Finite Group Databasebeta

← #365293

Subgroups — #365293

Order 512.

1282 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1935C212yesmaximal
2C2×#1803C211yesmaximal, characteristic
3C2×#1803C211yesmaximal, characteristic
4C2×#1767C211yesmaximal, characteristic
5#14712C211yesmaximal, characteristic
6C4×#391C211yesmaximal, characteristic
7#11315C212yesmaximal
8#11190C212yesmaximal
9#10881C212yesmaximal
10#10849C211yesmaximal, characteristic
11#10849C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#97C2212yes—
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#389—21no—
21C2×#389—21no—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C43C2211yescharacteristic
27#1935—28no—
28#1803—24no—
29#1803C418yescomplemented
30#1767—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1436—21no—
39#1436—21no—
40C4⋊#95—21no—
41C4⋊#95—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C4×#95—21no—
46C4×#95—21no—
47C23×Q8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8C2×C414yes—
56C22×C4⋊C4—21no—
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×C42—21no—
68C22×C42—21no—
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8C2×C414yes—
81C2×C4⋊C8D414yes—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—42no—
87C2×#103C2×C418yes—
88C2×#97—24no—
89C2×#97—24no—
90C2×#97—42no—
91#391—22no—
92#391—22no—
93#391—22no—
94#391D414yes—
95#389—24no—
96#389—24no—
97#389—44no—
98#389—44no—
99#384—22no—
100#384—22no—
101#384—22no—
102#384C2×C414yes—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117C43—22no—
118C43—22no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—42no—
124#340C2×C418yes—
125#336—24no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8C22×C412yes—
138C22×Q8C2×D412yes—
139C22×Q8#4112yes—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8C22×C414yes—
166C22×C8C2×D414yes—
167C4×Q8—28no—
168C4×Q8—28no—
169C4×Q8—44no—
170C4×Q8—44no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C414yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4#4112yes—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—24no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42C22×C412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42#4112yes—
251C2×C42#4112yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255C4⋊C8—28no—
256C4⋊C8—28no—
257#103—216no—
258#103—48no—
259#97—48no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—42no—
265#95—42no—
266#95—44no—
267#95—44no—
268#95—42no—
269#95—42no—
270#95—44no—
271#95—42no—
272C24C22×D411yescharacteristic
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8C4×D414yes—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4C4×D412yes—
372C22×C4C4×D412yes—
373C22×C4#12112yes—
374C22×C4#12112yes—
375C22×C4#12112yes—
376C22×C4#12112yes—
377C22×C4#12412yes—
378C22×C4#12412yes—
379C22×C4AGL(1,ℤ/8)12yes—
380C22×C4AGL(1,ℤ/8)12yes—
381C22×C4#13712yes—
382C22×C4#13712yes—
383C22×C4#13712yes—
384C22×C4#13712yes—
385C22×C4C23×C411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4H(5,𝔽2)11yescharacteristic
396C22×C42−1+411yescharacteristic
397C22×C42−1+411yescharacteristic
398C22×C42−1+411yescharacteristic
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—48no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—84no—
437C4⋊C4—88no—
438C4⋊C4—84no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—44no—
444C42—44no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—48no—
450C42—44no—
451C42—44no—
452C42—84no—
453C42#11714yes—
454C42C4×D414yes—
455C42#12314yes—
456C42#12614yes—
457C23C2×C4×D411yescharacteristic
458C23C2×#12111yescharacteristic
459C23C2×#12111yescharacteristic
460C23#52511yescharacteristic
461C23C2×#12411yescharacteristic
462C23#53611yescharacteristic
463C23#53711yescharacteristic
464C23C2×AGL(1,ℤ/8)11yescharacteristic
