Finite Group Databasebeta

← #6578963

Subgroups — #6578963

Order 512.

2073 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3117C212yesmaximal
2#36116C211yesmaximal, characteristic
3C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
4#33010C212yesmaximal, complemented
5#32906C211yesmaximal, characteristic, complemented
6#27910C211yesmaximal, characteristic, complemented
7#27645C211yesmaximal, characteristic
8#26395C214yesmaximal, complemented
9#26177C214yesmaximal, complemented
10#26047C212yesmaximal, complemented
11#24729C212yesmaximal, complemented
12#24606C212yesmaximal, complemented
13#24067C212yesmaximal
14#23004C212yesmaximal, complemented
15#22671C212yesmaximal, complemented
16#22603C211yesmaximal, characteristic, complemented
17C4×#448C211yesmaximal, characteristic
18C2×#529C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21#3117—216no—
22C2×#484C2212yes—
23C2×#484C2212yes—
24C2×#481C2212yes—
25C2×#479C2214yes—
26C2×#464C2214yes—
27C2×#462C2212yes—
28C2×#451C2212yes—
29C2×#448C2211yescharacteristic
30C2×#441C2214yes—
31C2×#440C2214yes—
32C2×C4×SD16C2212yes—
33C2×#432C2212yes—
34#2857C2211yescharacteristic
35#2855C2212yes—
36#2552C2211yescharacteristic
37#2547C2211yescharacteristic
38#2496—21no—
39#2486—21no—
40#2478—22no—
41#2453C2212yes—
42#2447C2212yes—
43#2419—22no—
44#2365C2211yescharacteristic
45Q8×C4⋊C4—21no—
46Q8×C4⋊C4—22no—
47Q8×C4⋊C4—21no—
48Q8×C4⋊C4C2214yes—
49Q8×C4⋊C4C2212yes—
50Q8×C4⋊C4C2212yes—
51D4×C4⋊C4—21no—
52D4×C4⋊C4C2211yescharacteristic
53#2344—21no—
54C4×#121C2211yescharacteristic
55C2×#389—22no—
56C2×#389C2212yes—
57C2×#384—21no—
58C2×#384C2211yescharacteristic
59C2×C4×C4⋊C4—21no—
60C2×C4×C4⋊C4C2211yescharacteristic
61#2095C2214yes—
62#2086C2214yes—
63#2084C2212yes—
64#2071C2212yes—
65#2066C2214yes—
66#2059C2214yes—
67#2042C2214yes—
68#1956C2212yes—
69#1948C2212yes—
70#1944C2214yes—
71#1939C2212yes—
72#1935C2214yes—
73#1917C2214yes—
74#1880C2212yes—
75#1878C2214yes—
76#1878C2214yes—
77#1878C2214yes—
78#1874C2211yescharacteristic
79#1872C2212yes—
80#1865C2214yes—
81#1865C2212yes—
82#1863C2214yes—
83#1863C2214yes—
84#1862C2214yes—
85#1853C2211yescharacteristic
86#1845C2212yes—
87#1824C2212yes—
88#1817C2212yes—
89#1803C2212yes—
90#1802C2212yes—
91#1797C2211yescharacteristic
92#1787C2211yescharacteristic
93#1770C2212yes—
94C4×#103C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×Q16C2314yes—
98C22×SD16C2312yes—
99C2×C4⋊Q8C2311yescharacteristic
100#529—28no—
101#529—44no—
102C2×#126—22no—
103C2×#125C2312yes—
104C2×#124C2312yes—
105C2×#123—22no—
106C2×C22⋊Q8C2312yes—
107C2×#121C2311yescharacteristic
108C2×C4×Q8—28no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—22no—
112C2×C4×Q8—22no—
113C2×C4×Q8—28no—
114C2×C4×Q8—44no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2314yes—
117C2×C4×Q8C2314yes—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C2×#117—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4D411yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C42—21no—
136C22×C42—22no—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139#484—28no—
140#484—44no—
141#481—28no—
142#479—216no—
143#464—216no—
144#462—28no—
145#451—28no—
146#448D418yes—
147#441—216no—
148#440—216no—
149C4×SD16—28no—
150#432—28no—
151C2×#107C2314yes—
152C2×#107C2314yes—
153C2×#106C2314yes—
154C2×#106C2312yes—
155C2×C4⋊C8C2314yes—
156C2×C4⋊C8C2312yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#102C2314yes—
