Finite Group Databasebeta

← #373786

Subgroups — #373786

Order 512.

1374 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic
2C2×#1947C211yesmaximal, characteristic
3C2×#1803C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5#14151C211yesmaximal, characteristic
6#11124C212yesmaximal
7#11067C212yesmaximal
8#10907C212yesmaximal
9#10885C212yesmaximal
10#10841C211yesmaximal, characteristic
11#10834C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2355—24no—
20#2355—24no—
21C2×#391C2212yes—
22C2×#389—21no—
23C2×#389C2212yes—
24C2×#387—22no—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#382—21no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1947—24no—
33#1803C418yescomplemented
34C4×#103—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1441—22no—
44C4⋊#95—21no—
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—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42—22no—
71C22×C42—21no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C2×#107—24no—
75C2×#107—24no—
76C2×#107—42no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8C2×C414yes—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#103C2×C418yes—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92#391—24no—
93#391—24no—
94#389—24no—
95#389—42no—
96#389—44no—
97#387—216no—
98#384—22no—
99#384—22no—
100#384C2×C414yes—
101#384D414yes—
102#382—28no—
103C4×C4⋊C4—28no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95D412yes—
118#340—24no—
119#340—24no—
120#340—22no—
121#340—22no—
122#340—42no—
123#340C2×C418yes—
124#336—24no—
125#336—42no—
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×Q8C2×Q812yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
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—21no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8C22×C414yes—
170C22×C8#4112yes—
171C22×C8#4112yes—
172C4×Q8—28no—
173C4×Q8—28no—
174C4×Q8—28no—
175C4×Q8C4⋊C4116yescomplemented
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—81no—
218C2×C4⋊C4—81no—
219C2×C4⋊C4—81no—
220C2×C4⋊C4—81no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4Q1614yes—
223C2×C4⋊C4C22×C414yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×Q812yes—
228C2×C4⋊C4#4112yes—
229C2×C4⋊C4#4112yes—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—28no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—82no—
244C2×C42—82no—
245C2×C42C22×C412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×Q812yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251#107—48no—
252C4⋊C8—28no—
253#103—28no—
254#103—28no—
255#103—48no—
256C4×C8—44no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—24no—
261#95—44no—
262#95—44no—
263#95—42no—
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×Q8C2×C4⋊C414yes—
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—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
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—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4C4×D412yes—
354C22×C4C4×D412yes—
355C22×C4#12012yes—
356C22×C4#12012yes—
357C22×C4C22⋊Q812yes—
358C22×C4C22⋊Q812yes—
359C22×C4C2×Q1614yes—
360C22×C4C2×Q1612yes—
361C22×C4#13512yes—
362C22×C4AGL(1,ℤ/8)12yes—
363C22×C4#13712yes—
364C22×C4C23×C411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×Q811yescharacteristic
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—24no—
379C2×C8—24no—
380C2×C8—24no—
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—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—216no—
403C4⋊C4—28no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—42no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—48no—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4—88no—
426C4⋊C4C2×C4⋊C414yes—
427C42—216no—
428C42—42no—
429C42—42no—
430C42—42no—
431C42—42no—
432C42—88no—
433C42—84no—
434C42C2×C4⋊C414yes—
435C42C4×D414yes—
436C42C22⋊Q814yes—
437C42#12314yes—
438C23C2×C4×D411yescharacteristic
439C23C2×#12011yescharacteristic
440C23C2×C22⋊Q811yescharacteristic
