Finite Group Databasebeta

← #357721

Subgroups — #357721

Order 512.

2706 conjugacy classes of subgroups.

The subgroup lattice has 740 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#419C211yesmaximal, characteristic
2C22×#419C211yesmaximal, characteristic
3#34052C211yesmaximal, characteristic
4C2×#2310C211yesmaximal, characteristic
5C2×#1742C211yesmaximal, characteristic
6C2×#1733C211yesmaximal, characteristic
7C2×#1733C211yesmaximal, characteristic
8#11064C218yesmaximal, complemented
9C24×Q8C2211yescharacteristic
10C23×M16C2211yescharacteristic
11C23×M16C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#419—24no—
17C2×#419—24no—
18C2×#419—24no—
19C2×#419—22no—
20C2×#419—22no—
21C2×#419C418yescomplemented
22C2×#419C2214yes—
23C2×#419C2214yes—
24C22×#103—22no—
25C22×#103—22no—
26#2317—22no—
27#2317C2214yes—
28#2316—22no—
29#2315—22no—
30#2310—24no—
31#2310—24no—
32#2296—22no—
33#2296—22no—
34C2×#391—22no—
35C2×#391C2214yes—
36#2285—22no—
37C2×#381—22no—
38#1742—24no—
39C2×#343C2214yes—
40#1733—24no—
41#1733—24no—
42C2×#339C2214yes—
43C2×#339C2214yes—
44C23×Q8—22no—
45C23×Q8—21no—
46C23×Q8—22no—
47C23×Q8—22no—
48C23×Q8—21no—
49C23×Q8—21no—
50C23×Q8—21no—
51C23×Q8—21no—
52C23×Q8—22no—
53C23×Q8—22no—
54C23×Q8—22no—
55C23×Q8—22no—
56C23×Q8—22no—
57C23×Q8—22no—
58C23×Q8C2×C414yes—
59C23×Q8D414yes—
60C23×Q8Q814yes—
61C23×Q8C2312yes—
62C24×C4—21no—
63C24×C4C2311yescharacteristic
64C22×M16—22no—
65C22×M16—22no—
66C22×M16—21no—
67C22×M16—21no—
68C22×M16—22no—
69C22×M16—22no—
70C22×M16—22no—
71C22×M16—22no—
72C22×M16—22no—
73C22×M16—22no—
74C22×M16—22no—
75C22×M16—22no—
76C22×M16C2×C414yes—
77C22×M16D414yes—
78C22×M16C2312yes—
79C22×M16C2312yes—
80C23×C8—21no—
81C23×C8—21no—
82C2×#117—21no—
83C2×#117—21no—
84C2×#117—22no—
85C2×#117—22no—
86C2×#117—22no—
87C2×#117—22no—
88C2×#117—22no—
89C2×#117—21no—
90C2×#117—21no—
91C2×#117—21no—
92C2×#117—22no—
93C2×#117—21no—
94C2×#117—21no—
95C2×#117—21no—
96C2×#117—22no—
97C2×#117—22no—
98C2×#117—22no—
99C2×#117—22no—
100C2×#117—22no—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117—22no—
104C2×#117—22no—
105C2×#117C2×C414yes—
106C2×#117D414yes—
107C2×#117D412yes—
108C2×#117D412yes—
109C2×#117D414yes—
110C2×#117C2312yes—
111C2×#117C2312yes—
112C22×C4⋊C4—21no—
113C22×C4⋊C4D412yes—
114C22×C4⋊C4D412yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4D412yes—
120C22×C42—21no—
121C22×C42—21no—
122C22×C42D412yes—
123C22×C42C2312yes—
124#419—216no—
125#419—28no—
126#419—28no—
127#419—216no—
128#419—28no—
129#419C2×C4116yescomplemented
130C2×#103—216no—
131C2×#103—216no—
132C2×#103—28no—
133C2×#103—28no—
134#391—216no—
135#391—28no—
136#391—28no—
137#389—28no—
138#389—28no—
139#385—28no—
140#385—28no—
141#384—24no—
142#382—24no—
143#381—216no—
144#380—28no—
145C4×C4⋊C4—28no—
146C2×#95—28no—
147#343—28no—
148#339—28no—
149#339—28no—
150C25C2×D411yescharacteristic
151C22×Q8—24no—
152C22×Q8—28no—
153C22×Q8—24no—
154C22×Q8—24no—
155C22×Q8—28no—
