Finite Group Databasebeta

← #356086

Subgroups — #356086

Order 512.

2594 conjugacy classes of subgroups.

The subgroup lattice has 696 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×#1787C211yesmaximal, characteristic
4C2×#1763C212yesmaximal, complemented
5C2×#1752C211yesmaximal, characteristic
6C22⋊C4×C4⋊C4C211yesmaximal, characteristic
7#10932C212yesmaximal, complemented
8#10928C212yesmaximal, complemented
9#10789C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C23×C4⋊C4C2211yescharacteristic
16C22×#103—28no—
17C22×#103C418yescomplemented
18C22×#103C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C22×#97C2212yes—
26Q8×C22⋊C4—24no—
27Q8×C22⋊C4C418yescomplemented
28C2×#391—22no—
29C2×#391C2212yes—
30C2×#380—21no—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C22⋊C4—21no—
35C2×C4×C22⋊C4—21no—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1787C418yescomplemented
38#1763—28no—
39#1752—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45#1436—22no—
46C4⋊#95C2212yes—
47C22⋊#95—21no—
48C23×Q8—22no—
49C23×Q8—21no—
50C23×Q8—22no—
51C23×Q8—22no—
52C23×Q8—24no—
53C23×Q8—24no—
54C23×Q8—42no—
55C23×Q8—42no—
56C23×Q8C2×C414yes—
57C23×Q8C2×C414yes—
58C23×Q8C2×C414yes—
59C23×Q8C2311yescharacteristic
60C23×Q8C2311yescharacteristic
61C24×C4—21no—
62C24×C4—21no—
63C24×C4C2311yescharacteristic
64C23×C8—22no—
65C23×C8—22no—
66C23×C8—24no—
67C23×C8—42no—
68C23×C8C2×C414yes—
69C23×C8C2312yes—
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C2×C4×Q8—24no—
73C2×C4×Q8—24no—
74C2×C4×Q8—24no—
75C2×C4×Q8—42no—
76C2×C4×Q8—42no—
77C2×C4×Q8C2×C418yescomplemented
78C22×C4⋊C4—24no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—42no—
83C22×C4⋊C4C2×C414yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4D411yescharacteristic
91C22×C22⋊C4D411yescharacteristic
92C22×C42—22no—
93C22×C42—22no—
94C22×C42C2312yes—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—232no—
101C2×#103—42no—
102C2×#103—42no—
103C2×#103C2×C414yes—
104C2×#103C2×C414yes—
105C2×#103C2×C414yes—
106C2×#103C2×C414yes—
107C2×#103C2×C418yescomplemented
108C2×#103C2×C418yescomplemented
109C2×#103D418yes—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8C2×C414yes—
113C2×#97—24no—
114C2×#97—24no—
115C2×#97—42no—
116#391—24no—
117#391—216no—
118#391C2×C418yescomplemented
119#380—22no—
120#380—22no—
121#380—28no—
122#380C2×C414yes—
123#380C2×C414yes—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—42no—
126C4×C22⋊C4—28no—
127C4×C22⋊C4—44no—
128C4×C22⋊C4C2×C414yes—
129C4×C22⋊C4D414yes—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95D412yes—
138#340—24no—
139#340—24no—
140#340—42no—
141#340—42no—
142#336—24no—
143C25C2×D411yescharacteristic
144C22×Q8—24no—
145C22×Q8—28no—
146C22×Q8—24no—
147C22×Q8—24no—
148C22×Q8—28no—
149C22×Q8—28no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—28no—
159C22×Q8—28no—
160C22×Q8—216no—
161C22×Q8—216no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—44no—
167C22×Q8—44no—
168C22×Q8—44no—
169C22×Q8—44no—
170C22×Q8—48no—
171C22×Q8—48no—
172C22×Q8—48no—
173C22×Q8—48no—
174C22×Q8—48no—
175C22×Q8—48no—
176C22×Q8—48no—
177C22×Q8—48no—
178C22×Q8C42116yescomplemented
179C22×Q8C22×C412yes—
180C22×Q8C22×C412yes—
181C22×Q8C22×C412yes—
182C22×Q8C22×C412yes—
183C22×Q8C22×C412yes—
184C22×Q8C22×C412yes—
185C22×Q8C2×D414yes—
186C22×Q8#4114yes—
187C23×C4—21no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—21no—
