Finite Group Databasebeta

← #189247

Subgroups — #189247

Order 512.

972 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60278C211yesmaximal, characteristic
2#21361C211yesmaximal, characteristic, complemented
3#20748C211yesmaximal, characteristic
4C2×#1806C211yesmaximal, characteristic
5#17817C211yesmaximal, characteristic, complemented
6#8669C212yesmaximal, complemented
7#8668C212yesmaximal, complemented
8#8667C212yesmaximal, complemented
9#8666C212yesmaximal, complemented
10#8098C211yesmaximal, characteristic, complemented
11#8093C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13#2991C412yescomplemented
14#2990—22no—
15#2990C412yescomplemented
16#2984C411yescharacteristic, complemented
17#2984C411yescharacteristic, complemented
18#2974—21no—
19#2974—21no—
20#2974C411yescharacteristic, complemented
21#2974C411yescharacteristic, complemented
22#2965—22no—
23#2965C2212yes—
24C8×#41C2212yes—
25#2920—21no—
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28C2×#431—21no—
29C2×#431C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C22×#104—21no—
32#2283C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1949—22no—
35#1853—21no—
36#1852—21no—
37#1826—21no—
38#1806—21no—
39#1806—21no—
40#1806C2211yescharacteristic
41#1806C2211yescharacteristic
42#1806C2211yescharacteristic
43#1806C2211yescharacteristic
44#1806C2212yes—
45#1806C2212yes—
46#1789—21no—
47C4×#107C2212yescomplemented
48C4×#106C2212yescomplemented
49#1759—21no—
50#1759—21no—
51C2×#339—21no—
52#1574—22no—
53#1573—21no—
54#1573—21no—
55#1567C2212yescomplemented
56#1566C2212yescomplemented
57#1303C2212yescomplemented
58#1303C2212yescomplemented
59#1303C2212yescomplemented
60#1303C2212yescomplemented
61C22×#41D411yescharacteristic
62C22×M16—21no—
63C4×#41—21no—
64C4×#41—21no—
65C4×#41—21no—
66C4×#41—21no—
67C4×#41C2×C411yescharacteristic
68C4×#41C2×C411yescharacteristic
69C4×#41D411yescharacteristic
70C4×#41D411yescharacteristic
71C4×#41Q811yescharacteristic
72C4×#41Q811yescharacteristic
73C4×#41C2311yescharacteristic
74C4×#41C2311yescharacteristic
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×Q8Q811yescharacteristic
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4C2×C411yescharacteristic
80C2×C4×D4Q811yescharacteristic
81C2×#117—21no—
82C2×#117D411yescharacteristic
83C22×C4⋊C4—21no—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#446—22no—
88#446—22no—
89#446C2×C412yescomplemented
90#446C2×C412yescomplemented
91C8×Q8C2×C412yescomplemented
92C8×Q8C2×C412yescomplemented
93#436—22no—
94#436—22no—
95#436C2×C412yescomplemented
96#436C2×C412yescomplemented
97#435—22no—
98#435—22no—
99#435—22no—
100#435—22no—
101#435—22no—
102#435—22no—
103C8×D4—22no—
104C8×D4—22no—
105C8×D4C2×C412yescomplemented
106C8×D4C2×C412yescomplemented
107#432—21no—
108#432—22no—
109#432—21no—
110#432—21no—
111#432—21no—
112#432—21no—
113#432C2×C411yescharacteristic
114#432C2×C411yescharacteristic
115#431—21no—
116#431—22no—
117#431—21no—
118#431—21no—
119#431—22no—
120#431—22no—
121#431—21no—
122#431—21no—
123#431C2×C411yescharacteristic
124#431C2×C411yescharacteristic
125#431C2×C412yes—
126#431D411yescharacteristic
127#431D411yescharacteristic
