Finite Group Databasebeta

← #379019

Subgroups — #379019

Order 512.

2034 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2011C211yesmaximal, characteristic
2C2×#1972C212yesmaximal, complemented
3C2×#1926C212yesmaximal, complemented
4C2×#1879C212yesmaximal, complemented
5#14713C211yesmaximal, characteristic, complemented
6C4×#390C211yesmaximal, characteristic, complemented
7#11323C211yesmaximal, characteristic, complemented
8#11049C211yesmaximal, characteristic, complemented
9#11008C211yesmaximal, characteristic, complemented
10#10927C212yesmaximal, complemented
11#10693C211yesmaximal, characteristic, complemented
12C23×D8C2212yes—
13C22×#127C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#106C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×#97C2212yes—
21C2×#390—22no—
22C2×#390—22no—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4—22no—
30C2×C43C2211yescharacteristic
31#2011C418yescomplemented
32#1972—28no—
33#1926—28no—
34#1879—28no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40#1437—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×D4—22no—
44C23×D4C2312yes—
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C22×D8—24no—
48C22×D8—24no—
49C22×D8—24no—
50C22×D8—24no—
51C22×D8—42no—
52C22×D8—42no—
53C22×D8C2×C418yes—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×#127—22no—
57C2×#127—42no—
58C2×#127—42no—
59C2×#127C2×C414yescomplemented
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—24no—
63C2×C4×D4—42no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—24no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C42—21no—
76C22×C42D412yes—
77C22×C42D412yes—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#106—24no—
82C2×#106—24no—
83C2×#106—42no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—42no—
93C2×#102—42no—
94C2×#102—42no—
95C2×#102C2×C418yescomplemented
96C2×#97—24no—
97C2×#97—24no—
98C2×#97—42no—
99#390—24no—
100#390—24no—
101#390—24no—
102#390—22no—
103#390—22no—
104#390—24no—
105#390—48no—
106#390—48no—
107#390C2×C414yes—
108#390D414yescomplemented
109#384—42no—
110C4×C4⋊C4—24no—
111C4×C22⋊C4—48no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C43—42no—
116#340—24no—
117#340—24no—
118#336—22no—
119#336C2×C414yescomplemented
120#336D418yescomplemented
121C25—21no—
122C25—22no—
123C22×D4—28no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
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—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—44no—
145C22×D4—44no—
146C22×D4—44no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4C22×C414yes—
151C22×D4C22×C412yes—
152C22×D4C2×D412yes—
153C22×D4#4112yes—
154C23×C4—24no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—42no—
160C23×C4—44no—
161C23×C4—42no—
162C23×C4—44no—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
168C2×D8—216no—
169C2×D8—216no—
170C2×D8—48no—
171C2×D8—48no—
172C2×D8—48no—
173C2×D8—48no—
174C2×D8—48no—
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×C8C22×C414yes—
186C22×C8C2×D414yes—
187C22×C8C2×D414yes—
188C22×C8#4114yes—
189#127—48no—
190#127—48no—
191C4×D4—48no—
192C4×D4—48no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4#4112yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—48no—
224C2×C22⋊C4—48no—
225C2×C22⋊C4—48no—
226C2×C22⋊C4—48no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—48no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—44no—
244C2×C42—42no—
245C2×C42C22×C412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D414yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250#106—48no—
251#102—216no—
252#102—48no—
253#102—48no—
254#102—48no—
255#97—48no—
256#95—44no—
257#95—42no—
258#95—44no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—24no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—44no—
278C24—44no—
279C24—44no—
280C24—44no—
281C24C22×D411yescharacteristic
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—416no—
298C2×D4—416no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
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—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—416no—
328C2×D4—88no—
329C2×D4—88no—
330C2×D4—88no—
331C2×D4—88no—
332C2×D4C4×D414yes—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—48no—
354C22×C4—416no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
