C22×Q8C2411yescharacteristic
317C22×D4C2412yes—
318C22×D4C2411yescharacteristic
319C22×D4C2414yes—
320C23×C4—24no—
321C23×C4—24no—
322C23×C4—24no—
323C23×C4C2414yes—
324C23×C4C2411yescharacteristic
325C23×C4C2412yes—
326C23×C4C2412yes—
327C23×C4C2412yes—
328C23×C4C2414yes—
329C23×C4C2412yes—
330C23×C4C2414yes—
331C23×C4C2418yes—
332C23×C4C2412yes—
333C23×C4C2411yescharacteristic
334C4⋊Q8—216no—
335C4⋊Q8—28no—
336C4⋊Q8—28no—
337C4⋊Q8—216no—
338#124—216no—
339#123—28no—
340#123—232no—
341C22⋊Q8—28no—
342C22⋊Q8—232no—
343C22⋊Q8—416no—
344C22⋊Q8—416no—
345#121—28no—
346#120—28no—
347C4×Q8—28no—
348C4×Q8—48no—
349C4×Q8—48no—
350C4×Q8—48no—
351C4×Q8—48no—
352C4×Q8—48no—
353C4×D4—216no—
354C4×D4—216no—
355C4×D4—28no—
356C4×D4—48no—
357C4×D4—416no—
358#117—24no—
359#117—24no—
360#117—28no—
361#117—24no—
362#117—28no—
363#117—28no—
364#117—28no—
365#117—216no—
366#117—28no—
367#117—416no—
368#117—48no—
369C2×C4⋊C4—28no—
370C2×C4⋊C4—216no—
371C2×C4⋊C4—28no—
372C2×C4⋊C4—48no—
373C2×C4⋊C4—48no—
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2414yes—
376C2×C4⋊C4C2414yes—
377C2×C4⋊C4C2414yes—
378C2×C4⋊C4C2414yes—
379C2×C4⋊C4C2418yes—
380C2×C4⋊C4C2418yes—
381C2×C4⋊C4C2412yes—
382C2×C4⋊C4C2412yes—
383C2×C4⋊C4C2414yes—
384C2×C4⋊C4C2412yes—
385C2×C4⋊C4C2412yes—
386C2×C4⋊C4C2414yes—
387C2×C4⋊C4C2414yes—
388C2×C4⋊C4C24116yes—
389C2×C4⋊C4C2418yes—
390C2×C4⋊C4C24116yes—
391C2×C4⋊C4C2418yes—
392C2×C4⋊C4C2418yes—
393C2×C4⋊C4C2418yes—
394C2×C4⋊C4C2418yes—
395C2×C4⋊C4C24116yes—
396C2×C4⋊C4C24116yes—
397C2×C4⋊C4C2418yes—
398C2×C4⋊C4C24116yes—
399C2×C22⋊C4—28no—
400C2×C22⋊C4—48no—
401C2×C22⋊C4C2414yes—
402C2×C22⋊C4C2414yes—
403C2×C22⋊C4C2412yes—
404C2×C22⋊C4C2414yes—
405C2×C22⋊C4C2412yes—
406C2×C22⋊C4C2414yes—
407C2×C22⋊C4C2412yes—
408C2×C22⋊C4C2414yes—
409C2×C22⋊C4C2414yes—
410C2×C22⋊C4C2414yes—
411C2×C22⋊C4C24116yes—
412C2×C22⋊C4C24116yes—
413C2×C22⋊C4C2418yes—
414C2×C22⋊C4C2418yes—
415C2×C22⋊C4C2418yes—
416C2×C22⋊C4C2418yes—
417C2×C22⋊C4C2418yes—
418C2×C42—24no—
419C2×C42—28no—
420C2×C42#4118yes—
421C2×C42C2412yes—
422C2×C42C2412yes—
423C2×C42C2412yes—
424C2×C42C2412yes—
425C2×C42C2414yes—
426C2×C42C2412yes—
427C2×C42C2414yes—
428C2×C42C2414yes—
429C2×C42C2418yes—
430C2×C42C2418yes—
431C2×C42C2418yes—
432C2×C42C2418yes—
433C2×C42C2418yes—
434C2×C42C2418yes—
435C2×C42C2411yescharacteristic
436C2×C42C2412yes—
437C2×C42C2411yescharacteristic
438#95C2414yes—
439#95C2418yes—
440#95C24116yes—
441#95C24116yes—
442#95C2418yes—
443#95C24116yes—
444#95C2418yes—
445#95C2418yes—
446#95C2418yes—
447#95C2418yes—
448#95C2418yes—
449#95C2418yes—
450#95C2418yes—
451#95C24116yes—
452#95C2414yes—
453#95C24116yes—
454#95C2414yes—
455#95C2414yes—
456#95C24116yes—
457#95C2418yes—
458#95C24116yes—
459#95C24116yes—
460#95C24116yes—