156C22×Q8—28no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—28no—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—28no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—28no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—28no—
176C22×Q8—28no—
177C22×Q8—28no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—41no—
187C22×Q8—41no—
188C22×Q8—42no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—42no—
194C22×Q8—44no—
195C22×Q8—44no—
196C22×Q8—42no—
197C22×Q8—42no—
198C22×Q8—44no—
199C22×Q8—44no—
200C22×Q8—42no—
201C22×Q8—42no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8—44no—
206C22×Q8—44no—
207C22×Q8—42no—
208C22×Q8—42no—
209C22×Q8C4⋊C4116yescomplemented
210C22×Q8C22×C414yes—
211C22×Q8C2×D414yes—
212C22×Q8C2×Q814yes—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—41no—
221C23×C4—41no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4C22×C412yes—
226C23×C4C2×D414yes—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D412yes—
230C23×C4C2×D412yes—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D412yes—
235C23×C4C2×D412yes—
236C23×C4C2×Q812yes—
237C23×C4#4112yes—
238C23×C4#4112yes—
239C23×C4#4112yes—
240C23×C4#4112yes—
241C23×C4#4112yes—
242C23×C4#4112yes—
243C23×C4C2411yescharacteristic
244C2×M16—24no—
245C2×M16—24no—
246C2×M16—22no—
247C2×M16—22no—
248C2×M16—28no—
249C2×M16—28no—
250C2×M16—28no—
251C2×M16—28no—
252C2×M16—22no—
253C2×M16—22no—
254C2×M16—22no—
255C2×M16—22no—
256C2×M16—22no—
257C2×M16—22no—
258C2×M16—22no—
259C2×M16—22no—
260C2×M16—42no—
261C2×M16—44no—
262C2×M16—44no—
263C2×M16—44no—
264C2×M16—44no—
265C2×M16—42no—
266C2×M16—42no—
267C2×M16—42no—
268C2×M16C22×C414yes—
269C2×M16C2×D414yes—
270C22×C8—24no—
271C22×C8—24no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—42no—
275C22×C8—42no—
276C22×C8—42no—
277C22×C8—42no—
278#117—24no—
279#117—24no—
280#117—24no—
281#117—24no—
282#117—24no—
283#117—22no—
284#117—22no—
285#117—28no—
286#117—28no—
287#117—24no—
288#117—24no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—42no—
294#117—42no—
295#117—42no—
296#117—44no—
297#117—42no—
298#117—42no—
299#117—42no—
300#117—42no—
301#117—42no—
302#117—42no—
303#117—44no—
304#117—42no—
305#117—42no—
306#117—42no—
307#117C22⋊C418yes—
308#117C22×C414yes—
309#117C2×D414yes—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—216no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—42no—
325C2×C4⋊C4C22⋊C418yes—
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2×D414yes—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—42no—
340C2×C22⋊C4—42no—
341C2×C22⋊C4—42no—
342C2×C22⋊C4—42no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—42no—
359C2×C42—42no—
360C2×C42C2×D414yes—
361#103—232no—
362#103—232no—
363#95—232no—
364#95—216no—
365C24—44no—
366C24C4×D412yes—
367C24#12011yescharacteristic
368C24#12011yescharacteristic
369C24#12012yes—
370C24#12112yes—
371C24#12112yes—
372C24C22⋊Q812yes—
373C24#12412yes—
374C24C22×D411yescharacteristic
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—216no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—216no—
392C2×Q8—216no—
393C2×Q8—216no—
