Finite Group Databasebeta

← #184749

Subgroups — #184749

Order 512.

3602 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#520C211yesmaximal, characteristic
2#21738C212yesmaximal, complemented
3C2×#1459C212yesmaximal, complemented
4C22×#324C212yesmaximal, complemented
5#8303C212yesmaximal, complemented
6#8301C212yesmaximal, complemented
7#8300C212yesmaximal, complemented
8#8299C212yesmaximal, complemented
9C23×#41C411yescharacteristic
10C2×#520—28no—
11C2×#520—28no—
12C2×#520—28no—
13C22×C4×Q8C411yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C411yescharacteristic
16C22×C4×D4C411yescharacteristic
17C22×C4×D4C411yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#117C411yescharacteristic
22C22×#117C411yescharacteristic
23C22×#117C2211yescharacteristic
24C23×C4⋊C4C411yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C22×C22⋊C8C2214yes—
28#1869—28no—
29#1817C2214yes—
30#1779C2214yes—
31#1459—28no—
32C2×#324—28no—
33C2×#324C2214yes—
34C2×#324C2214yes—
35C2×#324C2214yes—
36C2×#323C2214yes—
37C22×#41—24no—
38C22×#41—24no—
39C22×#41—24no—
40C22×#41—24no—
41C22×#41—24no—
42C22×#41—24no—
43C22×#41C818yescomplemented
44C23×Q8C2×C411yescharacteristic
45C23×D4C2×C411yescharacteristic
46C23×D4C2×C411yescharacteristic
47C23×D4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4C2×C411yescharacteristic
54C24×C4C2×C411yescharacteristic
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C23×C8—24no—
58#520—264no—
59#520—264no—
60C2×C4×Q8—24no—
61C2×C4×Q8—24no—
62C2×C4×Q8—24no—
63C2×C4×Q8—24no—
64C2×C4×Q8—24no—
65C2×C4×Q8—24no—
66C2×C4×D4—24no—
67C2×C4×D4—24no—
68C2×C4×D4—24no—
69C2×C4×D4—24no—
70C2×C4×D4—24no—
71C2×C4×D4—24no—
72C2×C4×D4—24no—
73C2×C4×D4—24no—
74C2×C4×D4—24no—
75C2×C4×D4—24no—
76C2×C4×D4—24no—
77C2×C4×D4—24no—
78C2×C4×D4—24no—
79C2×C4×D4—24no—
80C2×C4×D4—24no—
81C2×C4×D4—24no—
82C2×C4×D4D418yes—
83C2×C4×D4D418yes—
84C2×#117—24no—
85C2×#117—24no—
86C2×#117—24no—
87C2×#117—24no—
88C2×#117—24no—
89C2×#117—24no—
90C2×#117—24no—
91C2×#117—24no—
92C2×#117—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4C2×C411yescharacteristic
102C22×C4⋊C4C2×C411yescharacteristic
103C22×C4⋊C4C2×C411yescharacteristic
104C22×C4⋊C4C2×C411yescharacteristic
105C22×C4⋊C4C2×C411yescharacteristic
106C22×C4⋊C4C2×C411yescharacteristic
107C22×C4⋊C4C2×C411yescharacteristic
108C22×C4⋊C4D418yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2×C411yescharacteristic
113C22×C22⋊C4C2×C411yescharacteristic
114C22×C22⋊C4C2×C411yescharacteristic
115C22×C22⋊C4C2×C411yescharacteristic
116C22×C22⋊C4C2×C411yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C42C2×C411yescharacteristic
119C22×C42C2×C411yescharacteristic
120C22×C42C2×C411yescharacteristic
121C22×C42C2×C411yescharacteristic
122C22×C42C2×C411yescharacteristic
123C22×C42C2311yescharacteristic
124C2×C22⋊C8—28no—
125C2×C22⋊C8—28no—
126C2×C22⋊C8—24no—
127C2×C22⋊C8—28no—
128C2×C22⋊C8C2318yes—
129#336—28no—
130#324—28no—
131#324—28no—
132#324—28no—
133#323—28no—
134C25C22×C411yescharacteristic
135C25C22×C411yescharacteristic
136C25C22×C411yescharacteristic
137C2×#41—232no—
138C2×#41—232no—