165C2×#102C2312yes—
166C2×C22⋊C8C2312yes—
167C2×C22⋊C8C2311yescharacteristic
168C2×C22⋊C8C2311yescharacteristic
169C2×#97C2314yes—
170C2×C4×C8C2312yes—
171#400—22no—
172#398—22no—
173#398—22no—
174#397—22no—
175#397—22no—
176#397—22no—
177#391C2312yes—
178#391C2312yes—
179#391C2312yes—
180#390C2311yescharacteristic
181#389—21no—
182#389—22no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—22no—
187#389—22no—
188#389—24no—
189#389—24no—
190#389—24no—
191#389—24no—
192#389C2314yes—
193#389C2311yescharacteristic
194#389C2312yes—
195#389C2312yes—
196#389C2314yes—
197#388—21no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#386—21no—
203#385—22no—
204#385—22no—
205#385—22no—
206#384—21no—
207#384—22no—
208#384—22no—
209#384—24no—
210#384—21no—
211#384—21no—
212#384C2312yes—
213#384C2311yescharacteristic
214#384C2312yes—
215#381—22no—
216#381—22no—
217#380—21no—
218#380—21no—
219#380—21no—
220#380—21no—
221#380—21no—
222#380C2311yescharacteristic
223C4×C4⋊C4—21no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—24no—
227C4×C4⋊C4D411yescharacteristic
228C4×C4⋊C4D412yes—
229C4×C4⋊C4D411yescharacteristic
230C4×C4⋊C4C2311yescharacteristic
231C4×C4⋊C4C2311yescharacteristic
232C4×C4⋊C4C2312yes—
233C4×C4⋊C4C2311yescharacteristic
234C4×C22⋊C4—21no—
235C2×#95—22no—
236C2×#95D412yes—
237#340C2314yes—
238#340C2314yes—
239#340C2312yes—
240#340C2314yes—
241#340C2312yes—
242#336C2312yes—
243#336C2314yes—
244#336C2314yes—
245#336C2312yes—
246C22×Q8—24no—
247C22×Q8—24no—
248C22×Q8—21no—
249C22×Q8—22no—
250C22×Q8C2×D414yes—
251C22×Q8C2412yes—
252C22×Q8C2412yes—
253C22×D4—21no—
254C22×D4C2411yescharacteristic
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2411yescharacteristic
265C2×Q16—28no—
266C2×Q16—28no—
267C2×Q16—28no—
268C2×SD16—24no—
269C2×SD16—24no—
270C2×SD16C2×D418yes—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C4⋊Q8—24no—
275#126—48no—
276#125—24no—
277#125—42no—
278#124—24no—
279#124—42no—
280#123—48no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283#121—22no—
284#121C2×D414yes—
285C4×Q8—28no—
286C4×Q8—28no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—28no—
291C4×Q8—28no—
292C4×Q8—24no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×Q8—42no—
296C4×Q8—48no—
297C4×Q8—48no—
298C4×Q8#4118yes—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—42no—
302C4×D4—48no—
303C4×D4—44no—
304#117—48no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—42no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4—48no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4—82no—
338C2×C4⋊C4—84no—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2414yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2412yes—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C42—28no—
361C2×C42—24no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42—24no—
369C2×C42—24no—
370C2×C42—48no—
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2412yes—
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—28no—
380#107—44no—
381#106—28no—
382#106—24no—
383C4⋊C8—28no—
384C4⋊C8—24no—
385#103—28no—
386#103—24no—
387#103—24no—
388#103—28no—
389#103—28no—
390#103—44no—
391#103C2×D418yes—
392#102—28no—
393#102—24no—
394C22⋊C8—24no—
395C22⋊C8—22no—
396C22⋊C8C2×D414yes—
397#97—28no—
398C4×C8—24no—
399#95—24no—
400#95—24no—
401#95—21no—
402#95—21no—
403#95—22no—
404#95—21no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95C2×D412yes—