128#431D412yes—
129#431C2312yes—
130C2×#107—22no—
131C2×#106—22no—
132#423—22no—
133#423—21no—
134#423—21no—
135#423—21no—
136#423C2×C412yes—
137#423C2311yescharacteristic
138#423C2312yes—
139#423C2311yescharacteristic
140C2×C4⋊C8—22no—
141C2×C4⋊C8—21no—
142C2×C4⋊C8—21no—
143C2×C4⋊C8D412yes—
144C2×#104—22no—
145C2×#104—22no—
146C2×#104—21no—
147C2×#104—21no—
148C2×#104—21no—
149C2×#104—21no—
150C2×#104—44no—
151#407—21no—
152#407—21no—
153#407—21no—
154#407—21no—
155C2×C22⋊C8—21no—
156C2×C22⋊C8—21no—
157C2×C22⋊C8D411yescharacteristic
158C2×C22⋊C8D411yescharacteristic
159C4×M16—21no—
160C4×M16—22no—
161C4×M16—21no—
162C2×C4×C8—22no—
163C2×C4×C8C2312yes—
164#383—21no—
165#382—21no—
166#381—21no—
167#380—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2311yescharacteristic
173C4×C4⋊C4C2311yescharacteristic
174C4×C22⋊C4—21no—
175#376—21no—
176C2×#95—21no—
177#340—22no—
178#340—22no—
179#339—22no—
180#339—22no—
181#339—21no—
182#339—21no—
183#339—21no—
184#339—21no—
185#335—22no—
186#335—22no—
187#334—22no—
188#334—22no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—22no—
192C2×#41—22no—
193C2×#41—41no—
194C2×#41—41no—
195C2×#41C4⋊C411yescharacteristic
196C2×#41C4⋊C411yescharacteristic
197C2×#41C2×D411yescharacteristic
198C2×#41C2×D411yescharacteristic
199C22×Q8C4⋊C411yescharacteristic
200C22×D4—21no—
201C22×D4C4⋊C411yescharacteristic
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—41no—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4#4111yescharacteristic
209C2×M16—22no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—41no—
216C2×M16—42no—
217C2×M16—42no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—41no—
221C22×C8C2×D412yes—
222C22×C8C2×D412yes—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8C4⋊C411yescharacteristic
228C4×Q8C4⋊C411yescharacteristic
229C4×Q8C4⋊C411yescharacteristic
230C4×Q8C4⋊C411yescharacteristic
231C4×Q8C22×C411yescharacteristic
232C4×Q8C22×C411yescharacteristic
233C4×Q8C2×Q811yescharacteristic
234C4×Q8C2×Q811yescharacteristic
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4C4⋊C411yescharacteristic
256C4×D4C4⋊C411yescharacteristic
257C4×D4C4⋊C411yescharacteristic
258C4×D4C4⋊C411yescharacteristic
259C4×D4C22×C411yescharacteristic
260C4×D4C22×C411yescharacteristic
261C4×D4C2×Q811yescharacteristic
262C4×D4C2×Q811yescharacteristic
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117C4⋊C411yescharacteristic
274#117C4⋊C411yescharacteristic
275#117C2×D411yescharacteristic
276#117C2×D411yescharacteristic
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—41no—
284C2×C4⋊C4C4⋊C411yescharacteristic
285C2×C4⋊C4C4⋊C411yescharacteristic
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—41no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—22no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—41no—
307C2×C42C22×C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×Q811yescharacteristic
314C2×C42C2×Q811yescharacteristic
315C2×C42#4111yescharacteristic
316C2×C42#4111yescharacteristic
317C2×C42#4111yescharacteristic
318C2×C42C2411yescharacteristic
319#107—44no—
320#106—44no—
321C4⋊C8—22no—
322C4⋊C8—22no—
323C4⋊C8—22no—
324C4⋊C8—22no—
325C4⋊C8—22no—