139C2×#41—232no—
140C2×#41—232no—
141C2×#41—232no—
142C2×#41—232no—
143C2×#41—232no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8C2×C814yescomplemented
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4C22⋊C414yes—
166C22×D4C22⋊C414yes—
167C22×D4C2×C814yes—
168C22×D4C2×C814yes—
169C22×D4C2×C814yescomplemented
170C22×D4M1614yes—
171C22×D4M1614yes—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—82no—
201C23×C4—82no—
202C23×C4—82no—
203C23×C4—82no—
204C23×C4—82no—
205C23×C4—82no—
206C23×C4C22⋊C414yes—
207C23×C4C22⋊C414yes—
208C23×C4C22⋊C414yes—
209C23×C4C2×C814yes—
210C23×C4C2×C814yescomplemented
211C23×C4C2×C814yescomplemented
212C23×C4M1614yes—
213C23×C4C22×C411yescharacteristic
214C23×C4C22×C411yescharacteristic
215C23×C4C22×C411yescharacteristic
216C23×C4C22×C411yescharacteristic
217C23×C4C22×C411yescharacteristic
218C23×C4C22×C411yescharacteristic
219C23×C4C22×C411yescharacteristic
220C23×C4C22×C411yescharacteristic
221C23×C4C22×C411yescharacteristic
222C23×C4C22×C411yescharacteristic
223C23×C4C22×C411yescharacteristic
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2411yescharacteristic
229C22×C8—28no—
230C22×C8—48no—
231C22×C8—88no—
232C4×Q8—232no—
233C4×Q8—232no—
234C4×D4—232no—
235C4×D4—232no—
236C4×D4—232no—
237C4×D4—232no—
238C4×D4—216no—
239C4×D4—216no—
240C4×D4—216no—
241C4×D4—216no—
242#117—232no—
243#117—232no—
244#117—232no—
245C2×C4⋊C4—232no—
246C2×C4⋊C4—232no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—216no—
266C2×C4⋊C4—216no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4C22⋊C414yes—
274C2×C4⋊C4C22⋊C414yes—
275C2×C4⋊C4C22⋊C414yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D414yes—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4C22⋊C414yes—
294C2×C22⋊C4C22⋊C414yes—
295C2×C22⋊C4C22⋊C414yes—
296C2×C22⋊C4C2×D414yes—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42C22⋊C414yes—
314C2×C42C2×D414yes—
315C22⋊C8—28no—
316C22⋊C8—28no—
317C22⋊C8—48no—
318C22⋊C8—88no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24—42no—
332C24—42no—
333C24C2×C22⋊C412yes—
334C24C2×C22⋊C412yes—
335C24C2×C22⋊C412yes—
336C24C2×C22⋊C412yes—
337C24C2×C22⋊C412yes—
338C24C22×C812yes—
339C24C22×C812yes—
340C24C22×C812yes—
341C24C2×M1612yes—
342C24C2×M1612yes—
343C24C2×M1612yes—
344C24C2×M1612yes—
345C24C2×M1612yes—
346C24C23×C411yescharacteristic
347#41—2256no—
348C2×Q8—28no—
349C2×Q8—28no—
350C2×Q8—28no—
351C2×Q8—28no—
352C2×Q8—28no—
353C2×Q8—28no—
354C2×Q8—28no—
355C2×D4—28no—
356C2×D4—28no—
357C2×D4—28no—
358C2×D4—28no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—28no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—28no—
374C2×D4C22⋊C8116yes—
375C2×D4C22⋊C8116yescomplemented
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—48no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—48no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—84no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4C22⋊C8116yescomplemented
456C22×C4C2×C22⋊C412yes—
457C22×C4C2×C22⋊C412yes—
458C22×C4C2×C22⋊C412yes—
459C22×C4C2×C22⋊C412yes—
460C22×C4C2×C22⋊C412yes—
461C22×C4C2×C22⋊C412yes—
462C22×C4C2×C22⋊C412yes—
