Finite Group Databasebeta

← #356277

Subgroups — #356277

Order 512.

1508 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×#1803C211yesmaximal, characteristic
3C2×C8×C4⋊C4C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5C4⋊C42C211yesmaximal, characteristic
6#10932C212yesmaximal
7#10926C212yesmaximal
8#10849C211yesmaximal, characteristic
9#10841C211yesmaximal, characteristic
10#10807C212yesmaximal
11C4×#340C212yesmaximal
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17C22×C4×C8C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2355—24no—
20#2355C418yescomplemented
21C2×#391C2212yes—
22C2×#387—22no—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382—21no—
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1803C418yescomplemented
33C8×C4⋊C4—24no—
34C4×#103C418yescomplemented
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#336C2212yes—
43#1436—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×Q8C2×C414yes—
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⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—21no—
69C22×C42—21no—
70C22×C42—22no—
71C22×C42—22no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8C2×C414yes—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103C2×C414yes—
84C2×#103C2×C414yes—
85C2×#103C2×C418yes—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88C2×C4×C8—24no—
89C2×C4×C8—24no—
90C2×C4×C8—42no—
91C2×C4×C8C2×C414yes—
92#391—24no—
93#391C2×C418yes—
94#387—216no—
95#384—22no—
96#384—22no—
97#384C2×C414yes—
98#384C2×C414yes—
99#382—28no—
100C4×C4⋊C4—28no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—42no—
105C4×C4⋊C4—44no—
106C4×C4⋊C4C2×C418yes—
107C4×C4⋊C4C2×C414yes—
108C4×C4⋊C4C2×C414yes—
109C4×C4⋊C4D414yes—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D412yes—
118#340—24no—
119#340—24no—
120#340—42no—
121#340—42no—
122#340C2×C418yes—
123#340C2×C414yes—
124#340C2×C414yes—
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×Q8C22×C412yes—
139C22×Q8C22×C412yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8C22×C412yes—
168C22×C8C22×C414yes—
169C22×C8C22×C412yes—
170C22×C8#4112yes—
171C22×C8#4112yes—
172C4×Q8—28no—
173C4×Q8—28no—
174C4×Q8—28no—
175C4×Q8C42116yescomplemented
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
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—22no—
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—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—81no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4—81no—
213C2×C4⋊C4—81no—
214C2×C4⋊C4C22⋊C414yes—
215C2×C4⋊C4C22⋊C414yes—
216C2×C4⋊C4SD1614yes—
217C2×C4⋊C4Q1614yes—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C414yes—
220C2×C4⋊C4C22×C414yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4#4112yes—
226C2×C4⋊C4#4112yes—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—28no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—44no—
239C2×C42—44no—
240C2×C42—82no—
241C2×C42—82no—
242C2×C42—82no—
243C2×C42C22×C414yes—
244C2×C42C22×C412yes—
245C2×C42C22×C412yes—
246C2×C42C22×C412yes—
247C2×C42C2×D414yes—
248C2×C42C2×D412yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251C4⋊C8—28no—
252#103—28no—
253#103—28no—
254#103—48no—
255C4×C8—28no—
256C4×C8—48no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—24no—
261#95—44no—
262#95—44no—
263#95—82no—
264#95C22⋊C418yes—
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×C4214yes—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
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—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
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—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4C2×C22⋊C414yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C414yes—
353C22×C4C2×C22⋊C412yes—
354C22×C4C4×D412yes—
355C22×C4C4×D412yes—
356C22×C4C4×D412yes—
357C22×C4C4×D412yes—
358C22×C4C2×SD1614yes—
359C22×C4C2×SD1612yes—
360C22×C4C2×Q1614yes—
361C22×C4C2×Q1612yes—
362C22×C4#13512yes—
363C22×C4#13512yes—
364C22×C4C23×C411yescharacteristic
365C22×C4C23×C411yescharacteristic
366C22×C4C23×C411yescharacteristic
367C22×C4C22×D411yescharacteristic
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—22no—
385C2×C8—22no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—22no—
