Finite Group Databasebeta

← #6792523

Subgroups — #6792523

Order 512.

1739 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36133C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic
3#32906C211yesmaximal, characteristic
4#31640C214yesmaximal, complemented
5#31524C214yesmaximal, complemented
6#29909C212yesmaximal
7#28401C212yesmaximal, complemented
8#27815C212yesmaximal, complemented
9#25902C211yesmaximal, characteristic
10#25324C212yesmaximal, complemented
11#24817C212yesmaximal, complemented
12#24502C212yesmaximal
13#24302C211yesmaximal, characteristic
14#23934C212yesmaximal
15#22599C211yesmaximal, characteristic
16#21860C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C2×#533C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#487C2212yes—
22C2×#477C2212yes—
23C2×#462C2212yes—
24C2×#451C2211yescharacteristic
25C2×#451C2212yes—
26C2×C4×Q16C2212yes—
27C2×#431C2212yes—
28#2857C2211yescharacteristic
29#2855C2212yes—
30#2567—21no—
31#2567—22no—
32#2567—21no—
33#2567C2214yes—
34#2567C2212yes—
35#2567C2212yes—
36#2551C2211yescharacteristic
37#2546C2211yescharacteristic
38#2506—21no—
39#2462—22no—
40#2451C2212yes—
41#2446C2212yes—
42#2440—21no—
43#2438—22no—
44C2×#395—22no—
45#2365—21no—
46#2365C2211yescharacteristic
47Q8×C4⋊C4C2211yescharacteristic
48#2344—21no—
49C4×C22⋊Q8C2211yescharacteristic
50C2×#391C2212yes—
51C2×#389—21no—
52C2×#389—21no—
53C2×#384C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2100C2214yes—
56#2095C2214yes—
57#2083C2214yes—
58#2072C2214yes—
59#2068C2214yes—
60#2057C2212yes—
61#2056C2214yes—
62#2024C2214yes—
63#2024C2214yes—
64#2024C2212yes—
65#2016C2214yes—
66#1990C2214yes—
67#1990C2212yes—
68#1987C2212yes—
69#1969C2214yes—
70#1956C2214yes—
71#1939C2212yes—
72#1933C2214yes—
73#1916C2214yes—
74#1878C2214yes—
75#1878C2214yes—
76#1878C2214yes—
77#1878C2212yes—
78#1872C2212yes—
79#1872C2212yes—
80#1872C2211yescharacteristic
81#1872C2212yes—
82#1872C2212yes—
83#1863C2212yes—
84#1852C2212yes—
85#1852C2211yescharacteristic
86#1817C2212yes—
87#1803C2214yes—
88#1803C2214yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91#1787C2211yescharacteristic
92C8×C4⋊C4C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99#533—28no—
100#533—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×#125—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2311yescharacteristic
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—22no—
117C2×C4×Q8—44no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×#117—22no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C42—21no—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#487—44no—
136#477—44no—
137#462—44no—
138#451—24no—
139#451—44no—
140C4×Q16—44no—
141#431—44no—
142C2×#107C2314yes—
143C2×#107C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2314yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2311yescharacteristic
157C2×C22⋊C8C2311yescharacteristic
158C2×C4×C8C2312yes—
159C2×C4×C8C2314yes—
160#400—22no—
161#398—22no—
162#398—22no—
163#398—22no—
164#395—22no—
165#395—24no—
166#395—24no—
167#395—24no—
168#395—24no—
169#395—22no—
170#391—22no—
171#391—22no—
172#391—22no—
173#391C2314yes—
174#391C2314yes—
175#391C2312yes—
176#391C2312yes—
177#391C2312yes—
178#391C2311yescharacteristic
179#389—22no—
180#389—24no—
181#389—22no—
182#389—24no—
183#389—22no—
184#389—22no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—22no—
192#389C2311yescharacteristic
193#389C2312yes—
194#389C2312yes—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—21no—
200#387—22no—
201#387—22no—
202#385—22no—
203#385—21no—
204#384—21no—
205#384—22no—
206#384—22no—
207#384C2314yes—
208#384C2314yes—
209#384C2311yescharacteristic
210#381—22no—
211#380—21no—
212#380—21no—
213#380C2311yescharacteristic
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—21no—
225C2×#95—22no—
226C2×#95—22no—
227C2×#95D412yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—24no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—42no—
247C22×Q8—44no—
248C22×Q8—82no—
249C22×Q8—82no—
250C22×Q8C2412yes—
251C22×Q8C2411yescharacteristic
252C22×Q8C2412yes—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C2×Q16—24no—
262C2×Q16—24no—
263C2×Q16—28no—
264C2×Q16—42no—
265C2×Q16—44no—
