Finite Group Databasebeta

← #416001

Subgroups — #416001

Order 512.

1334 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#1803C211yesmaximal, characteristic
3C2×C4×#103C211yesmaximal, characteristic
4#13926C211yesmaximal, characteristic
5#11293C212yesmaximal
6#11267C212yesmaximal
7#11067C212yesmaximal
8#11050C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10841C211yesmaximal, characteristic
11#10834C211yesmaximal, characteristic
12C22×#125—21no—
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2552—24no—
20#2552—24no—
21C2×#398—22no—
22C2×#391C2212yes—
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#384C2212yes—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1803—24no—
31C4×#103—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1441—22no—
42C4⋊#95—21no—
43C4⋊#95—21no—
44C4⋊#95C2212yes—
45C23×Q8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#125—28no—
50C2×#125—44no—
51C2×#125—44no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8Q814yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—42no—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92#398—216no—
93#391—24no—
94#391—42no—
95#389—28no—
96#389—24no—
97#389—24no—
98#384—24no—
99#384—24no—
100#384—22no—
101#384—22no—
102#384—22no—
103#384D414yes—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4D414yes—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D412yes—
115#340—24no—
116#340—24no—
117#340—22no—
118#340—22no—
119#340—42no—
120#340—42no—
121#340—42no—
122#336—24no—
123#336—42no—
124C22×Q8—21no—
125C22×Q8—21no—
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×Q8C2×Q812yes—
136C22×Q8#4112yes—
137C22×Q8#4112yes—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
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—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8#4112yes—
169C22×C8#4112yes—
170#125—432no—
171C4×Q8—28no—
172C4×Q8—28no—
173C4×Q8—44no—
174C4×Q8—44no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—81no—
224C2×C4⋊C4—81no—
225C2×C4⋊C4—81no—
226C2×C4⋊C4—81no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4SD1614yes—
230C2×C4⋊C4Q1614yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×Q812yes—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×Q812yes—
250C2×C42#4112yes—
251C2×C42#4112yes—
252C2×C42#4112yes—
253C4⋊C8—44no—
254#103—28no—
255#103—28no—
256#103—48no—
257C4×C8—44no—
258#95—28no—
259#95—24no—
260#95—24no—
261#95—44no—
262#95—44no—
263#95—82no—
264#95—82no—
265C24C22×D411yescharacteristic
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
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—41no—
282C2×Q8—41no—
283C2×Q8—41no—
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#12514yescomplemented
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
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—24no—
316C22×C4—22no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4#12012yes—
353C22×C4#12012yes—
354C22×C4C22⋊Q812yes—
355C22×C4C22⋊Q812yes—
356C22×C4#12412yes—
357C22×C4#12412yes—
358C22×C4C2×SD1614yes—
359C22×C4C2×SD1612yes—
360C22×C4C2×Q1614yes—
361C22×C4C2×Q1612yes—
362C22×C4AGL(1,ℤ/8)12yes—
363C22×C4#13712yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×Q811yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4H(5,𝔽2)11yescharacteristic
375C22×C4H(5,𝔽2)11yescharacteristic
376C22×C42−1+411yescharacteristic
377C22×C42−1+411yescharacteristic
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C4⋊C4—28no—
400C4⋊C4—28no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—416no—
410C4⋊C4—416no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—84no—
426C4⋊C4—84no—
427C4⋊C4—88no—
428C4⋊C4—84no—
429C42—42no—
430C42—42no—
431C42—42no—
432C42—42no—
433C42—48no—
434C42—48no—
435C42C22⋊Q814yes—
436C42C22⋊Q814yes—
437C42#12414yes—
438C42#12414yes—
439C23C2×#12011yescharacteristic
440C23C2×C22⋊Q811yescharacteristic
441C23C2×#12411yescharacteristic