394C2×Q8—216no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—48no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—48no—
413C2×Q8—48no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—48no—
419C2×Q8—48no—
420C2×Q8—48no—
421C2×Q8—48no—
422C2×Q8—48no—
423C2×Q8—48no—
424C2×Q8—48no—
425C2×Q8—48no—
426C2×Q8—48no—
427C2×Q8—48no—
428C2×Q8C2×C4⋊C418yes—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—216no—
434C22×C4—28no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4C2×C22⋊C412yes—
461C22×C4C2×C22⋊C412yes—
462C22×C4C2×C4⋊C414yes—
463C22×C4#11714yes—
464C22×C4C4×D412yes—
465C22×C4C4×D412yes—
466C22×C4C4×D414yes—
467C22×C4C4×D414yes—
468C22×C4C4×D412yes—
469C22×C4#12012yes—
470C22×C4#12012yes—
471C22×C4#12012yes—
472C22×C4#12011yescharacteristic
473C22×C4#12011yescharacteristic
474C22×C4#12112yes—
475C22×C4#12112yes—
476C22×C4#12114yes—
477C22×C4#12112yes—
478C22×C4#12112yes—
479C22×C4#12112yes—
480C22×C4#12112yes—
481C22×C4#12114yes—
482C22×C4#12114yes—
483C22×C4#12112yes—
484C22×C4#12112yes—
485C22×C4C22⋊Q812yes—
486C22×C4C22⋊Q812yes—
487C22×C4C22⋊Q814yes—
488C22×C4C22⋊Q812yes—
489C22×C4C22⋊Q812yes—
490C22×C4C22⋊Q814yes—
491C22×C4C22⋊Q814yes—
492C22×C4C22⋊Q812yes—
493C22×C4#12314yes—
494C22×C4#12314yes—
495C22×C4#12312yes—
496C22×C4#12312yes—
497C22×C4#12314yes—
498C22×C4#12314yes—
499C22×C4#12412yes—
500C22×C4#12412yes—
501C22×C4#12412yes—
502C22×C4#12412yes—
503C22×C4#12412yes—
504C22×C4#12614yes—
505C22×C4#12614yes—
506C22×C4#12712yes—
507C22×C4#12712yes—
508C22×C4C23×C411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×Q811yescharacteristic
516C22×C4C2×#4111yescharacteristic
517C22×C4C2×#4111yescharacteristic
518C22×C4C2×#4111yescharacteristic
519C22×C4C2×#4111yescharacteristic
520C22×C4C2×#4111yescharacteristic
521C22×C4C2×#4111yescharacteristic
522M16—24no—
523M16—24no—
524M16—24no—
525M16—24no—
526M16—42no—
527M16—42no—
528M16—42no—
529M16—42no—
530M16—48no—
531M16—48no—
532C2×C8—24no—
533C2×C8—24no—
534C2×C8—42no—
535C2×C8—42no—
536C2×C8—42no—
537C2×C8—42no—
538C2×C8—48no—
539C2×C8—48no—
540C4⋊C4—24no—
541C4⋊C4—28no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—44no—
549C4⋊C4—48no—
550C4⋊C4—816no—
551C4⋊C4C2×C22⋊C418yes—
552C22⋊C4—24no—
553C22⋊C4—24no—
554C22⋊C4—28no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—48no—
561C42—24no—
562C42—24no—
563C42—28no—
564C42—24no—
565C42—24no—
566C42—42no—
567C42—42no—
568C42—42no—
569C42—42no—
570C42—44no—
571C42—44no—
572C42—44no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23#38514yes—
583C23#38612yes—
584C23#38612yes—
585C23#38814yes—
586C23#39212yes—
587C23#39212yes—
588C23#39312yes—
589C23#39312yes—
590C23#39414yes—
591C23#39614yes—
592C23C2×C4×D411yescharacteristic
593C23C2×#12011yescharacteristic
594C23C2×#12011yescharacteristic
595C23C2×#12111yescharacteristic
596C23C2×#12111yescharacteristic
597C23C2×C22⋊Q811yescharacteristic
598C23C2×#12411yescharacteristic
599Q8—28no—
600Q8—28no—
601Q8—44no—
602Q8—44no—
603Q8—44no—
604Q8—44no—
605Q8—416no—
606Q8—416no—
607Q8—44no—
