Finite Group Databasebeta

← #373885

Subgroups — #373885

Order 512.

1386 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2C2×#1952C211yesmaximal, characteristic, complemented
3C2×#1951C211yesmaximal, characteristic, complemented
4#20845C211yesmaximal, characteristic
5#20844C211yesmaximal, characteristic
6C2×#1811C211yesmaximal, characteristic
7#20386C211yesmaximal, characteristic
8#11331C211yesmaximal, characteristic, complemented
9#11330C211yesmaximal, characteristic, complemented
10#11271C211yesmaximal, characteristic, complemented
11#11270C211yesmaximal, characteristic, complemented
12#10870C211yesmaximal, characteristic, complemented
13#10869C211yesmaximal, characteristic, complemented
14#10728C211yesmaximal, characteristic
15#10727C211yesmaximal, characteristic
16C23×M16C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C22×#108C2211yescharacteristic
21C2×#428—24no—
22C2×#428—24no—
23C2×#428—22no—
24C2×#428—22no—
25C2×#428C2211yescharacteristic
26C2×#428C2211yescharacteristic
27C2×#428C2211yescharacteristic
28C2×#428C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×#107C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×#106C2211yescharacteristic
33C2×#407C2211yescharacteristic
34C2×#407C2211yescharacteristic
35C2×#407C2211yescharacteristic
36C2×#405C2211yescharacteristic
37C2×#405C2211yescharacteristic
38#2280C2211yescharacteristic
39#1952—24no—
40#1951—24no—
41#1854—22no—
42#1854—22no—
43#1853—22no—
44#1852—22no—
45#1811C418yescomplemented
46#1753—22no—
47#1753—22no—
48C2×#343C2211yescharacteristic
49C2×#343C2211yescharacteristic
50C2×#343C2211yescharacteristic
51C2×#343C2211yescharacteristic
52C2×#343C2211yescharacteristic
53C2×#343C2211yescharacteristic
54C2×#343C2211yescharacteristic
55C2×#343C2211yescharacteristic
56C2×#341C2211yescharacteristic
57C2×#341C2211yescharacteristic
58#1736C2211yescharacteristic
59#1736C2211yescharacteristic
60#1736C2211yescharacteristic
61#1736C2211yescharacteristic
62#1729C2211yescharacteristic
63#1729C2211yescharacteristic
64C24×C4—21no—
65C24×C4C2311yescharacteristic
66C22×M16—22no—
67C22×M16—22no—
68C22×M16—22no—
69C22×M16—22no—
70C22×M16C2×C414yes—
71C22×M16D412yes—
72C22×M16D412yes—
73C22×M16Q814yes—
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C22×M16C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C2×#117—22no—
83C2×#117—22no—
84C2×#117—22no—
85C2×#117—22no—
86C2×#117—41no—
87C2×#117—41no—
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C22×C4⋊C4—22no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—41no—
96C22×C4⋊C4—41no—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4—21no—
102C22×C42—21no—
103C2×#108—22no—
104C2×#108—22no—
105C2×#108—22no—
106C2×#108—22no—
107C2×#108C2×C414yescomplemented
108C2×#108C2×C414yescomplemented
109#428—216no—
110#428—216no—
111#428—22no—
112#428—22no—
113#428—22no—
114#428—22no—
115#428—41no—
116#428—41no—
117#428—41no—
118#428—41no—
119C2×#107—22no—
120C2×#107—21no—
121C2×#107—21no—
122C2×#107—22no—
123C2×#107—22no—
124C2×#107—22no—
125C2×#107—21no—
126C2×#107—21no—
127C2×#107—21no—
128C2×#107—21no—
129C2×#107D414yes—
130C2×#106—22no—
131C2×#106—21no—
132C2×#106—21no—
133C2×#106—22no—
134C2×#106—22no—
135C2×#106—22no—
136C2×#106—21no—
137C2×#106—21no—
138C2×#106—21no—
139C2×#106—21no—
140C2×#106D414yes—
141#407—22no—
142#407—22no—
143#407—22no—
