Finite Group Databasebeta

← #363733

Subgroups — #363733

Order 512.

1420 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic
2C2×#1939C211yesmaximal, characteristic
3C2×#1803C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5#13910C211yesmaximal, characteristic
6#11077C212yesmaximal
7#11050C212yesmaximal
8#10849C211yesmaximal, characteristic
9#10849C212yesmaximal
10#10841C211yesmaximal, characteristic
11#10807C212yesmaximal
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2355—24no—
20#2355C418yescomplemented
21C2×#391C2212yes—
22C2×#389—21no—
23C2×#389—21no—
24C2×#387—22no—
25C2×#384C2211yescharacteristic
26C2×#384C2212yes—
27C2×#384C2211yescharacteristic
28C2×#382—21no—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1939—24no—
34#1803—24no—
35C4×#103—24no—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1444—22no—
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C414yes—
54C2×C4×Q8D414yes—
55C2×C4×Q8Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C42—21no—
71C22×C42—22no—
72C22×C42—21no—
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75C2×C4⋊C8—24no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—42no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—22no—
87C2×#103—24no—
88C2×#103—42no—
89C2×#103—42no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8—22no—
93#391—24no—
94#391C2×C418yes—
95#389—44no—
96#389—44no—
97#387—216no—
98#384—24no—
99#384—22no—
100#384—22no—
101#384—22no—
102#384—42no—
103#384C2×C414yes—
104#382—28no—
105C4×C4⋊C4—28no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4C2×C418yes—
110C4×C4⋊C4C2×C414yes—
111C4×C4⋊C4D414yes—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95D412yes—
119#340—24no—
120#340—24no—
121#340—22no—
122#340—22no—
123#340—24no—
124#340—42no—
125#340—42no—
126#336—24no—
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×Q8—21no—
138C22×Q8C22×C412yes—
139C22×Q8C2×D412yes—
140C22×Q8C2×Q812yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8#4112yes—
172C22×C8#4112yes—
173C4×Q8—28no—
174C4×Q8—28no—
175C4×Q8—28no—
176C4×Q8C4⋊C4116yescomplemented
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—81no—
219C2×C4⋊C4—81no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—81no—
222C2×C4⋊C4—81no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4C22⋊C414yes—
225C2×C4⋊C4C22⋊C414yes—
226C2×C4⋊C4SD1614yes—
227C2×C4⋊C4Q1614yes—
228C2×C4⋊C4C22×C414yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×Q812yes—
232C2×C4⋊C4#4112yes—
233C2×C4⋊C4#4112yes—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—28no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—82no—
246C2×C42—82no—
247C2×C42C22×C414yes—
248C2×C42C22×C412yes—
249C2×C42C2×D414yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×Q812yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255C4⋊C8—48no—
256C4⋊C8—44no—
257#103—28no—
258#103—28no—
259#103—48no—
260C4×C8—44no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—44no—
265#95—42no—
266#95—44no—
267#95C22⋊C418yes—
268C24C22×D411yescharacteristic
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
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—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
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×Q8C2×C4⋊C414yes—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
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—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4C2×C22⋊C414yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C414yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4#12112yes—
358C22×C4#12112yes—
359C22×C4C22⋊Q812yes—
360C22×C4C22⋊Q812yes—
361C22×C4C2×SD1614yes—
362C22×C4C2×SD1612yes—
363C22×C4C2×Q1614yes—
364C22×C4C2×Q1612yes—
365C22×C4AGL(1,ℤ/8)12yes—
366C22×C4#13712yes—
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×Q811yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4H(5,𝔽2)11yescharacteristic
378C22×C4H(5,𝔽2)11yescharacteristic
379C22×C42−1+411yescharacteristic
380C22×C42−1+411yescharacteristic
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—216no—
406C4⋊C4—28no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—48no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—84no—
425C4⋊C4—88no—
426C4⋊C4—84no—
427C4⋊C4—88no—
