Finite Group Databasebeta

← #6578940

Subgroups — #6578940

Order 512.

2331 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3103C212yesmaximal, complemented
2#36126C211yesmaximal, characteristic
3C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
4#33005C212yesmaximal, complemented
5#32903C211yesmaximal, characteristic, complemented
6#27906C211yesmaximal, characteristic, complemented
7#27645C211yesmaximal, characteristic, complemented
8#26373C214yesmaximal, complemented
9#26163C214yesmaximal, complemented
10#26042C212yesmaximal, complemented
11#24728C212yesmaximal, complemented
12#24603C212yesmaximal, complemented
13#24061C212yesmaximal, complemented
14#22997C212yesmaximal, complemented
15C4⋊C4×SD16C212yesmaximal, complemented
16#22599C211yesmaximal, characteristic, complemented
17C4×#448C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#532C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22#3103—216no—
23C2×#481C2212yes—
24C2×#481C2212yes—
25C2×#481C2212yes—
26C2×#475C2214yes—
27C2×#460C2214yes—
28C2×#460C2212yes—
29C2×#448C2212yes—
30C2×#448C2211yescharacteristic
31C2×C4×SD16C2214yes—
32C2×C4×SD16C2214yes—
33C2×C4×SD16C2212yes—
34C2×#431C2212yes—
35#2856C2211yescharacteristic
36#2854C2212yes—
37#2551C2211yescharacteristic
38#2541C2211yescharacteristic
39#2491—21no—
40#2461—22no—
41#2452C2212yes—
42#2445C2212yes—
43#2430—22no—
44#2416—21no—
45#2366C2211yescharacteristic
46#2361—21no—
47Q8×C4⋊C4—21no—
48Q8×C4⋊C4—22no—
49Q8×C4⋊C4—21no—
50Q8×C4⋊C4C2214yes—
51Q8×C4⋊C4C2212yes—
52Q8×C4⋊C4C2212yes—
53D4×C4⋊C4C2211yescharacteristic
54C4×#121C2211yescharacteristic
55C2×#389—22no—
56C2×#389C2212yes—
57C2×#384—21no—
58C2×#384C2211yescharacteristic
59C2×C4×C4⋊C4—21no—
60C2×C4×C4⋊C4C2211yescharacteristic
61#2094C2214yes—
62#2088C2214yes—
63#2084C2212yes—
64#2071C2212yes—
65#2065C2214yes—
66#2058C2214yes—
67#2041C2214yes—
68#1957C2212yes—
69#1948C2212yes—
70#1941C2214yes—
71#1939C2212yes—
72#1934C2214yes—
73#1916C2214yes—
74#1880C2214yes—
75#1880C2214yes—
76#1880C2212yes—
77#1880C2214yes—
78#1874C2212yes—
79#1874C2211yescharacteristic
80#1865C2212yes—
81#1865C2214yes—
82#1865C2214yes—
83#1864C2214yes—
84#1864C2214yes—
85#1853C2211yescharacteristic
86#1844C2212yes—
87#1824C2212yes—
88#1817C2212yes—
89#1803C2212yes—
90#1802C2212yes—
91#1797C2211yescharacteristic
92#1787C2211yescharacteristic
93C8×C4⋊C4C2212yes—
94C4×#103C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C22×#41—22no—
97C23×Q8C2311yescharacteristic
98C22×SD16C2314yes—
99C22×SD16C2312yes—
100#547—48no—
101#532—28no—
102#532—44no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#127C2312yes—
107C2×#124—22no—
108C2×#123—22no—
109C2×#123—22no—
110C2×C22⋊Q8—22no—
111C2×#121—21no—
112C2×#121—22no—
113C2×#121—21no—
114C2×#121—21no—
115C2×#121—21no—
116C2×#121C2312yes—
117C2×#121C2311yescharacteristic
118C2×C4×Q8—28no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—22no—
122C2×C4×Q8—22no—
123C2×C4×Q8—28no—
124C2×C4×Q8—44no—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2314yes—
127C2×C4×Q8C2314yes—
128C2×C4×Q8C2312yes—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—22no—
132C2×C4×D4C2311yescharacteristic
133C2×C4×D4C2312yes—
134C2×C4×D4C2312yes—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—22no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4D411yescharacteristic