390C2×C8—42no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—41no—
396C2×C8—42no—
397C2×C8—41no—
398C2×C8#11714yes—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—216no—
406C4⋊C4—28no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—41no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C4⋊C4#103116yescomplemented
424C4⋊C4C2×C4214yes—
425C4⋊C4#11714yes—
426C42—24no—
427C42—24no—
428C42—24no—
429C42—24no—
430C42—216no—
431C42—48no—
432C42—88no—
433C42—88no—
434C42C2×C4214yes—
435C42#11714yes—
436C42C4×D414yes—
437C42C4×D414yes—
438C23C22×C22⋊C411yescharacteristic
439C23C2×C4×D411yescharacteristic
440C23C2×C4×D411yescharacteristic
441C23#54611yescharacteristic
442C23#54711yescharacteristic
443C23#54811yescharacteristic
444C23#55011yescharacteristic
445C23C22×SD1611yescharacteristic
446C23C22×Q1611yescharacteristic
447C23C2×#13511yescharacteristic
448C23C2×#13511yescharacteristic
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—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
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—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4C4×C22⋊C414yes—
523C2×C4C4×C22⋊C414yes—
524C2×C4C2×#10318yes—
525C2×C4C2×#10314yes—
526C2×C4#41614yes—
527C2×C4C4×SD1614yes—
528C2×C4C4×SD1614yes—
529C2×C4C4×Q1614yes—
530C2×C4C4×Q1614yes—
531C2×C4C22×C4211yescharacteristic
532C2×C4C2×#11711yescharacteristic
533C2×C4C2×#11711yescharacteristic
534C2×C4C2×#11711yescharacteristic
535C2×C4C2×C4×D411yescharacteristic
536C2×C4C2×C4×D411yescharacteristic
537C2×C4C4×#4111yescharacteristic
538C2×C4C4×#4111yescharacteristic
539C2×C4C4×#4111yescharacteristic
540C2×C4C4×#4111yescharacteristic
541C2×C4#51811yescharacteristic
542C2×C4#51911yescharacteristic
543C2×C4#52011yescharacteristic
544C2×C4#52011yescharacteristic
545C2×C4#52011yescharacteristic
546C2×C4#52011yescharacteristic
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—24no—
560C8—42no—
561C8—44no—
562C22C2×C4×C22⋊C411yescharacteristic
563C22#233911yescharacteristic
564C22#234111yescharacteristic
565C22#235211yescharacteristic
566C22#235411yescharacteristic
567C22#236211yescharacteristic
568C22#236311yescharacteristic
569C22C22×#10311yescharacteristic
570C22C2×#41611yescharacteristic
571C22#289711yescharacteristic
572C22#289811yescharacteristic
573C22C2×C4×SD1611yescharacteristic
574C22C2×C4×SD1611yescharacteristic
575C22C2×C4×Q1611yescharacteristic
576C22C2×C4×Q1611yescharacteristic
577C22#294611yescharacteristic
578C22#294711yescharacteristic
579C22#295111yescharacteristic
580C22#295211yescharacteristic
581C22#329511yescharacteristic
582C22#329911yescharacteristic
583C22#330211yescharacteristic
584C22#330411yescharacteristic
585C22#333511yescharacteristic
586C22#333811yescharacteristic
587C22#334011yescharacteristic
588C22#334211yescharacteristic
589C22#335911yescharacteristic
590C22#336111yescharacteristic
591C22#336311yescharacteristic
592C22#336511yescharacteristic
593C22#339211yescharacteristic
594C22#339411yescharacteristic
595C22#339511yescharacteristic
596C22#339611yescharacteristic
597C4—24no—
598C4—28no—
599C4—44no—
600C4—44no—
601C4—44no—
602C4—44no—
603C4—84no—
604C4—82no—
605C4—84no—
606C4C4×#10318yes—
607C4C4×#11711yescharacteristic
608C4#227411yescharacteristic
609C4#234611yescharacteristic
610C4#234711yescharacteristic
611C4#235211yescharacteristic
612C4#235411yescharacteristic
613C4#236211yescharacteristic
614C4#236311yescharacteristic
615C2#1380111yescharacteristic
616C2C2×C4×#10311yescharacteristic
617C2#2041511yescharacteristic
618C2#2248711yescharacteristic
619C2#2249011yescharacteristic
620C2#2254211yescharacteristic
621C2#2254311yescharacteristic
622C2#2277011yescharacteristic
623C2#2277111yescharacteristic
624C2#2280511yescharacteristic
625C2#2280811yescharacteristic
626C2#2306211yescharacteristic
627C2#2306311yescharacteristic
628C2#2309511yescharacteristic
629C2#2309611yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
LCSG ▷ C2×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
32. #1803C4semidirect
34. C4×#103C4semidirect
175. C4×Q8C42semidirect
423. 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×#18032
C2×C8×C4⋊C42
C2×C4×#1032
C4⋊C422
#109322
#109262
#108492
#108412
#108072
C4×#3402