Finite Group Databasebeta

← #393265

Subgroups — #393265

Order 512.

985 conjugacy classes of subgroups.

The subgroup lattice has 764 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1929C211yesmaximal, characteristic
2#21058C211yesmaximal, characteristic
3#21046C211yesmaximal, characteristic
4C2×#1922C211yesmaximal, characteristic, complemented
5C4×#405C211yesmaximal, characteristic, complemented
6#10934C211yesmaximal, characteristic
7#10934C211yesmaximal, characteristic
8#10934C211yesmaximal, characteristic
9#10933C211yesmaximal, characteristic
10#10911C211yesmaximal, characteristic, complemented
11#10910C211yesmaximal, characteristic, complemented
12#10909C211yesmaximal, characteristic, complemented
13#10887C211yesmaximal, characteristic, complemented
14#10808C211yesmaximal, characteristic, complemented
15#10797C211yesmaximal, characteristic, complemented
16C22×#108C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C2×C4×M16C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31#2297C2211yescharacteristic
32C4×#117C2211yescharacteristic
33#1929—24no—
34#1928C411yescharacteristic, complemented
35#1928C411yescharacteristic, complemented
36#1927C411yescharacteristic, complemented
37#1927C411yescharacteristic, complemented
38#1926C411yescharacteristic, complemented
39#1926C411yescharacteristic, complemented
40#1925C411yescharacteristic, complemented
41#1925C411yescharacteristic, complemented
42#1923—21no—
43#1923—21no—
44#1923—21no—
45#1923—21no—
46#1922—24no—
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50#1735C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57#1729C2211yescharacteristic
58#1728C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64#1726—21no—
65#1726—21no—
66C4×#97—21no—
67C42×C8—21no—
68C4×#95C2211yescharacteristic
69C22×#41C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#108—22no—
85C2×#108—22no—
86C2×#108—22no—
87#427—22no—
88#427—22no—
89C2×#107C2×C411yescharacteristic
90C2×#107C2×C411yescharacteristic
91C2×#106C2×C411yescharacteristic
92C2×#106C2×C411yescharacteristic
93#424—22no—
94#424—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×#104—22no—
98C2×#104—22no—
99C2×#104—22no—
100C2×#104—22no—
101C2×#104—22no—
102C2×#104—22no—
103#416—22no—
104#416—22no—
105#416—22no—
106#416—22no—
107C2×#103C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115#408—22no—
116#408—22no—
117#408—22no—
118#408—22no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123#405—22no—
124#405—22no—
125#405—22no—
126#405—22no—
127#405—22no—
128#405—22no—
129#405—22no—
130#405D414yes—
131C4×M16—22no—
132C4×M16—22no—
133C2×#97—21no—
134C2×#97—21no—
135C2×#97—21no—
136C2×#97—21no—
137C2×#97—21no—
138C2×C4×C8—21no—
139C2×C4×C8—21no—
140C2×C4×C8—21no—
141C2×C4×C8—21no—
142C2×C4×C8—22no—
143C2×C4×C8—22no—
144C2×C4×C8—21no—
145C2×C4×C8D411yescharacteristic
146C2×C4×C8D411yescharacteristic
147C2×C4×C8D411yescharacteristic
148C2×C4×C8D411yescharacteristic
149#391C2×C411yescharacteristic
150#391C2×C411yescharacteristic
151#390C2×C411yescharacteristic
152#390C2×C411yescharacteristic
153#386—21no—
154#386—21no—
155C4×C4⋊C4C2×C411yescharacteristic
156C4×C4⋊C4C2×C411yescharacteristic
157C4×C4⋊C4C2×C411yescharacteristic
158C4×C4⋊C4C2×C411yescharacteristic
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—21no—
163#376—21no—
164C2×#95—21no—
165C2×#95—21no—
166C43—21no—
167#341—22no—
168#341—22no—
169#341—22no—
170#340C2×C411yescharacteristic
171#340C2×C411yescharacteristic
172#340C2×C411yescharacteristic
173#340C2×C411yescharacteristic
174#339—22no—
175#339—22no—
176#339—22no—
177#339—22no—
178#339—22no—
179#339—22no—
180#336—22no—
