Finite Group Databasebeta

← #393331

Subgroups — #393331

Order 512.

1766 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#405C211yesmaximal, characteristic
2C2×#2628C211yesmaximal, characteristic
3C2×#1993C211yesmaximal, characteristic, complemented
4C2×#1990C211yesmaximal, characteristic, complemented
5#14340C211yesmaximal, characteristic
6#10881C212yesmaximal, complemented
7#10880C212yesmaximal, complemented
8#10843C211yesmaximal, characteristic, complemented
9#10842C211yesmaximal, characteristic, complemented
10#10722C212yesmaximal
11#10721C212yesmaximal
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#405—24no—
17C2×#405—24no—
18C2×#405—24no—
19C22×#97C2211yescharacteristic
20C22×#97C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2628—24no—
24#2628C418yescomplemented
25C2×#400—21no—
26C2×#399—21no—
27C2×#395—21no—
28C2×#393—21no—
29C2×#389C2212yes—
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2212yes—
35#1993—24no—
36#1990—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45#1441—21no—
46#1441—21no—
47#1439—21no—
48#1439—21no—
49C4⋊#95C2212yes—
50C23×Q8—21no—
51C24×C4C2311yescharacteristic
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—22no—
58C22×M16Q814yescomplemented
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—22no—
63C23×C8—22no—
64C23×C8—22no—
65C23×C8Q814yescomplemented
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8C2×C414yes—
76C2×#117—22no—
77C2×#117—22no—
78C2×#117—22no—
79C2×#117—22no—
80C2×#117C2×C414yes—
81C2×#117D414yes—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4D411yescharacteristic
87C22×C4⋊C4D411yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C22⋊C4—22no—
93C22×C22⋊C4—22no—
94C22×C22⋊C4D411yescharacteristic
95C22×C22⋊C4D411yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#405—232no—
99C2×#97—22no—
100C2×#97—22no—
101C2×#97—22no—
102C2×#97—22no—
103C2×#97—22no—
104C2×#97D414yes—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8D414yes—
111#400—28no—
112#399—28no—
113#395—28no—
114#393—28no—
115#389—24no—
116#389C2×C418yes—
117#384—22no—
118#384—22no—
119#382—22no—
120#382—22no—
121#382—22no—
122#382C2×C414yes—
123#380—24no—
124#380—24no—
125#380—24no—
126#380C2×C418yes—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136#340—24no—
137#340—24no—
138#340—24no—
139#340—24no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—24no—
144C25C2×D411yescharacteristic
145C22×Q8—24no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—82no—
149C22×Q8—82no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4C22×C412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×Q812yes—
170C23×C4C2×Q812yes—
171C23×C4#4112yes—
172C23×C4#4112yes—
173C23×C4C2411yescharacteristic
174C2×M16—28no—
175C2×M16—28no—
176C2×M16—28no—
177C2×M16—28no—
178C2×M16—28no—
179C2×M16—28no—
180C2×M16—28no—
181C22×C8—28no—
182C22×C8—28no—
183C22×C8—28no—
184C22×C8—28no—
185C22×C8—28no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—28no—
203C22×C8—28no—
204C22×C8C2×D412yes—
205C22×C8C2×D412yes—
206C22×C8C2×D412yes—
207C22×C8C2×D412yes—
208C22×C8C2×Q812yes—
209C22×C8C2×Q812yes—
210C22×C8C2×Q812yes—
211C22×C8C2×Q812yes—
212C22×C8#4112yes—
213C22×C8#4112yes—
214C22×C8#4112yes—
215C22×C8#4112yes—
216C4⋊Q8—28no—
217C4⋊Q8—28no—
218C4⋊Q8—44no—
219C4⋊Q8—44no—
220#117—28no—
221#117—28no—
222#117—28no—
223#117—28no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
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—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
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—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—82no—
257C2×C4⋊C4—82no—
258C2×C4⋊C4—82no—
259C2×C4⋊C4—82no—
260C2×C4⋊C4C22⋊C414yes—
261C2×C4⋊C4C22⋊C414yes—
262C2×C4⋊C4C22×C414yes—
263C2×C4⋊C4C22×C414yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—82no—
271C2×C22⋊C4—82no—
272C2×C22⋊C4—82no—
273C2×C22⋊C4—82no—
274C2×C22⋊C4C22⋊C414yes—
275C2×C22⋊C4C22⋊C414yes—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42C22×C412yes—
281C2×C42C22×C412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42#4112yes—
287C2×C42#4112yes—
288#97—28no—
289#97—28no—
290C4×C8—28no—
291C4×C8—28no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—42no—
297#95—42no—
298#95—44no—
299#95—82no—
300#95—82no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24C2×C22⋊C412yes—
310C24#12412yes—
311C24C4⋊Q812yes—
312C24C22×D411yescharacteristic
313C2×Q8—48no—
314C2×Q8—88no—
