Finite Group Databasebeta

← #6375752

Subgroups — #6375752

Order 512.

1924 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#34060C211yesmaximal, characteristic
4#22129C214yesmaximal, complemented
5#22127C214yesmaximal, complemented
6#22119C212yesmaximal, complemented
7#22117C212yesmaximal, complemented
8#22113C212yesmaximal, complemented
9#22111C212yesmaximal, complemented
10C2×#1987C211yesmaximal, characteristic, complemented
11#21170C211yesmaximal, characteristic, complemented
12#21160C212yesmaximal, complemented
13#21092C212yesmaximal
14#21062C212yesmaximal
15#21035C212yesmaximal, complemented
16#20645C211yesmaximal, characteristic
17#20431C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20#3079C4116yescomplemented
21C2×#477C2212yes—
22C2×#475C2212yes—
23C2×#476C2212yes—
24C2×#474C2212yes—
25C2×#428C2212yes—
26C2×#428C2212yes—
27C2×#423C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C2×#417C2212yes—
30C2×#416C2212yes—
31C2×#405C2212yes—
32#2369—21no—
33#2369—21no—
34#2365—21no—
35#2365—21no—
36#2345—22no—
37Q8×C22⋊C4—22no—
38#2331C2214yes—
39#2320C2212yes—
40#2319C2212yes—
41#2318—22no—
42#2317C2211yescharacteristic
43#2317C2211yescharacteristic
44#2316C2212yes—
45#2314—21no—
46#2314—21no—
47#2311C2211yescharacteristic
48#2311C2211yescharacteristic
49C4×C4⋊Q8C2212yes—
50C4×C4⋊Q8C2212yes—
51#2295C2212yes—
52#2293C2212yes—
53#2290—22no—
54#2285—21no—
55#2285C2211yescharacteristic
56#2283—21no—
57#2274C2212yes—
58C4×#117C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60#1987C2212yes—
61#1987C2212yes—
62#1987C2214yes—
63#1949C2214yes—
64#1948C2212yes—
65#1947C2212yes—
66#1944C2214yes—
67#1943C2214yes—
68#1942C2214yes—
69#1941C2214yes—
70#1936C2214yes—
71#1935C2214yes—
72#1934C2214yes—
73#1933C2214yes—
74#1928C2214yes—
75#1927C2214yes—
76#1926C2214yes—
77#1925C2214yes—
78#1921C2214yes—
79#1920C2214yes—
80#1919C2214yes—
81#1918C2214yes—
82#1805C2212yes—
83#1804C2212yes—
84#1803C2211yescharacteristic
85#1803C2211yescharacteristic
86#1802C2211yescharacteristic
87#1802C2211yescharacteristic
88#1768C2214yes—
89#1767C2212yes—
90C4×C4⋊C8C2212yes—
91#1736C2212yes—
92#1735C2212yes—
93C2×#340C2212yes—
94C2×#340C2212yes—
95#1729C2212yes—
96C22×#41C2311yescharacteristic
97C23×Q8—21no—
98C22×M16C2312yes—
99C23×C8C2312yes—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8C2×C418yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C4×#41—24no—
107C4×#41C2×C418yes—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×#117—21no—
117C2×#117—22no—
118C2×#117C2312yes—
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C42—21no—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129#477C2×C4116yescomplemented
130#476C2×C4116yescomplemented
131#475C2×C4116yescomplemented
132#474C2×C4116yescomplemented
133#428—44no—
134#428C2×C4116yes—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139#423C2×C418yes—
140C2×C4⋊C8—24no—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146#417C2×C4116yes—
147#416—44no—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154#405—44no—
155C2×#97C2314yes—
156C2×C4×C8C2314yes—
157#391—22no—
158#391—22no—
159#391—24no—
160#391—24no—
161#391C2312yes—
162#391C2312yes—
163#390C2312yes—
164#390C2312yes—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389C2314yes—
170#389C2314yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—24no—
184#387—24no—
185#386—22no—
186#386—22no—
187#385—22no—
188#385—22no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#383—22no—
194#382—21no—
195#382—21no—
196#382—22no—
197#382C2312yes—
198#381—22no—
199#380—21no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—24no—
206#380—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2314yes—
221C4×C4⋊C4C2314yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C22⋊C4—24no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4D412yes—
228#376—22no—
229#376C2312yes—
230C2×#95—21no—
231C43C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#340C2314yes—
242#336C2314yes—
243#336C2314yes—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41C22×C414yes—
250C22×Q8—28no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—82no—
254C22×Q8—82no—
255C22×Q8C2411yescharacteristic
256C22×D4—21no—
257C22×D4C2411yescharacteristic
258C23×C4—22no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2411yescharacteristic
267C2×M16—24no—
268C2×M16—24no—
269C2×M16C22×C418yes—
270C22×C8—24no—
271C22×C8—24no—
272C22×C8—24no—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2414yes—
276C4⋊Q8—28no—
277C4⋊Q8—28no—
278C4⋊Q8—24no—
279C4⋊Q8—24no—
280C4⋊Q8C22×C418yes—
281C4⋊Q8C22×C418yes—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—84no—
285C4×Q8—84no—
