Finite Group Databasebeta

← #393301

Subgroups — #393301

Order 512.

1798 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#405C211yesmaximal, characteristic
2C2×#2396C211yesmaximal, characteristic
3C2×#1985C211yesmaximal, characteristic, complemented
4C2×#1984C211yesmaximal, characteristic, complemented
5C2×#1798C211yesmaximal, characteristic
6#10875C212yesmaximal, complemented
7#10874C212yesmaximal, complemented
8#10835C211yesmaximal, characteristic, complemented
9#10834C211yesmaximal, characteristic, complemented
10#10715C212yesmaximal
11#10714C212yesmaximal
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19C2×#405—24no—
20C2×#405—24no—
21C2×#405—24no—
22C22×#97C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2396—24no—
27#2396—24no—
28C2×#400C411yescharacteristic
29C2×#399C411yescharacteristic
30C2×#398C412yes—
31C2×#396C412yes—
32C2×#395C411yescharacteristic
33C2×#393C411yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C22⋊C4C2212yes—
37#1985—24no—
38#1984—24no—
39#1798—24no—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47C2×#336C2212yes—
48C2×#336C2212yes—
49C23×Q8C2×C411yescharacteristic
50C24×C4C2311yescharacteristic
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16Q814yescomplemented
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—22no—
63C23×C8—22no—
64C23×C8Q814yescomplemented
65C23×C8C2312yes—
66C23×C8C2312yes—
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×#117—22no—
78C2×#117—22no—
79C2×#117—22no—
80C2×#117—22no—
81C2×#117D414yes—
82C2×#117D414yes—
83C22×C4⋊C4C2×C412yes—
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C4⋊C4C2×C412yes—
86C22×C4⋊C4C2×C412yes—
87C22×C4⋊C4C2×C412yes—
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C4⋊C4C2×C411yescharacteristic
90C22×C22⋊C4C2×C412yes—
91C22×C22⋊C4C2×C412yes—
92C22×C22⋊C4C2×C411yescharacteristic
93C22×C22⋊C4C2×C411yescharacteristic
94C22×C42C2312yes—
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—24no—
107#405—232no—
108C2×#97—22no—
109C2×#97—22no—
110C2×#97—22no—
111C2×#97—22no—
112C2×#97—22no—
113C2×#97D414yes—
114C2×C4×C8—22no—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117C2×C4×C8—22no—
118C2×C4×C8—22no—
119C2×C4×C8D414yes—
120#400—24no—
121#399—24no—
122#398—28no—
123#396—28no—
124#395—24no—
125#393—24no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C22⋊C4—24no—
131C4×C22⋊C4—24no—
132C2×#95C2×C412yes—
133C2×#95C2×C411yescharacteristic
134C2×#95C2×C412yes—
135C2×#95C2×C411yescharacteristic
136C2×#95C2×C412yes—
137C2×#95C2×C412yes—
138#336—24no—
139#336—24no—
140#336—24no—
141#336—24no—
142#336—24no—
143#336—24no—
144#336—24no—
145#336—24no—
146#336—24no—
147C25C22×C411yescharacteristic
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—24no—
152C22×Q8—42no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C22×C412yes—
162C23×C4C22×C412yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C412yes—
166C23×C4C22×C412yes—
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×Q812yes—
173C23×C4C2×Q812yes—
174C23×C4#4112yes—
175C23×C4#4112yes—
176C23×C4C2411yescharacteristic
177C2×M16—28no—
178C2×M16—28no—
179C2×M16—28no—
180C2×M16—28no—
181C2×M16—28no—
182C2×M16—28no—
183C2×M16—28no—
184C22×C8—28no—
185C22×C8—28no—
186C22×C8—28no—
187C22×C8—28no—
188C22×C8—28no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
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—28no—
206C22×C8—28no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8C2×D412yes—
222C22×C8C2×D412yes—
223C22×C8C2×D412yes—
224C22×C8C2×D412yes—
225C22×C8C2×Q812yes—
226C22×C8C2×Q812yes—
227C22×C8C2×Q812yes—
228C22×C8C2×Q812yes—
229C22×C8#4112yes—
230C22×C8#4112yes—
231C22×C8#4112yes—
232C22×C8#4112yes—
233C4×Q8—44no—
234C4×Q8—44no—
235C4×Q8—44no—
236C4×Q8—44no—
237#117—28no—
238#117—28no—
239#117—28no—
240#117—28no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4C22⋊C414yes—
261C2×C4⋊C4C22⋊C414yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4C22⋊C414yes—
273C2×C22⋊C4C22⋊C414yes—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42#4112yes—
297C2×C42#4112yes—
298C4⋊C8—44no—
299C4⋊C8—44no—
300C22⋊C8—48no—
301#97—28no—
302#97—28no—
303C4×C8—28no—
304C4×C8—28no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—22no—
309#95—22no—
310#95—24no—
311C24—21no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24C2×C22⋊C412yes—
320C24C4×Q812yes—
321C24C4×Q812yes—
322C24C23×C411yescharacteristic
323C2×Q8—28no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—44no—