315C2×Q8—84no—
316C2×Q8—84no—
317C2×Q8—88no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
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—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C22⋊C412yes—
388C22×C4C2×C22⋊C412yes—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4C4×Q814yes—
392C22×C4C4×Q814yes—
393C22×C4#12112yes—
394C22×C4#12112yes—
395C22×C4#12112yes—
396C22×C4#12112yes—
397C22×C4C22⋊Q814yes—
398C22×C4C22⋊Q814yes—
399C22×C4#12412yes—
400C22×C4C4⋊Q812yes—
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×Q811yescharacteristic
406C22×C4C22×Q811yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415M16—232no—
416C2×C8—232no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8C22⋊Q814yes—
442C2×C8C22⋊Q814yes—
443C2×C8C22⋊Q814yes—
444C2×C8C22⋊Q814yes—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—48no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—88no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C22⋊C4—28no—
468C22⋊C4—28no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42C4×D414yes—
476C42C4×D414yes—
477C42#12014yes—
478C42#12314yes—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23#39114yes—
502C23C22×C22⋊C411yescharacteristic
503C23C2×#12111yescharacteristic
504C23C2×#12111yescharacteristic
505C23C2×#12411yescharacteristic
506C23C2×C4⋊Q811yescharacteristic
507C23#53211yescharacteristic
508C23#53211yescharacteristic
509C23#57612yes—
510C23#57712yes—
511C23#57712yes—
512C23#57812yes—
513Q8—816no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4#38614yes—
575C2×C4#38614yes—
576C2×C4#39114yes—
577C2×C4C2×C4×D411yescharacteristic
578C2×C4C2×C4×D411yescharacteristic
579C2×C4C2×C4×Q811yescharacteristic
580C2×C4C2×C4×Q811yescharacteristic
581C2×C4C4×#4111yescharacteristic
582C2×C4C4×#4111yescharacteristic
583C2×C4C2×#12011yescharacteristic
584C2×C4C2×C22⋊Q811yescharacteristic
585C2×C4C2×C22⋊Q811yescharacteristic
586C2×C4C2×C22⋊Q811yescharacteristic
587C2×C4C2×C22⋊Q811yescharacteristic
588C2×C4C2×C22⋊Q811yescharacteristic
589C2×C4C2×C22⋊Q811yescharacteristic
590C2×C4C2×#12311yescharacteristic
591C2×C4#52511yescharacteristic
592C2×C4#52511yescharacteristic
593C2×C4#52511yescharacteristic
594C2×C4#52511yescharacteristic
595C2×C4#52511yescharacteristic
596C2×C4#52511yescharacteristic
597C2×C4#52911yescharacteristic
598C2×C4#52911yescharacteristic
599C2×C4#52911yescharacteristic
600C2×C4#52911yescharacteristic
601C2×C4#53311yescharacteristic
602C2×C4#53311yescharacteristic
603C2×C4#53311yescharacteristic
604C2×C4#53311yescharacteristic
605C8—24no—
606C8—24no—
607C8—24no—
608C8—24no—
609C22—28no—
610C22—24no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—44no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—42no—
619C22—42no—
620C22C2×#38611yescharacteristic
621C22C2×#39111yescharacteristic
622C22#229711yescharacteristic
623C22#285011yescharacteristic
624C22#285011yescharacteristic
625C22#285711yescharacteristic
626C22#285711yescharacteristic
627C22#289712yes—
628C22#289812yes—
629C22#305711yescharacteristic
630C22#305711yescharacteristic
631C22#305811yescharacteristic
632C22#305911yescharacteristic
633C22#306212yes—
634C22#306312yes—
635C22#313612yes—
636C22#313712yes—
637C22#316312yes—
638C22#316412yes—
639C22#317312yes—
640C22#317512yes—
641C22#317812yes—
642C22#317912yes—
643C4—24no—
644C4—24no—
645C4—24no—
646C4—42no—
647C4—42no—
648C4—84no—
649C4—84no—
650C4—162no—
651C4—162no—
652C4C4×C22⋊Q811yescharacteristic
653C4C4×C22⋊Q811yescharacteristic
654C4C4×C22⋊Q811yescharacteristic
655C4C4×C22⋊Q811yescharacteristic
656C4#284811yescharacteristic
657C4#284811yescharacteristic
658C4#285511yescharacteristic
659C4#285511yescharacteristic
660C2—44no—
661C2#1470211yescharacteristic
662C2#2097411yescharacteristic
663C2#2097511yescharacteristic
664C2#2132612yes—
665C2#2141212yes—
666C2#3304511yescharacteristic
667C2#3304611yescharacteristic
668C2#3305412yes—
669C2#3321311yescharacteristic
670C2#3321511yescharacteristic
671C2#3321912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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
3. C2×#1993C2semidirect
4. C2×#1990C2semidirect
6. #10881C2semidirect
7. #10880C2semidirect
8. #10843C2semidirect
9. #10842C2semidirect
24. #2628C4semidirect
58. C22×M16Q8semidirect
65. C23×C8Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4052
C2×#26282
C2×#19932
C2×#19902
#143402
#108812
#108802
#108432
#108422
#107222
#107212