Finite Group Databasebeta

← #360049

Subgroups — #360049

Order 512.

1614 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q16C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic
3C2×#1803C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5#13926C211yesmaximal, characteristic
6#11190C212yesmaximal
7#11024C212yesmaximal
8#10932C212yesmaximal
9#10907C212yesmaximal
10#10841C211yesmaximal, characteristic
11#10834C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×C4×Q16—24no—
16C2×C4×Q16—24no—
17C2×C4×Q16C418yescomplemented
18C22×#107C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2212yes—
24C22×C4×C8C2211yescharacteristic
25#2355—24no—
26#2355—24no—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#387—22no—
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#382—21no—
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1803—24no—
37#1803—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1441—22no—
45C4⋊#95—21no—
46C4⋊#95—21no—
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C23×Q8C2311yescharacteristic
50C22×Q16—22no—
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—22no—
54C22×Q16—22no—
55C22×Q16—22no—
56C22×Q16C2×C414yes—
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8C2×C414yes—
69C2×C4×Q8C2×C414yes—
70C2×C4×Q8D414yes—
71C2×C4×Q8D414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C42—21no—
85C22×C42—21no—
86C22×C42—22no—
87C22×C42C2312yes—
88C22×C42C2311yescharacteristic
89C4×Q16—232no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#107C2×C414yes—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—42no—
104C2×#103—42no—
105C2×#103—42no—
106C2×#103C2×C414yes—
107C2×#103C2×C414yes—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8C2×C414yes—
111#391—24no—
112#391—24no—
113#391—24no—
114#391—42no—
115#387—216no—
116#384—22no—
117#384—22no—
118#384—22no—
119#384D414yes—
120#382—28no—
121C4×C4⋊C4—28no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95D412yes—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—42no—
138#336—24no—
139#336—42no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8C22×C412yes—
163C22×Q8C22×C412yes—
164C22×Q8C2×D412yes—
165C22×Q8C2×D412yes—
166C22×Q8#4112yes—
167C22×Q8#4112yes—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2411yescharacteristic
179C2×Q16—28no—
180C2×Q16—28no—
181C2×Q16—28no—
182C2×Q16—28no—
183C2×Q16—28no—
184C2×Q16—28no—
185C2×Q16—28no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8C22×C412yes—
204C22×C8C22×C412yes—
205C22×C8#4112yes—
206C22×C8#4112yes—
207C4×Q8—28no—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—28no—
211C4×Q8—28no—
212C4×Q8—44no—
213C4×Q8—44no—
214C4×Q8C22⋊C4116yes—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—81no—
254C2×C4⋊C4—81no—
255C2×C4⋊C4—81no—
256C2×C4⋊C4—81no—
257C2×C4⋊C4Q1614yescomplemented
258C2×C4⋊C4C22×C412yes—
259C2×C4⋊C4C22×C412yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—28no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—82no—
277C2×C42—82no—
278C2×C42C22×C412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42#4112yes—
283C2×C42#4112yes—
284#107—28no—
285C4⋊C8—44no—
286#103—28no—
287#103—28no—
288#103—48no—
289#103—48no—
290C4×C8—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—24no—
295#95—44no—
296#95—44no—
297#95—82no—
298#95—82no—
299C24C22×D411yescharacteristic
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8C2×C22⋊C414yes—
355C2×Q8#11714yes—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4C4×D412yes—
416C22×C4C4×D412yes—
417C22×C4#12012yes—
418C22×C4#12012yes—
419C22×C4#12112yes—
420C22×C4#12112yes—
421C22×C4C2×Q1614yes—
422C22×C4C2×Q1612yes—
423C22×C4#13512yes—
424C22×C4#13712yes—
425C22×C4#13712yes—
426C22×C4C23×C411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4H(5,𝔽2)11yescharacteristic
437C22×C4H(5,𝔽2)11yescharacteristic
438C22×C42−1+411yescharacteristic
439C22×C42−1+411yescharacteristic
440Q16—232no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8#11714yes—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—216no—
466C4⋊C4—28no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4C2×C22⋊C414yes—
