Finite Group Databasebeta

← #362491

Subgroups — #362491

Order 512.

1656 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2C2×#1954C211yesmaximal, characteristic
3C2×#1825C212yesmaximal, complemented
4#13906C211yesmaximal, characteristic
5#11200C212yesmaximal, complemented
6#11199C212yesmaximal, complemented
7#10849C211yesmaximal, characteristic, complemented
8#10849C211yesmaximal, characteristic, complemented
9#10841C211yesmaximal, characteristic, complemented
10#10731C212yesmaximal, complemented
11C4×#340C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2354—24no—
19#2354C418yescomplemented
20C2×#390—21no—
21C2×#390—21no—
22C2×#388—22no—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1954—24no—
32#1825—28no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2211yescharacteristic
43#1439—22no—
44C4⋊#95—21no—
45C23×D4D411yescharacteristic
46C24×C4C2312yes—
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4C2×C414yes—
55C2×C4×D4D414yes—
56C2×#117—24no—
57C2×#117—24no—
58C2×#117—24no—
59C2×#117—42no—
60C2×#117—42no—
61C2×#117C2×C418yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4D412yes—
78C22×C42—22no—
79C22×C42—21no—
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—42no—
91C2×C4×C8—22no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94#390—28no—
95#390—44no—
96#388—216no—
97#384—22no—
98#384—22no—
99#384—22no—
100#384C2×C414yes—
101#380—24no—
102#380—24no—
103#380—24no—
104#380—42no—
105#380—42no—
106#380C2×C418yes—
107C4×C4⋊C4—28no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2×C414yes—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117#340—24no—
118#340—22no—
119#340—24no—
120#340—24no—
121#340—22no—
122#340—22no—
123#340—42no—
124#340—42no—
125#340—42no—
126#340D414yescomplemented
127C25C2×D412yes—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4C22⋊C414yes—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4C22×C414yes—
154C23×C4C2×D412yes—
155C23×C4C2×D414yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8C2×D412yes—
179C22×C8#4112yes—
180C4×D4—28no—
181C4×D4—28no—
182C4×D4—28no—
183C4×D4—28no—
184#117—216no—
185#117—48no—
186#117—48no—
187#117—48no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—81no—
228C2×C4⋊C4—81no—
229C2×C4⋊C4—81no—
230C2×C4⋊C4—81no—
231C2×C4⋊C4C22⋊C414yes—
232C2×C4⋊C4C22×C414yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4#4112yes—
238C2×C4⋊C4#4112yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—82no—
250C2×C22⋊C4—82no—
251C2×C22⋊C4—82no—
252C2×C22⋊C4—82no—
253C2×C22⋊C4C22⋊C418yes—
254C2×C42—28no—
255C2×C42—24no—
256C2×C42—44no—
257C2×C42—44no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42C22×C412yes—
264C2×C42C2×D412yes—
265C2×C42#4112yes—
266C2×C42#4112yes—
267C2×C42#4112yes—
268C2×C42#4112yes—
269C4⋊C8—44no—
270C22⋊C8—48no—
271C22⋊C8—48no—
272C4×C8—44no—
273#95—28no—
274#95—24no—
275#95—44no—
276#95—44no—
277#95—42no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24C2×C22⋊C414yes—
293C24C22×D411yescharacteristic
294C2×D4—28no—
295C2×D4—28no—
296C2×D4—28no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4C2×C22⋊C414yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4#12312yes—
385C22×C4#12312yes—
386C22×C4#12312yes—
387C22×C4#12312yes—
388C22×C4#13512yes—
389C22×C4#13512yes—
390C22×C4AGL(1,ℤ/8)12yes—
391C22×C4#13712yes—
392C22×C4C23×C411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4H(5,𝔽2)11yescharacteristic
403C22×C4H(5,𝔽2)11yescharacteristic
404C22×C4H(5,𝔽2)11yescharacteristic
405C22×C42−1+411yescharacteristic
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—28no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—41no—
443C4⋊C4—42no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—48no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—84no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C4⋊C4C4×D414yes—
456C22⋊C4—216no—
457C22⋊C4—416no—
458C22⋊C4—88no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C42—416no—
462C42—48no—
463C42—84no—
464C42#11714yes—
465C42#11714yes—
466C42#12314yes—
467C42#12314yes—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
