Finite Group Databasebeta

← #373824

Subgroups — #373824

Order 512.

1554 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34183C211yesmaximal, characteristic
2#21170C211yesmaximal, characteristic, complemented
3C2×#1859C212yesmaximal
4C2×#1806C211yesmaximal, characteristic
5C4×#417C211yesmaximal, characteristic
6#13910C211yesmaximal, characteristic, complemented
7#11340C212yesmaximal, complemented
8#11276C212yesmaximal, complemented
9#10857C211yesmaximal, characteristic, complemented
10#10851C211yesmaximal, characteristic, complemented
11#10781C212yesmaximal
12C2×#520C2212yes—
13C2×C4×#41C2211yescharacteristic
14C2×#428C2212yes—
15C2×#423C2211yescharacteristic
16C22×#104C2212yes—
17C2×#417C2211yescharacteristic
18C2×#417C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20Q8×C4⋊C4—21no—
21Q8×C4⋊C4—21no—
22D4×C4⋊C4—21no—
23D4×C4⋊C4—21no—
24D4×C4⋊C4—22no—
25D4×C4⋊C4—22no—
26#2317—22no—
27#2317C2212yes—
28C2×#389—22no—
29#2285—21no—
30#2285C2211yescharacteristic
31C2×#384—21no—
32C2×#384C2212yes—
33C2×#384C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1949—22no—
37#1948—21no—
38#1947—21no—
39#1859—28no—
40#1806C418yescomplemented
41#1763—21no—
42#1762—21no—
43C4×#103—21no—
44C4×#102—21no—
45C2×#343C2212yes—
46C2×#343C2212yes—
47#1736C2211yescharacteristic
48#1736C2212yes—
49#1736C2211yescharacteristic
50C2×#339C2212yes—
51C2×#339C2212yes—
52C2×#339C2212yes—
53#1444—22no—
54C4⋊#95C2212yes—
55C22×#41C2311yescharacteristic
56C22×M16C2312yes—
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59#520—24no—
60#520—24no—
61#520—42no—
62#520—42no—
63C4×#41—22no—
64C4×#41—22no—
65C4×#41C2×C414yes—
66C4×#41Q814yes—
67C2×C4×Q8—22no—
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—22no—
73C2×C4×D4—24no—
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×#117—22no—
77C2×#117—21no—
78C2×#117C2312yes—
79C2×#117C2311yescharacteristic
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4D412yes—
86C22×C4⋊C4D412yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C42—21no—
91C22×C42—21no—
92C22×C42C2312yes—
93C22×C42C2311yescharacteristic
94#428—24no—
95#428—42no—
96C2×#107—22no—
97C2×#106—22no—
98#423—22no—
99#423C2×C414yescomplemented
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×#104—24no—
103C2×#104—24no—
104C2×#104C2×C418yes—
105#417—22no—
106#417—22no—
107#417D414yes—
108#417Q814yes—
109C2×#103—21no—
110C2×#103—21no—
111C2×#102—21no—
112C2×#102—21no—
113C4×M16—22no—
114C4×M16—22no—
115C2×#97—21no—
116C2×C4×C8—21no—
117#389—24no—
118#389—24no—
119#389—22no—
120#389—22no—
121#389—48no—
122#385—24no—
123#385—24no—
124#385—24no—
125#384—22no—
126#384—22no—
127#384—22no—
128#384—24no—
129#384—21no—
130#384—21no—
131#384—21no—
132#384—21no—
133#384—42no—
134#384D411yescharacteristic
135#384D411yescharacteristic
136#382—22no—
137#382D412yes—
138#380—22no—
139#380—22no—
140#380—21no—
141#380—21no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4C2×C414yescomplemented
152C2×#95—21no—
153C2×#95—24no—
154C2×#95—21no—
155#343—24no—
156#343—42no—
157#340—22no—
158#340—22no—
159#340—21no—
160#340—21no—
161#340—21no—
162#340—21no—
163#339—24no—
164#339—42no—
165#339C2×C418yescomplemented
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41C22×C412yes—
173C2×#41C2×D412yes—
174C2×#41C2×Q812yes—
175C2×#41C2×Q812yes—
176C2×#41#4112yes—
177C2×#41#4112yes—
178C22×Q8C2×D411yescharacteristic
179C22×D4—21no—
180C22×D4C2×D411yescharacteristic
