Finite Group Databasebeta

← #189239

Subgroups — #189239

Order 512.

976 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1M16×#41C211yesmaximal, characteristic
2#20819C211yesmaximal, characteristic, complemented
3#20748C211yesmaximal, characteristic
4#20654C211yesmaximal, characteristic
5#17796C211yesmaximal, characteristic, complemented
6#8669C212yesmaximal, complemented
7#8668C212yesmaximal, complemented
8#8667C212yesmaximal, complemented
9#8666C212yesmaximal, complemented
10#8094C211yesmaximal, characteristic, complemented
11#8093C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13#2971C411yescharacteristic, complemented
14#2971C411yescharacteristic, complemented
15#2970—21no—
16#2970—21no—
17#2970C411yescharacteristic, complemented
18#2970C411yescharacteristic, complemented
19#2965—22no—
20#2965C2212yes—
21C8×#41C2212yes—
22Q8×M16C412yescomplemented
23D4×M16—22no—
24D4×M16C412yescomplemented
25#2920—21no—
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28C2×#423C2211yescharacteristic
29C2×#421—21no—
30C2×C4×M16—21no—
31C2×C4×M16C2211yescharacteristic
32#2285C2211yescharacteristic
33#2283C2211yescharacteristic
34#1847—21no—
35#1836—21no—
36#1826—21no—
37#1809—22no—
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—
51#1733—21no—
52#1569—21no—
53#1569—21no—
54#1567C2212yescomplemented
55#1566C2212yescomplemented
56#1565—22no—
57#1303C2212yescomplemented
58#1303C2212yescomplemented
59#1303C2212yescomplemented
60#1303C2212yescomplemented
61C22×#41D411yescharacteristic
62C22×M16—22no—
63C22×M16—21no—
64C22×M16D412yes—
65C4×#41—21no—
66C4×#41—21no—
67C4×#41—21no—
68C4×#41—21no—
69C4×#41C2×C411yescharacteristic
70C4×#41C2×C411yescharacteristic
71C4×#41D411yescharacteristic
72C4×#41D411yescharacteristic
73C4×#41Q811yescharacteristic
74C4×#41Q811yescharacteristic
75C4×#41C2311yescharacteristic
76C4×#41C2311yescharacteristic
77C2×C4×Q8C2×C411yescharacteristic
78C2×C4×Q8Q811yescharacteristic
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4Q811yescharacteristic
83C2×#117—21no—
84C2×#117—21no—
85C2×#117D411yescharacteristic
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#446—22no—
89#446—22no—
90#446C2×C412yes—
91#446C2×C412yes—
92C8×Q8C2×C412yes—
93C8×Q8C2×C412yes—
94#436—22no—
95#436—22no—
96#436C2×C412yes—
97#436C2×C412yes—
98#435—22no—
99#435—22no—
100#435—22no—
101#435—22no—
102#435—22no—
103#435—22no—
104C8×D4—22no—
105C8×D4—22no—
106C8×D4C2×C412yes—
107C8×D4C2×C412yes—
108#432—21no—
109#432—21no—
110#432—21no—
111#432—21no—
112#432—21no—
113#432—21no—
114#432—21no—
115#432C2×C411yescharacteristic
116#432C2×C411yescharacteristic
117#431—21no—
118#431—22no—
119#431—22no—
120#431—21no—
121#431—21no—
122#431C2312yes—
123C2×#107—22no—
124C2×#106—22no—
125#423—22no—
126#423—21no—
127#423—21no—
128#423—21no—
129#423C2×C412yes—
130#423C2311yescharacteristic
131#423C2312yes—
132#423C2311yescharacteristic
133C2×C4⋊C8—21no—
134C2×C4⋊C8—21no—
135#421—44no—
136C2×#104—22no—
137C2×#104—22no—
138C2×#104—21no—
139C2×#104—21no—
140C2×#104—21no—
141C2×#104—21no—
142#407—21no—
143#407—22no—
144#407—21no—
145C4×M16—22no—
146C4×M16—21no—
147C4×M16—21no—
148C4×M16—21no—
149C4×M16—22no—
150C4×M16—21no—
151C4×M16C2×C412yes—
152C4×M16C2×C411yescharacteristic
153C4×M16C2×C411yescharacteristic
154C4×M16D412yes—
155C4×M16D411yescharacteristic
156C4×M16D411yescharacteristic
157C2×#97—21no—
158C2×#97—21no—
159C2×#97D411yescharacteristic
160C2×#97D411yescharacteristic
161C2×C4×C8—22no—
162C2×C4×C8C2312yes—
163#384—21no—
164#383—21no—
165#382—21no—
166#382—21no—
167#381—21no—
168#380—21no—
169#380—21no—
170#380—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4C2311yescharacteristic
173C4×C4⋊C4C2311yescharacteristic
174C4×C4⋊C4C2312yes—
175C4×C22⋊C4—21no—
176#376—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×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4#4111yescharacteristic
208C2×M16—22no—
209C2×M16—22no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—44no—
215C2×M16—48no—
216C2×M16—41no—
217C2×M16—42no—
