Finite Group Databasebeta

← #364240

Subgroups — #364240

Order 512.

1754 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic
2C2×#1853C211yesmaximal, characteristic
3C2×#1852C211yesmaximal, characteristic
4#14685C211yesmaximal, characteristic
5C2×C4⋊#95C211yesmaximal, characteristic
6#11232C212yesmaximal, complemented
7#11231C212yesmaximal, complemented
8#10863C212yesmaximal, complemented
9#10821C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10730C211yesmaximal, characteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C2×#432—24no—
16C2×#432—24no—
17C2×#432C418yescomplemented
18C22×#107C2212yes—
19C22×#106C2212yes—
20C22×C4⋊C8C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×#97C2212yes—
24C2×#381—21no—
25C2×#380C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C22×#95—21no—
28C22×#95—21no—
29#1853—24no—
30#1852—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C4⋊#95—21no—
38C4⋊#95—21no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45#1434—21no—
46#1434—21no—
47C22⋊#95—21no—
48C24×C4—21no—
49C24×C4—21no—
50C24×C4D411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2312yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—42no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C42—22no—
74C22×C42—22no—
75C22×C42—22no—
76C22×C42—22no—
77C22×C42C2×C414yes—
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42Q814yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#432—232no—
84C2×#107—24no—
85C2×#107—24no—
86C2×#107—42no—
87C2×#106—24no—
88C2×#106—24no—
89C2×#106—42no—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—24no—
92C2×C4⋊C8C2×C418yescomplemented
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8C2×C414yes—
96C2×C22⋊C8C2×C414yes—
97C2×C22⋊C8D414yes—
98C2×C22⋊C8D414yes—
99C2×#97—24no—
100C2×#97—24no—
101C2×#97C2×C418yescomplemented
102#381—44no—
103#380—22no—
104#380—22no—
105#380—22no—
106#380—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—44no—
122C2×#95—44no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—42no—
128#340—42no—
129C25C2×D411yescharacteristic
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—41no—
153C23×C4—41no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4C4⋊C414yes—
167C23×C4C4⋊C414yes—
168C23×C4C22×C412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×Q812yes—
177C23×C4#4112yes—
178C23×C4#4112yes—
179C23×C4C2411yescharacteristic
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8C22×C414yes—
191C22×C8C22×C414yes—
192C22×C8C2×D414yes—
193C22×C8C2×D414yes—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—48no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4#4114yes—
231C2×C4⋊C4#4114yes—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C42—28no—
239C2×C42—28no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—28no—
245C2×C42—28no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42C4⋊C414yes—
255C2×C42C4⋊C414yes—
256C2×C42C22×C412yes—
257C2×C42C22×C412yes—
258C2×C42C2×Q812yes—
259C2×C42C2×Q812yes—
260#107—48no—
261#106—48no—
262C4⋊C8—216no—
263C22⋊C8—28no—
264C22⋊C8C22⋊C4116yes—
265#97—216no—
266#95—42no—
267#95—42no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—42no—
272#95—42no—
273#95—42no—
274#95—42no—
275#95—42no—
276#95—42no—
277#95—42no—
278#95—44no—
279#95—42no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24C2×C4⋊C412yes—
289C24#12112yes—
290C24#12112yes—
291C24AGL(1,ℤ/8)12yes—
292C24AGL(1,ℤ/8)12yes—
293C24#13712yes—
294C24#13712yes—
295C24C22×D411yescharacteristic
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4C2×C22⋊C414yes—
405C22×C4C2×C4⋊C412yes—
406C22×C4C2×C4⋊C412yes—
407C22×C4C2×C4⋊C412yes—
408C22×C4C2×C4⋊C412yes—
409C22×C4C2×C4⋊C412yes—
410C22×C4#11714yes—
411C22×C4#12112yes—
412C22×C4#12112yes—
413C22×C4C22⋊Q814yes—
414C22×C4C22⋊Q814yes—
415C22×C4AGL(1,ℤ/8)12yes—
