Finite Group Databasebeta

← #368069

Subgroups — #368069

Order 512.

2046 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic, complemented
2C2×#2352C211yesmaximal, characteristic
3C2×#1870C211yesmaximal, characteristic
4C2×#1853C212yesmaximal, complemented
5C2×#1853C211yesmaximal, characteristic, complemented
6#13872C211yesmaximal, characteristic
7#11325C212yesmaximal, complemented
8#11096C212yesmaximal, complemented
9#10908C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic, complemented
11#10722C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#428—28no—
18C2×#428—24no—
19C22×#107C2212yes—
20C22×#106C2212yes—
21C22×#106C2212yes—
22C22×C22⋊C8C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24#2352—24no—
25#2352—24no—
26C2×#388—21no—
27C2×#387—21no—
28C2×#386—21no—
29C2×#386—21no—
30C2×#382C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4—21no—
37#1870C418yescomplemented
38#1853—28no—
39#1853—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2212yes—
46#1441—21no—
47#1439—21no—
48C4⋊#95C2212yes—
49C22⋊#95—21no—
50C23×D4D411yescharacteristic
51C24×C4—21no—
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C24×C4C2311yescharacteristic
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—24no—
58C22×M16—24no—
59C22×M16C2×C414yescomplemented
60C23×C8C2312yes—
61C23×C8C2312yes—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4C2×C414yes—
67C2×C4×D4Q814yes—
68C2×#117—22no—
69C2×#117—24no—
70C2×#117—42no—
71C2×#117D414yes—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4—42no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—22no—
94C22×C22⋊C4D411yescharacteristic
95C22×C22⋊C4D411yescharacteristic
96C22×C42—21no—
97C22×C42—21no—
98C22×C42C2311yescharacteristic
99#428—232no—
100C2×#107—24no—
101C2×#107—24no—
102C2×#107—24no—
103C2×#106—24no—
104C2×#106—24no—
105C2×#106—24no—
106C2×#106—24no—
107C2×#106—42no—
108C2×#106D418yescomplemented
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—24no—
112C2×C22⋊C8—24no—
113C2×C22⋊C8C2×C414yescomplemented
114C2×C22⋊C8C2×C418yescomplemented
115#388—28no—
116#387—28no—
117#386—28no—
118#386—44no—
119#382—22no—
120#382—22no—
121#380—22no—
122#380—22no—
123#380—22no—
124#380—22no—
125#380—22no—
126#380—24no—
127#380—42no—
128#380D414yes—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C22⋊C4—28no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—21no—
140C2×#95—22no—
141#340—24no—
142#340—24no—
143#340—24no—
144#340—24no—
145#340—42no—
146#336C2×C418yescomplemented
147C25C2×D411yescharacteristic
148C25C2×D411yescharacteristic
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—41no—
154C22×D4—41no—
155C22×D4C4⋊C414yes—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—82no—
183C23×C4—82no—
184C23×C4C22×C412yes—
185C23×C4C22×C412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4C2×Q812yes—
193C23×C4C2×Q812yes—
194C23×C4#4112yes—
195C23×C4#4112yes—
196C23×C4C2411yescharacteristic
197C2×M16—216no—
198C2×M16—28no—
199C2×M16—28no—
200C2×M16—28no—
201C2×M16—216no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8C22×C414yes—
213C22×C8C22×C414yes—
214C22×C8C2×D414yes—
215C22×C8#4114yes—
216C4×D4—28no—
217C4×D4—28no—
218C4×D4—28no—
219C4×D4—28no—
220#117—28no—
221#117—44no—
222#117—48no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C4⋊C414yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D412yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—82no—
280C2×C22⋊C4—82no—
281C2×C22⋊C4C4⋊C414yes—
282C2×C22⋊C4C4⋊C414yes—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—44no—
290C2×C42—82no—
291C2×C42C22×C412yes—
292C2×C42C2×Q812yes—
293#107—216no—
294#106—216no—
295#106—48no—
296C22⋊C8—28no—
297C22⋊C8—216no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—44no—
304#95—44no—
305#95—42no—
306#95—82no—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24C2×C4⋊C412yes—
324C24C2×C4⋊C412yes—
325C24#12012yes—
326C24#12312yes—
327C24AGL(1,ℤ/8)12yes—
328C24#13712yes—
329C24C22×D411yescharacteristic
330C2×D4—28no—
331C2×D4—28no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—48no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
