Finite Group Databasebeta

← #365292

Subgroups — #365292

Order 512.

1478 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1936C212yesmaximal
2C2×#1802C211yesmaximal, characteristic
3C2×#1802C211yesmaximal, characteristic
4C2×#1767C211yesmaximal, characteristic, complemented
5#14712C211yesmaximal, characteristic, complemented
6C4×#390C211yesmaximal, characteristic
7#11316C212yesmaximal, complemented
8#11189C212yesmaximal
9#10881C212yesmaximal, complemented
10#10849C211yesmaximal, characteristic, complemented
11#10849C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#97C2212yes—
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#389—21no—
21C2×#389—21no—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4—21no—
27C2×C4×C22⋊C4—21no—
28C2×C43C2211yescharacteristic
29#1936—28no—
30#1802—24no—
31#1802C418yescomplemented
32#1767—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40#1437—21no—
41#1437—21no—
42C4⋊#95—21no—
43C4⋊#95—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4C2×C414yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C42—21no—
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8C2×C414yes—
85C2×C4⋊C8D414yes—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—42no—
91C2×#102C2×C418yes—
92C2×#97—24no—
93C2×#97—24no—
94C2×#97—42no—
95#390—22no—
96#390—22no—
97#390—22no—
98#390D414yes—
99#389—24no—
100#389—24no—
101#389—44no—
102#389—44no—
103#384—22no—
104#384—22no—
105#384—22no—
106#384C2×C414yes—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C22⋊C4—44no—
110C4×C22⋊C4—44no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C43—22no—
120C43—22no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—42no—
126#340C2×C418yes—
127#336—24no—
128C25—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4C22×C412yes—
141C22×D4C2×D412yes—
142C22×D4#4112yes—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8C22×C414yes—
175C22×C8C2×D414yes—
176C4×D4—28no—
177C4×D4—28no—
178C4×D4—44no—
179C4×D4—44no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C414yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4#4112yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—41no—
261C2×C42—41no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42C22×C412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42#4112yes—
269C2×C42#4112yes—
270C2×C42#4112yes—
271C2×C42#4112yes—
272C2×C42#4112yes—
273C4⋊C8—28no—
274C4⋊C8—28no—
275#102—216no—
276#102—48no—
277#97—48no—
278#95—42no—
279#95—42no—
280#95—42no—
281#95—42no—
282#95—44no—
283#95—44no—
284#95—42no—
285#95—44no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24C22×D411yescharacteristic
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4C4×D414yes—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4C4×D412yes—
401C22×C4C4×D412yes—
402C22×C4#12112yes—
403C22×C4#12112yes—
404C22×C4#12112yes—
405C22×C4#12112yes—
406C22×C4#12412yes—
407C22×C4#12412yes—
408C22×C4AGL(1,ℤ/8)12yes—
409C22×C4AGL(1,ℤ/8)12yes—
410C22×C4AGL(1,ℤ/8)12yes—
411C22×C4AGL(1,ℤ/8)12yes—
412C22×C4#13712yes—
413C22×C4#13712yes—
414C22×C4C23×C411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4H(5,𝔽2)11yescharacteristic
425C22×C42−1+411yescharacteristic
426C22×C42−1+411yescharacteristic
427C22×C42−1+411yescharacteristic
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—48no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—48no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—84no—
465C4⋊C4—88no—
466C4⋊C4—84no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—44no—
477C42—44no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42—44no—
483C42—44no—
484C42—84no—
485C42#11714yes—
486C42C4×D414yes—
487C42#12314yes—
488C42#12614yes—
489C23—24no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—81no—
509C23—81no—
510C23—81no—
511C23—81no—
512C23—81no—
513C23—81no—
514C23—81no—
515C23—81no—
516C23—81no—
517C23—81no—
518C23—81no—
519C23—81no—
520C23—81no—
521C23—81no—
522C23—81no—
523C23—81no—
524C23C2×C4×D411yescharacteristic
525C23C2×#12111yescharacteristic
526C23C2×#12111yescharacteristic
527C23#52511yescharacteristic
