Finite Group Databasebeta

← #379907

Subgroups — #379907

Order 512.

1894 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#482C211yesmaximal, characteristic
2C2×#2296C211yesmaximal, characteristic, complemented
3C2×#1875C211yesmaximal, characteristic
4C2×#1825C211yesmaximal, characteristic, complemented
5#14698C211yesmaximal, characteristic
6#11220C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#11066C212yesmaximal, complemented
9#11066C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic, complemented
11#10712C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C2×#482—24no—
17C2×#482—24no—
18C2×#482C418yescomplemented
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#102C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26#2296—24no—
27#2296—24no—
28C2×#390C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#386—22no—
32C2×#385—22no—
33C2×#380C2211yescharacteristic
34C2×C4×C4⋊C4—21no—
35C2×C4×C4⋊C4—21no—
36#1875—24no—
37#1825—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1437—21no—
46#1437—21no—
47C4⋊#95C2212yes—
48C23×D4—21no—
49C23×D4C2311yescharacteristic
50C24×C4D412yes—
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×#121—22no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121C2×C414yes—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—42no—
66C2×#117—42no—
67C2×#117C2×C414yes—
68C2×#117D414yes—
69C2×#117D414yes—
70C2×#117D414yes—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C42—21no—
85C22×C42—22no—
86C22×C42—22no—
87C22×C42—21no—
88C22×C42—42no—
89C22×C42—42no—
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42C2312yes—
93#482—232no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107C2×C414yescomplemented
97C2×#106—22no—
98C2×#106—22no—
99C2×#106C2×C414yescomplemented
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—24no—
105C2×#102—42no—
106C2×#102—42no—
107C2×#102C2×C414yescomplemented
108C2×#102C2×C414yescomplemented
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8C2×C414yes—
115#390—42no—
116#390—42no—
117#389—24no—
118#389—24no—
119#389—24no—
120#389—24no—
121#386—48no—
122#385—216no—
123#380—22no—
124#380—22no—
125C4×C4⋊C4—28no—
126C4×C4⋊C4—28no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—21no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—42no—
136#340—42no—
137#336—42no—
138#336—42no—
139C25—21no—
140C25C2×D411yescharacteristic
141C22×D4—24no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—44no—
152C22×D4—44no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4C22×C412yes—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4C22×C412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4#4112yes—
180C23×C4#4112yes—
181C23×C4C2411yescharacteristic
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8C22×C412yes—
198C22×C8C22×C412yes—
199#121—28no—
200#121—28no—
201#121—44no—
202#121—44no—
203#117—28no—
204#117—28no—
205#117—28no—
206#117—48no—
207#117—44no—
208#117—44no—
209#117C22⋊C4116yescomplemented
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
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—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D412yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—44no—
277C2×C42—48no—
278C2×C42—44no—
279C2×C42—44no—
280#107—28no—
281#106—28no—
282#102—28no—
283#102—28no—
284#102—48no—
285C22⋊C8—28no—
286C22⋊C8—44no—
287#95—28no—
288#95—28no—
289#95—44no—
290#95—42no—
291#95—44no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—41no—
300C24—41no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24C4×D412yes—
308C24#12012yes—
309C24#12112yes—
310C24C22×D411yescharacteristic
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—416no—
316C2×D4—48no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—42no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—88no—
332C2×D4—84no—
333C2×D4—84no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
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—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4C2×C22⋊C414yes—
399C22×C4C4×D414yes—
400C22×C4C4×D412yes—
401C22×C4#12012yes—
402C22×C4#12114yes—
403C22×C4#12112yes—
404C22×C4#12314yes—
405C22×C4#13514yes—
406C22×C4AGL(1,ℤ/8)14yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4H(5,𝔽2)12yes—
414C22×C4H(5,𝔽2)12yes—
415C22×C42−1+412yes—
416C22×C42−1+412yes—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—416no—
445C4⋊C4—416no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4C2×C22⋊C414yes—
455C4⋊C4C2×C22⋊C414yes—
456C22⋊C4—216no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C22⋊C4—84no—
462C42—216no—
463C42—44no—
464C42—44no—
465C42—48no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—42no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—42no—
495C23—81no—
496C23—81no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—81no—
510C23—81no—
511C23#38614yes—
512C23C2×C4×D411yescharacteristic
513C23C2×#12011yescharacteristic
514C23C2×#12111yescharacteristic
515C23#53412yes—
516C23#53612yes—
517C23C2×#13512yes—
518C23C2×AGL(1,ℤ/8)12yes—
519C23#57512yes—
520C23#57512yes—
521D4—42no—
522D4—42no—
523D4—44no—
524D4—816no—
525D4—82no—
526D4—81no—
527D4—81no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—432no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4#38614yes—
583C2×C4C22×C22⋊C411yescharacteristic
584C2×C4C2×C4×D411yescharacteristic
585C2×C4#52012yes—
586C2×C4#52012yes—
587C2×C4C2×#12111yescharacteristic
588C2×C4C2×#12311yescharacteristic
589C2×C4#53412yes—
590C2×C4#53511yescharacteristic
591C2×C4#53511yescharacteristic
592C2×C4#53611yescharacteristic
593C2×C4#53611yescharacteristic
594C2×C4#53612yes—
595C2×C4#53712yes—
596C2×C4#53712yes—
597C2×C4#53912yes—
598C2×C4#53912yes—
599C2×C4#53912yes—
600C2×C4#53912yes—
601C8—42no—
602C8—42no—
603C8—82no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—84no—
614C22—84no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—162no—
619C22—162no—
620C22—162no—
621C22—162no—
622C22C2×#38611yescharacteristic
623C22#232212yes—
624C22#261311yescharacteristic
625C22#261411yescharacteristic
626C22#261812yes—
627C22C4×#13511yescharacteristic
628C22#294311yescharacteristic
629C22C4×AGL(1,ℤ/8)12yes—
630C22#300212yes—
631C22#300612yes—
632C22#303512yes—
633C22#303811yescharacteristic
634C22#304011yescharacteristic
635C22#318812yes—
636C22#319112yes—
637C22#319412yes—
638C22#319712yes—
639C22#322812yes—
640C22#323312yes—
641C22#323512yes—
642C22#324012yes—
643C4—24no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—84no—
649C4—82no—
650C4—162no—
651C4#231111yescharacteristic
652C4#231111yescharacteristic
653C4#232212yes—
654C4#261812yes—
655C4#262311yescharacteristic
656C4#262311yescharacteristic
657C2—44no—
658C2—162no—
659C2#1425611yescharacteristic
660C2#2092411yescharacteristic
661C2#2092811yescharacteristic
662C2#2208012yes—
663C2#2208612yes—
664C2#2960012yes—
665C2#2964412yes—
666C2#2975812yes—
667C2#2976212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2296C2semidirect
4. C2×#1825C2semidirect
6. #11220C2semidirect
7. #11219C2semidirect
8. #11066C2semidirect
9. #11066C2semidirect
10. #10731C2semidirect
11. #10712C2semidirect
18. C2×#482C4semidirect
96. C2×#107C2×C4semidirect
99. C2×#106C2×C4semidirect
107. C2×#102C2×C4semidirect
108. C2×#102C2×C4semidirect
209. #117C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4822
C2×#22962
C2×#18752
C2×#18252
#146982
#112202
#112192
#110662
#107312
#107122