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—84no—
423C22×C4—88no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4C2×C4⋊C412yes—
430C22×C4C2×C4⋊C412yes—
431C22×C4C2×C4⋊C412yes—
432C22×C4C2×C4⋊C412yes—
433C22×C4C4×D414yes—
434C22×C4C4×D414yes—
435C22×C4#12012yes—
436C22×C4C22⋊Q814yes—
437C22×C4C22⋊Q814yes—
438C22×C4#12312yes—
439C22×C4C23×C411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×Q811yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4C2×#4112yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C42−1+412yes—
449M16—232no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8C4×D418yes—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—28no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—42no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4#12014yes—
487C22⋊C4—28no—
488C22⋊C4—28no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—84no—
492C22⋊C4—88no—
493C22⋊C4—84no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—48no—
499C42—84no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—41no—
534C23—41no—
535C23—42no—
536C23—41no—
537C23—42no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23#38514yes—
545C23#42814yescomplemented
546C23C22×C4⋊C411yescharacteristic
547C23C2×#12011yescharacteristic
548C23C2×#12311yescharacteristic
549C23D4212yes—
550C23#54712yes—
551C23C2×AGL(1,ℤ/8)11yescharacteristic
552C23C2×#13711yescharacteristic
553C23#57511yescharacteristic
554C23#57511yescharacteristic
555C23#57712yes—
556C23#57712yes—
557D4—416no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—816no—
609C2×C4—816no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4#38514yes—
628C2×C4C2×C4×D411yescharacteristic
629C2×C4C2×C4×D411yescharacteristic
630C2×C4C2×C4×D412yes—
631C2×C4#52012yes—
632C2×C4C2×#12011yescharacteristic
633C2×C4C2×C22⋊Q811yescharacteristic
634C2×C4C2×C22⋊Q811yescharacteristic
635C2×C4#52511yescharacteristic
636C2×C4#52512yes—
637C2×C4#53411yescharacteristic
638C2×C4#53611yescharacteristic
639C2×C4#53912yes—
640C2×C4D4212yes—
641C2×C4#54712yes—
642C2×C4#54812yes—
643C2×C4D4×Q812yes—
644C2×C4#55312yes—
645C2×C4#55412yes—
646C8—28no—
647C8—44no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—28no—
652C22—24no—
653C22—44no—
654C22—44no—
655C22—42no—
656C22—42no—
657C22—44no—
658C22—44no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—42no—
670C22C2×#38511yescharacteristic
671C22D4×C4⋊C412yes—
672C22#240311yescharacteristic
673C22#240611yescharacteristic
674C22#244312yes—
675C22C2×#42811yescharacteristic
676C22#291111yescharacteristic
677C22#291212yes—
678C22#300611yescharacteristic
679C22#300711yescharacteristic
680C22#300911yescharacteristic
681C22#301011yescharacteristic
682C22#309411yescharacteristic
683C22#309511yescharacteristic
684C22#309912yes—
685C22#326512yes—
686C22#326612yes—
687C22#327512yes—
688C22#327512yes—
689C22#334412yes—
690C22#334512yes—
691C22#335112yes—
692C22#335112yes—
693C4—24no—
694C4—44no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—84no—
702C4—162no—
703C4C4×#12011yescharacteristic
704C4#232211yescharacteristic
705C4#236112yes—
706C4#240911yescharacteristic
707C4#244912yes—
708C4#257911yescharacteristic
709C2—44no—
710C2—44no—
711C2#1401511yescharacteristic
712C2#2084611yescharacteristic
713C2#2084811yescharacteristic
714C2#2265912yes—
715C2#2266612yes—
716C2#2386211yescharacteristic
717C2#2386611yescharacteristic
718C2#2410111yescharacteristic
719C2#2411111yescharacteristic
720C2#2614312yes—
721C2#2614912yes—
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
1. C22×#428C2semidirect
4. C2×#1853C2semidirect
5. C2×#1853C2semidirect
7. #11325C2semidirect
8. #11096C2semidirect
9. #10908C2semidirect
10. #10731C2semidirect
11. #10722C2semidirect
37. #1870C4semidirect
59. C22×M16C2×C4semidirect
108. C2×#106D4semidirect
113. C2×C22⋊C8C2×C4semidirect
114. C2×C22⋊C8C2×C4semidirect
146. #336C2×C4semidirect
545. C23#428semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#23522
C2×#18702
C2×#18532
#138722
#113252
#110962
#109082
#107312
#107222