416C22×C4#13712yes—
417C22×C4C23×C411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×Q811yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C42−1+412yes—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8C2×C22⋊C418yes—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—28no—
470C42—28no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23#38014yes—
500C23#42814yes—
501C23#42814yes—
502C23#46814yes—
503C23#46814yes—
504C23#46914yes—
505C23#47014yes—
506C23C22×C4⋊C411yescharacteristic
507C23C2×#12111yescharacteristic
508C23C2×#12111yescharacteristic
509C23#52511yescharacteristic
510C23#52511yescharacteristic
511C23#53511yescharacteristic
512C23#53611yescharacteristic
513C23C2×AGL(1,ℤ/8)11yescharacteristic
514C23C2×AGL(1,ℤ/8)11yescharacteristic
515C23C2×AGL(1,ℤ/8)11yescharacteristic
516C23C2×#13711yescharacteristic
517C23C2×#13711yescharacteristic
518C23C2×#13711yescharacteristic
519C23#57511yescharacteristic
520C23#57511yescharacteristic
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—416no—
547C2×C4—416no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—816no—
567C2×C4—816no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4#38014yes—
605C2×C4#42814yes—
606C2×C4C22×C22⋊C411yescharacteristic
607C2×C4C2×#11711yescharacteristic
608C2×C4C4×#4112yes—
609C2×C4#52012yes—
610C2×C4C2×C22⋊Q811yescharacteristic
611C2×C4C2×C22⋊Q811yescharacteristic
612C2×C4#53312yes—
613C2×C4#54412yes—
614C2×C4#54612yes—
615C2×C4#54612yes—
616C2×C4#54712yes—
617C2×C4#54712yes—
618C2×C4#55112yes—
619C2×C4#55112yes—
620C2×C4#55212yes—
621C2×C4#55212yes—
622C8—28no—
623C8—44no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22#180918yes—
639C22C2×#38011yescharacteristic
640C22#229611yescharacteristic
641C22#231711yescharacteristic
642C22#241111yescharacteristic
643C22#241111yescharacteristic
644C22#241311yescharacteristic
645C22#241311yescharacteristic
646C22C2×#42811yescharacteristic
647C22C2×#42811yescharacteristic
648C22C2×#42811yescharacteristic
649C22#291111yescharacteristic
650C22C2×#46811yescharacteristic
651C22C2×#46811yescharacteristic
652C22C2×#46911yescharacteristic
653C22C2×#47011yescharacteristic
654C22#305511yescharacteristic
655C22#305511yescharacteristic
656C22#305611yescharacteristic
657C22#305611yescharacteristic
658C22#310911yescharacteristic
659C22#311011yescharacteristic
660C22#311211yescharacteristic
661C22#311311yescharacteristic
662C22#311411yescharacteristic
663C22#311411yescharacteristic
664C22#311711yescharacteristic
665C22#311711yescharacteristic
666C22#321311yescharacteristic
667C22#321511yescharacteristic
668C22#321711yescharacteristic
669C22#321811yescharacteristic
670C22#323411yescharacteristic
671C22#323611yescharacteristic
672C22#323811yescharacteristic
673C22#323911yescharacteristic
674C4—24no—
675C4—24no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—84no—
680C4—84no—
681C4—161no—
682C4—161no—
683C4—161no—
684C4—161no—
685C4#233912yes—
686C4#234312yes—
687C4#243812yes—
688C4#243812yes—
689C2—28no—
690C2#1396211yescharacteristic
691C2C2×#180911yescharacteristic
692C2#2066111yescharacteristic
693C2#2136311yescharacteristic
694C2#2136511yescharacteristic
695C2#2198611yescharacteristic
696C2#2198711yescharacteristic
697C2#2450611yescharacteristic
698C2#2450711yescharacteristic
699C2#2450811yescharacteristic
700C2#2450911yescharacteristic
701C2#2460611yescharacteristic
702C2#2460711yescharacteristic
703C2#2460811yescharacteristic
704C2#2460911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11232C2semidirect
7. #11231C2semidirect
8. #10863C2semidirect
9. #10821C2semidirect
17. C2×#432C4semidirect
92. C2×C4⋊C8C2×C4semidirect
101. C2×#97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
C2×#18532
C2×#18522
#146852
C2×C4⋊#952
#112322
#112312
#108632
#108212
#107302