218C2×M16—42no—
219C2×M16C22⋊C412yes—
220C2×M16C22⋊C412yes—
221C2×M16C22⋊C412yes—
222C2×M16C22⋊C412yes—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—41no—
226C22×C8C2×D412yes—
227C22×C8C2×D412yes—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8C4⋊C411yescharacteristic
233C4×Q8C4⋊C411yescharacteristic
234C4×Q8C4⋊C411yescharacteristic
235C4×Q8C4⋊C411yescharacteristic
236C4×Q8C22×C411yescharacteristic
237C4×Q8C22×C411yescharacteristic
238C4×Q8C2×Q811yescharacteristic
239C4×Q8C2×Q811yescharacteristic
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×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4C4⋊C411yescharacteristic
261C4×D4C4⋊C411yescharacteristic
262C4×D4C4⋊C411yescharacteristic
263C4×D4C4⋊C411yescharacteristic
264C4×D4C22×C411yescharacteristic
265C4×D4C22×C411yescharacteristic
266C4×D4C2×Q811yescharacteristic
267C4×D4C2×Q811yescharacteristic
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—42no—
279#117—42no—
280#117C4⋊C411yescharacteristic
281#117C4⋊C411yescharacteristic
282#117C2×D411yescharacteristic
283#117C2×D411yescharacteristic
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—41no—
290C2×C4⋊C4—41no—
291C2×C4⋊C4C4⋊C411yescharacteristic
292C2×C4⋊C4C4⋊C411yescharacteristic
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—41no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—22no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42C22×C411yescharacteristic
313C2×C42C22×C411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×Q811yescharacteristic
319C2×C42C2×Q811yescharacteristic
320C2×C42#4111yescharacteristic
321C2×C42#4111yescharacteristic
322C2×C42#4111yescharacteristic
323C2×C42C2411yescharacteristic
324#107—44no—
325#106—44no—
326C4⋊C8—22no—
327C4⋊C8—22no—
328C4⋊C8—22no—
329C4⋊C8—22no—
330C4⋊C8—22no—
331C4⋊C8—21no—
332C4⋊C8—22no—
333C4⋊C8—21no—
334C4⋊C8—22no—
335C4⋊C8—21no—
336C4⋊C8—21no—
337C4⋊C8—41no—
338C4⋊C8—41no—
339C4⋊C8C22×C412yes—
340C4⋊C8C22×C412yes—
341#104—42no—
342#104—42no—
343#104—44no—
344#104—42no—
345#104—44no—
346#104—42no—
347C22⋊C8—21no—
348C22⋊C8—21no—
349C22⋊C8—22no—
350C22⋊C8—22no—
351C22⋊C8—22no—
352C22⋊C8—22no—
353C22⋊C8—22no—
354C22⋊C8—41no—
355C22⋊C8—41no—
356#97—22no—
357#97—22no—
358#97—21no—
359#97—21no—
360#97—41no—
361#97C22⋊C411yescharacteristic
362#97C22⋊C411yescharacteristic
363#97C22⋊C411yescharacteristic
364#97C22⋊C411yescharacteristic
365#97C22⋊C412yes—
366#97C22⋊C412yes—
367C4×C8—22no—
368C4×C8—22no—
369C4×C8—22no—
370C4×C8—41no—
371C4×C8C22×C412yes—
372C4×C8C22×C412yes—
373C4×C8C2×D412yes—
374C4×C8C2×D412yes—
375#95—22no—
376#95—21no—
377#95—21no—
378#95—41no—
379#95—41no—
380#95—41no—
381#95—41no—
382C24—21no—
383C24#12111yescharacteristic
384#41—21no—
385#41—21no—
386#41—22no—
387#41—22no—
388#41—21no—
389#41—21no—
390#41—21no—
391#41—21no—
392#41—42no—
393#41—42no—
394#41—41no—
395#41—41no—
396#41—42no—
397#41—42no—
398#41AGL(1,ℤ/8)12yes—
399#41#13712yes—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—41no—
403C2×Q8C2×C4⋊C411yescharacteristic
404C2×Q8C2×C4⋊C411yescharacteristic
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—41no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4C2×C4⋊C411yescharacteristic
415C2×D4C2×C4⋊C411yescharacteristic
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—21no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—81no—
443C22×C4—82no—
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C4⋊C411yescharacteristic
447C22×C4C2×C4⋊C411yescharacteristic
448C22×C4#11711yescharacteristic
449C22×C4#11711yescharacteristic
450C22×C4#12111yescharacteristic
451C22×C4#12111yescharacteristic
452C22×C4#12111yescharacteristic
453C22×C4#12111yescharacteristic
454C22×C4#12111yescharacteristic
455C22×C4C22⋊Q811yescharacteristic
456C22×C4C22⋊Q811yescharacteristic
457C22×C4C22⋊Q811yescharacteristic
458C22×C4C22⋊Q811yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4111yescharacteristic
462M16—24no—
463M16—24no—
464M16—42no—
465M16—42no—
466M16—44no—
467M16—44no—
468M16—44no—
469M16—42no—
