Finite Group Databasebeta

← #367044

Subgroups — #367044

Order 512.

1850 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2548C211yesmaximal, characteristic
2C2×#1983C212yesmaximal, complemented
3C2×#1982C211yesmaximal, characteristic
4C2×#1961C211yesmaximal, characteristic
5C2×#1940C211yesmaximal, characteristic, complemented
6C2×#1805C211yesmaximal, characteristic, complemented
7C2×#1763C211yesmaximal, characteristic, complemented
8#13804C211yesmaximal, characteristic, complemented
9#11076C212yesmaximal, complemented
10#11009C212yesmaximal, complemented
11#10931C212yesmaximal, complemented
12C22×#127C2211yescharacteristic
13C22×#124C2212yes—
14C22×C4×Q8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2211yescharacteristic
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×#97C2211yescharacteristic
25#2548—24no—
26#2548C418yescomplemented
27C2×#394—21no—
28C2×#394—21no—
29C2×#390—22no—
30C2×#390C2212yes—
31C2×#382C2211yescharacteristic
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1983—28no—
36#1982—24no—
37#1961—24no—
38#1940—24no—
39#1805—24no—
40#1763—24no—
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#340C2211yescharacteristic
44C2×#336C2212yes—
45#1436—22no—
46C4⋊#95C2212yes—
47#1434—21no—
48C4×#95—21no—
49C23×Q8C2311yescharacteristic
50C23×D4—21no—
51C23×D4—21no—
52C23×D4C2312yes—
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×#127—22no—
57C2×#127—22no—
58C2×#127—22no—
59C2×#127—22no—
60C2×#127—22no—
61C2×#127C2×C414yes—
62C2×#124—24no—
63C2×#124—24no—
64C2×#124—24no—
65C2×#124—42no—
66C2×#124—42no—
67C2×#124C2×C418yes—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8C2×C414yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D412yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42—22no—
87C22×C42C2312yes—
88C22×C42C2311yescharacteristic
89C2×C4⋊C8—24no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—24no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—42no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—42no—
103C2×#103D414yescomplemented
104C2×#102—24no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102—24no—
109C2×#102—24no—
110C2×#102—24no—
111C2×#102—24no—
112C2×#102—42no—
113C2×#102—42no—
114C2×#102—42no—
115C2×#102—42no—
116C2×#97—22no—
117C2×#97—22no—
118C2×#97—22no—
119#394—28no—
120#394—28no—
121#390—216no—
122#390—24no—
123#390C2×C418yes—
124#382—22no—
125#382D414yes—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—42no—
129C4×C4⋊C4C2×C414yes—
130C4×C4⋊C4D414yes—
131C4×C4⋊C4D414yes—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95D411yescharacteristic
139C2×#95D411yescharacteristic
140#340—24no—
141#340—22no—
142#340—22no—
143#336—24no—
144C25—22no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8C22×C412yes—
157C22×Q8C2×D412yes—
158C22×Q8#4112yes—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—44no—
175C22×D4—44no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4—82no—
181C22×D4—82no—
182C22×D4C22×C414yes—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—22no—
187C23×C4—21no—
188C23×C4C2×D412yes—
189C23×C4C2×D412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2411yescharacteristic
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—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8C2×D412yes—
213C22×C8#4112yes—
214#127—28no—
215#127—28no—
216#127—44no—
217#127—44no—
218#124—216no—
219#124—48no—
220#124—48no—
221#124—48no—
222C4×Q8—28no—
223C4×Q8—28no—
224C4×Q8—44no—
225C4×Q8—44no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4C22⋊C418yes—
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4#4112yes—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—82no—
265C2×C22⋊C4—82no—
266C2×C22⋊C4—82no—
267C2×C22⋊C4—82no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—44no—
277C2×C42—44no—
278C2×C42—82no—
279C2×C42—82no—
280C2×C42C22×C414yes—
281C2×C42C22×C412yes—
282C2×C42C2×D414yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42#4112yes—
287C2×C42#4112yes—
288C4⋊C8—48no—
289C4⋊C8—44no—
290#103—28no—
291#103—28no—
292#103—48no—
293#102—48no—
294#102—48no—
295#102—48no—
296#102—48no—
297#97—44no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—44no—
302#95—44no—
303#95—82no—
304#95C22⋊C414yes—
305#95C22⋊C414yes—
306C24—24no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24C22×D411yescharacteristic
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8C4×D414yes—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—416no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—84no—
384C2×D4—84no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4—88no—