181#336—21no—
182#336—21no—
183#336—22no—
184#336—22no—
185#336—22no—
186#336—21no—
187#336—21no—
188#336C2×C411yescharacteristic
189#336C2×C411yescharacteristic
190#336C2×C411yescharacteristic
191#336C2×C411yescharacteristic
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41—21no—
197C2×#41—21no—
198C2×#41—21no—
199C2×#41—21no—
200C2×#41—21no—
201C2×#41—21no—
202C2×#41—21no—
203C2×#41—21no—
204C22×Q8C22×C411yescharacteristic
205C22×D4—21no—
206C22×D4C22×C411yescharacteristic
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4#4111yescharacteristic
211C23×C4#4111yescharacteristic
212C23×C4C2411yescharacteristic
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16C2×D412yes—
232C2×M16#4112yes—
233C2×M16#4112yes—
234C2×M16#4112yes—
235C2×M16#4112yes—
236C2×M16#4112yes—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—21no—
261C22×C8C22×C411yescharacteristic
262C22×C8C22×C411yescharacteristic
263C22×C8C22×C411yescharacteristic
264C22×C8C22×C411yescharacteristic
265C22×C8C2×D412yes—
266C22×C8C2×D411yescharacteristic
267C22×C8C2×D411yescharacteristic
268C22×C8C2×D411yescharacteristic
269C22×C8C2×D411yescharacteristic
270C22×C8#4112yes—
271C22×C8#4112yes—
272C22×C8#4112yes—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117C2×D412yes—
282#117#4112yes—
283#117#4112yes—
284#117#4112yes—
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
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—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42C22×C411yescharacteristic
322C2×C42C22×C411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325#108—44no—
326#107—22no—
327#107—22no—
328#106—22no—
329#106—22no—
330C4⋊C8—42no—
331C4⋊C8—42no—
332#104—44no—
333#104—44no—
334#103—22no—
335#103—22no—
336#103—22no—
337#103—22no—
338#102—22no—
339#102—22no—
340#102—22no—
341#102—22no—
342C22⋊C8—42no—
343C22⋊C8—42no—
344C22⋊C8—42no—
345C22⋊C8—42no—
346#97—24no—
347#97—24no—
348#97—24no—
349#97—24no—
350#97—24no—
351C4×C8—24no—
352C4×C8—24no—
353C4×C8—24no—
354C4×C8—24no—
355C4×C8—24no—
356C4×C8—22no—
357C4×C8—22no—
358C4×C8—22no—
359C4×C8—22no—
360#95—24no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—42no—
370C24—21no—
371C24C2×#4111yescharacteristic
372#41—41no—
373#41—41no—
374#41—41no—
375#41—41no—
376#41—41no—
377#41—41no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386#41—41no—
387#41—41no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4C4×D411yescharacteristic
442C22×C4C4×D411yescharacteristic
443C22×C4C4×D411yescharacteristic
444C22×C4C4×D411yescharacteristic
445C22×C4C23×C411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4C2×#4111yescharacteristic
456C22×C4C2×#4111yescharacteristic
457C22×C4C2×#4111yescharacteristic
458C22×C4C2×#4111yescharacteristic
459M16—22no—
460M16—22no—
461M16—22no—
462M16—22no—
463M16—22no—
464M16—22no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470M16—41no—
471M16—41no—
472M16—41no—
473M16#12414yes—
474M16#12614yes—
475C2×C8—22no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8—41no—
514C2×C8—41no—
515C2×C8—41no—
516C2×C8—41no—
517C2×C8—41no—
518C2×C8—41no—
519C2×C8—41no—
520C2×C8—41no—
521C2×C8—41no—
522C2×C8—41no—
523C2×C8#12112yes—
524C2×C8#12112yes—
525C2×C8#12312yes—
526C2×C8#12312yes—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4#11712yes—
536C4⋊C4#11712yes—
537C4⋊C4C4×D412yes—
538C4⋊C4C4×D412yes—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—41no—
546C22⋊C4—41no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—22no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—21no—
561C42—42no—
562C42—42no—
563C42—41no—
564C42—41no—
565C42—42no—
566C42—42no—
567C42—41no—
