Finite Group Databasebeta

← #6621063

Subgroups — #6621063

Order 512.

1847 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3369C212yesmaximal
2Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
3C4×#556C211yesmaximal, characteristic
4#33197C211yesmaximal, characteristic
5#33196C212yesmaximal, complemented
6#28726C212yesmaximal
7#28717C211yesmaximal, characteristic, complemented
8#28255C211yesmaximal, characteristic, complemented
9#28254C212yesmaximal
10#25336C214yesmaximal
11#25324C212yesmaximal, complemented
12#24698C214yesmaximal, complemented
13#24680C212yesmaximal
14#22722C211yesmaximal, characteristic, complemented
15#22525C212yesmaximal, complemented
16#21506C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic
18C2×Q82C2212yes—
19C2×#556C2211yescharacteristic
20#3369—216no—
21C2×#489C2214yes—
22C2×#487C2212yes—
23C2×#475C2214yes—
24C2×#476C2212yes—
25C2×#464C2214yes—
26C2×#462C2212yes—
27C2×#462C2212yes—
28C2×#461C2211yescharacteristic
29C2×#446C2212yes—
30C2×#441C2214yes—
31C2×C4×SD16C2212yes—
32#2868C2211yescharacteristic
33#2868C2212yes—
34#2567C2212yes—
35#2567C2214yes—
36#2551—22no—
37#2551—21no—
38#2551—22no—
39#2551—21no—
40#2373—22no—
41#2367C2212yes—
42#2365C2211yescharacteristic
43#2362—21no—
44#2356C2212yes—
45#2355C2211yescharacteristic
46#2354C2211yescharacteristic
47Q8×C4⋊C4—21no—
48Q8×C4⋊C4—21no—
49#2347—21no—
50C4×C4⋊Q8C2212yes—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×#126—22no—
54C4×#124C2212yes—
55C4×C22⋊Q8—21no—
56C42×Q8C2212yes—
57C42×Q8C2211yescharacteristic
58C42×D4C2211yescharacteristic
59C4×#117—21no—
60#2072C2214yes—
61#2072C2214yes—
62#2071C2214yes—
63#2056C2214yes—
64#2056C2214yes—
65#2055C2214yes—
66#1993C2214yes—
67#1990C2212yes—
68#1988C2214yes—
69#1987C2212yes—
70#1981C2211yescharacteristic
71#1948C2212yes—
72#1948C2214yes—
73#1948C2212yes—
74#1948C2211yescharacteristic
75#1940C2214yes—
76#1939C2212yes—
77#1927C2214yes—
78#1927C2212yes—
79#1878C2214yes—
80#1877C2212yes—
81#1842C2212yes—
82#1842C2211yescharacteristic
83#1805C2214yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2214yes—
89#1802C2211yescharacteristic
90#1802C2212yes—
91#1802C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94#1763C2214yes—
95C4×#103C2212yes—
96C22×Q16C2314yes—
97C22×SD16C2312yes—
98Q82—28no—
99Q82—28no—
100#556—24no—
101#556D418yes—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—22no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2314yes—
111C2×C4⋊Q8C2312yes—
112C2×#126—22no—
113C2×#124C2312yes—
114C2×C22⋊Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×#117—21no—
128C2×#117—21no—
129C22×C42—21no—
130#489—216no—
131#487—28no—
132#476—28no—
133#475—216no—
134#464—216no—
135#462—28no—
136#462—28no—
137#461—24no—
138#446—28no—
139#441—216no—
140C4×SD16—28no—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2314yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×#97C2314yes—
160C2×C4×C8C2312yes—
161#395—24no—
162#395—24no—
163#395—24no—
164#391—21no—
165#391C2314yes—
166#391C2312yes—
167#391C2312yes—
168#391C2312yes—
169#390C2312yes—
170#389—22no—
171#389—24no—
172#389—24no—
173#389—22no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—21no—
179#389—21no—
180#389—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#385—22no—
191#384—22no—
192#384—21no—
193#384—22no—
194#384—21no—
195#384C2312yes—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2314yes—
200#383—22no—
201#382C2312yes—
202#380—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4D412yes—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—21no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—21no—
229C4×C22⋊C4—21no—
230C4×C22⋊C4—22no—
231#376—22no—
232C43—22no—
233C43C2311yescharacteristic
234C43C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2312yes—
240#340C2312yes—
241#340C2314yes—
242#336C2314yes—
243#336C2312yes—
244C22×Q8—24no—
245C22×Q8—22no—
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2412yes—
248C22×Q8C2412yes—
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C2×Q16—28no—
254C2×Q16—28no—
255C2×Q16—44no—
256C2×SD16—24no—
257C2×SD16—24no—
258C2×SD16—42no—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C4⋊Q8—28no—
263C4⋊Q8—24no—
264C4⋊Q8—22no—
265C4⋊Q8—24no—
266C4⋊Q8—28no—
267C4⋊Q8—22no—
268C4⋊Q8—48no—
269C4⋊Q8—416no—
270C4⋊Q8—84no—
271C4⋊Q8—82no—
272C4⋊Q8—84no—