327C2×Q8—44no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
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×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C412yes—
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C412yes—
398C22×C4C2×C22⋊C412yes—
399C22×C4C4×D412yes—
400C22×C4C4×D412yes—
401C22×C4C4×D412yes—
402C22×C4C4×D412yes—
403C22×C4C4×Q812yes—
404C22×C4C4×Q812yes—
405C22×C4C22⋊Q814yes—
406C22×C4C22⋊Q814yes—
407C22×C4#12414yes—
408C22×C4C4⋊Q814yes—
409C22×C4#13114yes—
410C22×C4#13114yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×Q811yescharacteristic
416C22×C4C22×Q811yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425M16—232no—
426C2×C8—232no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8C22⋊Q814yes—
466C2×C8C22⋊Q814yes—
467C2×C8C22⋊Q814yes—
468C2×C8C22⋊Q814yes—
469C4⋊C4—28no—
470C4⋊C4—28no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C22⋊C4—28no—
477C22⋊C4—28no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—42no—
485C42—42no—
486C42—42no—
487C42—42no—
488C42—42no—
489C42—42no—
490C42—42no—
491C42—42no—
492C42#12014yes—
493C42#12114yes—
494C42#12114yes—
495C42#12314yes—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23#39114yes—
519C23C22×C22⋊C411yescharacteristic
520C23C2×C4×D411yescharacteristic
521C23C2×C4×D411yescharacteristic
522C23C2×C4×Q811yescharacteristic
523C23C2×C4×Q811yescharacteristic
524C23C4×#4111yescharacteristic
525C23C4×#4111yescharacteristic
526C23C2×#13112yes—
527C23C2×#13112yes—
528C23#56812yes—
529C23#56812yes—
530Q8—416no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4#38614yes—
592C2×C4#38614yes—
593C2×C4#39114yes—
594C2×C4C2×#12011yescharacteristic
595C2×C4C2×#12111yescharacteristic
596C2×C4C2×#12111yescharacteristic
597C2×C4C2×C22⋊Q811yescharacteristic
598C2×C4C2×C22⋊Q811yescharacteristic
599C2×C4C2×C22⋊Q811yescharacteristic
600C2×C4C2×C22⋊Q811yescharacteristic
601C2×C4C2×C22⋊Q811yescharacteristic
602C2×C4C2×C22⋊Q811yescharacteristic
603C2×C4C2×#12311yescharacteristic
604C2×C4#52511yescharacteristic
605C2×C4#52511yescharacteristic
606C2×C4#52511yescharacteristic
607C2×C4#52511yescharacteristic
608C2×C4#52511yescharacteristic
609C2×C4#52511yescharacteristic
610C2×C4C2×#12411yescharacteristic
611C2×C4#52911yescharacteristic
612C2×C4#52911yescharacteristic
613C2×C4#52911yescharacteristic
614C2×C4#52911yescharacteristic
615C2×C4C2×C4⋊Q811yescharacteristic
616C2×C4#53211yescharacteristic
617C2×C4#53211yescharacteristic
618C2×C4#53311yescharacteristic
619C2×C4#53311yescharacteristic
620C2×C4#53311yescharacteristic
621C2×C4#53311yescharacteristic
622C8—24no—
623C8—24no—
624C8—24no—
625C8—24no—
626C8—82no—
627C8—82no—
628C22—28no—
629C22—24no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—42no—
638C22—42no—
639C22C2×#38611yescharacteristic
640C22C2×#39111yescharacteristic
641C22#229711yescharacteristic
642C22C4×C22⋊Q811yescharacteristic
643C22C4×C22⋊Q811yescharacteristic
644C22C4×C22⋊Q811yescharacteristic
645C22C4×C22⋊Q811yescharacteristic
646C22#288012yes—
647C22#288012yes—
648C22#292911yescharacteristic
649C22#293011yescharacteristic
650C22#293111yescharacteristic
651C22#293211yescharacteristic
652C22#293312yes—
653C22#293512yes—
654C22#296012yes—
655C22#296112yes—
656C22#296212yes—
657C22#296212yes—
658C22#296612yes—
659C22#296712yes—
660C22#296812yes—
661C22#296912yes—
662C4—24no—
663C4—24no—
664C4—24no—
665C4—44no—
666C4—42no—
667C4—44no—
668C4—42no—
669C4—82no—
670C4—82no—
671C4#284811yescharacteristic
672C4#284811yescharacteristic
673C4#285011yescharacteristic
674C4#285011yescharacteristic
675C4#285511yescharacteristic
676C4#285511yescharacteristic
677C4#285711yescharacteristic
678C4#285711yescharacteristic
679C2—44no—
680C2#1470211yescharacteristic
681C2#2089911yescharacteristic
682C2#2090011yescharacteristic
683C2#2130412yes—
684C2#2138112yes—
685C2#2160111yescharacteristic
686C2#2160211yescharacteristic
687C2#2160311yescharacteristic
688C2#2160411yescharacteristic
689C2#2160512yes—
690C2#2160612yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C24 ▷ 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×#1985C2semidirect
4. C2×#1984C2semidirect
6. #10875C2semidirect
7. #10874C2semidirect
8. #10835C2semidirect
9. #10834C2semidirect
57. C22×M16Q8semidirect
64. C23×C8Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4052
C2×#23962
C2×#19852
C2×#19842
C2×#17982
#108752
#108742
#108352
#108342
#107152
#107142