568C42—41no—
569C42—41no—
570C42—41no—
571C42—41no—
572C42—41no—
573C42—41no—
574C42—41no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23C2×C4×D411yescharacteristic
587C23#52911yescharacteristic
588C23#52911yescharacteristic
589C23#52911yescharacteristic
590C23#52911yescharacteristic
591C23#52911yescharacteristic
592C23#53211yescharacteristic
593Q8—41no—
594Q8—41no—
595Q8—41no—
596Q8—41no—
597D4—41no—
598D4—41no—
599D4—41no—
600D4—41no—
601D4—81no—
602D4—81no—
603D4—81no—
604D4—81no—
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—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—81no—
650C2×C4—81no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—81no—
656C2×C4—81no—
657C2×C4—81no—
658C2×C4#44312yes—
659C2×C4#44412yes—
660C2×C4C2×#11711yescharacteristic
661C2×C4C2×C4×D411yescharacteristic
662C2×C4C4×#4111yescharacteristic
663C2×C4C4×#4111yescharacteristic
664C2×C4C4×#4111yescharacteristic
665C2×C4C4×#4111yescharacteristic
666C2×C4C2×#12111yescharacteristic
667C2×C4C2×#12311yescharacteristic
668C2×C4#52511yescharacteristic
669C2×C4#52511yescharacteristic
670C2×C4#52511yescharacteristic
671C2×C4#52511yescharacteristic
672C2×C4C2×#12411yescharacteristic
673C2×C4C2×#12611yescharacteristic
674C2×C4#52911yescharacteristic
675C2×C4#52911yescharacteristic
676C2×C4#52911yescharacteristic
677C2×C4#52911yescharacteristic
678C2×C4#52911yescharacteristic
679C2×C4#52911yescharacteristic
680C2×C4#52911yescharacteristic
681C2×C4#52911yescharacteristic
682C2×C4#52911yescharacteristic
683C2×C4#52911yescharacteristic
684C2×C4#52911yescharacteristic
685C2×C4#53211yescharacteristic
686C2×C4#53211yescharacteristic
687C2×C4#53211yescharacteristic
688C8—24no—
689C8—24no—
690C8—22no—
691C8—22no—
692C8—22no—
693C8—22no—
694C8—41no—
695C8—41no—
696C8—41no—
697C8—41no—
698C8—81no—
699C8—81no—
700C22—21no—
701C22—21no—
702C22—21no—
703C22—21no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—21no—
709C22—21no—
710C22—21no—
711C22—21no—
712C22—81no—
713C22—81no—
714C22—81no—
715C22—81no—
716C22—81no—
717C22—81no—
718C22—81no—
719C22#229011yescharacteristic
720C22C4×#12111yescharacteristic
721C22C4×#12311yescharacteristic
722C22#286011yescharacteristic
723C22#286111yescharacteristic
724C22#286511yescharacteristic
725C22#286611yescharacteristic
726C22C2×#44311yescharacteristic
727C22C2×#44411yescharacteristic
728C22#295511yescharacteristic
729C22#295511yescharacteristic
730C4—22no—
731C4—22no—
732C4—42no—
733C4—42no—
734C4—41no—
735C4—41no—
736C4—42no—
737C4—42no—
738C4—82no—
739C4—82no—
740C4—81no—
741C4—81no—
742C4—81no—
743C4#229311yescharacteristic
744C4#229811yescharacteristic
745C4C4×#12411yescharacteristic
746C4C4×#12611yescharacteristic
747C4#285011yescharacteristic
748C4#285211yescharacteristic
749C4#285311yescharacteristic
750C4#285411yescharacteristic
751C2—21no—
752C2—21no—
753C2—21no—
754C2—21no—
755C2—161no—
756C2#1470111yescharacteristic
757C2#2113711yescharacteristic
758C2#2113811yescharacteristic
759C2#2151711yescharacteristic
760C2#2151811yescharacteristic
761C2#2159211yescharacteristic
762C2#2159311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1922C2semidirect
5. C4×#405C2semidirect
10. #10911C2semidirect
11. #10910C2semidirect
12. #10909C2semidirect
13. #10887C2semidirect
14. #10808C2semidirect
15. #10797C2semidirect
34. #1928C4semidirect
35. #1928C4semidirect
36. #1927C4semidirect
37. #1927C4semidirect
38. #1926C4semidirect
39. #1926C4semidirect
40. #1925C4semidirect
41. #1925C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19292
#210582
#210462
C2×#19222
C4×#4052
#109342
#109332
#109112
#109102
#109092
#108872
#108082
#107972