142C22×C4⋊C4D411yescharacteristic
143C22×C4⋊C4C2312yes—
144C22×C4⋊C4C2311yescharacteristic
145C22×C42—21no—
146C22×C42—22no—
147C22×C42C2311yescharacteristic
148C22×C42C2312yes—
149#481—28no—
150#481—28no—
151#481—44no—
152#475—216no—
153#460—216no—
154#460—28no—
155#448—28no—
156#448D418yescomplemented
157C4×SD16—216no—
158C4×SD16—28no—
159C4×SD16—216no—
160#431—28no—
161C2×#106C2314yes—
162C2×#106C2314yes—
163C2×#106C2312yes—
164C2×#106C2314yes—
165C2×C4⋊C8C2314yes—
166C2×C4⋊C8C2312yes—
167C2×#103C2314yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2312yes—
171C2×#102C2314yes—
172C2×#102C2314yes—
173C2×#102C2314yes—
174C2×#102C2312yes—
175C2×#102C2314yes—
176C2×C22⋊C8C2312yes—
177C2×C22⋊C8C2311yescharacteristic
178C2×C22⋊C8C2311yescharacteristic
179C2×C4×C8C2312yes—
180C2×C4×C8C2314yes—
181#396—22no—
182#395—22no—
183#394—22no—
184#394—22no—
185#393—22no—
186#392—22no—
187#390—21no—
188#390—21no—
189#390—21no—
190#390C2312yes—
191#390C2311yescharacteristic
192#390C2312yes—
193#390C2312yes—
194#389—21no—
195#389—22no—
196#389—22no—
197#389—24no—
198#389—24no—
199#389—24no—
200#389—24no—
201#389C2314yes—
202#389C2311yescharacteristic
203#389C2312yes—
204#389C2312yes—
205#389C2314yes—
206#388—22no—
207#388—22no—
208#388—21no—
209#386—22no—
210#386—21no—
211#386—22no—
212#386—22no—
213#385—22no—
214#385—22no—
215#384—22no—
216#384—22no—
217#384—24no—
218#384—21no—
219#384—21no—
220#384C2312yes—
221#384C2311yescharacteristic
222#384C2312yes—
223#380—21no—
224#380C2311yescharacteristic
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—24no—
228C4×C4⋊C4D411yescharacteristic
229C4×C4⋊C4D412yes—
230C4×C4⋊C4D411yescharacteristic
231C4×C4⋊C4C2311yescharacteristic
232C4×C4⋊C4C2311yescharacteristic
233C4×C4⋊C4C2312yes—
234C4×C4⋊C4C2311yescharacteristic
235C4×C22⋊C4—21no—
236C2×#95—22no—
237C2×#95D412yes—
238#340C2314yes—
239#340C2312yes—
240#340C2314yes—
241#340C2314yes—
242#340C2312yes—
243#336C2312yes—
244#336C2314yes—
245#336C2314yes—
246#336C2312yes—
247C2×#41—48no—
248C2×#41—48no—
249C2×#41—84no—
250C22×Q8—24no—
251C22×Q8—24no—
252C22×Q8—21no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8C2×D414yes—
256C22×Q8C2412yes—
257C22×D4—21no—
258C22×D4—22no—
259C22×D4—21no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—21no—
263C22×D4C2412yes—
264C22×D4C2411yescharacteristic
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—21no—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D412yes—
274C23×C4C2×D412yes—
275C23×C4C2411yescharacteristic
276C2×SD16—24no—
277C2×SD16—24no—
278C2×SD16—28no—
279C2×SD16—28no—
280C2×SD16—28no—
281C2×SD16C2×D418yes—
282C22×C8C2414yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C4⋊Q8—24no—
286C4⋊Q8—42no—
287C4⋊Q8—44no—
288#127—24no—
289#127—24no—
290#127—42no—
291#124—48no—
292#123—48no—
293#123—48no—
294C22⋊Q8—48no—
295#121—22no—
296#121—24no—
297#121—24no—
298#121—44no—
299#121—48no—
300#121—44no—
301#121—44no—
302#121—82no—
303#121C2×D414yes—
304C4×Q8—28no—
305C4×Q8—28no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—28no—
310C4×Q8—28no—
311C4×Q8—44no—
312C4×Q8—44no—
313C4×Q8—48no—
314C4×Q8—48no—
315C4×Q8#4118yescomplemented
316C4×D4—24no—
317C4×D4—24no—
318C4×D4—24no—