486C42—216no—
487C42—42no—
488C42—42no—
489C42—42no—
490C42—42no—
491C42—48no—
492C42—88no—
493C42C2×C22⋊C414yes—
494C42C4×D414yes—
495C42#12114yes—
496C42#12314yes—
497C23C2×C4×D411yescharacteristic
498C23C2×#12011yescharacteristic
499C23C2×#12111yescharacteristic
500C23#54611yescharacteristic
501C23#54711yescharacteristic
502C23#54811yescharacteristic
503C23#55011yescharacteristic
504C23C22×Q1611yescharacteristic
505C23C2×#13511yescharacteristic
506C23C2×#13711yescharacteristic
507C23C2×#13711yescharacteristic
508C23#57611yescharacteristic
509C23#57711yescharacteristic
510C23#57711yescharacteristic
511C23#57811yescharacteristic
512Q8—24no—
513Q8—24no—
514Q8—24no—
515Q8—24no—
516Q8—42no—
517Q8—42no—
518Q8—42no—
519Q8—42no—
520Q8—42no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—42no—
526Q8—42no—
527Q8—42no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4#38614yes—
591C2×C4#38614yes—
592C2×C4C4×Q1614yes—
593C2×C4#44114yes—
594C2×C4#45114yes—
595C2×C4#45214yes—
596C2×C4#46214yes—
597C2×C4#46414yes—
598C2×C4C22×C22⋊C411yescharacteristic
599C2×C4C2×#11711yescharacteristic
600C2×C4C2×#11711yescharacteristic
601C2×C4C2×C4×D411yescharacteristic
602C2×C4#52011yescharacteristic
603C2×C4#52011yescharacteristic
604C2×C4C2×#12111yescharacteristic
605C2×C4C2×#12311yescharacteristic
606C2×C4#52511yescharacteristic
607C2×C4#52511yescharacteristic
608C2×C4#52511yescharacteristic
609C2×C4#52511yescharacteristic
610C2×C4#53511yescharacteristic
611C2×C4#53611yescharacteristic
612C2×C4#53711yescharacteristic
613C2×C4#53711yescharacteristic
614C2×C4#54611yescharacteristic
615C2×C4#54711yescharacteristic
616C2×C4#54811yescharacteristic
617C2×C4#55011yescharacteristic
618C2×C4#55211yescharacteristic
619C2×C4#55211yescharacteristic
620C2×C4#55311yescharacteristic
621C2×C4#55311yescharacteristic
622C2×C4#55511yescharacteristic
623C2×C4#55511yescharacteristic
624C2×C4#55611yescharacteristic
625C2×C4#55611yescharacteristic
626C8—24no—
627C8—42no—
628C8—82no—
629C22C2×#38611yescharacteristic
630C22#235211yescharacteristic
631C22#235411yescharacteristic
632C22#240711yescharacteristic
633C22#240711yescharacteristic
634C22#241411yescharacteristic
635C22#241511yescharacteristic
636C22C2×C4×Q1611yescharacteristic
637C22C2×#44111yescharacteristic
638C22#295111yescharacteristic
639C22#295211yescharacteristic
640C22C2×#45111yescharacteristic
641C22C2×#45211yescharacteristic
642C22#301211yescharacteristic
643C22#301311yescharacteristic
644C22C2×#46211yescharacteristic
645C22C2×#46411yescharacteristic
646C22#304511yescharacteristic
647C22#304611yescharacteristic
648C22#329911yescharacteristic
649C22#330411yescharacteristic
650C22#332811yescharacteristic
651C22#333311yescharacteristic
652C22#333811yescharacteristic
653C22#334211yescharacteristic
654C22#335611yescharacteristic
655C22#335611yescharacteristic
656C22#336311yescharacteristic
657C22#336511yescharacteristic
658C22#337411yescharacteristic
659C22#337511yescharacteristic
660C22#339511yescharacteristic
661C22#339611yescharacteristic
662C22#340011yescharacteristic
663C22#340011yescharacteristic
664C4—24no—
665C4—44no—
666C4—44no—
667C4—44no—
668C4—44no—
669C4—82no—
670C4—84no—
671C4—82no—
672C4—162no—
673C4#187818yes—
674C4#228011yescharacteristic
675C4#231111yescharacteristic
676C4#235211yescharacteristic
677C4#235411yescharacteristic
678C4#241411yescharacteristic
679C4#241511yescharacteristic
680C4#248511yescharacteristic
681C4#248511yescharacteristic
682C2#1387111yescharacteristic
683C2C2×#187811yescharacteristic
684C2#2093711yescharacteristic
685C2#2276311yescharacteristic
686C2#2277111yescharacteristic
687C2#2280811yescharacteristic
688C2#2281411yescharacteristic
689C2#2412211yescharacteristic
690C2#2412311yescharacteristic
691C2#2419011yescharacteristic
692C2#2419111yescharacteristic
693C2#2463211yescharacteristic
694C2#2463311yescharacteristic
695C2#2468211yescharacteristic
696C2#2468311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. C2×C4×Q16C4semidirect
257. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q162
C2×#23552
C2×#18032
#139262
#111902
#110242
#109322
#109072
#108412
#108342