461C24—22no—
462C24—22no—
463C24—22no—
464C24—44no—
465C24C2512yes—
466C24C2512yes—
467C24C2511yescharacteristic
468#41—232no—
469C2×Q8—22no—
470C2×Q8—28no—
471C2×Q8—28no—
472C2×Q8—28no—
473C2×Q8—28no—
474C2×Q8—22no—
475C2×Q8—28no—
476C2×Q8—28no—
477C2×Q8—432no—
478C2×Q8C2×#4114yes—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—28no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—48no—
485C2×D4C2×#4118yes—
486C2×D4C2×#4114yes—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—28no—
491C22×C4—22no—
492C22×C4—28no—
493C22×C4—28no—
494C22×C4—22no—
495C22×C4—24no—
496C22×C4—28no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—24no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—28no—
504C22×C4—44no—
505C22×C4—48no—
506C22×C4—44no—
507C22×C4—48no—
508C22×C4—48no—
509C22×C4—416no—
510C22×C4—416no—
511C22×C4C2×#4118yes—
512C22×C4C2×#4118yes—
513C22×C4C2×#4114yes—
514C22×C4C2×#4114yes—
515C22×C4H(5,𝔽2)14yes—
516C22×C4C2514yes—
517C22×C4C2514yes—
518C22×C4C2514yes—
519C22×C4C2514yes—
520C22×C4C2518yes—
521C22×C4C2518yes—
522C22×C4C2518yes—
523C22×C4C2512yes—
524C22×C4C2514yes—
525C22×C4C2514yes—
526C22×C4C2512yes—
527C22×C4C2512yes—
528C22×C4C2512yes—
529C22×C4C2511yescharacteristic
530C22×C4C2511yescharacteristic
531C4⋊C4—28no—
532C4⋊C4—28no—
533C4⋊C4—216no—
534C4⋊C4—216no—
535C4⋊C4—28no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—28no—
539C4⋊C4—28no—
540C4⋊C4—232no—
541C4⋊C4—28no—
542C4⋊C4—28no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—24no—
546C4⋊C4—48no—
547C4⋊C4—416no—
548C4⋊C4—48no—
549C4⋊C4—48no—
550C4⋊C4—48no—
551C4⋊C4—416no—
552C4⋊C4—416no—
553C4⋊C4—416no—
554C4⋊C4—48no—
555C4⋊C4—48no—
556C22⋊C4—24no—
557C22⋊C4—24no—
558C22⋊C4—28no—
559C22⋊C4—28no—
560C22⋊C4—232no—
561C22⋊C4—24no—
562C22⋊C4—28no—
563C22⋊C4—28no—
564C22⋊C4—28no—
565C22⋊C4—28no—
566C22⋊C4—28no—
567C22⋊C4—416no—
568C22⋊C4—48no—
569C22⋊C4—48no—
570C22⋊C4—48no—
571C22⋊C4—48no—
572C22⋊C4—48no—
573C42—24no—
574C42—24no—
575C42—24no—
576C42—28no—
577C42—24no—
578C42—28no—
579C42—28no—
580C42—24no—
581C42—48no—
582C42—48no—
583C42—48no—
584C42—48no—
585C42—48no—
586C42—48no—
587C42C2×#4114yes—
588C42C2×#4114yes—
589C42C2×#4118yes—
590C23—24no—
591C23—22no—
592C23—22no—
593C23—28no—
594C23—22no—
595C23—22no—
596C23—22no—
597C23—24no—
598C23—24no—
599C23—48no—
600C23—44no—
601C23—84no—
602C23—84no—
603C23C22×#4114yes—
604C23C22×#4112yes—
605C23C22×#4112yes—
606C23C2×H(5,𝔽2)12yes—
607C23C2611yescharacteristic
608Q8—28no—
609Q8—416no—
610Q8—416no—
611D4—216no—
612D4—28no—
613D4—88no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—216no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—28no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—28no—
625C2×C4—22no—
