Finite Group Databasebeta

← #380111

Subgroups — #380111

Order 512.

1724 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1893C211yesmaximal, characteristic
2C2×#1893C211yesmaximal, characteristic
3C22×#336C211yesmaximal, characteristic, complemented
4#14698C211yesmaximal, characteristic
5#14697C211yesmaximal, characteristic, complemented
6#11316C212yesmaximal, complemented
7#11293C212yesmaximal, complemented
8#11189C212yesmaximal, complemented
9#11066C212yesmaximal, complemented
10#10721C211yesmaximal, characteristic, complemented
11#10721C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C2×#390C2212yes—
20C2×#389C2212yes—
21C2×#386—21no—
22C2×#386—21no—
23C2×#385—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×#380C2211yescharacteristic
27#1893—24no—
28#1893C418yescomplemented
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336—24no—
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1437—22no—
40#1436—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C22⋊#95—21no—
44C23×D4—21no—
45C23×D4C2311yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—42no—
54C23×C8—42no—
55C23×C8—42no—
56C23×C8—42no—
57C23×C8C2×C414yes—
58C23×C8D414yes—
59C23×C8C2312yes—
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#121—22no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121—22no—
67C2×#121C2×C414yes—
68C22×C4⋊C4—22no—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C42—22no—
81C22×C42—22no—
82C22×C42—42no—
83C22×C42—42no—
84C22×C42D412yes—
85C22×C42C2312yes—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—42no—
89C2×#102—42no—
90C2×#102—42no—
91C2×#102—42no—
92C2×#102C2×C418yescomplemented
93#390—42no—
94#390—42no—
95#389—42no—
96#389—42no—
97#386—44no—
98#386—44no—
99#385—44no—
100#380—22no—
101#380—22no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380C2×C414yes—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—21no—
114#340—22no—
115#340—22no—
116#340—42no—
117#340—42no—
118#340—42no—
119#340—42no—
120#340C2×C418yescomplemented
121#336—24no—
122#336—24no—
123C25—21no—
124C25C2×D411yescharacteristic
125C22×D4—24no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4C22×C412yes—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4C22×C412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4#4112yes—
167C23×C4#4112yes—
168C23×C4#4112yes—
169C23×C4#4112yes—
170C23×C4C2411yescharacteristic
171C22×C8—28no—
172C22×C8—28no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—48no—
188C22×C8—44no—
189C22×C8—44no—
190C22×C8—44no—
191C22×C8—44no—
192C22×C8—48no—
193C22×C8—48no—
194C22×C8—48no—
195C22×C8C22×C414yes—
196C22×C8C2×D412yes—
197C22×C8C2×D412yes—
198#121—28no—
199#121—28no—
200#121—44no—
201#121—44no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C414yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—44no—
259C2×C42—44no—
260C2×C42—48no—
261#102—48no—
262#102—44no—
263#102—44no—
264#95—24no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—42no—
270#95—42no—
271#95—44no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—41no—
278C24—41no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24C4×D412yes—
288C24#12112yes—
289C24#12412yes—
290C24AGL(1,ℤ/8)12yes—
291C24AGL(1,ℤ/8)12yes—
292C24AGL(1,ℤ/8)12yes—
293C24#13712yes—
294C24C22×D411yescharacteristic
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—48no—
300C2×D4—48no—
301C2×D4—48no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—84no—
317C2×D4—84no—
318C2×D4—84no—
319C2×D4—84no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4#11714yes—
390C22×C4C4×D414yes—
391C22×C4C4×D412yes—
392C22×C4#12112yes—
393C22×C4#12314yes—
394C22×C4#12412yes—
395C22×C4#12614yes—