200C23×C4—24no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—82no—
207C23×C4C22×C412yes—
208C23×C4C22×C412yes—
209C23×C4C22×C412yes—
210C23×C4C2×D412yes—
211C23×C4C2×D412yes—
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D412yes—
214C23×C4#4112yes—
215C23×C4#4112yes—
216C23×C4C2411yescharacteristic
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—22no—
220C22×C8—28no—
221C22×C8—21no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—21no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—28no—
232C22×C8—44no—
233C22×C8—48no—
234C22×C8—48no—
235C22×C8C22×C412yes—
236C22×C8C22×C414yes—
237C22×C8C22×C412yes—
238C22×C8C2×D414yes—
239C4×Q8—216no—
240C4×Q8—48no—
241C4×Q8—48no—
242C4×Q8—48no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
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—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C22×C412yes—
267C2×C4⋊C4C22×C414yes—
268C2×C4⋊C4C22×C414yes—
269C2×C4⋊C4C22×C412yes—
270C2×C4⋊C4C2×D414yes—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4C22⋊C414yes—
282C2×C22⋊C4C22⋊C414yes—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—28no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—82no—
293C2×C42C22×C414yes—
294C2×C42C2×D414yes—
295#103—28no—
296#103—216no—
297#103—28no—
298#103—28no—
299#103—48no—
300#103C22⋊C4116yescomplemented
301C22⋊C8—28no—
302#97—48no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—44no—
308#95—44no—
309#95—82no—
310#95C22⋊C418yes—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24C2×C22⋊C412yes—
316C24C4×D412yes—
317C24C4×D412yes—
318C24AGL(1,ℤ/8)12yes—
319C24#13712yes—
320C24#13712yes—
321C24#13712yes—
322C24C22×D411yescharacteristic
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—24no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—216no—
340C2×Q8—232no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—416no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—416no—
353C2×Q8—416no—
354C2×Q8—416no—
355C2×Q8—432no—
356C2×Q8—416no—
357C2×Q8—432no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8C2×C4214yes—
365C2×Q8C2×C4214yes—
366C2×Q8C2×C22⋊C414yes—
367C2×Q8#11714yes—
368C2×Q8C4×D418yes—
369C2×Q8C4×D418yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
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—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—216no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—416no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—416no—
416C22×C4—48no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—48no—
439C22×C4—84no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4C2×C4214yes—
443C22×C4C2×C22⋊C414yes—
444C22×C4C2×C22⋊C414yes—
445C22×C4C2×C22⋊C412yes—
446C22×C4#11714yes—
447C22×C4C4×D414yes—
448C22×C4C4×D414yes—
449C22×C4C4×D412yes—
450C22×C4C4×D412yes—
451C22×C4C23×C411yescharacteristic
452C22×C4C23×C411yescharacteristic
453C22×C4C23×C411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C2×#4112yes—
457C22×C4C2×#4111yescharacteristic
458C22×C4C2×#4111yescharacteristic
459C22×C4H(5,𝔽2)12yes—
460C22×C4H(5,𝔽2)12yes—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—416no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8C2×C22⋊C414yes—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—48no—
488C4⋊C4—416no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4C2×C22⋊C414yes—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C42—24no—