144#407C2×C414yes—
145#407C2×C414yes—
146#407Q814yes—
147C2×C22⋊C8—22no—
148C2×C22⋊C8—22no—
149C2×C22⋊C8—22no—
150#405—22no—
151#405—22no—
152#405—22no—
153#405—22no—
154C2×#97—21no—
155C2×#97—21no—
156C2×C4×C8—21no—
157C2×C4×C8—21no—
158#381—21no—
159#380D412yes—
160#380D412yes—
161#379—21no—
162C4×C22⋊C4—22no—
163#343—22no—
164#343—22no—
165#343—22no—
166#343—22no—
167#343—41no—
168#343—41no—
169#343—41no—
170#343—41no—
171#341C2×C414yescomplemented
172#341C2×C414yescomplemented
173#340—22no—
174#340—22no—
175#340—22no—
176#340—22no—
177#336—22no—
178#336—22no—
179C25#4111yescharacteristic
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—84no—
196C23×C4C22×C412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D412yes—
199C23×C4C2×Q812yes—
200C23×C4C2×Q812yes—
201C23×C4#4112yes—
202C23×C4#4111yescharacteristic
203C23×C4C2411yescharacteristic
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—28no—
211C2×M16—28no—
212C2×M16—24no—
213C2×M16—24no—
214C2×M16—28no—
215C2×M16—28no—
216C2×M16—24no—
217C2×M16—24no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—21no—
241C2×M16—21no—
242C2×M16C4⋊C418yes—
243C2×M16C4⋊C418yes—
244C2×M16C22×C412yes—
245C2×M16C22×C412yes—
246C2×M16C22×C412yes—
247C2×M16C22×C412yes—
248C2×M16C2×Q812yes—
249C2×M16C2×Q812yes—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8C22×C412yes—
261C22×C8C22×C412yes—
262C22×C8C2×D412yes—
263C22×C8C2×D411yescharacteristic
264C22×C8C2×D411yescharacteristic
265C22×C8C2×D412yes—
266C22×C8C2×D412yes—
267C22×C8C2×Q812yes—
268C22×C8C2×Q812yes—
269C22×C8#4112yes—
270C22×C8#4112yes—
271#117—28no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—21no—
279#117—21no—
280#117—44no—
281#117—44no—
282#117—44no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—42no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—42no—
329#108—28no—
330#108—28no—
331#107—28no—
332#107—24no—
333#107—24no—
334#107—42no—
335#107—42no—
336#107—42no—
337#107—42no—
338#106—28no—
339#106—24no—
340#106—24no—
341#106—42no—
342#106—42no—
343#106—42no—
344#106—42no—
345C22⋊C8—28no—
346C22⋊C8—28no—
347C22⋊C8—28no—
348#97—42no—
349#97—42no—
350C4×C8—42no—
351C4×C8—42no—
352#95—22no—
353#95—22no—
354C24—21no—
355C24—21no—
356C24—21no—
357C24—21no—
358C24—42no—
359C24—42no—
360C24C4×Q812yes—
361C24#12112yes—
362C24#12512yes—
363C24C2×#4111yescharacteristic
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—48no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—88no—
411C22×C4—88no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4C2×C4⋊C414yes—
417C22×C4C4×D414yes—
418C22×C4C4×Q812yes—
419C22×C4#12112yes—
420C22×C4C22⋊Q814yes—
421C22×C4#12512yes—
422C22×C4C4⋊Q814yes—
423C22×C4C23×C411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×Q811yescharacteristic
429C22×C4C22×Q811yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4H(5,𝔽2)11yescharacteristic
434C22×C4H(5,𝔽2)11yescharacteristic
435C22×C42−1+411yescharacteristic
436C22×C42−1+411yescharacteristic
437M16—216no—
438M16—216no—
439M16—22no—
440M16—22no—
441M16—22no—
442M16—22no—
443M16—22no—
444M16—22no—
445M16—22no—
446M16—22no—
447M16—22no—
448M16—22no—
449M16—22no—
450M16—22no—
451M16—41no—
452M16—41no—
453M16—41no—
454M16—41no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—41no—
459M16—41no—
460M16—41no—
461M16—41no—