266C2×Q16—44no—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—44no—
276C4⋊Q8—42no—
277C4⋊Q8—42no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281#125—48no—
282C22⋊Q8—22no—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285C22⋊Q8—48no—
286C22⋊Q8#4114yes—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—42no—
293C4×Q8—42no—
294C4×Q8—42no—
295C4×Q8—42no—
296C4×Q8—48no—
297C4×Q8—44no—
298C4×Q8—44no—
299#117—48no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—84no—
331C2×C4⋊C4—84no—
332C2×C4⋊C4Q1618yescomplemented
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2414yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—44no—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2412yes—
364C2×C42C2412yes—
365C2×C42C2411yescharacteristic
366#107—42no—
367#107—44no—
368C4⋊C8—24no—
369C4⋊C8—44no—
370#103—24no—
371#103—24no—
372#103—42no—
373#103—42no—
374#103—44no—
375#103—44no—
376#103—44no—
377#103—44no—
378#103—44no—
379C22⋊C8—22no—
380C22⋊C8—22no—
381C22⋊C8—24no—
382C4×C8—24no—
383C4×C8—44no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—21no—
394#95—21no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405C24C22×D411yescharacteristic
406C2×Q8—24no—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—24no—
419C2×Q8—44no—
420C2×Q8—48no—
421C2×Q8—48no—
422C2×Q8—42no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—42no—
426C2×Q8—44no—
427C2×Q8—82no—
428C2×Q8—82no—
429C2×Q8—82no—
430C2×Q8—82no—
431C2×Q8—82no—
432C2×Q8—816no—
433C2×Q8C2×#4112yes—
434C2×Q82−1+414yes—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—42no—
456C22×C4—41no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4C2×Q1618yes—
466C22×C4C2×Q1614yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C2×#4112yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4C2511yescharacteristic
479Q16—48no—
480Q16—88no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—81no—
520C4⋊C4—82no—
521C4⋊C4—81no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4C2×Q1618yes—
527C4⋊C4C2×Q1614yes—
528C4⋊C4C2×Q1614yes—
529C4⋊C4#13714yes—
530C4⋊C4C2×#4112yes—
531C22⋊C4—24no—
532C22⋊C4—44no—
533C22⋊C4—42no—
534C42—22no—
535C42—22no—
536C42—24no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—44no—
544C42—44no—
545C42—44no—
546C42—44no—
547C42—42no—
548C42—42no—
549C42H(5,𝔽2)14yes—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23#54612yes—
555C23C22×Q1612yes—
556C23#57712yes—
557C23C23×D411yescharacteristic
558Q8—24no—
559Q8—24no—
560Q8—24no—
561Q8—48no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—44no—
566Q8—44no—
567Q8—88no—
568Q8—88no—
569Q8—88no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—216no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—84no—
623C2×C4—88no—
624C2×C4#53512yes—
625C2×C4#53512yes—
626C2×C4#54612yes—
627C2×C4C22×Q1612yes—
628C2×C4C22×Q1614yes—
629C2×C4C22×Q1614yes—
630C2×C4C22×Q1612yes—
631C2×C4C2×#13714yes—
632C2×C4C2×#13712yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C2×H(5,𝔽2)12yes—
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×2−1+412yes—
637C8—42no—
638C8—42no—
639C8—44no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22#329914yes—
647C22C2×#53511yescharacteristic
648C22C2×#54611yescharacteristic
649C22#348611yescharacteristic
650C22C23×Q1611yescharacteristic
651C22C22×#13711yescharacteristic
652C22C2×#57711yescharacteristic
653C22#358511yescharacteristic
654C4—22no—
655C4—22no—
656C4—24no—
657C4—24no—
658C4—24no—
659C4—22no—
660C4—41no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—42no—
665C4—81no—
666C4—82no—
667C4—84no—
668C4—81no—
669C4—82no—
670C4—82no—
671C4#320914yes—
672C4#352012yes—
673C4#353612yes—
674C2—24no—
675C2#3890511yescharacteristic
676C2C2×#320911yescharacteristic
677C2#6081211yescharacteristic
678C2C2×#329911yescharacteristic
679C2#6116211yescharacteristic
680C2#6182111yescharacteristic
681C2#6183811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #31640C2semidirect
5. #31524C2semidirect
7. #28401C2semidirect
8. #27815C2semidirect
10. #25324C2semidirect
11. #24817C2semidirect
332. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#361332
C2×#25672
#329062
#316402
#315242
#299092
#284012
#278152
#259022
#253242
#248172
#245022
#243022
#239342
#225992
#218602
C4×#4512