Finite Group Databasebeta

← #357130

Subgroups — #357130

Order 512.

1558 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1924C211yesmaximal, characteristic
2C2×#1923C211yesmaximal, characteristic
3C2×#1771C211yesmaximal, characteristic, complemented
4C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
5C2×#1749C211yesmaximal, characteristic
6C2×C4×C22⋊C8C211yesmaximal, characteristic
7C4×#385C211yesmaximal, characteristic
8#10922C211yesmaximal, characteristic
9#10921C211yesmaximal, characteristic
10#10877C211yesmaximal, characteristic, complemented
11#10876C211yesmaximal, characteristic, complemented
12#10875C211yesmaximal, characteristic, complemented
13#10874C211yesmaximal, characteristic, complemented
14#10696C211yesmaximal, characteristic, complemented
15#10696C211yesmaximal, characteristic, complemented
16C23×C42C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C2×#385C412yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36#1924C418yescomplemented
37#1923C418yescomplemented
38#1771—24no—
39C8×C4⋊C4—24no—
40#1749—24no—
41C4×C22⋊C8—24no—
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44C2×#336C2211yescharacteristic
45C2×#336C2211yescharacteristic
46C2×#336C2211yescharacteristic
47C2×#336C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52C2×#336C2211yescharacteristic
53C2×#336C2211yescharacteristic
54C2×#336C2211yescharacteristic
55C2×#336C2211yescharacteristic
56#1436C412yes—
57#1436C412yes—
58C22⋊#95C412yes—
59C4×#95C2211yescharacteristic
60C4×#95C2211yescharacteristic
61C24×C4C2×C412yes—
62C24×C4C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C42—22no—
77C22×C42—22no—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42—41no—
81C22×C42—41no—
82C22×C42C2×C412yes—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8C2×C414yescomplemented
94C2×C4⋊C8C2×C414yescomplemented
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8C2×C414yes—
104C2×C22⋊C8C2×C414yes—
105C2×C22⋊C8C2×C414yes—
106C2×C22⋊C8C2×C414yes—
107C2×#97—22no—
108C2×#97—22no—
109C2×#97—22no—
110C2×#97—22no—
111C2×#97—22no—
112C2×#97—22no—
113C2×#97—22no—
114C2×#97—22no—
115C2×#97—41no—
116C2×#97—41no—
117C2×#97—41no—
118C2×#97C2×C414yescomplemented
119C2×C4×C8—22no—
120C2×C4×C8—22no—
121C2×C4×C8—22no—
122C2×C4×C8—22no—
123C2×C4×C8—22no—
124C2×C4×C8—22no—
125C2×C4×C8—22no—
126C2×C4×C8—22no—
127C2×C4×C8—41no—
128C2×C4×C8—41no—
129C2×C4×C8—41no—
130C2×C4×C8C2×C414yescomplemented
131#385—28no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C22⋊C4—41no—
137C4×C22⋊C4—41no—
138C4×C22⋊C4C2×C414yes—
139C4×C22⋊C4C2×C414yes—
140C2×#95C2×C412yes—
141C2×#95C2×C412yes—
142C2×#95C2×C412yes—
143C2×#95C2×C412yes—
144C2×#95C2×C412yes—
145C2×#95C2×C412yes—
146C2×#95C2×C412yes—
147C2×#95C2×C411yescharacteristic
148C2×#95C2×C411yescharacteristic
149#336—22no—
150#336—22no—
151#336—22no—
152#336—22no—
153#336—22no—
154#336—22no—
155#336—41no—
156#336—41no—
157#336C2×C414yescomplemented
158#336C2×C414yescomplemented
159#336C2×C414yescomplemented
160#336C2×C414yescomplemented
161#336D414yes—
162#336D414yes—
163C25C22×C411yescharacteristic
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4C22×C412yes—
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C412yes—
189C23×C4C22×C412yes—
190C23×C4C22×C412yes—
191C23×C4C22×C412yes—
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C412yes—
194C23×C4C2411yescharacteristic
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8C22×C412yes—
236C22×C8C22×C412yes—
237C22×C8C22×C412yes—
238C22×C8C22×C412yes—
239C22×C8C22×C412yes—
240C22×C8C22×C412yes—
241C22×C8C22×C412yes—
242C22×C8C22×C412yes—
243C22×C8C2×D412yes—
244C22×C8C2×D412yes—
245C22×C8C2×D412yes—
246C22×C8C2×D412yes—
247C22×C8#4112yes—
248C22×C8#4112yes—
249C22×C8#4112yes—
250C22×C8#4112yes—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4M1618yes—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4C4214yes—
263C2×C22⋊C4C4214yes—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—42no—
299C2×C42C22×C412yes—
300C2×C42C22×C412yes—
301C2×C42C22×C412yes—
302C2×C42C22×C412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42#4112yes—
306C2×C42#4112yes—
307C4⋊C8—28no—
308C4⋊C8—28no—
309C22⋊C8—28no—
310C22⋊C8—28no—
311C22⋊C8—28no—
312C22⋊C8—28no—
313#97—28no—
314#97—44no—
315#97—44no—
316#97—44no—
317C4×C8—28no—
318C4×C8—44no—
319C4×C8—44no—
320C4×C8—44no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—42no—
327#95—42no—
328#95C4214yes—
329#95C4214yes—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—21no—
334C24—21no—
335C24—21no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24C2×C4212yes—