470M16—41no—
471M16—41no—
472M16—41no—
473M16—41no—
474M16—42no—
475M16—82no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—42no—
480C2×C8—44no—
481C2×C8—44no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—42no—
485C2×C8—82no—
486C2×C8C2×C22⋊C412yes—
487C2×C8C2×C22⋊C412yes—
488C2×C8C2×C22⋊C412yes—
489C2×C8C2×C22⋊C412yes—
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—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4C2×C4⋊C411yescharacteristic
507C4⋊C4C2×C4⋊C411yescharacteristic
508C4⋊C4C2×C4⋊C411yescharacteristic
509C4⋊C4C2×C4⋊C411yescharacteristic
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—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—42no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—42no—
542C42C2×C22⋊C411yescharacteristic
543C42C2×C22⋊C411yescharacteristic
544C42C2×C4⋊C411yescharacteristic
545C42C2×C4⋊C411yescharacteristic
546C42#11711yescharacteristic
547C42#11711yescharacteristic
548C42#12111yescharacteristic
549C42#12111yescharacteristic
550C42C22⋊Q811yescharacteristic
551C42C22⋊Q811yescharacteristic
552C42C22⋊Q811yescharacteristic
553C42C22⋊Q811yescharacteristic
554C42C23×C411yescharacteristic
555C42C22×D411yescharacteristic
556C42C22×Q811yescharacteristic
557C42C2×#4111yescharacteristic
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23—81no—
567C23#38011yescharacteristic
568C23#38011yescharacteristic
569C23C2×#12111yescharacteristic
570Q8—21no—
571Q8—21no—
572Q8—21no—
573Q8—42no—
574Q8#42812yes—
575D4—22no—
576D4—22no—
577D4—21no—
578D4—21no—
579D4—21no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—41no—
584D4—41no—
585D4#42812yes—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—82no—
635C2×C4—81no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4#38011yescharacteristic
641C2×C4#38011yescharacteristic
642C2×C4#38011yescharacteristic
643C2×C4#38011yescharacteristic
644C2×C4#38011yescharacteristic
645C2×C4#38011yescharacteristic
646C2×C4C22×C22⋊C411yescharacteristic
647C2×C4C22×C4⋊C411yescharacteristic
648C2×C4C2×#11711yescharacteristic
649C2×C4C2×#12111yescharacteristic
650C2×C4C2×C22⋊Q811yescharacteristic
651C2×C4C2×C22⋊Q811yescharacteristic
652C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
653C2×C4C2×#13711yescharacteristic
654C2×C4#57511yescharacteristic
655C2×C4#57511yescharacteristic
656C8—22no—
657C8—22no—
658C8—22no—
659C8—42no—
660C8—42no—
661C8—82no—
662C8—81no—
663C8—81no—
664C8#42114yescomplemented
665C22—21no—
666C22—21no—
667C22—21no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22—22no—
672C22—21no—
673C22—21no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—81no—
680C22—81no—
681C22—84no—
682C22C2×#38011yescharacteristic
683C22#305511yescharacteristic
684C22#305611yescharacteristic
685C4—21no—
686C4—21no—
687C4—21no—
688C4—21no—
689C4—21no—
690C4—22no—
691C4—21no—
692C4—21no—
693C4—21no—
694C4—21no—
695C4—41no—
696C4—41no—
697C4—41no—
698C4—41no—
699C4—41no—
700C4—161no—
701C4—162no—
702C4—161no—
703C4C2×#42112yes—
704C4C2×#42811yescharacteristic
705C4#291111yescharacteristic
706C4#305511yescharacteristic
707C4#305611yescharacteristic
708C2—21no—
709C2—21no—
710C2—41no—
711C2—41no—
712C2—42no—
713C2—81no—
714C2#2065411yescharacteristic
715C2#2066111yescharacteristic
716C2#2067511yescharacteristic
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. #20819C2semidirect
5. #17796C2semidirect
6. #8669C2semidirect
7. #8668C2semidirect
8. #8667C2semidirect
9. #8666C2semidirect
10. #8094C2semidirect
11. #8093C2semidirect
13. #2971C4semidirect
14. #2971C4semidirect
17. #2970C4semidirect
18. #2970C4semidirect
22. Q8×M16C4semidirect
24. D4×M16C4semidirect
47. C4×#107C22semidirect
48. C4×#106C22semidirect
54. #1567C22semidirect
55. #1566C22semidirect
57. #1303C22semidirect
58. #1303C22semidirect
59. #1303C22semidirect
60. #1303C22semidirect
664. C8#421semidirect

Maximal subgroups

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

subgroupindex
M16×#412
#208192
#207482
#206542
#177962
#86692
#86682
#86672
#86662
#80942
#80932