462M16—41no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8C2×C4⋊C414yes—
488C2×C8C2×C4⋊C412yes—
489C2×C8C2×C4⋊C412yes—
490C2×C8C4×D414yes—
491C2×C8C4×D414yes—
492C2×C8#12012yes—
493C2×C8#12012yes—
494C2×C8C22⋊Q814yes—
495C2×C8C22⋊Q814yes—
496C2×C8#12312yes—
497C2×C8#12312yes—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—48no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C22⋊C4—48no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C42—48no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—42no—
552C23—42no—
553C23#38914yes—
554C23C2×C4×Q811yescharacteristic
555C23C2×#12111yescharacteristic
556C23C2×#12511yescharacteristic
557C23#54611yescharacteristic
558C23#54611yescharacteristic
559C23#55211yescharacteristic
560C23#55211yescharacteristic
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—816no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4#38914yes—
600C2×C4C22×C4⋊C411yescharacteristic
601C2×C4C2×C4×D411yescharacteristic
602C2×C4C2×C4×D411yescharacteristic
603C2×C4C2×C4×D411yescharacteristic
604C2×C4#52011yescharacteristic
605C2×C4#52011yescharacteristic
606C2×C4C2×#12011yescharacteristic
607C2×C4C2×C22⋊Q811yescharacteristic
608C2×C4C2×C22⋊Q811yescharacteristic
609C2×C4C2×C22⋊Q811yescharacteristic
610C2×C4C2×#12311yescharacteristic
611C2×C4C2×C4⋊Q811yescharacteristic
612C2×C4#53511yescharacteristic
613C2×C4#53611yescharacteristic
614C2×C4#54411yescharacteristic
615C2×C4#54411yescharacteristic
616C2×C4D4211yescharacteristic
617C2×C4D4211yescharacteristic
618C2×C4#54711yescharacteristic
619C2×C4#54711yescharacteristic
620C2×C4#54811yescharacteristic
621C2×C4#54811yescharacteristic
622C2×C4D4×Q811yescharacteristic
623C2×C4D4×Q811yescharacteristic
624C2×C4#55311yescharacteristic
625C2×C4#55311yescharacteristic
626C2×C4#55411yescharacteristic
627C2×C4#55411yescharacteristic
628C8—24no—
629C8—42no—
630C8—42no—
631C8—42no—
632C8—42no—
633C8—42no—
634C8—42no—
635C8#38514yes—
636C8#38514yes—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22C2×#38911yescharacteristic
657C22#236011yescharacteristic
658C22#236011yescharacteristic
659C22#240811yescharacteristic
660C22#244211yescharacteristic
661C22#244211yescharacteristic
662C22#248611yescharacteristic
663C22#306812yes—
664C22#306912yes—
665C4—24no—
666C4—82no—
667C4—81no—
668C4—82no—
669C4—81no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4C2×#38511yescharacteristic
674C4#231711yescharacteristic
675C4#236111yescharacteristic
676C4#236111yescharacteristic
677C4#240911yescharacteristic
678C4#244011yescharacteristic
679C4#244011yescharacteristic
680C4#248711yescharacteristic
681C2—21no—
682C2—21no—
683C2—21no—
684C2—21no—
685C2—44no—
686C2#1413011yescharacteristic
687C2#2118212yes—
688C2#2428311yescharacteristic
689C2#2428411yescharacteristic
690C2#2767612yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
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
2. C2×#1952C2semidirect
3. C2×#1951C2semidirect
8. #11331C2semidirect
9. #11330C2semidirect
10. #11271C2semidirect
11. #11270C2semidirect
12. #10870C2semidirect
13. #10869C2semidirect
45. #1811C4semidirect
107. C2×#108C2×C4semidirect
108. C2×#108C2×C4semidirect
171. #341C2×C4semidirect
172. #341C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#19522
C2×#19512
#208452
#208442
C2×#18112
#203862
#113312
#113302
#112712
#112702
#108702
#108692
#107282
#107272