181C23×C4—21no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—24no—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D414yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2411yescharacteristic
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—22no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—22no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—22no—
207C2×M16C22×C414yes—
208C2×M16C2×D412yes—
209C2×M16C2×Q812yes—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—44no—
216C4×Q8—44no—
217C4×Q8C4⋊C414yescomplemented
218C4×Q8C4⋊C414yescomplemented
219C4×D4—24no—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—48no—
226C4×D4—48no—
227C4×D4—44no—
228C4×D4—44no—
229C4×D4C4⋊C414yescomplemented
230C4×D4C4⋊C414yescomplemented
231#117—21no—
232#117—21no—
233#117—24no—
234#117—24no—
235#117—22no—
236#117—22no—
237#117—22no—
238#117—22no—
239#117—44no—
240#117—44no—
241#117C22×C412yes—
242#117C2×Q812yes—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4—48no—
280C2×C4⋊C4—84no—
281C2×C4⋊C4—84no—
282C2×C4⋊C4—82no—
283C2×C4⋊C4—84no—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×Q812yes—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—44no—
308C2×C42—41no—
309C2×C42—41no—
310C2×C42C22×C414yes—
311C2×C42C22×C412yes—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×Q812yes—
317C2×C42#4112yes—
318C2×C42#4112yes—
319#107—44no—
320#106—44no—
321C4⋊C8—24no—
322C4⋊C8—24no—
323#104—216no—
324#103—24no—
325#103—24no—
326#102—24no—
327#102—24no—
328#97—42no—
329C4×C8—42no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—22no—
334#95—22no—
335#95—48no—
336#95—41no—
337#95—41no—
338C24—21no—
339C24C22×D411yescharacteristic
340#41—22no—
341#41—22no—
342#41—22no—
343#41—22no—
344#41—22no—
345#41—22no—
346#41—22no—
347#41—22no—
348#41—22no—
349#41—22no—
350#41—22no—
351#41—22no—
352#41C4×D414yes—
353#41C4×Q814yes—
354#41#12514yes—
355#41C4⋊Q814yes—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8C2×C4⋊C412yes—
360C2×Q8#12112yes—
361C2×Q8C22⋊Q812yes—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4C2×C4⋊C412yes—
372C2×D4#12112yes—
373C2×D4C22⋊Q812yes—
374C22×C4—21no—
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—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4C4×D412yes—
421C22×C4C4×D412yes—
422C22×C4#12011yescharacteristic
423C22×C4#12011yescharacteristic
424C22×C4#12011yescharacteristic
425C22×C4#12011yescharacteristic
426C22×C4C22⋊Q812yes—
427C22×C4C22⋊Q812yes—
428C22×C4C23×C411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×Q811yescharacteristic
433C22×C4C22×Q811yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4H(5,𝔽2)11yescharacteristic
439C22×C4H(5,𝔽2)11yescharacteristic
440C22×C42−1+411yescharacteristic
441C22×C42−1+411yescharacteristic
442M16—24no—
443M16—24no—
444M16—22no—
445M16—22no—
446M16—22no—
447M16—22no—
448M16—42no—
449M16—41no—
450M16—41no—
451M16—42no—
452M16—41no—
453M16—41no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—42no—
462C2×C8—42no—
463C4⋊C4—216no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4—41no—
484C4⋊C4—42no—
485C4⋊C4—41no—
486C4⋊C4—42no—
487C4⋊C4—41no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4—816no—
496C4⋊C4C2×C4⋊C412yes—
497C4⋊C4C2×C4⋊C412yes—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—48no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C42—22no—
506C42—22no—
507C42—24no—
508C42—24no—
509C42—24no—
510C42—24no—
511C42—48no—
512C42—42no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—82no—
517C42—82no—
518C42C2×C4⋊C412yes—
519C42C2×C4⋊C412yes—
520C42C4×D412yes—