319C4×D4—48no—
320C4×D4—42no—
321C4×D4—42no—
322C4×D4—44no—
323C4×D4—44no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—42no—
347C2×C4⋊C4—48no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4—82no—
351C2×C4⋊C4—84no—
352C2×C4⋊C4SD1618yescomplemented
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D412yes—
359C2×C4⋊C4C2414yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2411yescharacteristic
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C42—28no—
380C2×C42—24no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—24no—
384C2×C42—24no—
385C2×C42—21no—
386C2×C42—24no—
387C2×C42—24no—
388C2×C42—48no—
389C2×C42C2×D412yes—
390C2×C42C2×D412yes—
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2412yes—
394C2×C42C2412yes—
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397#106—28no—
398#106—28no—
399#106—24no—
400#106—44no—
401C4⋊C8—28no—
402C4⋊C8—24no—
403#103—28no—
404#103—24no—
405#103—24no—
406#103C2×D418yes—
407#102—28no—
408#102—28no—
409#102—24no—
410#102—28no—
411#102—44no—
412C22⋊C8—24no—
413C22⋊C8—22no—
414C22⋊C8C2×D414yes—
415C4×C8—28no—
416C4×C8—24no—
417#95—24no—
418#95—24no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—22no—
423#95—22no—
424#95C2×D412yes—
425#95C2×D412yes—
426#95C2×D412yes—
427#95C2×D412yes—
428C24—21no—
429C24—22no—
430C24C22×D411yescharacteristic
431#41—832no—
432C2×Q8—24no—
433C2×Q8—216no—
434C2×Q8—24no—
435C2×Q8—216no—
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—82no—
443C2×Q8C22×D414yes—
444C2×Q8C2×#4114yes—
445C2×Q82−1+414yes—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—24no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—44no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—44no—
466C2×D4—42no—
467C2×D4—82no—
468C2×D4—82no—
469C2×D4—81no—
470C2×D4—81no—
471C2×D4C22×D412yes—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—41no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—42no—
507C22×C4—82no—
508C22×C4—82no—
509C22×C4C2×SD1614yes—
510C22×C4C2×SD1618yes—
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4H(5,𝔽2)12yes—
522C22×C4C2511yescharacteristic
523SD16—216no—
524SD16—416no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—24no—
528C2×C8—22no—
529C2×C8—24no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—24no—
533C2×C8C22×D414yes—
534C4⋊C4—28no—
535C4⋊C4—24no—
536C4⋊C4—28no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—21no—
541C4⋊C4—24no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—24no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—42no—
548C4⋊C4—48no—
549C4⋊C4—48no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4—44no—
555C4⋊C4—42no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—82no—
559C4⋊C4—84no—
560C4⋊C4—82no—
561C4⋊C4—84no—
562C4⋊C4—81no—
563C4⋊C4C2×SD1614yes—
564C4⋊C4C2×SD1614yes—
565C4⋊C4C2×SD1618yes—
566C4⋊C4C22×D412yes—
567C4⋊C4C2×#4114yes—
568C22⋊C4—24no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—42no—
573C22⋊C4—42no—
574C22⋊C4—44no—
575C22⋊C4—42no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—44no—
579C22⋊C4—81no—
580C22⋊C4—82no—
581C22⋊C4—82no—
582C42—28no—
583C42—28no—
584C42—22no—
585C42—22no—
586C42—24no—
587C42—24no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—48no—
594C42—48no—