428C4⋊C4—84no—
429C4⋊C4#103116yescomplemented
430C4⋊C4C2×C4⋊C414yes—
431C42—24no—
432C42—24no—
433C42—24no—
434C42—24no—
435C42—216no—
436C42—88no—
437C42—84no—
438C42C2×C4⋊C414yes—
439C42#11714yes—
440C42#12114yes—
441C42C22⋊Q814yes—
442C23C22×C22⋊C411yescharacteristic
443C23C2×#12111yescharacteristic
444C23C2×C22⋊Q811yescharacteristic
445C23#54611yescharacteristic
446C23#54711yescharacteristic
447C23#54811yescharacteristic
448C23#55011yescharacteristic
449C23C22×SD1611yescharacteristic
450C23C22×Q1611yescharacteristic
451C23C2×AGL(1,ℤ/8)11yescharacteristic
452C23C2×#13711yescharacteristic
453C23#57611yescharacteristic
454C23#57711yescharacteristic
455C23#57711yescharacteristic
456C23#57811yescharacteristic
457Q8—24no—
458Q8—24no—
459Q8—42no—
460Q8—42no—
461Q8—42no—
462Q8—42no—
463Q8—42no—
464Q8—42no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4#38014yes—
527C2×C4#38014yes—
528C2×C4C2×#10318yes—
529C2×C4C2×#10314yes—
530C2×C4#41714yes—
531C2×C4#46014yes—
532C2×C4#46214yes—
533C2×C4#47514yes—
534C2×C4#47714yes—
535C2×C4C22×C4⋊C411yescharacteristic
536C2×C4C2×#11711yescharacteristic
537C2×C4C4×#4111yescharacteristic
538C2×C4C4×#4111yescharacteristic
539C2×C4#52011yescharacteristic
540C2×C4#52011yescharacteristic
541C2×C4C2×#12111yescharacteristic
542C2×C4C2×C22⋊Q811yescharacteristic
543C2×C4#52511yescharacteristic
544C2×C4#52511yescharacteristic
545C2×C4#53311yescharacteristic
546C2×C4#53311yescharacteristic
547C2×C4#53511yescharacteristic
548C2×C4#53611yescharacteristic
549C2×C4#54411yescharacteristic
550C2×C4#54411yescharacteristic
551C2×C4#54611yescharacteristic
552C2×C4#54711yescharacteristic
553C2×C4#54811yescharacteristic
554C2×C4#55011yescharacteristic
555C2×C4#55211yescharacteristic
556C2×C4#55211yescharacteristic
557C2×C4#55311yescharacteristic
558C2×C4#55311yescharacteristic
559C2×C4#55511yescharacteristic
560C2×C4#55511yescharacteristic
561C2×C4#55611yescharacteristic
562C2×C4#55611yescharacteristic
563C8—42no—
564C8—42no—
565C8—82no—
566C22C2×#38011yescharacteristic
567C22#233911yescharacteristic
568C22#234111yescharacteristic
569C22#241411yescharacteristic
570C22#241511yescharacteristic
571C22#243911yescharacteristic
572C22#244111yescharacteristic
573C22C22×#10311yescharacteristic
574C22C2×#41711yescharacteristic
575C22#289711yescharacteristic
576C22#289811yescharacteristic
577C22C2×#46011yescharacteristic
578C22C2×#46211yescharacteristic
579C22#304511yescharacteristic
580C22#304611yescharacteristic
581C22C2×#47511yescharacteristic
582C22C2×#47711yescharacteristic
583C22#308311yescharacteristic
584C22#308411yescharacteristic
585C22#329511yescharacteristic
586C22#329911yescharacteristic
587C22#332411yescharacteristic
588C22#332811yescharacteristic
589C22#333511yescharacteristic
590C22#333811yescharacteristic
591C22#335311yescharacteristic
592C22#335611yescharacteristic
593C22#335911yescharacteristic
594C22#336311yescharacteristic
595C22#337311yescharacteristic
596C22#337511yescharacteristic
597C22#339211yescharacteristic
598C22#339511yescharacteristic
599C22#339911yescharacteristic
600C22#340011yescharacteristic
601C4—24no—
602C4—28no—
603C4—44no—
604C4—44no—
605C4—44no—
606C4—84no—
607C4—82no—
608C4—82no—
609C4—162no—
610C4#180318yes—
611C4#229611yescharacteristic
612C4#231711yescharacteristic
613C4#234611yescharacteristic
614C4#234711yescharacteristic
615C4#241411yescharacteristic
616C4#241511yescharacteristic
617C4#243911yescharacteristic
618C4#244111yescharacteristic
619C2#1395111yescharacteristic
620C2C2×#180311yescharacteristic
621C2#2065211yescharacteristic
622C2#2248711yescharacteristic
623C2#2250411yescharacteristic
624C2#2254211yescharacteristic
625C2#2254611yescharacteristic
626C2#2462811yescharacteristic
627C2#2463211yescharacteristic
628C2#2467811yescharacteristic
629C2#2468211yescharacteristic
630C2#2602211yescharacteristic
631C2#2602611yescharacteristic
632C2#2608811yescharacteristic
633C2#2609211yescharacteristic
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
20. #2355C4semidirect
176. C4×Q8C4⋊C4semidirect
429. C4⋊C4#103semidirect

Maximal subgroups

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

subgroupindex
C2×#23552
C2×#19392
C2×#18032
C2×C4×#1032
#139102
#110772
#110502
#108492
#108412
#108072