Finite Group Databasebeta

← #378103

Subgroups — #378103

Order 512.

1838 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1955C212yesmaximal, complemented
2C2×#1870C212yesmaximal, complemented
3C2×#1870C212yesmaximal, complemented
4C2×#1753C212yesmaximal, complemented
5C4×#392C211yesmaximal, characteristic
6#11147C211yesmaximal, characteristic, complemented
7#11042C212yesmaximal, complemented
8#10921C211yesmaximal, characteristic, complemented
9#10877C212yesmaximal, complemented
10C23×M16C2212yes—
11C22×C4×D4C2211yescharacteristic
12C22×C4×D4C2212yes—
13C22×C4⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C22×#97C2212yes—
21C22×C4×C8C2211yescharacteristic
22C2×#392C411yescharacteristic
23C2×C4×C22⋊C4C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1955—28no—
26#1870—28no—
27#1870—28no—
28#1753—28no—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35#1444C411yescharacteristic
36#1439C412yes—
37#1439C411yescharacteristic
38#1438C411yescharacteristic
39#1438C412yes—
40C4×#95C2211yescharacteristic
41C23×D4C2×C412yes—
42C23×D4C2×C411yescharacteristic
43C24×C4C2311yescharacteristic
44C24×C4C2312yes—
45C22×M16—24no—
46C22×M16—24no—
47C22×M16—24no—
48C22×M16—24no—
49C22×M16—24no—
50C22×M16—42no—
51C22×M16—42no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4—24no—
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—42no—
63C2×C4×D4—42no—
64C2×C4×D4—41no—
65C2×C4×D4—41no—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2312yes—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—42no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—41no—
93C2×C22⋊C8—41no—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8D418yescomplemented
97C2×#97—24no—
98C2×#97—24no—
99C2×#97—42no—
100C2×C4×C8—24no—
101C2×C4×C8—41no—
102#392—24no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—24no—
105C4×C22⋊C4—41no—
106C4×C22⋊C4—41no—
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C412yes—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114#336—24no—
115#336—24no—
116#336—24no—
117#336—24no—
118#336—41no—
119#336D418yescomplemented
120C25C22×C412yes—
121C25C22×C411yescharacteristic
122C22×D4—24no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—24no—
126C22×D4—22no—
127C22×D4—41no—
128C22×D4—42no—
129C22×D4—42no—
130C22×D4—41no—
131C22×D4—42no—
132C22×D4—42no—
133C22×D4—42no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C412yes—
157C23×C4C22×C412yes—
158C23×C4C22×C412yes—
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D414yes—
163C23×C4#4114yes—
164C23×C4C2411yescharacteristic
165C2×M16—216no—
166C2×M16—48no—
167C2×M16—48no—
168C2×M16—48no—
169C2×M16—48no—
170C2×M16—48no—
171C2×M16—48no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
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—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8C2×D414yes—
196C22×C8C2×D414yes—
197C22×C8C2×D414yes—
198C22×C8C2×D414yes—
199C22×C8#4114yes—
200C4×D4—48no—
201C4×D4—48no—
202C4×D4—48no—
203C4×D4—48no—
204C4×D4—44no—
205C4×D4—48no—
206C4×D4—44no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—42no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4M1614yescomplemented
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42C2×D414yes—
246C2×C42#4114yes—
247C4⋊C8—48no—
248C22⋊C8—216no—
249C22⋊C8—44no—
250C22⋊C8—44no—
251C22⋊C8—48no—
252C22⋊C8—48no—
253C22⋊C8—48no—
254#97—48no—
255C4×C8—44no—
256#95—24no—
257#95—24no—
258#95—22no—
259#95—41no—
260#95—42no—
261#95—41no—
262#95—41no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—21no—
269C24—22no—
270C24—21no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—21no—
277C24—22no—
278C24—21no—
279C24—22no—
280C24—21no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24C4×D414yes—