413#95C2×D412yes—
414#95C2×D412yes—
415#95C2×D412yes—
416C24—21no—
417C24C22×D411yescharacteristic
418C2×Q8—24no—
419C2×Q8—216no—
420C2×Q8—24no—
421C2×Q8—216no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8C22×D414yes—
431C2×Q8C2×#4114yes—
432C2×Q82−1+414yes—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—21no—
436C2×D4—22no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4C22×D412yes—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—41no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—82no—
476C22×C4#13714yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4H(5,𝔽2)12yes—
488C22×C4C2511yescharacteristic
489Q16—416no—
490SD16—216no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8C22×D414yes—
500C4⋊C4—28no—
501C4⋊C4—24no—
502C4⋊C4—28no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—42no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—81no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—84no—
536C4⋊C4—84no—
537C4⋊C4C22×D412yes—
538C4⋊C4C2×#4114yes—
539C22⋊C4—24no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—81no—
548C42—28no—
549C42—28no—
550C42—22no—
551C42—22no—
552C42—24no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—48no—
560C42—48no—
561C42—44no—
562C42—42no—
563C42C2×#4114yes—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23D4212yes—
576C23C2×#13712yes—
577C23#57812yes—
578C23C23×D411yescharacteristic
579Q8—216no—
580Q8—24no—
581Q8—24no—
582Q8—24no—
583Q8—44no—
584Q8—44no—
585Q8#54718yes—
586D4—24no—
587D4—48no—
588D4—42no—
589C2×C4—216no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—24no—
593C2×C4—21no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—44no—
620C2×C4—48no—
621C2×C4—42no—
622C2×C4—48no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—84no—
642C2×C4—88no—
643C2×C4—88no—
644C2×C4—84no—
645C2×C4#52512yes—
646C2×C4#52512yes—
647C2×C4D4212yes—
648C2×C4C2×#13712yes—
649C2×C4C2×#13712yes—
650C2×C4#57512yes—
651C2×C4C23×D411yescharacteristic
652C2×C4C22×#4112yes—
653C2×C4C2×H(5,𝔽2)11yescharacteristic
654C2×C4C2×2−1+412yes—
655C8—24no—
656C8—44no—
657C8—42no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22#327614yes—
672C22C2×#52511yescharacteristic
673C22C2×D4211yescharacteristic
674C22#348611yescharacteristic
675C22C22×#13711yescharacteristic
676C22C2×#57511yescharacteristic
677C22C2×#57811yescharacteristic
678C22#358511yescharacteristic
679C4—28no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—41no—
689C4—44no—
690C4—82no—
691C4—81no—
692C4—81no—
693C4—84no—
694C4—82no—
695C4#311714yescomplemented
696C4C2×#54712yes—
697C4#349212yes—
698C2—24no—
699C2—82no—
700C2#3610311yescharacteristic
701C2C2×#311711yescharacteristic
702C2#6058911yescharacteristic
703C2C2×#327611yescharacteristic
704C2#6107011yescharacteristic
705C2#6183011yescharacteristic
706C2#6184211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
3. C2×Q8×C4⋊C4C2semidirect
4. #33010C2semidirect
5. #32906C2semidirect
6. #27910C2semidirect
8. #26395C2semidirect
9. #26177C2semidirect
10. #26047C2semidirect
11. #24729C2semidirect
12. #24606C2semidirect
14. #23004C2semidirect
15. #22671C2semidirect
16. #22603C2semidirect
695. C4#3117semidirect

Maximal subgroups

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

subgroupindex
C2×#31172
#361162
C2×Q8×C4⋊C42
#330102
#329062
#279102
#276452
#263952
#261772
#260472
#247292
#246062
#240672
#230042
#226712
#226032
C4×#4482