326C4⋊C8—21no—
327C4⋊C8—22no—
328C4⋊C8—21no—
329C4⋊C8—22no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—41no—
333C4⋊C8—41no—
334C4⋊C8C22⋊C412yes—
335C4⋊C8C22⋊C412yes—
336C4⋊C8C22⋊C412yes—
337C4⋊C8C22⋊C412yes—
338C4⋊C8C22×C412yes—
339C4⋊C8C22×C412yes—
340#104—42no—
341#104—42no—
342#104—44no—
343#104—42no—
344#104—44no—
345#104—42no—
346C22⋊C8—21no—
347C22⋊C8—21no—
348C22⋊C8—22no—
349C22⋊C8—22no—
350C22⋊C8—22no—
351C22⋊C8—22no—
352C22⋊C8—22no—
353C22⋊C8—41no—
354C22⋊C8—41no—
355C22⋊C8C22⋊C411yescharacteristic
356C22⋊C8C22⋊C411yescharacteristic
357C22⋊C8C22⋊C411yescharacteristic
358C22⋊C8C22⋊C411yescharacteristic
359C22⋊C8C22⋊C412yes—
360C22⋊C8C22⋊C412yes—
361#97—22no—
362#97—22no—
363#97—21no—
364#97—21no—
365#97—41no—
366C4×C8—22no—
367C4×C8—22no—
368C4×C8—22no—
369C4×C8—41no—
370C4×C8C22×C412yes—
371C4×C8C22×C412yes—
372C4×C8C2×D412yes—
373C4×C8C2×D412yes—
374#95—22no—
375#95—21no—
376#95—21no—
377#95—41no—
378#95—41no—
379#95—41no—
380#95—41no—
381C24—21no—
382C24#12111yescharacteristic
383#41—21no—
384#41—21no—
385#41—22no—
386#41—22no—
387#41—21no—
388#41—21no—
389#41—21no—
390#41—21no—
391#41—42no—
392#41—42no—
393#41—41no—
394#41—41no—
395#41—42no—
396#41—42no—
397#41AGL(1,ℤ/8)12yes—
398#41#13712yes—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—41no—
402C2×Q8C2×C4⋊C411yescharacteristic
403C2×Q8C2×C4⋊C411yescharacteristic
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—41no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4C2×C4⋊C411yescharacteristic
414C2×D4C2×C4⋊C411yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C4⋊C411yescharacteristic
449C22×C4C2×C4⋊C411yescharacteristic
450C22×C4#11711yescharacteristic
451C22×C4#11711yescharacteristic
452C22×C4#12111yescharacteristic
453C22×C4#12111yescharacteristic
454C22×C4#12111yescharacteristic
455C22×C4#12111yescharacteristic
456C22×C4#12111yescharacteristic
457C22×C4C22⋊Q811yescharacteristic
458C22×C4C22⋊Q811yescharacteristic
459C22×C4C22⋊Q811yescharacteristic
460C22×C4C22⋊Q811yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2×#4111yescharacteristic
464M16—42no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—42no—
470M16—82no—
471M16—82no—
472M16—82no—
473M16—82no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—42no—
481C2×C8—82no—
482C2×C8—82no—
483C2×C8—82no—
484C2×C8C2×C22⋊C412yes—
485C2×C8C2×C22⋊C412yes—
486C2×C8C2×C22⋊C412yes—
487C2×C8C2×C22⋊C412yes—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—24no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4C2×C4⋊C411yescharacteristic
504C4⋊C4C2×C4⋊C411yescharacteristic
505C4⋊C4C2×C4⋊C411yescharacteristic
506C4⋊C4C2×C4⋊C411yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—82no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—42no—
533C42—41no—
534C42—41no—
535C42—41no—
536C42—41no—
537C42—42no—
538C42—82no—
539C42C2×C22⋊C411yescharacteristic
540C42C2×C22⋊C411yescharacteristic
541C42C2×C4⋊C411yescharacteristic
542C42C2×C4⋊C411yescharacteristic
543C42#11711yescharacteristic
544C42#11711yescharacteristic
545C42#12111yescharacteristic
546C42#12111yescharacteristic
547C42C22⋊Q811yescharacteristic
548C42C22⋊Q811yescharacteristic
549C42C22⋊Q811yescharacteristic