286C4×Q8C22×C414yes—
287C4×Q8C22×C414yes—
288C4×D4—28no—
289C4×D4—28no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4C22×C414yes—
293C4×D4C22×C414yes—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—28no—
300#117—24no—
301#117—24no—
302#117—84no—
303#117C22×C414yes—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4C22×C418yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—28no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—82no—
359C2×C42C22×C414yes—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42#4114yes—
367C2×C42C2412yes—
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107—44no—
372#107C22×C4116yes—
373#106—44no—
374#106C22×C4116yes—
375C4⋊C8—22no—
376C4⋊C8—22no—
377C4⋊C8—24no—
378C4⋊C8C22×C418yes—
379C4⋊C8C22×C418yes—
380#103—42no—
381#103—42no—
382#103C22×C4116yes—
383#102—42no—
384#102—42no—
385#102C22×C4116yes—
386#97—44no—
387C4×C8—44no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408C24—21no—
409C24C22×D411yescharacteristic
410#41—28no—
411#41—28no—
412#41—44no—
413#41—44no—
414C2×Q8—28no—
415C2×Q8—28no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—82no—
424C2×Q8—82no—
425C2×Q8—816no—
426C2×Q8C23×C412yes—
427C2×D4—24no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4C23×C412yes—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—28no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4C4×D414yes—
467C22×C4C4×D414yes—
468C22×C4C23×C412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C42−1+412yes—
481C22×C42−1+412yes—
482C22×C42−1+412yes—
483C22×C4C2511yescharacteristic
484M16—216no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—48no—
494C2×C8C23×C418yes—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C4⋊C4C23×C418yes—
528C4⋊C4C23×C412yes—
529C4⋊C4C23×C412yes—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—44no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C42—24no—
541C42—24no—
542C42—24no—
543C42—24no—
544C42—24no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—42no—
550C42—42no—
551C42—44no—
552C42—44no—
553C42—82no—
554C42—84no—
555C42—84no—
556C42C4×D414yes—
557C42C4×D414yes—
558C42C23×C412yes—
559C42C23×C412yes—
560C42C2×#4112yes—
561C42C2×#4112yes—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—22no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23C2×C4×D412yes—
574C23#53612yes—
575C23#53712yes—
576C23C23×D411yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—416no—
580Q8—41no—
581Q8—41no—
582Q8—88no—
583D4—22no—
584D4—22no—
585D4—48no—
586D4—41no—
587D4—41no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—48no—
621C2×C4—48no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—84no—
635C2×C4—88no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4C2×C4×D412yes—
639C2×C4C2×C4×D412yes—
640C2×C4C2×C4×D412yes—
641C2×C4C2×C4×D412yes—
642C2×C4C2×C4×D412yes—
643C2×C4#51914yes—
644C2×C4#52014yes—
645C2×C4#53612yes—
646C2×C4#53712yes—
647C2×C4#54114yes—
648C2×C4#54214yes—
649C2×C4#57712yes—
650C2×C4#57812yes—
651C2×C4C24×C411yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×H(5,𝔽2)11yescharacteristic
654C2×C4C2×2−1+411yescharacteristic
655C2×C4C2×2−1+411yescharacteristic
656C2×C4C2×2−1+411yescharacteristic
657C8—28no—
658C8—42no—
659C8—42no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22#232314yes—
674C22C22×C4×D411yescharacteristic
675C22C2×#53611yescharacteristic
676C22C2×#53711yescharacteristic
677C22C2×#57711yescharacteristic
678C22C2×#57811yescharacteristic
679C22#358511yescharacteristic
680C22#358511yescharacteristic
681C4—22no—
682C4—22no—
683C4—22no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—48no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4#232314yes—
697C4#294714yes—
698C4C2×#51911yescharacteristic
699C4C2×#52011yescharacteristic
700C4C2×#54111yescharacteristic
701C4C2×#54211yescharacteristic
702C2—24no—
703C2—82no—
704C2C2×#232311yescharacteristic
705C2C2×#294711yescharacteristic
706C2#6016811yescharacteristic
707C2#6089211yescharacteristic
708C2#6089311yescharacteristic
709C2#6092911yescharacteristic
710C2#6093011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34216C2semidirect
4. #22129C2semidirect
5. #22127C2semidirect
6. #22119C2semidirect
7. #22117C2semidirect
8. #22113C2semidirect
9. #22111C2semidirect
10. C2×#1987C2semidirect
11. #21170C2semidirect
12. #21160C2semidirect
15. #21035C2semidirect
17. #20431C2semidirect
20. #3079C4semidirect
129. #477C2×C4semidirect
130. #476C2×C4semidirect
131. #475C2×C4semidirect
132. #474C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
#342162
#340602
#221292
#221272
#221192
#221172
#221132
#221112
C2×#19872
#211702
#211602
#210922
#210622
#210352
#206452
#204312