626C2×C4—28no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—28no—
631C2×C4—28no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—28no—
636C2×C4—28no—
637C2×C4—28no—
638C2×C4—28no—
639C2×C4—28no—
640C2×C4—28no—
641C2×C4—28no—
642C2×C4—216no—
643C2×C4—48no—
644C2×C4—44no—
645C2×C4—416no—
646C2×C4—48no—
647C2×C4—48no—
648C2×C4—88no—
649C2×C4—84no—
650C2×C4—88no—
651C2×C4—168no—
652C2×C4#553116yes—
653C2×C4C22×#4114yes—
654C2×C4C22×#4114yes—
655C2×C4C22×#4114yes—
656C2×C4C22×#4112yes—
657C2×C4C22×#4112yes—
658C2×C4C22×#4112yes—
659C2×C4C22×#4112yes—
660C2×C4C2×H(5,𝔽2)12yes—
661C2×C4C2×H(5,𝔽2)12yes—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—28no—
666C22—24no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—42no—
671C22—42no—
672C22—44no—
673C22—42no—
674C22—88no—
675C22C2×#55314yes—
676C22C23×#4111yescharacteristic
677C22C23×#4111yescharacteristic
678C22C22×H(5,𝔽2)11yescharacteristic
679C222−1+614yes—
680C4—24no—
681C4—24no—
682C4—24no—
683C4—44no—
684C4—44no—
685C4—48no—
686C4—44no—
687C4—44no—
688C4—44no—
689C4—48no—
690C4—44no—
691C4—42no—
692C4—48no—
693C4C2×#55314yes—
694C4C2×#55314yes—
695C2—24no—
696C2—44no—
697C2—82no—
698C2C22×#55311yescharacteristic
699C2#6286012yes—
700C2#6286112yes—
701C2#6288712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC2×#525C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #44465C2semidirect
2. #43515C2semidirect
3. #39553C2semidirect
4. #39286C2semidirect
5. #39283C2semidirect
6. #37897C2semidirect
7. #37894C2semidirect
8. #35672C2semidirect
10. #35339C2semidirect
11. #35316C2semidirect
12. #35313C2semidirect
13. C2×#2621C2semidirect
14. #34276C2semidirect
15. #34209C2semidirect
16. #34184C2semidirect
18. #34048C2semidirect
27. #2864C22semidirect
28. #2863C22semidirect
29. #2857C22semidirect
30. #2802C22semidirect
32. #2787C22semidirect
35. #2766C22semidirect
36. #2747C22semidirect
38. #2647C22semidirect
39. #2645C22semidirect
43. #2621C22semidirect
47. #2590C22semidirect
48. #2585C22semidirect
49. #2577C22semidirect
50. #2570C22semidirect
52. #2569C22semidirect
53. #2547C22semidirect
54. #2546C22semidirect
56. #2514C22semidirect
57. #2511C22semidirect
74. #2365C22semidirect
75. #2362C22semidirect
76. #2360C22semidirect
78. #2351C22semidirect
80. #2349C22semidirect
82. #2327C22semidirect
83. #2327C22semidirect
84. #2327C22semidirect
85. #2326C22semidirect
92. #2317C22semidirect
101. C4×C4⋊Q8C22semidirect
102. C4×C4⋊Q8C22semidirect
103. C4×#124C22semidirect
112. #2285C22semidirect
114. C2×#382C22semidirect

Maximal subgroups

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

subgroupindex
#444652
#435152
#395532
#392862
#392832
#378972
#378942
#356722
#356332
#353392
#353162
#353132
C2×#26212
#342762
#342092
#341842
#340652
#340482