463C22×C4C2×C22⋊C412yes—
464C22×C4C2×C22⋊C412yes—
465C22×C4C2×C22⋊C412yes—
466C22×C4C2×C22⋊C412yes—
467C22×C4C2×C22⋊C412yes—
468C22×C4C2×C22⋊C412yes—
469C22×C4#12018yes—
470C22×C4#12018yes—
471C22×C4C22×C812yes—
472C22×C4C22×C812yes—
473C22×C4C22×C812yes—
474C22×C4C22×C812yes—
475C22×C4C2×M1612yes—
476C22×C4C2×M1612yes—
477C22×C4C2×M1612yes—
478C22×C4C2×M1612yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C2×C8—416no—
483C2×C8—48no—
484C2×C8—816no—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C42—28no—
502C42—28no—
503C42—28no—
504C42—28no—
505C42—28no—
506C42—28no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—22no—
524C23—22no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23#37914yes—
552C23C2×C22⋊C814yes—
553C23C2×C22⋊C814yes—
554C23C2×C22⋊C814yes—
555C23C2×C22⋊C814yes—
556C23C2×C22⋊C814yes—
557C23#40714yes—
558C23#40714yes—
559C23#40714yes—
560C23C22×C22⋊C411yescharacteristic
561C23C22×C22⋊C411yescharacteristic
562C23C22×C22⋊C411yescharacteristic
563C23C23×C811yescharacteristic
564C23C22×M1611yescharacteristic
565C23C22×M1611yescharacteristic
566C23C22×M1611yescharacteristic
567Q8—232no—
568D4—232no—
569D4—232no—
570D4—232no—
571C2×C4—232no—
572C2×C4—232no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—232no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—48no—
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×C4—42no—
629C2×C4—42no—
630C2×C4—88no—
631C2×C4—88no—
632C2×C4—88no—
633C2×C4—88no—
634C2×C4—88no—
635C2×C4#37914yes—
636C2×C4C2×C22⋊C814yes—
637C2×C4C2×C22⋊C814yes—
638C2×C4C2×C22⋊C814yes—
639C2×C4C2×C22⋊C814yes—
640C2×C4C2×#12014yes—
641C8—88no—
642C22—21no—
643C22—21no—
644C22—24no—
645C22—21no—
646C22—21no—
647C22—24no—
648C22—22no—
649C22—22no—
650C22—28no—
651C22—22no—
652C22—22no—
653C22—28no—
654C22—24no—
655C22—24no—
656C22—28no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22#177918yes—
677C22C2×#37911yescharacteristic
678C22C22×C22⋊C811yescharacteristic
679C22C22×C22⋊C811yescharacteristic
680C22C22×C22⋊C811yescharacteristic
681C22C2×#40711yescharacteristic
682C22C2×#40711yescharacteristic
683C22C2×#40711yescharacteristic
684C22#288312yes—
685C22#288912yes—
686C4—24no—
687C4—24no—
688C4—24no—
689C4—24no—
690C4—24no—
691C4—24no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—42no—
698C2—21no—
699C2—21no—
700C2—21no—
701C2—21no—
702C2—28no—
703C2—44no—
704C2—44no—
705C2#1735112yes—
706C2#1736812yes—
707C2C2×#177911yescharacteristic
708C2#2058511yescharacteristic
709C2#2059311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC23×#41C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21738C2semidirect
3. C2×#1459C2semidirect
4. C22×#324C2semidirect
5. #8303C2semidirect
6. #8301C2semidirect
7. #8300C2semidirect
8. #8299C2semidirect
43. C22×#41C8semidirect
150. C22×Q8C2×C8semidirect
169. C22×D4C2×C8semidirect
210. C23×C4C2×C8semidirect
211. C23×C4C2×C8semidirect
375. C2×D4C22⋊C8semidirect
455. C22×C4C22⋊C8semidirect

Maximal subgroups

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

subgroupindex
C22×#5202
#217382
C2×#14592
C22×#3242
#83032
#83012
#83002
#82992