550C42C22⋊Q811yescharacteristic
551C42C23×C411yescharacteristic
552C42C22×D411yescharacteristic
553C42C22×Q811yescharacteristic
554C42C2×#4111yescharacteristic
555C23—22no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—41no—
561C23—42no—
562C23—42no—
563C23—81no—
564C23#38011yescharacteristic
565C23#38011yescharacteristic
566C23C2×#12111yescharacteristic
567Q8—21no—
568Q8—21no—
569Q8—21no—
570Q8—42no—
571Q8#42812yes—
572D4—22no—
573D4—22no—
574D4—21no—
575D4—21no—
576D4—21no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—41no—
581D4—41no—
582D4#42812yes—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—44no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—82no—
630C2×C4—88no—
631C2×C4—82no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4#38011yescharacteristic
639C2×C4#38011yescharacteristic
640C2×C4#38011yescharacteristic
641C2×C4#38011yescharacteristic
642C2×C4#38011yescharacteristic
643C2×C4#38011yescharacteristic
644C2×C4C22×C22⋊C411yescharacteristic
645C2×C4C22×C4⋊C411yescharacteristic
646C2×C4C2×#11711yescharacteristic
647C2×C4C2×#12111yescharacteristic
648C2×C4C2×C22⋊Q811yescharacteristic
649C2×C4C2×C22⋊Q811yescharacteristic
650C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
651C2×C4C2×#13711yescharacteristic
652C2×C4#57511yescharacteristic
653C2×C4#57511yescharacteristic
654C8—22no—
655C8—22no—
656C8—22no—
657C8—22no—
658C8—42no—
659C8—42no—
660C8—82no—
661C8—81no—
662C8—81no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—21no—
669C22—22no—
670C22—21no—
671C22—42no—
672C22—42no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—84no—
679C22#181412yes—
680C22C2×#38011yescharacteristic
681C22#305511yescharacteristic
682C22#305611yescharacteristic
683C4—21no—
684C4—21no—
685C4—21no—
686C4—21no—
687C4—21no—
688C4—22no—
689C4—21no—
690C4—21no—
691C4—21no—
692C4—21no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—41no—
697C4—41no—
698C4—84no—
699C4—161no—
700C4—161no—
701C4#290112yes—
702C4C2×#42811yescharacteristic
703C4#291111yescharacteristic
704C4#305511yescharacteristic
705C4#305611yescharacteristic
706C2—21no—
707C2—21no—
708C2—22no—
709C2—41no—
710C2—42no—
711C2—81no—
712C2#2065511yescharacteristic
713C2#2066111yescharacteristic
714C2C2×#181411yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#380
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21361C2semidirect
5. #17817C2semidirect
6. #8669C2semidirect
7. #8668C2semidirect
8. #8667C2semidirect
9. #8666C2semidirect
10. #8098C2semidirect
11. #8093C2semidirect
13. #2991C4semidirect
15. #2990C4semidirect
16. #2984C4semidirect
17. #2984C4semidirect
20. #2974C4semidirect
21. #2974C4semidirect
47. C4×#107C22semidirect
48. C4×#106C22semidirect
55. #1567C22semidirect
56. #1566C22semidirect
57. #1303C22semidirect
58. #1303C22semidirect
59. #1303C22semidirect
60. #1303C22semidirect
89. #446C2×C4semidirect
90. #446C2×C4semidirect
91. C8×Q8C2×C4semidirect
92. C8×Q8C2×C4semidirect
95. #436C2×C4semidirect
96. #436C2×C4semidirect
105. C8×D4C2×C4semidirect
106. C8×D4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602782
#213612
#207482
C2×#18062
#178172
#86692
#86682
#86672
#86662
#80982
#80932