595C42—42no—
596C42C2×#4114yes—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—22no—
602C23—41no—
603C23—41no—
604C23—41no—
605C23—41no—
606C23—41no—
607C23—41no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23D4212yes—
616C23C22×SD1612yes—
617C23#57712yes—
618C23C23×D411yescharacteristic
619Q8—216no—
620Q8—24no—
621Q8—24no—
622Q8—24no—
623Q8—88no—
624Q8#54718yescomplemented
625D4—24no—
626D4—48no—
627D4—44no—
628D4—42no—
629D4—44no—
630D4—88no—
631D4—88no—
632D4—88no—
633D4—84no—
634D4—162no—
635C2×C4—216no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—24no—
639C2×C4—21no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—21no—
645C2×C4—216no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—28no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—22no—
654C2×C4—21no—
655C2×C4—22no—
656C2×C4—21no—
657C2×C4—24no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—21no—
664C2×C4—44no—
665C2×C4—48no—
666C2×C4—42no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—44no—
672C2×C4—41no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—42no—
679C2×C4—41no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—88no—
683C2×C4—88no—
684C2×C4—84no—
685C2×C4—88no—
686C2×C4—84no—
687C2×C4#52512yes—
688C2×C4#52512yes—
689C2×C4D4212yes—
690C2×C4C22×SD1612yes—
691C2×C4C22×SD1614yes—
692C2×C4C22×SD1614yes—
693C2×C4C22×SD1612yes—
694C2×C4#57512yes—
695C2×C4C23×D411yescharacteristic
696C2×C4C22×#4112yes—
697C2×C4C2×H(5,𝔽2)11yescharacteristic
698C2×C4C2×2−1+412yes—
699C8—24no—
700C8—44no—
701C8—42no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—81no—
712C22—81no—
713C22—81no—
714C22—81no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22D4×SD1614yes—
723C22C2×#52511yescharacteristic
724C22C2×D4211yescharacteristic
725C22#348611yescharacteristic
726C22C23×SD1611yescharacteristic
727C22C2×#57511yescharacteristic
728C22C2×#57711yescharacteristic
729C22#358511yescharacteristic
730C4—28no—
731C4—22no—
732C4—24no—
733C4—24no—
734C4—22no—
735C4—22no—
736C4—42no—
737C4—42no—
738C4—42no—
739C4—41no—
740C4—44no—
741C4—82no—
742C4—81no—
743C4—81no—
744C4—84no—
745C4#310314yescomplemented
746C4C2×#54712yes—
747C4#349212yes—
748C2—24no—
749C2—82no—
750C2—162no—
751C2#3610311yescharacteristic
752C2C2×#310311yescharacteristic
753C2#6058811yescharacteristic
754C2C2×D4×SD1611yescharacteristic
755C2#6107011yescharacteristic
756C2#6182011yescharacteristic
757C2#6183611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36126C2
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
1. C2×#3103C2semidirect
3. C2×Q8×C4⋊C4C2semidirect
4. #33005C2semidirect
5. #32903C2semidirect
6. #27906C2semidirect
7. #27645C2semidirect
8. #26373C2semidirect
9. #26163C2semidirect
10. #26042C2semidirect
11. #24728C2semidirect
12. #24603C2semidirect
13. #24061C2semidirect
14. #22997C2semidirect
15. C4⋊C4×SD16C2semidirect
16. #22599C2semidirect
17. C4×#448C2semidirect
156. #448D4semidirect
315. C4×Q8#41semidirect
352. C2×C4⋊C4SD16semidirect
624. Q8#547semidirect
745. C4#3103semidirect

Maximal subgroups

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

subgroupindex
C2×#31032
#361262
C2×Q8×C4⋊C42
#330052
#329032
#279062
#276452
#263732
#261632
#260422
#247282
#246032
#240612
#229972
C4⋊C4×SD162
#225992
C4×#4482