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—44no—
371C22×C4—44no—
372C22×C4—84no—
373C22×C4—88no—
374C22×C4—84no—
375C22×C4C4×D414yes—
376C22×C4#12114yes—
377C22×C4#12712yes—
378C22×C4#12712yes—
379C22×C4AGL(1,ℤ/8)14yes—
380C22×C4AGL(1,ℤ/8)14yes—
381C22×C4AGL(1,ℤ/8)14yes—
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4112yes—
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4H(5,𝔽2)12yes—
391C22×C4H(5,𝔽2)12yes—
392D8—432no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8C4×D418yes—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—88no—
418C22⋊C4—416no—
419C22⋊C4—48no—
420C22⋊C4—416no—
421C22⋊C4—88no—
422C22⋊C4—816no—
423C22⋊C4—88no—
424C42—24no—
425C42—24no—
426C42—24no—
427C42—24no—
428C42—24no—
429C42—24no—
430C42—48no—
431C42—48no—
432C42—84no—
433C23—24no—
434C23—42no—
435C23—42no—
436C23—44no—
437C23—44no—
438C23—42no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—44no—
447C23—44no—
448C23—48no—
449C23—48no—
450C23—44no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—81no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—84no—
462C23—84no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—84no—
469C23—84no—
470C23—84no—
471C23—84no—
472C23—84no—
473C23—84no—
474C23—84no—
475C23—84no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23C2×C4×D411yescharacteristic
480C23C2×#12111yescharacteristic
481C23#52511yescharacteristic
482C23C2×#12711yescharacteristic
483C23#53211yescharacteristic
484C23#53411yescharacteristic
485C23#53511yescharacteristic
486C23C2×AGL(1,ℤ/8)12yes—
487C23C2×AGL(1,ℤ/8)12yes—
488C23C2×AGL(1,ℤ/8)12yes—
489C23#57512yes—
490D4—28no—
491D4—44no—
492D4—44no—
493D4—44no—
494D4—44no—
495D4—44no—
496D4—44no—
497D4—44no—
498D4—832no—
499D4—82no—
500D4—82no—
501D4—82no—
502D4—82no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—816no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4—82no—
546C2×C4#44218yes—
547C2×C4#49614yes—
548C2×C4C2×C4×D411yescharacteristic
549C2×C4C2×C4×D412yes—
550C2×C4C4×#4111yescharacteristic
551C2×C4#51812yes—
552C2×C4#52911yescharacteristic
553C2×C4#53212yes—
554C2×C4#53211yescharacteristic
555C2×C4#53812yes—
556C2×C4D4212yes—
557C2×C4#54612yes—
558C2×C4#54612yes—
559C2×C4#54612yes—
560C2×C4#55012yes—
561C2×C4#55112yes—
562C2×C4#55312yes—
563C2×C4#55512yes—
564C8—28no—
565C8—44no—
566C22—44no—
567C22—48no—
568C22—82no—
569C22—84no—
570C22—82no—
571C22—84no—
572C22—82no—
573C22—82no—
574C22—84no—
575C22—82no—
576C22—82no—
577C22—88no—
578C22—84no—
579C22—84no—
580C22—84no—
581C22—88no—
582C22—162no—
583C22—162no—
584C22—164no—
585C22—162no—
586C22—162no—
587C22—164no—
588C22C4×#12111yescharacteristic
589C22C4×#12711yescharacteristic
590C22#232111yescharacteristic
591C22#253911yescharacteristic
592C22#254111yescharacteristic
593C22#254311yescharacteristic
594C22#261011yescharacteristic
595C22C2×#44212yes—
596C22C4×AGL(1,ℤ/8)12yes—
597C22#303612yes—
598C22#303812yes—
599C22#310912yes—
600C22#311512yes—
601C22C2×#49611yescharacteristic
602C22#315011yescharacteristic
603C22#315111yescharacteristic
604C22#315111yescharacteristic
605C22#317112yes—
606C22#318412yes—
607C22#319212yes—
608C22#319412yes—
609C22#321312yes—
610C22#321712yes—
611C4—28no—
612C4—24no—
613C4—42no—
614C4—44no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—162no—
619C4#235812yes—
620C4#236312yes—
621C4#255412yes—
622C4#255512yes—
623C2—84no—
624C2—164no—
625C2#1422811yescharacteristic
626C2#2150312yes—
627C2#2169011yescharacteristic
628C2C4×#49611yescharacteristic
629C2#2204612yes—
630C2#2831011yescharacteristic
631C2#2831611yescharacteristic
632C2#2844511yescharacteristic
633C2#2844811yescharacteristic
634C2#2857812yes—
635C2#2955812yes—
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
2. C2×#1972C2semidirect
3. C2×#1926C2semidirect
4. C2×#1879C2semidirect
5. #14713C2semidirect
6. C4×#390C2semidirect
7. #11323C2semidirect
8. #11049C2semidirect
9. #11008C2semidirect
10. #10927C2semidirect
11. #10693C2semidirect
31. #2011C4semidirect
59. C2×#127C2×C4semidirect
95. C2×#102C2×C4semidirect
108. #390D4semidirect
119. #336C2×C4semidirect
120. #336D4semidirect

Maximal subgroups

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

subgroupindex
C2×#20112
C2×#19722
C2×#19262
C2×#18792
#147132
C4×#3902
#113232
#110492
#110082
#109272
#106932