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—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—42no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4C2×C22⋊C414yes—
434C22×C4C2×C22⋊C414yes—
435C22×C4C2×C22⋊C412yes—
436C22×C4C2×C22⋊C412yes—
437C22×C4#12112yes—
438C22×C4#12112yes—
439C22×C4#12112yes—
440C22×C4#12112yes—
441C22×C4C23×C411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4H(5,𝔽2)11yescharacteristic
452C22×C4H(5,𝔽2)11yescharacteristic
453C22×C4H(5,𝔽2)11yescharacteristic
454C22×C42−1+411yescharacteristic
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—48no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—48no—
493C4⋊C4#12114yes—
494C22⋊C4—416no—
495C22⋊C4—416no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C42—24no—
499C42—24no—
500C42—24no—
501C42—24no—
502C42—48no—
503C42—84no—
504C42—88no—
505C42C4×D414yes—
506C42C4×D414yes—
507C42#12414yes—
508C42#12714yes—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—82no—
517C23—82no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23—82no—
525C23—82no—
526C23—82no—
527C23—82no—
528C23—82no—
529C23—82no—
530C23—82no—
531C23—82no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23—82no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23—82no—
544C23C22×C22⋊C411yescharacteristic
545C23C2×#12111yescharacteristic
546C23C2×#12111yescharacteristic
547C23D4211yescharacteristic
548C23#54611yescharacteristic
549C23#54811yescharacteristic
550C23#55011yescharacteristic
551C23#57512yes—
552C23#57512yes—
553C23#57612yes—
554C23#57712yes—
555Q8—24no—
556Q8—24no—
557Q8—42no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—42no—
562Q8—42no—
563D4—44no—
564D4—44no—
565D4—44no—
566D4—44no—
567D4—816no—
568D4—816no—
569D4—82no—
570D4—82no—
571D4—82no—
572D4—82no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
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—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4#39014yes—
625C2×C4#39014yes—
626C2×C4C2×C4×D411yescharacteristic
627C2×C4C2×C4×D411yescharacteristic
628C2×C4C2×C4×D411yescharacteristic
629C2×C4C4×#4111yescharacteristic
630C2×C4#51811yescharacteristic
631C2×C4#52011yescharacteristic
632C2×C4C2×#12111yescharacteristic
633C2×C4#52511yescharacteristic
634C2×C4C2×#12411yescharacteristic
635C2×C4#52911yescharacteristic
636C2×C4C2×#12711yescharacteristic
637C2×C4#53211yescharacteristic
638C2×C4#53511yescharacteristic
639C2×C4#53711yescharacteristic
640C2×C4#54011yescharacteristic
641C2×C4#54211yescharacteristic
642C2×C4D4211yescharacteristic
643C2×C4D4211yescharacteristic
644C2×C4#54611yescharacteristic
645C2×C4#54611yescharacteristic
646C2×C4#54711yescharacteristic
647C2×C4#54811yescharacteristic
648C2×C4#54811yescharacteristic
649C2×C4#55011yescharacteristic
650C2×C4#55011yescharacteristic
651C2×C4#55311yescharacteristic
652C2×C4#55311yescharacteristic
653C2×C4#55511yescharacteristic
654C8—42no—
655C8—42no—
656C8—82no—
657C22—84no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—84no—
664C22—162no—
665C22—162no—
666C22—162no—
667C22—162no—
668C22—162no—
669C22—162no—
670C22—162no—
671C22—162no—
672C22C2×#39011yescharacteristic
673C22D4×C22⋊C411yescharacteristic
674C22#234111yescharacteristic
675C22#241411yescharacteristic
676C22#241611yescharacteristic
677C22#242611yescharacteristic
678C22#242811yescharacteristic
679C22#289612yes—
680C22#289712yes—
681C22#304012yes—
682C22#304011yescharacteristic
683C22#304011yescharacteristic
684C22#304111yescharacteristic
685C22#304311yescharacteristic
686C22#304512yes—
687C22#326812yes—
688C22#327112yes—
689C22#330912yes—
690C22#331712yes—
691C22#337012yes—
692C22#337112yes—
693C22#340112yes—
694C22#340312yes—
695C4—24no—
696C4—24no—
697C4—44no—
698C4—44no—
699C4—44no—
700C4—44no—
701C4—82no—
702C4—84no—
703C4—82no—
704C4C4×#12111yescharacteristic
705C4#232411yescharacteristic
706C4#236111yescharacteristic
707C4#236311yescharacteristic
708C4#249011yescharacteristic
709C4#249111yescharacteristic
710C4#254011yescharacteristic
711C4#254111yescharacteristic
712C2—164no—
713C2#1399811yescharacteristic
714C2#2126111yescharacteristic
715C2#2126311yescharacteristic
716C2#2242212yes—
717C2#2254912yes—
718C2#2464711yescharacteristic
719C2#2464911yescharacteristic
720C2#2474711yescharacteristic
721C2#2474911yescharacteristic
722C2#2529112yes—
723C2#2538312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1983C2semidirect
5. C2×#1940C2semidirect
6. C2×#1805C2semidirect
7. C2×#1763C2semidirect
8. #13804C2semidirect
9. #11076C2semidirect
10. #11009C2semidirect
11. #10931C2semidirect
26. #2548C4semidirect
103. C2×#103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25482
C2×#19832
C2×#19822
C2×#19612
C2×#19402
C2×#18052
C2×#17632
#138042
#110762
#110092
#109312