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—
503C23C22×C22⋊C411yescharacteristic
504C23C2×#12311yescharacteristic
505C23C2×#12311yescharacteristic
506C23D4211yescharacteristic
507C23#54611yescharacteristic
508C23#54611yescharacteristic
509C23#54711yescharacteristic
510C23C2×#13511yescharacteristic
511C23C2×#13511yescharacteristic
512C23C2×AGL(1,ℤ/8)11yescharacteristic
513C23C2×#13711yescharacteristic
514C23#57611yescharacteristic
515C23#57711yescharacteristic
516C23#57711yescharacteristic
517C23#57811yescharacteristic
518D4—416no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4#38114yes—
583C2×C4#38114yes—
584C2×C4#41614yes—
585C2×C4#41714yes—
586C2×C4#48214yes—
587C2×C4#48214yes—
588C2×C4#48514yes—
589C2×C4#48514yes—
590C2×C4C2×#11711yescharacteristic
591C2×C4C2×#11711yescharacteristic
592C2×C4C2×C4×D411yescharacteristic
593C2×C4C4×#4111yescharacteristic
594C2×C4#51811yescharacteristic
595C2×C4#52011yescharacteristic
596C2×C4C2×#12311yescharacteristic
597C2×C4C2×#12311yescharacteristic
598C2×C4#52911yescharacteristic
599C2×C4#52911yescharacteristic
600C2×C4#53211yescharacteristic
601C2×C4#53211yescharacteristic
602C2×C4#54011yescharacteristic
603C2×C4#54011yescharacteristic
604C2×C4#54211yescharacteristic
605C2×C4#54211yescharacteristic
606C2×C4D4211yescharacteristic
607C2×C4#54611yescharacteristic
608C2×C4#54611yescharacteristic
609C2×C4#54711yescharacteristic
610C2×C4#54711yescharacteristic
611C2×C4#55011yescharacteristic
612C2×C4#55311yescharacteristic
613C2×C4#55311yescharacteristic
614C2×C4#55511yescharacteristic
615C2×C4#55611yescharacteristic
616C2×C4#55911yescharacteristic
617C2×C4#55911yescharacteristic
618C8—42no—
619C8—42no—
620C8—82no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22C2×#38111yescharacteristic
637C22D4×C22⋊C411yescharacteristic
638C22#233911yescharacteristic
639C22#242011yescharacteristic
640C22#242211yescharacteristic
641C22#244511yescharacteristic
642C22#244711yescharacteristic
643C22C2×#41611yescharacteristic
644C22C2×#41711yescharacteristic
645C22#289711yescharacteristic
646C22#289811yescharacteristic
647C22C2×#48211yescharacteristic
648C22C2×#48211yescharacteristic
649C22C2×#48511yescharacteristic
650C22C2×#48511yescharacteristic
651C22#309611yescharacteristic
652C22#309711yescharacteristic
653C22#309811yescharacteristic
654C22#309911yescharacteristic
655C22#327211yescharacteristic
656C22#327611yescharacteristic
657C22#328411yescharacteristic
658C22#328711yescharacteristic
659C22#330211yescharacteristic
660C22#330411yescharacteristic
661C22#330411yescharacteristic
662C22#330511yescharacteristic
663C22#332511yescharacteristic
664C22#332911yescharacteristic
665C22#333011yescharacteristic
666C22#333111yescharacteristic
667C22#334011yescharacteristic
668C22#334211yescharacteristic
669C22#335411yescharacteristic
670C22#335711yescharacteristic
671C4—24no—
672C4—44no—
673C4—44no—
674C4—82no—
675C4—84no—
676C4—82no—
677C4—82no—
678C4—162no—
679C4#182518yescomplemented
680C4#229311yescharacteristic
681C4#231911yescharacteristic
682C4#235411yescharacteristic
683C4#235711yescharacteristic
684C4#242011yescharacteristic
685C4#242211yescharacteristic
686C4#244511yescharacteristic
687C4#244711yescharacteristic
688C2—84no—
689C2#1391611yescharacteristic
690C2C2×#182511yescharacteristic
691C2#2070311yescharacteristic
692C2#2241811yescharacteristic
693C2#2243111yescharacteristic
694C2#2249011yescharacteristic
695C2#2250311yescharacteristic
696C2#2496811yescharacteristic
697C2#2497211yescharacteristic
698C2#2501311yescharacteristic
699C2#2501711yescharacteristic
700C2#2623311yescharacteristic
701C2#2623711yescharacteristic
702C2#2629911yescharacteristic
703C2#2630311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1825C2semidirect
5. #11200C2semidirect
6. #11199C2semidirect
7. #10849C2semidirect
8. #10849C2semidirect
9. #10841C2semidirect
10. #10731C2semidirect
11. C4×#340C2semidirect
19. #2354C4semidirect
126. #340D4semidirect
679. C4#1825semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#19542
C2×#18252
#139062
#112002
#111992
#108492
#108412
#107312
C4×#3402