Finite Group Databasebeta

← #377041

Subgroups — #377041

Order 512.

2456 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#127C211yesmaximal, characteristic, complemented
2C2×#2011C211yesmaximal, characteristic
3C2×#1970C212yesmaximal, complemented
4C2×#1879C212yesmaximal, complemented
5C2×#1862C212yesmaximal, complemented
6#14700C211yesmaximal, characteristic, complemented
7#11324C211yesmaximal, characteristic, complemented
8#11049C211yesmaximal, characteristic, complemented
9#11008C211yesmaximal, characteristic, complemented
10#10925C212yesmaximal, complemented
11C4×#336C211yesmaximal, characteristic, complemented
12C23×D8C2212yes—
13C22×#127C2211yescharacteristic
14C22×C4×D4—22no—
15C22×C4×D4C2212yes—
16C22×#107C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×C4×C8C2212yes—
22C4×#127—28no—
23C2×#390—22no—
24C2×#390C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#390C2212yes—
28C2×#389C2211yescharacteristic
29C2×#388—22no—
30C2×#384C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#2011C418yescomplemented
33#1970—28no—
34#1879—28no—
35#1862—28no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1437—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C23×D4—22no—
45C23×D4C2311yescharacteristic
46C23×D4C2312yes—
47C24×C4—22no—
48C22×D8—24no—
49C22×D8—24no—
50C22×D8—24no—
51C22×D8—24no—
52C22×D8—42no—
53C22×D8—42no—
54C22×D8C2×C418yescomplemented
55C23×C8C2312yes—
56C23×C8C2312yes—
57C2×#127—24no—
58C2×#127—24no—
59C2×#127—22no—
60C2×#127C2×C414yes—
61C2×C4×D4—216no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—42no—
67C2×C4×D4—42no—
68C2×C4×D4—48no—
69C2×C4×D4—48no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—24no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—21no—
82C22×C42D412yes—
83C22×C42D412yes—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×#107—24no—
88C2×#107—24no—
89C2×#107—42no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—42no—
99C2×#102—42no—
100C2×#102—42no—
101C2×#102C2×C418yescomplemented
102C2×C4×C8—24no—
103C2×C4×C8—24no—
104C2×C4×C8—42no—
105#390—216no—
106#390—24no—
107#390—24no—
108#390—24no—
109#390—22no—
110#390C2×C414yescomplemented
111#390D418yes—
112#389—24no—
113#388—48no—
114#384—24no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C43—24no—
119#340—24no—
120#340—24no—
121#336—22no—
122#336C2×C414yescomplemented
123#336D418yescomplemented
124C25—22no—
125C25—21no—
126C22×D4—28no—
127C22×D4—28no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—22no—
148C22×D4—21no—
149C22×D4—44no—
150C22×D4—44no—
151C22×D4—44no—
152C22×D4—44no—
153C22×D4C22×C412yes—
154C22×D4C22×C414yes—
155C22×D4C2×D414yes—
156C22×D4#4114yes—
157C23×C4—28no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—44no—
163C23×C4—48no—
164C23×C4—44no—
165C23×C4—48no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2411yescharacteristic
171C2×D8—216no—
172C2×D8—216no—
173C2×D8—48no—
174C2×D8—48no—
175C2×D8—48no—
176C2×D8—48no—
177C2×D8—48no—
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×C8C22×C414yes—
189C22×C8C2×D414yes—
190C22×C8C2×D414yes—
191C22×C8#4114yes—
192#127—216no—
193#127—216no—
194C4×D4—48no—
195C4×D4—48no—
196C4×D4—48no—
197C4×D4—48no—
198C4×D4—464no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C2×D414yes—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—216no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—48no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—48no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—42no—
251C2×C42D818yescomplemented
252C2×C42C22×C412yes—
253C2×C42C2×D414yes—
254C2×C42C2×D414yes—
255C2×C42#4112yes—
256#107—48no—
257#102—216no—
258#102—48no—
259#102—48no—
260#102—48no—
261C4×C8—48no—
262#95—42no—
263#95—44no—
264#95—44no—
265C24—24no—
266C24—24no—
267C24—24no—
268C24—22no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—44no—
274C24—44no—
275C24—44no—
276C24—44no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24C22×D411yescharacteristic
288C2×D4—232no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—22no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—416no—
309C2×D4—416no—
310C2×D4—416no—
311C2×D4—416no—
312C2×D4—416no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—41no—
319C2×D4—42no—
320C2×D4—41no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—416no—
338C2×D4C4×D418yes—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—416no—
357C22×C4—432no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—88no—
377C22×C4—816no—
378C22×C4—88no—
379C22×C4C4×D414yes—
380C22×C4#12014yes—
381C22×C4#12712yes—