396C22×C4#13514yes—
397C22×C4#13514yes—
398C22×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4H(5,𝔽2)12yes—
405C22×C4H(5,𝔽2)12yes—
406C22×C4H(5,𝔽2)12yes—
407C22×C42−1+412yes—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—416no—
418C2×C8—416no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—88no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C22⋊C4—84no—
451C22⋊C4—84no—
452C42—48no—
453C42—42no—
454C42—42no—
455C42—42no—
456C42—42no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—81no—
479C23—81no—
480C23—82no—
481C23—81no—
482C23—81no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—81no—
493C23—81no—
494C23—81no—
495C23—81no—
496C23#38814yes—
497C23#44014yes—
498C23#44214yes—
499C23#46814yes—
500C23#46914yes—
501C23#48914yes—
502C23#49014yes—
503C23C2×C4×D411yescharacteristic
504C23C2×#12111yescharacteristic
505C23C2×#12411yescharacteristic
506C23#53412yes—
507C23#53712yes—
508C23C2×#13512yes—
509C23C2×#13512yes—
510C23C2×AGL(1,ℤ/8)11yescharacteristic
511C23C2×AGL(1,ℤ/8)11yescharacteristic
512C23C2×AGL(1,ℤ/8)11yescharacteristic
513C23C2×#13711yescharacteristic
514D4—44no—
515D4—42no—
516D4—42no—
517D4—816no—
518D4—81no—
519D4—81no—
520D4—82no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—416no—
538C2×C4—416no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4#38814yes—
579C2×C4C2×#11711yescharacteristic
580C2×C4C2×C4×D411yescharacteristic
581C2×C4#51812yes—
582C2×C4#52012yes—
583C2×C4C2×#12311yescharacteristic
584C2×C4C2×#12611yescharacteristic
585C2×C4#53411yescharacteristic
586C2×C4#53411yescharacteristic
587C2×C4#53511yescharacteristic
588C2×C4#53611yescharacteristic
589C2×C4#53812yes—
590C2×C4#53812yes—
591C2×C4#53812yes—
592C2×C4#53812yes—
593C2×C4#53912yes—
594C2×C4#53912yes—
595C2×C4#53912yes—
596C2×C4#53912yes—
597C8—44no—
598C8—42no—
599C8—42no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—28no—
606C22—28no—
607C22—24no—
608C22—28no—
609C22—28no—
610C22—82no—
611C22—82no—
612C22—84no—
613C22—84no—
614C22—82no—
615C22—161no—
616C22—161no—
617C22—161no—
618C22—161no—
619C22—162no—
620C22—162no—
621C22#193618yes—
622C22C2×#38811yescharacteristic
623C22#232512yes—
624C22#261711yescharacteristic
625C22#261811yescharacteristic
626C22#263912yes—
627C22C4×#13512yes—
628C22C2×#44011yescharacteristic
629C22C2×#44211yescharacteristic
630C22#305011yescharacteristic
631C22#305011yescharacteristic
632C22C2×#46811yescharacteristic
633C22C2×#46911yescharacteristic
634C22#313112yes—
635C22C2×#48911yescharacteristic
636C22C2×#49011yescharacteristic
637C22#318812yes—
638C22#318812yes—
639C22#318812yes—
640C22#319012yes—
641C22#324712yes—
642C22#324712yes—
643C22#324712yes—
644C22#324812yes—
645C4—24no—
646C4—44no—
647C4—44no—
648C4—82no—
649C4—84no—
650C4—161no—
651C4—161no—
652C4—162no—
653C4#231812yes—
654C4#232111yescharacteristic
655C4#232211yescharacteristic
656C4#262211yescharacteristic
657C4#262311yescharacteristic
658C4#264312yes—
659C2—28no—
660C2—162no—
661C2#1425911yescharacteristic
662C2#2108311yescharacteristic
663C2C2×#193611yescharacteristic
664C2#2218412yes—
665C2#2218512yes—
666C2#2973211yescharacteristic
667C2#2973511yescharacteristic
668C2#2977211yescharacteristic
669C2#2977511yescharacteristic
670C2#3018812yes—
671C2#3018912yes—
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
3. C22×#336C2semidirect
5. #14697C2semidirect
6. #11316C2semidirect
7. #11293C2semidirect
8. #11189C2semidirect
9. #11066C2semidirect
10. #10721C2semidirect
11. #10721C2semidirect
28. #1893C4semidirect
92. C2×#102C2×C4semidirect
120. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18932
C22×#3362
#146982
#146972
#113162
#112932
#111892
#110662
#107212