Finite Group Databasebeta

← #364995

Subgroups — #364995

Order 512.

2218 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2536C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3C2×#1955C211yesmaximal, characteristic
4C2×#1872C212yesmaximal, complemented
5C2×#1803C211yesmaximal, characteristic, complemented
6C2×#1763C211yesmaximal, characteristic, complemented
7#11199C212yesmaximal, complemented
8#10970C212yesmaximal, complemented
9#10850C211yesmaximal, characteristic, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal, complemented
12C22×C22⋊Q8C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4—21no—
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2211yescharacteristic
21C22×C22⋊C8C2212yes—
22C22×C22⋊C8C2212yes—
23C22×#97C2211yescharacteristic
24#2536—24no—
25#2536C418yescomplemented
26C2×#398—21no—
27C2×#394—21no—
28D4×C4⋊C4—24no—
29D4×C4⋊C4—24no—
30C2×#389—21no—
31C2×#388—22no—
32C2×#385—22no—
33C2×#384C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38#1955—24no—
39#1872—28no—
40#1803—24no—
41#1763—24no—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2212yes—
48C2×#340C2212yes—
49C23×Q8C2311yescharacteristic
50C23×D4D411yescharacteristic
51C24×C4—22no—
52C24×C4C2312yes—
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—42no—
60C2×C22⋊Q8—42no—
61C2×C22⋊Q8C2×C418yescomplemented
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8C2×C414yescomplemented
68C2×C4×D4—28no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—48no—
74C2×C4×D4C2×C414yes—
75C2×C4×D4D414yes—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4D411yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4—22no—
94C22×C22⋊C4—22no—
95C22×C22⋊C4—22no—
96C22×C22⋊C4D412yes—
97C22×C42—21no—
98C22×C42—21no—
99C22×C42C2311yescharacteristic
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—42no—
111C2×#103D414yescomplemented
112C2×C22⋊C8—24no—
113C2×C22⋊C8—24no—
114C2×C22⋊C8—24no—
115C2×C22⋊C8—24no—
116C2×C22⋊C8—42no—
117C2×C22⋊C8—42no—
118C2×#97—22no—
119C2×#97—22no—
120C2×#97—22no—
121#398—28no—
122#394—28no—
123#389—28no—
124#388—216no—
125#385—216no—
126#384—22no—
127#384D414yes—
128#382—22no—
129#382C2×C414yescomplemented
130#380—24no—
131#380—24no—
132#380—24no—
133#380C2×C418yescomplemented
134C4×C4⋊C4—22no—
135C4×C4⋊C4D414yes—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—21no—
139C2×#95—22no—
140C2×#95—21no—
141#340—24no—
142#340—24no—
143#340—24no—
144#340—22no—
145#340—22no—
146#340—42no—
147#340—42no—
148C25C2×D412yes—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8C22×C412yes—
161C22×Q8C2×D412yes—
162C22×Q8#4112yes—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4C22⋊C414yes—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—22no—
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—21no—
184C23×C4—21no—
185C23×C4—22no—
186C23×C4—28no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—82no—
192C23×C4—82no—
193C23×C4C22×C414yes—
194C23×C4C2×D412yes—
195C23×C4C2×D414yes—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2411yescharacteristic
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—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8C2×D412yes—
219C22×C8#4112yes—
220C22⋊Q8—216no—
221C22⋊Q8—48no—
222C22⋊Q8—48no—
223C22⋊Q8—48no—
224C4×Q8—28no—
225C4×Q8—28no—
226C4×Q8—44no—
227C4×Q8—44no—
228C4×D4—28no—
229C4×D4—28no—
230C4×D4—28no—
231C4×D4—28no—
232C4×D4—432no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—48no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—48no—
274C2×C4⋊C4—82no—
275C2×C4⋊C4C22⋊C414yes—
276C2×C4⋊C4C22×C412yes—
277C2×C4⋊C4C22×C414yes—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4#4112yes—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4C22⋊C418yes—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—44no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42C22×C412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42#4112yes—
305C2×C42#4112yes—
306C4⋊C8—44no—
307#103—28no—
308#103—28no—
309#103—48no—
310C22⋊C8—48no—
311C22⋊C8—48no—
312#97—44no—
313#95—28no—
314#95—28no—
315#95—24no—
316#95—24no—
317#95—24no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24C2×C22⋊C414yes—
331C24AGL(1,ℤ/8)14yes—
332C24#13714yes—
333C24C22×D411yescharacteristic
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8C4×D414yescomplemented
362C2×D4—28no—
363C2×D4—216no—
364C2×D4—28no—
365C2×D4—216no—
366C2×D4—28no—
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—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—416no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—416no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4C2×C22⋊C414yes—
