Finite Group Databasebeta

← #6389762

Subgroups — #6389762

Order 512.

1806 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#34225C211yesmaximal, characteristic
3#34216C211yesmaximal, characteristic
4#23169C214yesmaximal, complemented
5#23167C214yesmaximal, complemented
6#22623C214yesmaximal
7#22621C214yesmaximal
8#22248C211yesmaximal, characteristic, complemented
9#22247C211yesmaximal, characteristic, complemented
10#22246C211yesmaximal, characteristic, complemented
11#22246C211yesmaximal, characteristic, complemented
12#21835C211yesmaximal, characteristic
13#21833C211yesmaximal, characteristic
14#21727C211yesmaximal, characteristic, complemented
15#21726C211yesmaximal, characteristic, complemented
16C4×#500C211yesmaximal, characteristic, complemented
17C4×#498C211yesmaximal, characteristic, complemented
18#20824C211yesmaximal, characteristic
19#20472C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22#3165C4116yescomplemented
23C2×#501C2214yes—
24C2×#500C2211yescharacteristic
25C2×#500C2211yescharacteristic
26C2×#498C2211yescharacteristic
27C2×#498C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#428C2214yes—
30C2×C4×M16C2211yescharacteristic
31#2370C2212yes—
32#2370C2212yes—
33#2369C2212yes—
34#2369C2212yes—
35#2365—21no—
36#2365—21no—
37#2362—21no—
38#2362—21no—
39#2356—21no—
40#2351—21no—
41#2347C2212yes—
42#2346C2212yes—
43#2345C2212yes—
44#2345C2212yes—
45Q8×C22⋊C4—21no—
46Q8×C22⋊C4—21no—
47#2331—21no—
48#2331—21no—
49#2323—21no—
50#2323—21no—
51#2320—21no—
52#2316—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61#2285C2211yescharacteristic
62#2285C2211yescharacteristic
63#2276—21no—
64#2274—21no—
65C4×#117C2211yescharacteristic
66#1990C2211yescharacteristic
67#1990C2211yescharacteristic
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1989C2214yes—
75#1988C2214yes—
76#1944C2214yes—
77#1943C2214yes—
78#1942C2214yes—
79#1941C2214yes—
80#1921C2214yes—
81#1921C2214yes—
82#1920C2214yes—
83#1920C2214yes—
84#1919C2214yes—
85#1919C2214yes—
86#1918C2214yes—
87#1918C2214yes—
88#1845C2211yescharacteristic
89#1844C2211yescharacteristic
90#1834C2211yescharacteristic
91#1834C2211yescharacteristic
92#1825C2214yes—
93#1823C2214yes—
94#1809C2214yes—
95#1776C2214yes—
96#1776C2214yes—
97#1776C2214yes—
98C4×#107C2214yes—
99C4×#106C2214yes—
100#1771C2212yes—
101#1771C2212yes—
102#1753C2212yes—
103#1753C2212yes—
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C42×C8C2211yescharacteristic
108C23×Q8—21no—
109C22×M16C2312yes—
110#533—24no—
111#533C2×C418yes—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—24no—
115C2×C4⋊Q8C2×C418yes—
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2312yes—
121C2×#125C2312yes—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×#117—21no—
135C2×#117—21no—
136C2×#117D411yescharacteristic
137C2×#117D411yescharacteristic
138C2×#117C2312yes—
139C22×C4⋊C4—21no—
140C22×C4⋊C4C2312yes—
141C22×C42C2311yescharacteristic
142#501C2×C4132yes—
143#500C2×C418yes—
144#500C2×C418yes—
145#498C2×C418yes—
146#498C2×C418yes—
147#431—24no—
148#428C2×C4132yes—
149C2×#107C2314yes—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×#106C2314yes—
153C2×#106C2314yes—
154C2×#106C2314yes—
155C2×C4⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C4×M16C2×C418yes—
158C2×#97C2312yes—
159C2×#97C2312yes—
160C2×#97C2312yes—
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2312yes—
164C2×C4×C8C2312yes—
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167#391—21no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—22no—
176#391—22no—
177#391—22no—
178#391—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#388—22no—
200#388—22no—
201#388—22no—
202#388—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#383—22no—
216#382—21no—
217#382—21no—
218#382—22no—
219#380—21no—
220#380—21no—
221#380—22no—
222#380—22no—
223#380C2312yes—
224#380C2312yes—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4D411yescharacteristic
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2312yes—
252C4×C4⋊C4C2311yescharacteristic
253C4×C4⋊C4C2311yescharacteristic
254C4×C22⋊C4—22no—
255C4×C22⋊C4—22no—
256C4×C22⋊C4D411yescharacteristic
257C4×C22⋊C4D411yescharacteristic
258C4×C22⋊C4C2311yescharacteristic
259C4×C22⋊C4C2311yescharacteristic
260#376—22no—
261#376D411yescharacteristic
262#376D411yescharacteristic
263C43C2311yescharacteristic
264C43C2311yescharacteristic
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#340C2314yes—
272#340C2314yes—
273#340C2314yes—
274#340C2314yes—
275#336C2312yes—
276#336C2312yes—
277#336C2312yes—
278#336C2312yes—
279C22×Q8—28no—
280C22×Q8—21no—
281C22×Q8—21no—
282C22×Q8—21no—