382C22×C4#12712yes—
383C22×C4C2×D814yes—
384C22×C4C2×D814yes—
385C22×C4C2×D814yes—
386C22×C4#13514yes—
387C22×C4AGL(1,ℤ/8)14yes—
388C22×C4C23×C411yescharacteristic
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C2×#4112yes—
392C22×C4C2×#4112yes—
393C22×C4C2×#4111yescharacteristic
394C22×C4H(5,𝔽2)11yescharacteristic
395C22×C4H(5,𝔽2)11yescharacteristic
396C22×C4H(5,𝔽2)11yescharacteristic
397C22×C42−1+411yescharacteristic
398D8—432no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8C4×D418yescomplemented
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—416no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—84no—
425C22⋊C4—416no—
426C22⋊C4—416no—
427C22⋊C4—432no—
428C22⋊C4—84no—
429C22⋊C4—88no—
430C22⋊C4—84no—
431C42—24no—
432C42—24no—
433C42—24no—
434C42—24no—
435C42—24no—
436C42—24no—
437C42—216no—
438C42—48no—
439C23—28no—
440C23—44no—
441C23—44no—
442C23—48no—
443C23—48no—
444C23—44no—
445C23—48no—
446C23—44no—
447C23—48no—
448C23—44no—
449C23—48no—
450C23—48no—
451C23—44no—
452C23—42no—
453C23—42no—
454C23—44no—
455C23—44no—
456C23—42no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—84no—
462C23—84no—
463C23—84no—
464C23—84no—
465C23—84no—
466C23—84no—
467C23—82no—
468C23—82no—
469C23—81no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—82no—
475C23—82no—
476C23—81no—
477C23—81no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—81no—
485C23C2×C4×D411yescharacteristic
486C23C2×#12011yescharacteristic
487C23#52511yescharacteristic
488C23C2×#12711yescharacteristic
489C23#53211yescharacteristic
490C23#53511yescharacteristic
491C23#53711yescharacteristic
492C23C22×D812yes—
493C23C2×#13512yes—
494C23C2×AGL(1,ℤ/8)12yes—
495C23#57512yes—
496D4—28no—
497D4—28no—
498D4—464no—
499D4—44no—
500D4—44no—
501D4—44no—
502D4—44no—
503D4—44no—
504D4—44no—
505D4—44no—
506D4—44no—
507D4—82no—
508D4—82no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—48no—
534C2×C4—832no—
535C2×C4—81no—
536C2×C4—82no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—84no—
541C2×C4—82no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4C4×D818yes—
552C2×C4#44718yes—
553C2×C4#49314yes—
554C2×C4C2×C4×D412yes—
555C2×C4C2×C4×D412yes—
556C2×C4#51811yescharacteristic
557C2×C4#52011yescharacteristic
558C2×C4#52512yes—
559C2×C4#53212yes—
560C2×C4#54011yescharacteristic
561C2×C4#54211yescharacteristic
562C2×C4D4212yes—
563C2×C4D4212yes—
564C2×C4#54612yes—
565C2×C4#54612yes—
566C2×C4#54712yes—
567C2×C4#54812yes—
568C2×C4#55112yes—
569C2×C4#55312yes—
570C8—28no—
571C8—44no—
572C22—48no—
573C22—416no—
574C22—88no—
575C22—84no—
576C22—84no—
577C22—88no—
578C22—88no—
579C22—84no—
580C22—84no—
581C22—84no—
582C22—84no—
583C22—82no—
584C22—84no—
585C22—82no—
586C22—82no—
587C22—84no—
588C22—162no—
589C22—161no—
590C22—161no—
591C22—162no—
592C22—161no—
593C22—161no—
594C22C4×#12011yescharacteristic
595C22C4×#12711yescharacteristic
596C22#232411yescharacteristic
597C22#243011yescharacteristic
598C22#243311yescharacteristic
599C22#254011yescharacteristic
600C22#260311yescharacteristic
601C22C2×C4×D812yes—
602C22C4×AGL(1,ℤ/8)12yes—
603C22C2×#44712yes—
604C22#300212yes—
605C22#310012yes—
606C22#310512yes—
607C22C2×#49311yescharacteristic
608C22#314711yescharacteristic
609C22#315111yescharacteristic
610C22#315311yescharacteristic
611C22#316912yes—
612C22#318012yes—
613C22#320512yes—
614C22#321212yes—
615C22#324512yes—
616C22#324712yes—
617C4—24no—
618C4—28no—
619C4—28no—
620C4—42no—
621C4—84no—
622C4—82no—
623C4—161no—
624C4—161no—
625C4#235812yes—
626C4#236112yes—
627C4#246112yes—
628C4#246412yes—
629C2—88no—
630C2—162no—
631C2#1418911yescharacteristic
632C2C4×#44712yes—
633C2C4×#49311yescharacteristic
634C2#2170111yescharacteristic
635C2#2214512yes—
636C2#2549612yes—
637C2#2572111yescharacteristic
638C2#2572311yescharacteristic
639C2#2836911yescharacteristic
640C2#2837511yescharacteristic
641C2#2949312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2011C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×#127C2semidirect
3. C2×#1970C2semidirect
4. C2×#1879C2semidirect
5. C2×#1862C2semidirect
6. #14700C2semidirect
7. #11324C2semidirect
8. #11049C2semidirect
9. #11008C2semidirect
10. #10925C2semidirect
11. C4×#336C2semidirect
32. #2011C4semidirect
54. C22×D8C2×C4semidirect
101. C2×#102C2×C4semidirect
110. #390C2×C4semidirect
122. #336C2×C4semidirect
123. #336D4semidirect
251. C2×C42D8semidirect
412. C2×C8C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1272
C2×#20112
C2×#19702
C2×#18792
C2×#18622
#147002
#113242
#110492
#110082
#109252
C4×#3362