521C42C4×D412yes—
522C42C22⋊Q812yes—
523C42C22⋊Q812yes—
524C42#12312yes—
525C42#12312yes—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C2×C4×D411yescharacteristic
538C23C2×#12011yescharacteristic
539C23C2×C22⋊Q811yescharacteristic
540C23D4211yescharacteristic
541C23#54611yescharacteristic
542C23#54811yescharacteristic
543C23D4×Q811yescharacteristic
544Q8—22no—
545Q8—22no—
546Q8—22no—
547Q8#38914yescomplemented
548D4—24no—
549D4—24no—
550D4—24no—
551D4—24no—
552D4—22no—
553D4—22no—
554D4—22no—
555D4#38914yescomplemented
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—48no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—168no—
610C2×C4#38512yes—
611C2×C4#38512yes—
612C2×C4#38512yes—
613C2×C4#38512yes—
614C2×C4#45312yes—
615C2×C4#45612yes—
616C2×C4C22×C4⋊C411yescharacteristic
617C2×C4C2×C4×D411yescharacteristic
618C2×C4C2×C4×D411yescharacteristic
619C2×C4C2×C4×Q811yescharacteristic
620C2×C4#52011yescharacteristic
621C2×C4#52011yescharacteristic
622C2×C4C2×#12111yescharacteristic
623C2×C4C2×C22⋊Q811yescharacteristic
624C2×C4C2×C22⋊Q811yescharacteristic
625C2×C4C2×#12311yescharacteristic
626C2×C4C2×#12511yescharacteristic
627C2×C4C2×C4⋊Q811yescharacteristic
628C2×C4#53511yescharacteristic
629C2×C4#53611yescharacteristic
630C2×C4#54411yescharacteristic
631C2×C4#54411yescharacteristic
632C2×C4D4211yescharacteristic
633C2×C4#54611yescharacteristic
634C2×C4#54711yescharacteristic
635C2×C4#54711yescharacteristic
636C2×C4#54811yescharacteristic
637C2×C4D4×Q811yescharacteristic
638C2×C4#55211yescharacteristic
639C2×C4#55211yescharacteristic
640C2×C4#55311yescharacteristic
641C2×C4#55311yescharacteristic
642C2×C4#55411yescharacteristic
643C2×C4#55411yescharacteristic
644C8—44no—
645C8—42no—
646C8—42no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—24no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22C2×#38511yescharacteristic
667C22#236011yescharacteristic
668C22#236111yescharacteristic
669C22#240811yescharacteristic
670C22#240911yescharacteristic
671C22#244011yescharacteristic
672C22#244211yescharacteristic
673C22C2×#45311yescharacteristic
674C22C2×#45611yescharacteristic
675C22#301911yescharacteristic
676C22#302011yescharacteristic
677C4—24no—
678C4—22no—
679C4—22no—
680C4—22no—
681C4—48no—
682C4—42no—
683C4—42no—
684C4—82no—
685C4—82no—
686C4—84no—
687C4—82no—
688C4—82no—
689C4#185914yescomplemented
690C4C2×#38911yescharacteristic
691C4#231711yescharacteristic
692C4#236011yescharacteristic
693C4#236111yescharacteristic
694C4#244011yescharacteristic
695C4#244211yescharacteristic
696C4#248611yescharacteristic
697C4#248711yescharacteristic
698C2—21no—
699C2—21no—
700C2—21no—
701C2—21no—
702C2—44no—
703C2#1413011yescharacteristic
704C2C2×#185911yescharacteristic
705C2#2086411yescharacteristic
706C2#2423011yescharacteristic
707C2#2423311yescharacteristic
708C2#2432011yescharacteristic
709C2#2432311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21170C2semidirect
6. #13910C2semidirect
7. #11340C2semidirect
8. #11276C2semidirect
9. #10857C2semidirect
10. #10851C2semidirect
40. #1806C4semidirect
99. #423C2×C4semidirect
151. C4×C4⋊C4C2×C4semidirect
165. #339C2×C4semidirect
217. C4×Q8C4⋊C4semidirect
218. C4×Q8C4⋊C4semidirect
229. C4×D4C4⋊C4semidirect
230. C4×D4C4⋊C4semidirect
547. Q8#389semidirect
555. D4#389semidirect
689. C4#1859semidirect

Maximal subgroups

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

subgroupindex
#341832
#211702
C2×#18592
C2×#18062
C4×#4172
#139102
#113402
#112762
#108572
#108512
#107812