449C22×C4C2×C22⋊C412yes—
450C22×C4C2×C22⋊C412yes—
451C22×C4#12012yes—
452C22×C4#12012yes—
453C22×C4#12312yes—
454C22×C4#12312yes—
455C22×C4C23×C411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4111yescharacteristic
461C22×C4C2×#4111yescharacteristic
462C22×C4C2×#4111yescharacteristic
463C22×C4C2×#4111yescharacteristic
464C22×C4C2×#4111yescharacteristic
465C22×C4H(5,𝔽2)11yescharacteristic
466C22×C4H(5,𝔽2)11yescharacteristic
467C22×C4H(5,𝔽2)11yescharacteristic
468C22×C42−1+411yescharacteristic
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—42no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—42no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—42no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—28no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—416no—
503C4⋊C4—416no—
504C4⋊C4—48no—
505C4⋊C4—42no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—48no—
514C4⋊C4—84no—
515C4⋊C4#12114yes—
516C22⋊C4—216no—
517C22⋊C4—416no—
518C22⋊C4—416no—
519C22⋊C4—416no—
520C42—48no—
521C42—48no—
522C42C4×D414yes—
523C42C4×D414yes—
524C42#12114yes—
525C42#12114yes—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23#41718yes—
561C23C22×C22⋊C411yescharacteristic
562C23C2×#12011yescharacteristic
563C23C2×#12311yescharacteristic
564C23D4211yescharacteristic
565C23#54611yescharacteristic
566C23#54611yescharacteristic
567C23#54711yescharacteristic
568C23C2×AGL(1,ℤ/8)12yes—
569C23C2×#13712yes—
570C23#57712yes—
571C23#57812yes—
572Q8—24no—
573Q8—24no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—42no—
579Q8—42no—
580D4—232no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—832no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4#38614yes—
642C2×C4#38614yes—
643C2×C4C2×C4×D411yescharacteristic
644C2×C4C2×C4×D411yescharacteristic
645C2×C4C2×C4×D411yescharacteristic
646C2×C4C4×#4111yescharacteristic
647C2×C4#51811yescharacteristic
648C2×C4#52011yescharacteristic
649C2×C4C2×#12111yescharacteristic
650C2×C4C2×#12111yescharacteristic
651C2×C4C2×#12111yescharacteristic
652C2×C4#52511yescharacteristic
653C2×C4#52911yescharacteristic
654C2×C4#53211yescharacteristic
655C2×C4#53511yescharacteristic
656C2×C4#53711yescharacteristic
657C2×C4#54011yescharacteristic
658C2×C4#54211yescharacteristic
659C2×C4D4211yescharacteristic
660C2×C4D4211yescharacteristic
661C2×C4#54611yescharacteristic
662C2×C4#54611yescharacteristic
663C2×C4#54611yescharacteristic
664C2×C4#54711yescharacteristic
665C2×C4#54711yescharacteristic
666C2×C4#54811yescharacteristic
667C2×C4#55011yescharacteristic
668C2×C4#55311yescharacteristic
669C2×C4#55311yescharacteristic
670C2×C4#55511yescharacteristic
671C8—42no—
672C8—42no—
673C8—82no—
674C22—28no—
675C22—28no—
676C22—28no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22C2×#38611yescharacteristic
690C22D4×C22⋊C411yescharacteristic
691C22#233911yescharacteristic
692C22#240711yescharacteristic
693C22#240811yescharacteristic
694C22#242011yescharacteristic
695C22#242211yescharacteristic
696C22C2×#41712yes—
697C22#289812yes—
698C22#300311yescharacteristic
699C22#300411yescharacteristic
700C22#300911yescharacteristic
701C22#301111yescharacteristic
702C22#309212yes—
703C22#309712yes—
704C22#326512yes—
705C22#327012yes—
706C22#330812yes—
707C22#331012yes—
708C22#331212yes—
709C22#331612yes—
710C22#334312yes—
711C22#334412yes—
712C4—24no—
713C4—24no—
714C4—44no—
715C4—44no—
716C4—44no—
717C4—82no—
718C4—84no—
719C4—82no—
720C4—84no—
721C4C4×#12111yescharacteristic
722C4#232411yescharacteristic
723C4#236111yescharacteristic
724C4#236311yescharacteristic
725C4#242711yescharacteristic
726C4#242811yescharacteristic
727C4#253511yescharacteristic
728C4#253711yescharacteristic
729C2—48no—
730C2#1397011yescharacteristic
731C2#2090911yescharacteristic
732C2#2091311yescharacteristic
733C2#2242112yes—
734C2#2249112yes—
735C2#2413211yescharacteristic
736C2#2413411yescharacteristic
737C2#2425211yescharacteristic
738C2#2425511yescharacteristic
739C2#2497912yes—
740C2#2502412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1872C2semidirect
5. C2×#1803C2semidirect
6. C2×#1763C2semidirect
7. #11199C2semidirect
8. #10970C2semidirect
9. #10850C2semidirect
10. #10843C2semidirect
11. #10730C2semidirect
25. #2536C4semidirect
61. C2×C22⋊Q8C2×C4semidirect
67. C2×C4×Q8C2×C4semidirect
111. C2×#103D4semidirect
129. #382C2×C4semidirect
133. #380C2×C4semidirect
361. C2×Q8C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25362
C2×D4×C4⋊C42
C2×#19552
C2×#18722
C2×#18032
C2×#17632
#111992
#109702
#108502
#108432
#107302