Finite Group Databasebeta

← #354079

Subgroups — #354079

Order 512.

2986 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#419C211yesmaximal, characteristic
2C23×#103C211yesmaximal, characteristic
3C22×#405C211yesmaximal, characteristic
4C2×#2310C211yesmaximal, characteristic
5C2×#1736C211yesmaximal, characteristic
6C2×#1735C211yesmaximal, characteristic
7#13795C211yesmaximal, characteristic
8#10928C214yesmaximal, complemented
9#10926C214yesmaximal, complemented
10C24×Q8C2211yescharacteristic
11C23×M16C2211yescharacteristic
12C24×C8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#419—24no—
18C2×#419—24no—
19C2×#419C418yescomplemented
20C22×#103—24no—
21C22×#103—24no—
22C22×#103C418yescomplemented
23C22×#103C2212yes—
24C22×#103C2212yes—
25C22×#103C2214yes—
26C2×#405—24no—
27C2×#405—24no—
28C2×#405—24no—
29C22×#97C2212yes—
30C22×C4×C8C2212yes—
31#2310—24no—
32#2310C418yescomplemented
33C2×#391—22no—
34C2×#391C2214yes—
35C2×#381—22no—
36#1736—24no—
37#1735—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2214yes—
41C4⋊#95—22no—
42C4⋊#95C2214yes—
43C22⋊#95—22no—
44C23×Q8—22no—
45C23×Q8—21no—
46C23×Q8—22no—
47C23×Q8—22no—
48C23×Q8—22no—
49C23×Q8—22no—
50C23×Q8—22no—
51C23×Q8—22no—
52C23×Q8—22no—
53C23×Q8—22no—
54C23×Q8—22no—
55C23×Q8—22no—
56C23×Q8C2×C414yes—
57C23×Q8C2×C414yes—
58C23×Q8C2×C414yes—
59C23×Q8C2312yes—
60C24×C4—21no—
61C24×C4—21no—
62C24×C4C2311yescharacteristic
63C22×M16—22no—
64C22×M16—22no—
65C22×M16—22no—
66C22×M16—22no—
67C22×M16C2×C414yes—
68C22×M16D414yes—
69C22×M16D414yes—
70C23×C8—22no—
71C23×C8—22no—
72C23×C8—22no—
73C23×C8—22no—
74C23×C8C2×C414yes—
75C23×C8D414yes—
76C23×C8D414yes—
77C23×C8C2311yescharacteristic
78C23×C8C2312yes—
79C23×C8C2311yescharacteristic
80C2×#117—22no—
81C2×#117—22no—
82C2×#117—22no—
83C2×#117—22no—
84C2×#117—22no—
85C2×#117—22no—
86C2×#117—22no—
87C2×#117—22no—
88C2×#117—22no—
89C2×#117—22no—
90C2×#117—22no—
91C2×#117—22no—
92C2×#117—22no—
93C2×#117—22no—
94C2×#117C2×C414yes—
95C2×#117C2×C414yes—
96C2×#117C2×C414yes—
97C2×#117D414yes—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—22no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4C2×C414yes—
105C22×C4⋊C4D412yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4D412yes—
112C22×C42—21no—
113C22×C42—21no—
114C22×C42C2312yes—
115#419—232no—
116C2×#103—232no—
117C2×#103—24no—
118C2×#103—24no—
119C2×#103—24no—
120C2×#103—24no—
121C2×#103—28no—
122C2×#103—28no—
123C2×#103C2×C418yes—
124C2×#103C2×C418yes—
125C2×#103C2×C4116yescomplemented
126#405—232no—
127C2×#97—24no—
128C2×#97—24no—
129C2×#97—24no—
130C2×C4×C8—24no—
131C2×C4×C8—24no—
132C2×C4×C8—24no—
133#391—216no—
134#391—28no—
135#391C2×C4116yescomplemented
136#381—216no—
137C2×#95—28no—
138C2×#95—28no—
139#340—24no—
140#340—24no—
141#340—28no—
142C25C2×D411yescharacteristic
143C22×Q8—24no—
144C22×Q8—28no—
145C22×Q8—24no—
146C22×Q8—24no—
147C22×Q8—28no—
148C22×Q8—28no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—28no—
153C22×Q8—28no—
154C22×Q8—28no—
155C22×Q8—28no—
156C22×Q8—28no—
157C22×Q8—28no—
158C22×Q8—28no—
159C22×Q8—28no—
160C22×Q8—28no—
161C22×Q8—28no—
162C22×Q8—28no—
163C22×Q8—28no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—44no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8—44no—
176C22×Q8—44no—
177C22×Q8—44no—
178C22×Q8—44no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×Q8—44no—
182C22×Q8—42no—
183C22×Q8—44no—
184C22×Q8—44no—
185C22×Q8—44no—
186C22×Q8—44no—
187C22×Q8—42no—
188C22×Q8C42116yes—
189C22×Q8C22×C414yes—
190C22×Q8C22×C414yes—
191C22×Q8C22×C414yes—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—84no—
207C23×C4C22×C412yes—
