Finite Group Databasebeta

← #373667

Subgroups — #373667

Order 512.

2476 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#103C211yesmaximal, characteristic
2C2×Q8×C22⋊C4C211yesmaximal, characteristic
3C2×#1863C212yesmaximal, complemented
4C2×#1852C211yesmaximal, characteristic
5C2×#1787C211yesmaximal, characteristic
6#14130C211yesmaximal, characteristic
7#11124C212yesmaximal, complemented
8#11067C212yesmaximal, complemented
9#10789C212yesmaximal, complemented
10#10721C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#107C2212yes—
18C22×#103—28no—
19C22×#103—24no—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25Q8×C22⋊C4—24no—
26Q8×C22⋊C4—24no—
27C2×#391—21no—
28C2×#391—21no—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#389C2212yes—
32C2×#385—21no—
33C2×#380—21no—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×C4×C22⋊C4—21no—
37C2×C4×C22⋊C4C2211yescharacteristic
38#1863—28no—
39#1852—24no—
40#1787C418yescomplemented
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1444—21no—
47#1442—21no—
48C4⋊#95C2212yes—
49C22⋊#95—21no—
50C23×Q8—22no—
51C23×Q8—21no—
52C23×Q8—22no—
53C23×Q8—22no—
54C23×Q8—24no—
55C23×Q8—24no—
56C23×Q8—42no—
57C23×Q8—42no—
58C23×Q8C2×C414yes—
59C23×Q8D414yes—
60C23×Q8Q814yes—
61C23×Q8C2311yescharacteristic
62C23×Q8C2311yescharacteristic
63C24×C4—21no—
64C24×C4—21no—
65C24×C4C2311yescharacteristic
66C23×C8—22no—
67C23×C8—24no—
68C23×C8—22no—
69C23×C8—22no—
70C23×C8—42no—
71C23×C8C2312yes—
72C23×C8C2312yes—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—24no—
78C2×C4×Q8—24no—
79C2×C4×Q8—42no—
80C2×C4×Q8—42no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4—21no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4D411yescharacteristic
96C22×C22⋊C4D411yescharacteristic
97C22×C42—21no—
98C22×C42—21no—
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C2×#107—24no—
102C2×#107—24no—
103C2×#107—42no—
104C2×#103—232no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103—24no—
112C2×#103—42no—
113C2×#103—42no—
114C2×#103C2×C418yescomplemented
115C2×#103C2×C418yescomplemented
116C2×#103D418yes—
117C2×C22⋊C8—22no—
118C2×C22⋊C8—22no—
119C2×C22⋊C8C2×C414yes—
120#391—28no—
121#391—28no—
122#391—22no—
123#391—22no—
124#391—22no—
125#391—22no—
126#389—24no—
127#389—42no—
128#385—44no—
129#380—28no—
130#380—22no—
131#380—22no—
132#380C2×C414yes—
133#380D414yes—
134C4×C22⋊C4—28no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95D411yescharacteristic
147C2×#95D411yescharacteristic
148#340—24no—
149#340—24no—
150#340—42no—
151#336—24no—
152#336—42no—
153C25C2×D411yescharacteristic
154C22×Q8—24no—
155C22×Q8—28no—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—28no—
159C22×Q8—28no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—216no—
169C22×Q8—28no—
170C22×Q8—216no—
171C22×Q8—28no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—44no—
177C22×Q8—44no—
178C22×Q8—48no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×Q8—48no—
182C22×Q8—48no—
183C22×Q8—48no—
184C22×Q8—48no—
185C22×Q8—48no—
186C22×Q8—48no—
187C22×Q8—48no—
188C22×Q8C4⋊C4116yescomplemented
189C22×Q8C22×C412yes—
190C22×Q8C22×C412yes—
191C22×Q8C2×D412yes—
192C22×Q8C2×D412yes—
193C22×Q8C2×D414yes—
194C22×Q8C2×Q812yes—
195C22×Q8C2×Q812yes—
196C22×Q8#4114yes—
197C23×C4—21no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—21no—
206C23×C4—22no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—22no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—82no—
217C23×C4C22×C412yes—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D412yes—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D412yes—
225C23×C4C2×Q812yes—
226C23×C4#4112yes—
227C23×C4#4112yes—
228C23×C4C2411yescharacteristic
229C22×C8—28no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—48no—
243C22×C8—48no—
244C22×C8—44no—
245C22×C8—44no—
246C22×C8C22×C414yes—
247C22×C8C2×D414yes—
248C4×Q8—48no—
249C4×Q8—48no—
250C4×Q8—44no—
251C4×Q8—44no—
252C4×Q8—44no—
253C4×Q8—44no—
254C2×C4⋊C4—28no—
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—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—48no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4C22×C414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4Q1614yes—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—44no—
309C2×C42—44no—
310#107—48no—
311#103—216no—
312#103—216no—
313#103—216no—
314#103—48no—
315C22⋊C8—28no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—22no—
323#95—22no—
324#95—44no—
325#95—44no—
326#95—82no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24C4×D412yes—
332C24#12012yes—
333C24C22⋊Q812yes—
334C24C2×Q1612yes—
335C24#13512yes—
336C24AGL(1,ℤ/8)12yes—
337C24#13712yes—
338C24C22×D411yescharacteristic
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—24no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—216no—
356C2×Q8—232no—
357C2×Q8—24no—
358C2×Q8—24no—