608Q8—44no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—416no—
616C2×C4—416no—
617C2×C4—48no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—48no—
621C2×C4—48no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—88no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—88no—
631C2×C4—84no—
632C2×C4#38014yes—
633C2×C4#38014yes—
634C2×C4#38514yes—
635C2×C4#38514yes—
636C2×C4#38514yes—
637C2×C4#38612yes—
638C2×C4#38612yes—
639C2×C4#38814yes—
640C2×C4#38814yes—
641C2×C4#38814yes—
642C2×C4#39014yes—
643C2×C4#39014yes—
644C2×C4#39212yes—
645C2×C4#39212yes—
646C2×C4#39312yes—
647C2×C4#39312yes—
648C2×C4#39414yes—
649C2×C4#39414yes—
650C2×C4#39414yes—
651C2×C4#39614yes—
652C2×C4#39614yes—
653C2×C4#39614yes—
654C2×C4#39714yes—
655C2×C4#39714yes—
656C2×C4#39914yes—
657C2×C4#39914yes—
658C2×C4C22×C22⋊C411yescharacteristic
659C2×C4C22×C4⋊C411yescharacteristic
660C2×C4C2×#11711yescharacteristic
661C2×C4C2×C4×D411yescharacteristic
662C2×C4C2×C4×D411yescharacteristic
663C2×C4C2×C4×D411yescharacteristic
664C2×C4C2×#12011yescharacteristic
665C2×C4C2×#12111yescharacteristic
666C2×C4C2×#12111yescharacteristic
667C2×C4C2×#12111yescharacteristic
668C2×C4C2×#12111yescharacteristic
669C2×C4C2×#12111yescharacteristic
670C2×C4C2×#12111yescharacteristic
671C2×C4C2×C22⋊Q811yescharacteristic
672C2×C4C2×C22⋊Q811yescharacteristic
673C2×C4C2×C22⋊Q811yescharacteristic
674C2×C4C2×C22⋊Q811yescharacteristic
675C2×C4C2×C22⋊Q811yescharacteristic
676C2×C4C2×#12311yescharacteristic
677C2×C4C2×#12311yescharacteristic
678C2×C4C2×#12311yescharacteristic
679C2×C4C2×#12311yescharacteristic
680C2×C4C2×#12311yescharacteristic
681C2×C4C2×#12411yescharacteristic
682C2×C4C2×#12411yescharacteristic
683C2×C4C2×#12611yescharacteristic
684C2×C4C2×#12611yescharacteristic
685C2×C4C2×#12711yescharacteristic
686C8—44no—
687C8—44no—
688C22—22no—
689C22—22no—
690C22—24no—
691C22—24no—
692C22—22no—
693C22—22no—
694C22—24no—
695C22—24no—
696C22—22no—
697C22—24no—
698C22—22no—
699C22—24no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22#143914yes—
705C22#143914yes—
706C22C2×#38511yescharacteristic
707C22C2×#38611yescharacteristic
708C22C2×#38811yescharacteristic
709C22C2×#39211yescharacteristic
710C22C2×#39311yescharacteristic
711C22C2×#39411yescharacteristic
712C22C2×#39611yescharacteristic
713C22#302214yes—
714C4—48no—
715C4—48no—
716C4—84no—
717C4—84no—
718C4—84no—
719C4#143914yes—
720C4#143914yes—
721C4C2×#38011yescharacteristic
722C4C2×#38511yescharacteristic
723C4C2×#38811yescharacteristic
724C4C2×#39011yescharacteristic
725C4C2×#39411yescharacteristic
726C4C2×#39611yescharacteristic
727C4C2×#39711yescharacteristic
728C4C2×#39911yescharacteristic
729C2—21no—
730C2—21no—
731C2—24no—
732C2—24no—
733C2—21no—
734C2—21no—
735C2C2×#143911yescharacteristic
736C2#2094412yes—
737C2#2340112yes—
738C2#2346412yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11064C2semidirect
21. C2×#419C4semidirect
129. #419C2×C4semidirect
209. C22×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4192
#340522
C2×#23102
C2×#17422
C2×#17332
#110642