283C22×Q8—21no—
284C22×Q8—22no—
285C22×Q8—82no—
286C22×Q8—82no—
287C23×C4—22no—
288C23×C4C2×D411yescharacteristic
289C23×C4C2×D411yescharacteristic
290C23×C4C2411yescharacteristic
291C2×M16—24no—
292C2×M16—24no—
293C2×M16C22×C418yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C4⋊Q8—28no—
299C4⋊Q8—28no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—24no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8C22×C414yes—
306C4⋊Q8C22×C414yes—
307C4⋊Q8C22×C418yes—
308C4⋊Q8C22×C414yes—
309C4⋊Q8C22×C414yes—
310#125—24no—
311#125C22×C418yes—
312C22⋊Q8—28no—
313C22⋊Q8—28no—
314C4×Q8—28no—
315C4×Q8—28no—
316C4×Q8—44no—
317C4×Q8—44no—
318C4×Q8—82no—
319C4×Q8—82no—
320C4×Q8—82no—
321C4×Q8—82no—
322C4×Q8—82no—
323C4×Q8—82no—
324#117—24no—
325#117—24no—
326#117—42no—
327#117—44no—
328#117—82no—
329#117C22×C418yes—
330C2×C4⋊C4—28no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4C22×C418yes—
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42C22×C414yes—
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42#4114yes—
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405#107—28no—
406#107C22×C4116yes—
407#107C22×C4116yes—
408#106—28no—
409#106C22×C4116yes—
410#106C22×C4116yes—
411C4⋊C8—24no—
412C22⋊C8—24no—
413#97—24no—
414#97—24no—
415#97C22×C418yes—
416C4×C8—24no—
417C4×C8—24no—
418C4×C8—22no—
419C4×C8—22no—
420C4×C8C22×C414yes—
421C4×C8C22×C414yes—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450C24C22×D411yescharacteristic
451C2×Q8—24no—
452C2×Q8—24no—
453C2×Q8—24no—
454C2×Q8—24no—
455C2×Q8—28no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—42no—
464C2×Q8—82no—
465C2×Q8—82no—
466C2×Q8—816no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—28no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4C4×D414yes—
485C22×C4C4×D414yes—
486C22×C4C23×C412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C2×#4112yes—
501C22×C4C2511yescharacteristic
502M16—216no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8C23×C414yes—
512C2×C8C23×C414yes—
513C2×C82−1+414yes—
514C2×C82−1+414yes—
515C4⋊C4—28no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—28no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4C23×C414yes—
552C4⋊C4C23×C414yes—
553C4⋊C4C23×C414yes—
554C4⋊C4C23×C414yes—
555C22⋊C4—28no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—41no—
559C22⋊C4—41no—
560C22⋊C4—44no—
561C22⋊C4—82no—
562C42—28no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—44no—
575C42—44no—
576C42—44no—
577C42—82no—
578C42—82no—
579C42—82no—
580C42—82no—
581C42—82no—
582C42—82no—
583C42—82no—
584C42C4×D414yes—
585C42C4×D414yes—
586C42#13714yes—
587C42C23×C412yes—
588C42C23×C412yes—
589C42C2×#4112yes—
590C42C2×#4112yes—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23C2×C4×D412yes—
598C23C23×D411yescharacteristic
599Q8—48no—
600Q8—48no—
601Q8—48no—
602Q8—48no—
603Q8—88no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—24no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—88no—
661C2×C4C2×C4×D412yes—
662C2×C4C2×C4×D412yes—
663C2×C4C2×C4×D412yes—
664C2×C4C2×C4×D412yes—
665C2×C4C2×C4×D412yes—
666C2×C4C2×#13712yes—
667C2×C4C2×#13712yes—
668C2×C4C2×#13712yes—
669C2×C4#57512yes—
670C2×C4#57612yes—
671C2×C4#57712yes—
672C2×C4C24×C411yescharacteristic
673C2×C4C22×#4111yescharacteristic
674C2×C4C2×2−1+412yes—
675C2×C4C2×2−1+412yes—
676C8—24no—
677C8—42no—
678C8—42no—
679C8#51918yes—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22C22×C4×D411yescharacteristic
688C22#349911yescharacteristic
689C22#349911yescharacteristic
690C22C22×#13711yescharacteristic
691C22C2×#57511yescharacteristic
692C22C2×#57611yescharacteristic
693C22C2×#57711yescharacteristic
694C4—22no—
695C4—22no—
696C4—22no—
697C4—22no—
698C4—22no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—44no—
706C4—44no—
707C4—44no—
708C4—44no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4C4×#13714yes—
714C4#294614yes—
715C4C2×#51912yes—
716C4#350912yes—
717C2—42no—
718C2#3420911yescharacteristic
719C2C2×C4×#13711yescharacteristic
720C2C2×#294611yescharacteristic
721C2#6247811yescharacteristic
722C2#6248011yescharacteristic
723C2#6248111yescharacteristic
724C2#6248311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
4. #23169C2semidirect
5. #23167C2semidirect
8. #22248C2semidirect
9. #22247C2semidirect
10. #22246C2semidirect
11. #22246C2semidirect
14. #21727C2semidirect
15. #21726C2semidirect
16. C4×#500C2semidirect
17. C4×#498C2semidirect
22. #3165C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#342252
#342162
#231692
#231672
#226232
#226212
#222482
#222472
#222462
#218352
#218332
#217272
#217262
C4×#5002
C4×#4982
#208242
#204722