465C23C2×AGL(1,ℤ/8)11yescharacteristic
466C23C2×#13711yescharacteristic
467C23C2×#13711yescharacteristic
468C23C2×#13711yescharacteristic
469C23C2×#13711yescharacteristic
470C23#57511yescharacteristic
471C23#57511yescharacteristic
472Q8—24no—
473Q8—24no—
474Q8—42no—
475Q8—42no—
476Q8—42no—
477Q8—42no—
478Q8—42no—
479Q8—42no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4#38814yes—
557C2×C4#38814yes—
558C2×C4#44014yes—
559C2×C4#44114yes—
560C2×C4#46814yes—
561C2×C4#47014yes—
562C2×C4#48914yes—
563C2×C4#49114yes—
564C2×C4C2×#11711yescharacteristic
565C2×C4C2×C4×D411yescharacteristic
566C2×C4C2×C4×D411yescharacteristic
567C2×C4C4×#4111yescharacteristic
568C2×C4#51911yescharacteristic
569C2×C4#52011yescharacteristic
570C2×C4C2×#12311yescharacteristic
571C2×C4C2×#12611yescharacteristic
572C2×C4#52911yescharacteristic
573C2×C4#53211yescharacteristic
574C2×C4#54111yescharacteristic
575C2×C4#54211yescharacteristic
576C2×C4D4211yescharacteristic
577C2×C4#54711yescharacteristic
578C2×C4#54711yescharacteristic
579C2×C4#54711yescharacteristic
580C2×C4#54711yescharacteristic
581C2×C4#54811yescharacteristic
582C2×C4#54811yescharacteristic
583C2×C4#55211yescharacteristic
584C2×C4#55211yescharacteristic
585C2×C4#55211yescharacteristic
586C2×C4#55311yescharacteristic
587C2×C4#55311yescharacteristic
588C2×C4#55311yescharacteristic
589C2×C4#55611yescharacteristic
590C2×C4#55611yescharacteristic
591C2×C4#55611yescharacteristic
592C8—44no—
593C8—44no—
594C22C2×#38811yescharacteristic
595C22C4×#12111yescharacteristic
596C22#232311yescharacteristic
597C22#241511yescharacteristic
598C22#241611yescharacteristic
599C22#254711yescharacteristic
600C22#256211yescharacteristic
601C22C2×#44011yescharacteristic
602C22C2×#44111yescharacteristic
603C22C4×AGL(1,ℤ/8)11yescharacteristic
604C22C4×#13711yescharacteristic
605C22#303711yescharacteristic
606C22#303711yescharacteristic
607C22#303811yescharacteristic
608C22#303911yescharacteristic
609C22C2×#46811yescharacteristic
610C22C2×#47011yescharacteristic
611C22#305511yescharacteristic
612C22#305611yescharacteristic
613C22#311211yescharacteristic
614C22#311311yescharacteristic
615C22#311511yescharacteristic
616C22#311611yescharacteristic
617C22C2×#48911yescharacteristic
618C22C2×#49111yescharacteristic
619C22#314111yescharacteristic
620C22#314211yescharacteristic
621C22#323711yescharacteristic
622C22#323711yescharacteristic
623C22#323811yescharacteristic
624C22#323911yescharacteristic
625C22#325111yescharacteristic
626C22#325211yescharacteristic
627C22#325411yescharacteristic
628C22#325511yescharacteristic
629C4—24no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4—42no—
634C4—42no—
635C4—84no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—162no—
640C4#193518yescomplemented
641C4#234411yescharacteristic
642C4#234711yescharacteristic
643C4#236111yescharacteristic
644C4#236211yescharacteristic
645C4#242111yescharacteristic
646C4#242211yescharacteristic
647C4#249411yescharacteristic
648C4#249611yescharacteristic
649C2#1397411yescharacteristic
650C2C2×#193511yescharacteristic
651C2#2109311yescharacteristic
652C2#2150511yescharacteristic
653C2#2150611yescharacteristic
654C2#2211811yescharacteristic
655C2#2211911yescharacteristic
656C2#2471011yescharacteristic
657C2#2471111yescharacteristic
658C2#2476011yescharacteristic
659C2#2476111yescharacteristic
660C2#2872511yescharacteristic
661C2#2872611yescharacteristic
662C2#2913111yescharacteristic
663C2#2913211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
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
29. #1803C4semidirect
640. C4#1935semidirect

Maximal subgroups

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

subgroupindex
C2×#19352
C2×#18032
C2×#17672
#147122
C4×#3912
#113152
#111902
#108812
#108492