288C24C2×M1612yes—
289C24C23×C411yescharacteristic
290C2×D4—48no—
291C2×D4—48no—
292C2×D4—48no—
293C2×D4—48no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4—48no—
299C2×D4—48no—
300C2×D4—44no—
301C2×D4—48no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4C4×D414yes—
387C22×C4C4×D414yes—
388C22×C4C4×D414yes—
389C22×C4C2×M1612yes—
390C22×C4C2×M1612yes—
391C22×C4#13112yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C2×#4112yes—
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4H(5,𝔽2)11yescharacteristic
398C22×C4H(5,𝔽2)12yes—
399C22×C4H(5,𝔽2)11yescharacteristic
400M16—432no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8#12118yes—
429C4⋊C4—48no—
430C4⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—44no—
438C42—24no—
439C42—24no—
440C42—24no—
441C42—24no—
442C42—41no—
443C42—41no—
444C42—42no—
445C42—42no—
446C42—41no—
447C42—41no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—24no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—41no—
464C23—42no—
465C23—42no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23C2×C4×D412yes—
507C23C2×C4×D412yes—
508C23C4×#4111yescharacteristic
509C23#51811yescharacteristic
510C23#51811yescharacteristic
511C23C22×M1611yescharacteristic
512C23C2×#13111yescharacteristic
513C23#56811yescharacteristic
514C23#56811yescharacteristic
515C23#56812yes—
516C23#56812yes—
517D4—416no—
518D4—432no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
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—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4C2×#12112yes—
578C2×C4#52512yes—
579C2×C4#52912yes—
580C2×C4#53212yes—
581C2×C4#53812yes—
582C2×C4#54012yes—
583C2×C4D4211yescharacteristic
584C2×C4D4212yes—
585C2×C4D4211yescharacteristic
586C2×C4#54612yes—
587C2×C4#54612yes—
588C2×C4#54612yes—
589C2×C4#55012yes—
590C2×C4#55111yescharacteristic
591C2×C4#55112yes—
592C2×C4#55911yescharacteristic
593C8—28no—
594C8—44no—
595C8—44no—
596C8—82no—
597C22—44no—
598C22—42no—
599C22—44no—
600C22—44no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—42no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—81no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—81no—
615C22—82no—
616C22—81no—
617C22—82no—
618C22—81no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22C4×#12112yes—
624C22#233311yescharacteristic
625C22#235811yescharacteristic
626C22#235812yes—
627C22#236611yescharacteristic
628C22#293012yes—
629C22#293012yes—
630C22#293012yes—
631C22#293212yes—
632C22#293312yes—
633C22D4×M1612yes—
634C22D4×M1612yes—
635C22#293512yes—
636C22#296511yescharacteristic
637C22#296711yescharacteristic
638C22#296712yes—
639C22#297311yescharacteristic
640C22#297511yescharacteristic
641C22#297611yescharacteristic
642C22#297812yes—
643C22#297811yescharacteristic
644C22#298011yescharacteristic
645C22#298111yescharacteristic
646C4—44no—
647C4—42no—
648C4—44no—
649C4—44no—
650C4—42no—
651C4—42no—
652C4—81no—
653C4—81no—
654C4#242412yes—
655C4#242812yes—
656C4#247112yes—
657C4#255712yes—
658C2—84no—
659C2—82no—
660C2#1420711yescharacteristic
661C2#2149012yes—
662C2#2149312yes—
663C2#2233711yescharacteristic
664C2#2233811yescharacteristic
665C2#2284511yescharacteristic
666C2#2284912yes—
667C2#2285011yescharacteristic
668C2#2285212yes—
669C2#2317511yescharacteristic
670C2#2317811yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC2×#392C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1955C2semidirect
2. C2×#1870C2semidirect
3. C2×#1870C2semidirect
4. C2×#1753C2semidirect
6. #11147C2semidirect
7. #11042C2semidirect
8. #10921C2semidirect
9. #10877C2semidirect
96. C2×C22⋊C8D4semidirect
119. #336D4semidirect
232. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#19552
C2×#18702
C2×#17532
C4×#3922
#111472
#110422
#109212
#108772