Finite Group Databasebeta

← #374162

Subgroups — #374162

Order 512.

1890 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2C2×#2854C211yesmaximal, characteristic
3C2×#1900C212yesmaximal
4C2×#1728C211yesmaximal, characteristic, complemented
5#11204C212yesmaximal, complemented
6#11044C212yesmaximal, complemented
7#10962C212yesmaximal, complemented
8#10913C212yesmaximal, complemented
9#10739C212yesmaximal, complemented
10C23×M16C2212yes—
11C22×#123C412yes—
12C22×#121C2211yescharacteristic
13C23×C42C2211yescharacteristic
14#2920—28no—
15C2×#432—21no—
16C2×#432—21no—
17C2×#431—21no—
18C2×#431—21no—
19C2×#423C2212yes—
20C22×#100C2214yes—
21C2×#407C2212yes—
22C2×C4×M16C2212yes—
23#2854—21no—
24#2854—21no—
25#2854—21no—
26#2854—21no—
27#2854—21no—
28#2854—21no—
29#2854—21no—
30#2854—21no—
31C2×#390C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#388C412yes—
36C2×#388C412yes—
37C2×#386C412yes—
38C2×#380C2211yescharacteristic
39#1900—28no—
40C2×#343C2214yes—
41C2×#343C2214yes—
42C2×#343C2214yes—
43#1728—21no—
44#1728—21no—
45#1728C2212yes—
46#1728C2212yes—
47C2×#336—21no—
48C2×#336—21no—
49C23×D4C2×C412yes—
50C23×D4C2311yescharacteristic
51C24×C4C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4C2311yescharacteristic
54C22×M16—22no—
55C22×M16—28no—
56C22×M16—24no—
57C22×M16—24no—
58C22×M16—44no—
59C22×M16C2314yes—
60C22×M16C2314yes—
61C23×C8—22no—
62C2×#123—22no—
63C2×#123—22no—
64C2×#123—22no—
65C2×#123—22no—
66C2×#123—48no—
67C2×#121—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121—24no—
72C2×#121—24no—
73C22×C4⋊C4C2×C412yes—
74C22×C4⋊C4C2×C412yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C412yes—
80C22×C22⋊C4C2×C412yes—
81C22×C22⋊C4C2×C412yes—
82C22×C22⋊C4C2×C412yes—
83C22×C22⋊C4C2×C412yes—
84C22×C42—24no—
85C22×C42—24no—
86C22×C42C2×C411yescharacteristic
87C22×C42C2×C411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#432—28no—
99#432—28no—
100#431—28no—
101#431—28no—
102#423—28no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×#100—28no—
106C2×#100—28no—
107C2×#100—44no—
108#407—24no—
109#407D418yes—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C4×M16—28no—
113C2×#97—22no—
114C2×C4×C8—22no—
115#390—21no—
116#390—21no—
117#390—21no—
118#390—21no—
119#390—21no—
120#390—21no—
121#390—21no—
122#390—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#388—22no—
132#388—22no—
133#388—22no—
134#388—22no—
135#388—22no—
136#388—22no—
137#388—22no—
138#388—22no—
139#386—22no—
140#386—22no—
141#386—22no—
142#386—22no—
143#380—21no—
144#380—21no—
145#380—21no—
146#380—21no—
147C2×#95C2×C412yes—
148C2×#95C2×C412yes—
149C2×#95C2×C412yes—
150#343—28no—
151#343—28no—
152#343—44no—
153#336—22no—
154#336—22no—
155#336—22no—
156#336—22no—
157C25C22×C411yescharacteristic
158C25C22×C412yes—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—28no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—24no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—44no—
170C22×D4—44no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C412yes—
194C23×C4C22×C412yes—
195C23×C4C22×C412yes—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D414yes—
201C23×C4#4114yes—
202C23×C4C2411yescharacteristic
203C2×M16—28no—
204C2×M16—24no—
205C2×M16—24no—
206C2×M16—24no—
207C2×M16—24no—
208C2×M16—24no—
209C2×M16—24no—
210C2×M16—24no—
211C2×M16—24no—
212C2×M16—24no—
213C2×M16—24no—
214C2×M16—416no—
215C2×M16—416no—
216C2×M16—44no—
217C2×M16—44no—
218C2×M16C2×D418yes—
219C2×M16#4118yes—
220C22×C8—28no—
221C22×C8—24no—
222C22×C8—24no—
223C22×C8—44no—
224C22×C8—44no—
225#123—48no—
226#123—48no—
227#123—48no—
228#123—48no—
229#121—42no—
230#121—42no—
231#121—44no—
232#121—44no—
233#121—42no—
234#121—42no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22⋊C411yescharacteristic
292C2×C42C22⋊C411yescharacteristic