359C2×Q8—22no—
360C2×Q8—24no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—24no—
365C2×Q8—416no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—416no—
369C2×Q8—416no—
370C2×Q8—432no—
371C2×Q8—416no—
372C2×Q8—432no—
373C2×Q8—416no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8C2×C4⋊C414yes—
381C2×Q8C2×C4⋊C414yes—
382C2×Q8C4×D418yes—
383C2×Q8#12014yes—
384C2×Q8C22⋊Q818yes—
385C2×Q8#12314yes—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—216no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—416no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—416no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—48no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—88no—
461C22×C4—88no—
462C22×C4—84no—
463C22×C4—84no—
464C22×C4C2×C4⋊C414yes—
465C22×C4C4×D414yes—
466C22×C4C4×D412yes—
467C22×C4#12012yes—
468C22×C4C22⋊Q814yes—
469C22×C4C22⋊Q812yes—
470C22×C4#12314yes—
471C22×C4C2×Q1612yes—
472C22×C4C2×Q1612yes—
473C22×C4C23×C411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×Q811yescharacteristic
478C22×C4C2×#4112yes—
479C22×C4C2×#4111yescharacteristic
480C22×C4C2×#4111yescharacteristic
481C22×C4H(5,𝔽2)12yes—
482C22×C4H(5,𝔽2)12yes—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—416no—
490C2×C8—42no—
491C2×C8—42no—
492C2×C8—42no—
493C2×C8—42no—
494C2×C8—42no—
495C2×C8—42no—
496C2×C8—42no—
497C2×C8—42no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—42no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—88no—
516C22⋊C4—28no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—48no—
520C42—48no—
521C42—48no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—42no—
527C42—42no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23#38514yes—
545C23C4×Q1614yes—
546C23#44014yes—
547C23#44914yes—
548C23#45114yes—
549C23#47714yes—
550C23#47914yes—
551C23C2×C4×D411yescharacteristic
552C23C2×#12011yescharacteristic
553C23C2×C22⋊Q811yescharacteristic
554C23D4212yes—
555C23#54612yes—
556C23C22×Q1611yescharacteristic
557C23C2×#13511yescharacteristic
558C23C2×AGL(1,ℤ/8)11yescharacteristic
559C23C2×#13711yescharacteristic
560C23#57712yes—
561C23#57812yes—
562Q8—24no—
563Q8—28no—
564Q8—24no—
565Q8—28no—
566Q8—24no—
567Q8—24no—
568Q8—24no—
569Q8—432no—
570Q8—44no—
571Q8—42no—
572Q8—44no—
573Q8—42no—
574Q8#38518yescomplemented
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—432no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—816no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
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#38514yes—
642C2×C4C22×C4⋊C411yescharacteristic
643C2×C4C2×C4×D411yescharacteristic
644C2×C4C2×C4×D412yes—
645C2×C4#51812yes—
646C2×C4C2×#12011yescharacteristic
647C2×C4C2×C22⋊Q811yescharacteristic
648C2×C4C2×C22⋊Q812yes—
649C2×C4C2×#12311yescharacteristic
650C2×C4C2×#12311yescharacteristic
651C2×C4#53411yescharacteristic
652C2×C4#53411yescharacteristic
653C2×C4#54312yes—
654C2×C4D4212yes—
655C2×C4#54612yes—
656C2×C4#54612yes—
657C2×C4#54612yes—
658C2×C4#55112yes—
659C2×C4#55112yes—
660C8—44no—
661C8—44no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—28no—
666C22—28no—
667C22—28no—
668C22—28no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22#186318yescomplemented
673C22C2×#38511yescharacteristic
674C22#235812yes—
675C22#240311yescharacteristic
676C22#240511yescharacteristic
677C22#243712yes—
678C22C2×C4×Q1611yescharacteristic
679C22C2×#44011yescharacteristic
680C22#294712yes—
681C22C2×#45111yescharacteristic
682C22C2×#44911yescharacteristic
683C22#301011yescharacteristic
684C22#301111yescharacteristic
685C22C2×#47711yescharacteristic
686C22C2×#47911yescharacteristic
687C22#308212yes—
688C22#326512yes—
689C22#327012yes—
690C22#328212yes—
691C22D4×Q1612yes—
692C22#329912yes—
693C22#330012yes—
694C22#330812yes—
695C22#332012yes—
696C4—24no—
697C4—28no—
698C4—44no—
699C4—42no—
700C4—42no—
701C4—81no—
702C4—82no—
703C4—84no—
704C4—84no—
705C4—81no—
706C4—82no—
707C4C2×#38511yescharacteristic
708C4#231511yescharacteristic
709C4#235812yes—
710C4#240511yescharacteristic
711C4#243712yes—
712C4#248311yescharacteristic
713C2—28no—
714C2#1412711yescharacteristic
715C2C2×#186311yescharacteristic
716C2#2087711yescharacteristic
717C2#2285512yes—
718C2#2287112yes—
719C2#2385611yescharacteristic
720C2#2385811yescharacteristic
721C2#2403511yescharacteristic
722C2#2404211yescharacteristic
723C2#2595712yes—
724C2#2596312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1863C2semidirect
7. #11124C2semidirect
8. #11067C2semidirect
9. #10789C2semidirect
40. #1787C4semidirect
114. C2×#103C2×C4semidirect
115. C2×#103C2×C4semidirect
188. C22×Q8C4⋊C4semidirect
574. Q8#385semidirect
672. C22#1863semidirect

Maximal subgroups

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

subgroupindex
C23×#1032
C2×Q8×C22⋊C42
C2×#18632
C2×#18522
C2×#17872
#141302
#111242
#110672
#107892
#107212
#107142