273C4⋊Q8—82no—
274C4⋊Q8C2×D414yes—
275C4⋊Q8#4114yes—
276#126—216no—
277#124—24no—
278#124C2×D418yes—
279C22⋊Q8—28no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—28no—
286C4×Q8—24no—
287C4×Q8—28no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—42no—
291C4×Q8—48no—
292C4×Q8C2×D418yes—
293C4×Q8C2×D414yes—
294C4×Q8#4114yes—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—41no—
300C4×D4C2×D414yes—
301#117—28no—
302#117—44no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—44no—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2412yes—
366C2×C42C2411yescharacteristic
367#106—24no—
368#106—28no—
369C4⋊C8—24no—
370C4⋊C8—28no—
371C4⋊C8—22no—
372C4⋊C8—24no—
373C4⋊C8—22no—
374C4⋊C8—42no—
375C4⋊C8—44no—
376#103—28no—
377#103—28no—
378#103—28no—
379#103—24no—
380#103—28no—
381#103—24no—
382#103—24no—
383#102—24no—
384#102—28no—
385#97—28no—
386C4×C8—24no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—24no—
400#95—22no—
401#95—24no—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405C24—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—48no—
419C2×Q8—44no—
420C2×Q8—84no—
421C2×Q8—82no—
422C2×Q8—84no—
423C2×Q8—82no—
424C2×Q8#12114yes—
425C2×Q8C22×D414yes—
426C2×Q8C2×#4114yes—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4C22×D412yes—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—81no—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2511yescharacteristic
461Q16—416no—
462SD16—48no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C4⋊C4—24no—
473C4⋊C4—28no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4—82no—
506C4⋊C4#12114yes—
507C4⋊C4#12118yes—
508C4⋊C4#13714yes—
509C4⋊C4C22×D412yes—
510C4⋊C42−1+412yes—
511C22⋊C4—28no—
512C22⋊C4—24no—
513C22⋊C4—28no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—82no—
518C42—28no—
519C42—24no—
520C42—22no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—21no—
525C42—22no—
526C42—21no—
527C42—22no—
528C42—42no—
529C42—41no—
530C42—44no—
531C42—48no—
532C42—44no—
533C42—44no—
534C42—42no—
535C42—42no—
536C42—42no—
537C42—44no—
538C42—82no—
539C42#13714yes—
540C42C22×D412yes—
541C42C22×D414yes—
542C42C2×#4112yes—
543C42C2×#4114yes—
544C42C2×#4112yes—
545C422−1+412yes—
546C23—22no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23C23×D411yescharacteristic
554Q8—28no—
555Q8—24no—
556Q8—24no—
557Q8—24no—
558Q8—24no—
559Q8—42no—
560Q8—42no—
561Q8—42no—
562Q8—42no—
563Q8—168no—
564Q8—164no—
565D4—22no—
566D4—22no—
567D4—41no—
568D4—41no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4C2×#12114yes—
615C2×C4C2×#12114yes—
616C2×C4C2×#12112yes—
617C2×C4C2×#12112yes—
618C2×C4#54812yes—
619C2×C4#54812yes—
620C2×C4C2×#13714yes—
621C2×C4C2×#13712yes—
622C2×C4C2×#13712yes—
623C2×C4C2×#13712yes—
624C2×C4C2×#13712yes—
625C2×C4#57712yes—
626C2×C4C23×D411yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×2−1+411yescharacteristic
630C2×C4C2×2−1+411yescharacteristic
631C2×C4#58511yescharacteristic
632C8—42no—
633C8—44no—
634C8—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22C22×#12111yescharacteristic
643C22C2×#54811yescharacteristic
644C22#348911yescharacteristic
645C22C22×#13711yescharacteristic
646C22C22×#13711yescharacteristic
647C22C2×#57711yescharacteristic
648C22#358511yescharacteristic
649C4—24no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—24no—
654C4—24no—
655C4—44no—
656C4—44no—
657C4—42no—
658C4—42no—
659C4—41no—
660C4—42no—
661C4—84no—
662C4—82no—
663C4—81no—
664C4—82no—
665C4#303914yes—
666C4#336914yescomplemented
667C4D4×#4111yescharacteristic
668C4#348211yescharacteristic
669C4#349911yescharacteristic
670C4#351811yescharacteristic
671C2—82no—
672C2#3644411yescharacteristic
673C2C2×#303911yescharacteristic
674C2#6043111yescharacteristic
675C2C2×#336911yescharacteristic
676C2#6131911yescharacteristic
677C2#6210811yescharacteristic
678C2#6213111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×C4⋊Q8C2semidirect
5. #33196C2semidirect
7. #28717C2semidirect
8. #28255C2semidirect
11. #25324C2semidirect
12. #24698C2semidirect
14. #22722C2semidirect
15. #22525C2semidirect
16. #21506C2semidirect
666. C4#3369semidirect

Maximal subgroups

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

subgroupindex
C2×#33692
Q8×C4⋊Q82
C4×#5562
#331972
#331962
#287262
#287172
#282552
#282542
#253362
#253242
#246982
#246802
#227222
#225252
#215062
C4×#4612