293C2×C42C22⋊C411yescharacteristic
294C2×C42C22⋊C411yescharacteristic
295C2×C42C22⋊C411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C4⋊C8—28no—
305C4⋊C8—28no—
306#100—416no—
307C22⋊C8—28no—
308C22⋊C8—28no—
309#97—28no—
310C4×C8—28no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—41no—
326C24—41no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24C2×C22⋊C411yescharacteristic
332C24C2×C22⋊C411yescharacteristic
333C24C4×D414yes—
334C24C23×C411yescharacteristic
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—41no—
342C2×D4—42no—
343C2×D4—41no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—42no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—816no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
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—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—416no—
412C22×C4—416no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4#10014yescomplemented
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C4×D414yes—
431C22×C4#12012yes—
432C22×C4#12012yes—
433C22×C4#12114yes—
434C22×C4#12114yes—
435C22×C4#12312yes—
436C22×C4#12312yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4112yes—
441C22×C4C2×#4114yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C42−1+412yes—
444M16—28no—
445M16—28no—
446M16—28no—
447M16—28no—
448M16—416no—
449M16—44no—
450M16—44no—
451M16—44no—
452M16—44no—
453C2×C8—28no—
454C2×C8—28no—
455C2×C8—416no—
456C2×C8—44no—
457C2×C8—44no—
458C2×C8—44no—
459C2×C8—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—41no—
463C4⋊C4—42no—
464C4⋊C4—41no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—28no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—28no—
519C23—21no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—48no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23—84no—
537C23—84no—
538C23—84no—
539C23#38612yes—
540C23#38612yes—
541C23C2×#10012yes—
542C23#41312yes—
543C23C22×C22⋊C411yescharacteristic
544C23C2×C4×D412yes—
545C23C4×#4112yes—
546C23#52012yes—
547D4—88no—
548D4—88no—
549D4—82no—
550D4—82no—
551D4—82no—
552D4—82no—
553C2×C4—28no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—28no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—28no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4#38612yes—
611C2×C4#38612yes—
612C2×C4C2×#10012yes—
613C2×C4C2×#10012yes—
614C2×C4#41312yes—
615C2×C4#41312yes—
616C2×C4C2×#12011yescharacteristic
617C2×C4C2×#12112yes—
618C2×C4C2×#12311yescharacteristic
619C2×C4#52512yes—
620C2×C4#53212yes—
621C2×C4#53511yescharacteristic
622C2×C4#53611yescharacteristic
623C2×C4#53712yes—
624C2×C4#54614yes—
625C2×C4#54714yes—
626C2×C4#55114yes—
627C2×C4#55214yes—
628C8—48no—
629C8—48no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—416no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—168no—
658C22#190014yes—
659C22C2×#38611yescharacteristic
660C22#229711yescharacteristic
661C22#231111yescharacteristic
662C22#235914yes—
663C22C22×#10011yescharacteristic
664C22C2×#41311yescharacteristic
665C22C2×#41311yescharacteristic
666C22C2×#41311yescharacteristic
667C4—22no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4#241112yes—
681C4#241312yes—
682C4#243012yes—
683C4#249812yes—
684C2—24no—
685C2—24no—
686C2—42no—
687C2—162no—
688C2#1413511yescharacteristic
689C2C2×#190011yescharacteristic
690C2#2098911yescharacteristic
691C2#2142411yescharacteristic
692C2#2142611yescharacteristic
693C2#2187811yescharacteristic
694C2#2188011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2920C2semidirect
4. C2×#1728C2semidirect
5. #11204C2semidirect
6. #11044C2semidirect
7. #10962C2semidirect
8. #10913C2semidirect
9. #10739C2semidirect
419. C22×C4#100semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C2×#28542
C2×#19002
C2×#17282
#112042
#110442
#109622
#109132
#107392