340C24#13114yes—
341C24C23×C411yescharacteristic
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—216no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—48no—
421C22×C4C2×C4212yes—
422C22×C4C2×C4212yes—
423C22×C4C2×C4212yes—
424C22×C4C2×C4212yes—
425C22×C4C2×C4212yes—
426C22×C4C4×D414yes—
427C22×C4C4×D414yes—
428C22×C4C2×M1614yes—
429C22×C4C2×M1614yes—
430C22×C4C2×M1614yes—
431C22×C4#13114yes—
432C22×C4C23×C411yescharacteristic
433C22×C4C23×C411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4H(5,𝔽2)11yescharacteristic
443C22×C4H(5,𝔽2)11yescharacteristic
444C22×C42−1+411yescharacteristic
445C22×C42−1+411yescharacteristic
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8C4×D414yes—
499C2×C8C4×D414yes—
500C2×C8C4×D414yes—
501C2×C8C4×D414yes—
502C4⋊C4—216no—
503C4⋊C4—48no—
504C22⋊C4—28no—
505C22⋊C4—28no—
506C22⋊C4—48no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—48no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23#40514yes—
555C23C22×C4211yescharacteristic
556C23C2×C4×D411yescharacteristic
557C23C2×C4×D411yescharacteristic
558C23C4×#4111yescharacteristic
559C23C4×#4111yescharacteristic
560C23#51811yescharacteristic
561C23#51911yescharacteristic
562C23C22×M1612yes—
563C23C2×#13112yes—
564C23C2×#13112yes—
565C23#56812yes—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—432no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4C4×M1614yes—
629C2×C4#43518yes—
630C2×C4C2×C4×D411yescharacteristic
631C2×C4C2×C4×D411yescharacteristic
632C2×C4C2×C4×D411yescharacteristic
633C2×C4C2×C4×D411yescharacteristic
634C2×C4C4×#4111yescharacteristic
635C2×C4C4×#4111yescharacteristic
636C2×C4C4×#4111yescharacteristic
637C2×C4C4×#4111yescharacteristic
638C2×C4#52011yescharacteristic
639C2×C4#52011yescharacteristic
640C2×C4#52011yescharacteristic
641C2×C4#52011yescharacteristic
642C2×C4D4211yescharacteristic
643C2×C4#54611yescharacteristic
644C2×C4#54611yescharacteristic
645C2×C4#54711yescharacteristic
646C2×C4#54711yescharacteristic
647C2×C4#54811yescharacteristic
648C2×C4#54811yescharacteristic
649C2×C4#55111yescharacteristic
650C2×C4#55111yescharacteristic
651C2×C4#55211yescharacteristic
652C2×C4#55211yescharacteristic
653C2×C4#55311yescharacteristic
654C2×C4#55311yescharacteristic
655C2×C4#55611yescharacteristic
656C2×C4#55611yescharacteristic
657C2×C4#55911yescharacteristic
658C8—24no—
659C8—24no—
660C8—24no—
661C8—24no—
662C8—42no—
663C8—42no—
664C8—42no—
665C8—42no—
666C22—24no—
667C22—24no—
668C22—24no—
669C22—44no—
670C22—42no—
671C22—44no—
672C22—44no—
673C22—42no—
674C22—42no—
675C22—44no—
676C22—42no—
677C22C42×D411yescharacteristic
678C22C42×D411yescharacteristic
679C22#227411yescharacteristic
680C22#235811yescharacteristic
681C22#236211yescharacteristic
682C22#236211yescharacteristic
683C22#237411yescharacteristic
684C22C2×C4×M1611yescharacteristic
685C22C2×#40511yescharacteristic
686C22C4×#13111yescharacteristic
687C22#287511yescharacteristic
688C22C2×#43512yes—
689C22#292912yes—
690C22#293312yes—
691C22D4×M1612yes—
692C22#296512yes—
693C22#296612yes—
694C22#296812yes—
695C22#297112yes—
696C22#297412yes—
697C22#297612yes—
698C22#298412yes—
699C22#298512yes—
700C4—24no—
701C4—28no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—44no—
706C4—44no—
707C4—42no—
708C4—82no—
709C4#235911yescharacteristic
710C4#235911yescharacteristic
711C4#236111yescharacteristic
712C4#236111yescharacteristic
713C4#236711yescharacteristic
714C4#236711yescharacteristic
715C4#237311yescharacteristic
716C4#237311yescharacteristic
717C2—44no—
718C2#1382311yescharacteristic
719C2#2048411yescharacteristic
720C2C4×#43511yescharacteristic
721C2#2049311yescharacteristic
722C2#2049411yescharacteristic
723C2#2053511yescharacteristic
724C2#2054111yescharacteristic
725C2#2284612yes—
726C2#2305112yes—
727C2#2305212yes—
728C2#2329812yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C22×C42
socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#1771C2semidirect
4. C2×C8×C4⋊C4C2semidirect
10. #10877C2semidirect
11. #10876C2semidirect
12. #10875C2semidirect
13. #10874C2semidirect
14. #10696C2semidirect
15. #10696C2semidirect
36. #1924C4semidirect
37. #1923C4semidirect
93. C2×C4⋊C8C2×C4semidirect
94. C2×C4⋊C8C2×C4semidirect
118. C2×#97C2×C4semidirect
130. C2×C4×C8C2×C4semidirect
157. #336C2×C4semidirect
158. #336C2×C4semidirect
159. #336C2×C4semidirect
160. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19242
C2×#19232
C2×#17712
C2×C8×C4⋊C42
C2×#17492
C2×C4×C22⋊C82
C4×#3852
#109222
#109212
#108772
#108762
#108752
#108742
#106962