441C23#54611yescharacteristic
442C23#54711yescharacteristic
443C23#54811yescharacteristic
444C23#55011yescharacteristic
445C23C22×Q1611yescharacteristic
446C23C2×#13511yescharacteristic
447C23C2×AGL(1,ℤ/8)11yescharacteristic
448C23C2×#13711yescharacteristic
449C23#57611yescharacteristic
450C23#57711yescharacteristic
451C23#57711yescharacteristic
452C23#57811yescharacteristic
453Q8—24no—
454Q8—24no—
455Q8—42no—
456Q8—42no—
457Q8—42no—
458Q8—42no—
459Q8—42no—
460Q8—42no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
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—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—81no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4#38514yes—
524C2×C4#38514yes—
525C2×C4C4×Q1614yes—
526C2×C4#44014yes—
527C2×C4#44914yes—
528C2×C4#45114yes—
529C2×C4#47714yes—
530C2×C4#47914yes—
531C2×C4C22×C4⋊C411yescharacteristic
532C2×C4C2×C4×D411yescharacteristic
533C2×C4C4×#4111yescharacteristic
534C2×C4C4×#4111yescharacteristic
535C2×C4#51811yescharacteristic
536C2×C4#51911yescharacteristic
537C2×C4C2×C22⋊Q811yescharacteristic
538C2×C4C2×#12311yescharacteristic
539C2×C4#52511yescharacteristic
540C2×C4#52511yescharacteristic
541C2×C4#53311yescharacteristic
542C2×C4#53311yescharacteristic
543C2×C4#53711yescharacteristic
544C2×C4#53711yescharacteristic
545C2×C4#54411yescharacteristic
546C2×C4#54411yescharacteristic
547C2×C4#54611yescharacteristic
548C2×C4#54711yescharacteristic
549C2×C4#54811yescharacteristic
550C2×C4#55011yescharacteristic
551C2×C4#55211yescharacteristic
552C2×C4#55211yescharacteristic
553C2×C4#55311yescharacteristic
554C2×C4#55311yescharacteristic
555C2×C4#55511yescharacteristic
556C2×C4#55511yescharacteristic
557C2×C4#55611yescharacteristic
558C2×C4#55611yescharacteristic
559C8—44no—
560C8—42no—
561C8—42no—
562C22C2×#38511yescharacteristic
563C22#236211yescharacteristic
564C22#236311yescharacteristic
565C22#240711yescharacteristic
566C22#240711yescharacteristic
567C22#243911yescharacteristic
568C22#244111yescharacteristic
569C22C2×C4×Q1611yescharacteristic
570C22C2×#44011yescharacteristic
571C22#294611yescharacteristic
572C22#294711yescharacteristic
573C22C2×#45111yescharacteristic
574C22C2×#44911yescharacteristic
575C22#301211yescharacteristic
576C22#301311yescharacteristic
577C22C2×#47711yescharacteristic
578C22C2×#47911yescharacteristic
579C22#308311yescharacteristic
580C22#308411yescharacteristic
581C22#329911yescharacteristic
582C22#330211yescharacteristic
583C22#332411yescharacteristic
584C22#333311yescharacteristic
585C22#333811yescharacteristic
586C22#334011yescharacteristic
587C22#335311yescharacteristic
588C22#335611yescharacteristic
589C22#336111yescharacteristic
590C22#336311yescharacteristic
591C22#337311yescharacteristic
592C22#337411yescharacteristic
593C22#339411yescharacteristic
594C22#339511yescharacteristic
595C22#339911yescharacteristic
596C22#340011yescharacteristic
597C4—24no—
598C4—44no—
599C4—44no—
600C4—44no—
601C4—82no—
602C4—82no—
603C4—84no—
604C4—82no—
605C4—84no—
606C4#186318yes—
607C4#229611yescharacteristic
608C4#231611yescharacteristic
609C4#236211yescharacteristic
610C4#236311yescharacteristic
611C4#243911yescharacteristic
612C4#244111yescharacteristic
613C4#248511yescharacteristic
614C4#248511yescharacteristic
615C2#1412911yescharacteristic
616C2C2×#186311yescharacteristic
617C2#2087811yescharacteristic
618C2#2306311yescharacteristic
619C2#2306811yescharacteristic
620C2#2309611yescharacteristic
621C2#2310111yescharacteristic
622C2#2411911yescharacteristic
623C2#2412211yescharacteristic
624C2#2418611yescharacteristic
625C2#2419111yescharacteristic
626C2#2602311yescharacteristic
627C2#2602611yescharacteristic
628C2#2608911yescharacteristic
629C2#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
33. #1803C4semidirect
175. C4×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23552
C2×#19472
C2×#18032
C2×C4×#1032
#141512
#111242
#110672
#109072
#108852
#108412
#108342