528C23C2×#12411yescharacteristic
529C23#53611yescharacteristic
530C23#53711yescharacteristic
531C23C2×AGL(1,ℤ/8)11yescharacteristic
532C23C2×AGL(1,ℤ/8)11yescharacteristic
533C23C2×AGL(1,ℤ/8)11yescharacteristic
534C23C2×AGL(1,ℤ/8)11yescharacteristic
535C23C2×#13711yescharacteristic
536C23C2×#13711yescharacteristic
537C23#57511yescharacteristic
538C23#57511yescharacteristic
539D4—24no—
540D4—24no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—42no—
546D4—42no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—816no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4#38814yes—
618C2×C4#38814yes—
619C2×C4#44014yes—
620C2×C4#44214yes—
621C2×C4#46814yes—
622C2×C4#46914yes—
623C2×C4#48914yes—
624C2×C4#49014yes—
625C2×C4C2×#11711yescharacteristic
626C2×C4C2×C4×D411yescharacteristic
627C2×C4C2×C4×D411yescharacteristic
628C2×C4C4×#4111yescharacteristic
629C2×C4#51911yescharacteristic
630C2×C4#52011yescharacteristic
631C2×C4C2×#12311yescharacteristic
632C2×C4C2×#12611yescharacteristic
633C2×C4#52911yescharacteristic
634C2×C4#53211yescharacteristic
635C2×C4#54111yescharacteristic
636C2×C4#54211yescharacteristic
637C2×C4D4211yescharacteristic
638C2×C4#54711yescharacteristic
639C2×C4#54711yescharacteristic
640C2×C4#54711yescharacteristic
641C2×C4#54711yescharacteristic
642C2×C4#54811yescharacteristic
643C2×C4#54811yescharacteristic
644C2×C4#55211yescharacteristic
645C2×C4#55211yescharacteristic
646C2×C4#55211yescharacteristic
647C2×C4#55311yescharacteristic
648C2×C4#55311yescharacteristic
649C2×C4#55311yescharacteristic
650C2×C4#55611yescharacteristic
651C2×C4#55611yescharacteristic
652C2×C4#55611yescharacteristic
653C8—44no—
654C8—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22C2×#38811yescharacteristic
671C22C4×#12111yescharacteristic
672C22#232311yescharacteristic
673C22#241511yescharacteristic
674C22#241611yescharacteristic
675C22#254711yescharacteristic
676C22#256211yescharacteristic
677C22C2×#44011yescharacteristic
678C22C2×#44211yescharacteristic
679C22C4×AGL(1,ℤ/8)11yescharacteristic
680C22C4×#13711yescharacteristic
681C22#303611yescharacteristic
682C22#303611yescharacteristic
683C22#303811yescharacteristic
684C22#303911yescharacteristic
685C22C2×#46811yescharacteristic
686C22C2×#46911yescharacteristic
687C22#305511yescharacteristic
688C22#305611yescharacteristic
689C22#310911yescharacteristic
690C22#311011yescharacteristic
691C22#311511yescharacteristic
692C22#311611yescharacteristic
693C22C2×#48911yescharacteristic
694C22C2×#49011yescharacteristic
695C22#314111yescharacteristic
696C22#314211yescharacteristic
697C22#323511yescharacteristic
698C22#323511yescharacteristic
699C22#323811yescharacteristic
700C22#323911yescharacteristic
701C22#325111yescharacteristic
702C22#325211yescharacteristic
703C22#325311yescharacteristic
704C22#325411yescharacteristic
705C4—24no—
706C4—42no—
707C4—42no—
708C4—42no—
709C4—42no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—84no—
714C4—162no—
715C4#193618yescomplemented
716C4#234411yescharacteristic
717C4#234711yescharacteristic
718C4#236111yescharacteristic
719C4#236211yescharacteristic
720C4#242111yescharacteristic
721C4#242211yescharacteristic
722C4#249411yescharacteristic
723C4#249611yescharacteristic
724C2—84no—
725C2#1397411yescharacteristic
726C2C2×#193611yescharacteristic
727C2#2109311yescharacteristic
728C2#2150311yescharacteristic
729C2#2150411yescharacteristic
730C2#2211611yescharacteristic
731C2#2211711yescharacteristic
732C2#2470811yescharacteristic
733C2#2470911yescharacteristic
734C2#2475811yescharacteristic
735C2#2475911yescharacteristic
736C2#2872411yescharacteristic
737C2#2872611yescharacteristic
738C2#2913011yescharacteristic
739C2#2913111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1767C2semidirect
5. #14712C2semidirect
7. #11316C2semidirect
9. #10881C2semidirect
10. #10849C2semidirect
11. #10849C2semidirect
31. #1802C4semidirect
715. C4#1936semidirect

Maximal subgroups

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

subgroupindex
C2×#19362
C2×#18022
C2×#17672
#147122
C4×#3902
#113162
#111892
#108812
#108492