500C42—24no—
501C42—24no—
502C42—24no—
503C42—416no—
504C42—88no—
505C23—22no—
506C23—24no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23C4×C22⋊C414yes—
522C23#41714yes—
523C23#41914yes—
524C23#44014yes—
525C23#44014yes—
526C23#44114yes—
527C23#44114yes—
528C23C22×C22⋊C411yescharacteristic
529C23C2×C4×D411yescharacteristic
530C23C2×C4×D411yescharacteristic
531C23D4212yes—
532C23#54612yes—
533C23C2×AGL(1,ℤ/8)11yescharacteristic
534C23C2×#13711yescharacteristic
535C23C2×#13711yescharacteristic
536C23C2×#13711yescharacteristic
537C23#57712yes—
538C23#57812yes—
539Q8—24no—
540Q8—24no—
541Q8—28no—
542Q8—28no—
543Q8—24no—
544Q8—24no—
545Q8—24no—
546Q8—432no—
547Q8—42no—
548Q8—44no—
549Q8—42no—
550Q8—44no—
551Q8C4×C22⋊C418yescomplemented
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—432no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—816no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4C4×C22⋊C414yes—
613C2×C4C22×C4211yescharacteristic
614C2×C4C22×C22⋊C411yescharacteristic
615C2×C4C2×#11711yescharacteristic
616C2×C4C2×#11711yescharacteristic
617C2×C4C2×C4×D411yescharacteristic
618C2×C4C2×C4×D412yes—
619C2×C4C2×C4×D411yescharacteristic
620C2×C4C2×C4×D412yes—
621C2×C4#51811yescharacteristic
622C2×C4#51812yes—
623C2×C4#51811yescharacteristic
624C2×C4#51812yes—
625C2×C4D4212yes—
626C2×C4#54612yes—
627C2×C4#54612yes—
628C2×C4#54612yes—
629C2×C4#55112yes—
630C2×C4#55112yes—
631C8—24no—
632C8—42no—
633C8—44no—
634C22—24no—
635C22—24no—
636C22—28no—
637C22—28no—
638C22—28no—
639C22—28no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22#176318yescomplemented
645C22C2×C4×C22⋊C411yescharacteristic
646C22D4×C22⋊C412yes—
647C22D4×C22⋊C411yescharacteristic
648C22#234211yescharacteristic
649C22#235812yes—
650C22C2×#41911yescharacteristic
651C22C2×#41711yescharacteristic
652C22#289812yes—
653C22C2×#44011yescharacteristic
654C22C2×#44011yescharacteristic
655C22C2×#44111yescharacteristic
656C22C2×#44111yescharacteristic
657C22#294712yes—
658C22#294911yescharacteristic
659C22#295011yescharacteristic
660C22#326512yes—
661C22#326612yes—
662C22#327012yes—
663C22#327012yes—
664C22#330812yes—
665C22#331212yes—
666C22#331612yes—
667C22#332012yes—
668C4—24no—
669C4—28no—
670C4—44no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—84no—
675C4—84no—
676C4—82no—
677C4C2×C4×C22⋊C411yescharacteristic
678C4#227311yescharacteristic
679C4D4×C22⋊C411yescharacteristic
680C4#234212yes—
681C4#234211yescharacteristic
682C4#235812yes—
683C2—28no—
684C2C22⋊C4211yescharacteristic
685C2C2×#176311yescharacteristic
686C2#2041711yescharacteristic
687C2#2242112yes—
688C2#2242512yes—
689C2#2243711yescharacteristic
690C2#2244011yescharacteristic
691C2#2256311yescharacteristic
692C2#2256711yescharacteristic
693C2#2285512yes—
694C2#2285912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C2×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
4. C2×#1763C2semidirect
7. #10932C2semidirect
8. #10928C2semidirect
9. #10789C2semidirect
17. C22×#103C4semidirect
27. Q8×C22⋊C4C4semidirect
37. #1787C4semidirect
77. C2×C4×Q8C2×C4semidirect
107. C2×#103C2×C4semidirect
108. C2×#103C2×C4semidirect
118. #391C2×C4semidirect
178. C22×Q8C42semidirect
300. #103C22⋊C4semidirect
551. Q8C4×C22⋊C4semidirect
644. C22#1763semidirect

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×#17872
C2×#17632
C2×#17522
C22⋊C4×C4⋊C42
#109322
#109282
#107892
#107302
#107212