208C23×C4C22×C412yes—
209C23×C4C22×C412yes—
210C23×C4C2×D414yes—
211C23×C4C2×D412yes—
212C23×C4C2×D412yes—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D412yes—
215C23×C4C2×D412yes—
216C23×C4C2×D412yes—
217C23×C4#4112yes—
218C23×C4#4112yes—
219C23×C4#4112yes—
220C23×C4#4112yes—
221C23×C4#4112yes—
222C23×C4#4112yes—
223C23×C4C2411yescharacteristic
224C2×M16—28no—
225C2×M16—28no—
226C2×M16—28no—
227C2×M16—28no—
228C2×M16—28no—
229C2×M16—28no—
230C2×M16C22⋊C4116yes—
231C22×C8—28no—
232C22×C8—28no—
233C22×C8—28no—
234C22×C8—28no—
235C22×C8—28no—
236C22×C8—28no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8C22⋊C4116yes—
250C22×C8C22×C412yes—
251C22×C8C22×C412yes—
252C22×C8C22×C414yes—
253C22×C8C2×D412yes—
254C22×C8C2×D412yes—
255C22×C8C2×D412yes—
256C22×C8C2×D412yes—
257C22×C8C2×D414yes—
258C22×C8C2×D414yes—
259#117—28no—
260#117—28no—
261#117—28no—
262#117—28no—
263#117—28no—
264#117—28no—
265#117—28no—
266#117—28no—
267#117—44no—
268#117—44no—
269#117—44no—
270#117—44no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4C22⋊C418yes—
296C2×C4⋊C4C22×C414yes—
297C2×C4⋊C4C22×C414yes—
298C2×C4⋊C4C22×C414yes—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4C22⋊C418yes—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42C22×C414yes—
319C2×C42C2×D414yes—
320#103—216no—
321#103—216no—
322#103—232no—
323#97—216no—
324C4×C8—216no—
325#95—232no—
326#95—88no—
327C24—24no—
328C24—24no—
329C24C2×C22⋊C412yes—
330C24C4×D412yes—
331C24C4×D412yes—
332C24#12112yes—
333C24#12112yes—
334C24#12412yes—
335C24#12712yes—
336C24C22×D411yescharacteristic
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—216no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—232no—
354C2×Q8—232no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—416no—
368C2×Q8—416no—
369C2×Q8—48no—
370C2×Q8—416no—
371C2×Q8—416no—
372C2×Q8—416no—
373C2×Q8—416no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8C2×C4218yes—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—216no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—48no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—48no—
413C22×C4—88no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4C2×C4214yes—
417C22×C4C2×C22⋊C418yes—
418C22×C4C2×C22⋊C414yes—
419C22×C4C2×C22⋊C414yes—
420C22×C4C2×C22⋊C412yes—
421C22×C4#11714yes—
422C22×C4#11714yes—
423C22×C4#11714yes—
424C22×C4C4×D414yes—
425C22×C4C4×D414yes—
426C22×C4C4×D414yes—
427C22×C4C4×D414yes—
428C22×C4C4×D414yes—
429C22×C4C4×D414yes—
430C22×C4C4×D414yes—
431C22×C4C4×D414yes—
432C22×C4C4×D414yes—
433C22×C4C4×D414yes—
434C22×C4C4×D412yes—
435C22×C4C4×D412yes—
436C22×C4#12014yes—
437C22×C4#12114yes—
438C22×C4#12114yes—
439C22×C4#12114yes—
440C22×C4#12114yes—
441C22×C4#12112yes—
442C22×C4#12112yes—
443C22×C4#12314yes—
444C22×C4#12314yes—
445C22×C4#12314yes—
446C22×C4#12414yes—
447C22×C4#12414yes—
448C22×C4#12412yes—
449C22×C4#12614yes—
450C22×C4#12614yes—
451C22×C4#12712yes—
452C22×C4C23×C411yescharacteristic
453C22×C4C23×C411yescharacteristic
454C22×C4C23×C411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4111yescharacteristic
461C22×C4C2×#4111yescharacteristic
462C22×C4C2×#4111yescharacteristic
463C22×C4C2×#4111yescharacteristic
464C22×C4C2×#4111yescharacteristic
465C22×C4C2×#4111yescharacteristic
466M16—232no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—232no—
486C2×C8C2×C22⋊C414yes—
487C2×C8C2×C22⋊C414yes—
488C2×C8C2×C22⋊C418yes—
489C4⋊C4—216no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—48no—
503C22⋊C4—216no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C42—24no—
507C42—24no—
508C42—24no—
509C42—24no—
510C42—48no—
511C42—48no—
512C23—28no—