442C23#54611yescharacteristic
443C23#54711yescharacteristic
444C23#54811yescharacteristic
445C23#55011yescharacteristic
446C23C22×SD1611yescharacteristic
447C23C22×Q1611yescharacteristic
448C23C2×AGL(1,ℤ/8)11yescharacteristic
449C23C2×#13711yescharacteristic
450C23#57611yescharacteristic
451C23#57711yescharacteristic
452C23#57711yescharacteristic
453C23#57811yescharacteristic
454Q8—24no—
455Q8—24no—
456Q8—42no—
457Q8—42no—
458Q8—42no—
459Q8—42no—
460Q8—42no—
461Q8—42no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
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—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4#39314yes—
525C2×C4#39314yes—
526C2×C4#45014yes—
527C2×C4#45114yes—
528C2×C4#47514yes—
529C2×C4#47714yes—
530C2×C4#48714yes—
531C2×C4#48914yes—
532C2×C4C2×C22⋊Q811yescharacteristic
533C2×C4C2×C22⋊Q811yescharacteristic
534C2×C4#52511yescharacteristic
535C2×C4#52511yescharacteristic
536C2×C4C2×#12411yescharacteristic
537C2×C4C2×#12411yescharacteristic
538C2×C4C2×#12511yescharacteristic
539C2×C4#52911yescharacteristic
540C2×C4#52911yescharacteristic
541C2×C4#53311yescharacteristic
542C2×C4#53711yescharacteristic
543C2×C4#53711yescharacteristic
544C2×C4#54011yescharacteristic
545C2×C4#54111yescharacteristic
546C2×C4#54411yescharacteristic
547C2×C4#54411yescharacteristic
548C2×C4#54611yescharacteristic
549C2×C4#54711yescharacteristic
550C2×C4#54811yescharacteristic
551C2×C4#55011yescharacteristic
552C2×C4#55211yescharacteristic
553C2×C4#55211yescharacteristic
554C2×C4#55211yescharacteristic
555C2×C4#55211yescharacteristic
556C2×C4#55311yescharacteristic
557C2×C4#55311yescharacteristic
558C2×C4#55311yescharacteristic
559C2×C4#55311yescharacteristic
560C8—42no—
561C8—42no—
562C8—82no—
563C22C2×#39311yescharacteristic
564C22#240711yescharacteristic
565C22#240711yescharacteristic
566C22#244111yescharacteristic
567C22#244211yescharacteristic
568C22#248911yescharacteristic
569C22#249011yescharacteristic
570C22C2×#45011yescharacteristic
571C22C2×#45111yescharacteristic
572C22#301211yescharacteristic
573C22#301311yescharacteristic
574C22C2×#47511yescharacteristic
575C22C2×#47711yescharacteristic
576C22#308311yescharacteristic
577C22#308411yescharacteristic
578C22C2×#48711yescharacteristic
579C22C2×#48911yescharacteristic
580C22#313611yescharacteristic
581C22#313711yescharacteristic
582C22#329811yescharacteristic
583C22#329911yescharacteristic
584C22#333211yescharacteristic
585C22#333311yescharacteristic
586C22#333511yescharacteristic
587C22#333811yescharacteristic
588C22#335311yescharacteristic
589C22#335611yescharacteristic
590C22#336211yescharacteristic
591C22#336311yescharacteristic
592C22#336811yescharacteristic
593C22#337411yescharacteristic
594C22#339211yescharacteristic
595C22#339511yescharacteristic
596C22#339911yescharacteristic
597C22#340011yescharacteristic
598C4—24no—
599C4—44no—
600C4—44no—
601C4—44no—
602C4—82no—
603C4—82no—
604C4—84no—
605C4—82no—
606C4—162no—
607C4#202418yes—
608C4#244111yescharacteristic
609C4#244211yescharacteristic
610C4#248911yescharacteristic
611C4#249011yescharacteristic
612C4#256211yescharacteristic
613C4#256211yescharacteristic
614C4#262911yescharacteristic
615C4#285111yescharacteristic
616C2#1521611yescharacteristic
617C2C2×#202411yescharacteristic
618C2#2345811yescharacteristic
619C2#2412011yescharacteristic
620C2#2412211yescharacteristic
621C2#2418811yescharacteristic
622C2#2419111yescharacteristic
623C2#2608811yescharacteristic
624C2#2609211yescharacteristic
625C2#2610811yescharacteristic
626C2#2611211yescharacteristic
627C2#2772911yescharacteristic
628C2#2773711yescharacteristic
629C2#2774911yescharacteristic
630C2#2775811yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×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
293. C2×Q8#125semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#18032
C2×C4×#1032
#139262
#112932
#112672
#110672
#110502
#108492
#108412
#108342