513C23—28no—
514C23—24no—
515C23—24no—
516C23—28no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—28no—
522C23C4×C22⋊C414yes—
523C23#38614yes—
524C23#38814yes—
525C23#38814yes—
526C23#39014yes—
527C23#39014yes—
528C23#39014yes—
529C23C22×C22⋊C411yescharacteristic
530C23C2×C4×D411yescharacteristic
531C23C2×C4×D411yescharacteristic
532C23C2×#12111yescharacteristic
533C23C2×#12111yescharacteristic
534C23C2×#12411yescharacteristic
535C23C2×#12711yescharacteristic
536C23#57714yes—
537C23#57814yes—
538Q8—28no—
539Q8—28no—
540Q8—432no—
541Q8—44no—
542Q8—44no—
543Q8—44no—
544Q8—44no—
545Q8—44no—
546Q8—44no—
547C2×C4—216no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—432no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—816no—
567C2×C4—84no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—84no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4C4×C22⋊C414yes—
576C2×C4C4×C22⋊C418yes—
577C2×C4C4×C22⋊C418yes—
578C2×C4#38614yes—
579C2×C4#38618yes—
580C2×C4#38618yes—
581C2×C4#38814yes—
582C2×C4#38814yes—
583C2×C4#38818yes—
584C2×C4#38818yes—
585C2×C4#38818yes—
586C2×C4#38818yes—
587C2×C4#39014yes—
588C2×C4#39014yes—
589C2×C4#39014yes—
590C2×C4C22×C4211yescharacteristic
591C2×C4C22×C22⋊C411yescharacteristic
592C2×C4C22×C22⋊C411yescharacteristic
593C2×C4C2×#11711yescharacteristic
594C2×C4C2×#11711yescharacteristic
595C2×C4C2×#11711yescharacteristic
596C2×C4C2×C4×D411yescharacteristic
597C2×C4C2×C4×D411yescharacteristic
598C2×C4C2×C4×D411yescharacteristic
599C2×C4C2×C4×D411yescharacteristic
600C2×C4C2×C4×D411yescharacteristic
601C2×C4C2×C4×D411yescharacteristic
602C2×C4C2×C4×D411yescharacteristic
603C2×C4C2×C4×D411yescharacteristic
604C2×C4C2×C4×D411yescharacteristic
605C2×C4C2×C4×D411yescharacteristic
606C2×C4C2×#12011yescharacteristic
607C2×C4C2×#12111yescharacteristic
608C2×C4C2×#12111yescharacteristic
609C2×C4C2×#12111yescharacteristic
610C2×C4C2×#12111yescharacteristic
611C2×C4C2×#12311yescharacteristic
612C2×C4C2×#12311yescharacteristic
613C2×C4C2×#12311yescharacteristic
614C2×C4C2×#12411yescharacteristic
615C2×C4C2×#12411yescharacteristic
616C2×C4C2×#12611yescharacteristic
617C2×C4C2×#12611yescharacteristic
618C8—24no—
619C8—24no—
620C8—28no—
621C22—28no—
622C22—28no—
623C22—28no—
624C22—28no—
625C22—28no—
626C22—24no—
627C22—24no—
628C22—28no—
629C22#143718yes—
630C22C2×C4×C22⋊C411yescharacteristic
631C22C2×#38611yescharacteristic
632C22C2×#38811yescharacteristic
633C22C2×#38811yescharacteristic
634C22C2×#39011yescharacteristic
635C22C2×#39011yescharacteristic
636C22C2×#39011yescharacteristic
637C22#289814yes—
638C22#294714yes—
639C22#294912yes—
640C22#295012yes—
641C22#305712yes—
642C22#305912yes—
643C22#306314yes—
644C22#313714yes—
645C22#315412yes—
646C22#315612yes—
647C4—48no—
648C4—48no—
649C4—84no—
650C4—84no—
651C4—84no—
652C4#143718yes—
653C4C2×C4×C22⋊C411yescharacteristic
654C4C2×C4×C22⋊C411yescharacteristic
655C4C2×#38611yescharacteristic
656C4C2×#38611yescharacteristic
657C4C2×#38811yescharacteristic
658C4C2×#38811yescharacteristic
659C4C2×#38811yescharacteristic
660C4C2×#38811yescharacteristic
661C2—28no—
662C2C2×#143711yescharacteristic
663C2#2041712yes—
664C2#2097412yes—
665C2#2108812yes—
666C2#2109012yes—
667C2#2120512yes—
668C2#2123111yescharacteristic
669C2#2123211yescharacteristic
670C2#2123412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #10928C2semidirect
9. #10926C2semidirect
19. C2×#419C4semidirect
22. C22×#103C4semidirect
32. #2310C4semidirect
125. C2×#103C2×C4semidirect
135. #391C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4192
C23×#1032
C22×#4052
C2×#23102
C2×#17362
C2×#17352
#137952
#109282
#109262