Finite Group Databasebeta

← #392791

Subgroups — #392791

Order 512.

1888 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2274C211yesmaximal, characteristic, complemented
2C2×#1925C212yesmaximal, complemented
3C2×#1875C211yesmaximal, characteristic
4C2×#1874C211yesmaximal, characteristic
5C2×#1852C211yesmaximal, characteristic, complemented
6#14257C211yesmaximal, characteristic
7#10931C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10807C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic, complemented
11#10713C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#107C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#380C2211yescharacteristic
24#2274—28no—
25C2×C4×C4⋊C4—22no—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4—22no—
29#1925—28no—
30#1875C418yescomplemented
31#1874—24no—
32#1852—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1439—22no—
41#1438—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C23×D4—21no—
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#121—22no—
51C2×#121—22no—
52C2×#121—22no—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121C2×C414yes—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—24no—
60C2×#117—24no—
61C2×#117—24no—
62C2×#117—24no—
63C2×#117C2×C414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4D414yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D412yes—
81C22×C42—21no—
82C22×C42—22no—
83C22×C42—22no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#107—24no—
87C2×#107—24no—
88C2×#107—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102C2×C418yescomplemented
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8C2×C414yes—
101#390—24no—
102#390—24no—
103#380—22no—
104#380—22no—
105C4×C4⋊C4—216no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4D418yescomplemented
110C4×C22⋊C4—216no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340C2×C418yescomplemented
121#336—24no—
122#336—24no—
123C25—21no—
124C25C2×D411yescharacteristic
125C22×D4—24no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—82no—
138C22×D4—82no—
139C22×D4C22×C412yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—82no—
153C23×C4—82no—
154C23×C4C22×C412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4#4112yes—
162C23×C4#4112yes—
163C23×C4C2411yescharacteristic
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8C22×C414yes—
178#121—28no—
179#121—28no—
180#121—44no—
181#121—44no—
182#117—216no—
183#117—216no—
184#117—216no—
185#117—216no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—216no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
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—216no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C414yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4#4114yes—
219C2×C4⋊C4#4114yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4—82no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—28no—
247C2×C42—28no—
248C2×C42—82no—
249C2×C42—82no—
250C2×C42—82no—
251C2×C42—82no—
252C2×C42C2×D414yes—
253C2×C42#4114yes—
254#107—48no—
255#102—216no—
256#102—48no—
257C22⋊C8—28no—
258C22⋊C8—44no—
259#95—28no—
260#95—28no—
261#95—42no—
262#95—82no—
263#95—82no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24C4×D412yes—
284C24#12112yes—
285C24#12312yes—
286C24C22×D411yescharacteristic
287C2×D4—22no—
288C2×D4—24no—
289C2×D4—22no—
290C2×D4—24no—
291C2×D4—48no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—88no—
305C2×D4—88no—
306C2×D4—84no—
307C2×D4—84no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—216no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4#11714yes—
371C22×C4C4×D414yes—
372C22×C4C4×D412yes—
373C22×C4#12014yes—
374C22×C4#12112yes—
375C22×C4#12314yes—
376C22×C4#12312yes—
377C22×C4#12414yes—
378C22×C4#12614yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D412yes—
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4C2×#4112yes—
388C22×C4C2×#4111yescharacteristic
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C4⋊C4—216no—
404C4⋊C4—216no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4C4×D418yes—
420C4⋊C4#12118yes—
421C22⋊C4—216no—
422C22⋊C4—216no—
423C22⋊C4—48no—
424C22⋊C4—88no—
425C22⋊C4—88no—
426C22⋊C4—84no—
427C42—216no—
428C42—24no—
429C42—24no—
430C42—24no—
431C42—24no—
432C42—88no—
433C42—88no—
434C23—24no—
435C23—24no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—24no—
441C23—24no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—44no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—81no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23#38814yes—
480C23C2×C4×D411yescharacteristic
481C23C2×#12011yescharacteristic
482C23C2×#12111yescharacteristic
483C23C2×#12311yescharacteristic
484C23C2×#12311yescharacteristic
485C23#52511yescharacteristic
486C23#52511yescharacteristic
487C23#57612yes—
488C23#57612yes—
489C23#57712yes—
490C23#57812yes—
491D4—44no—
492D4—44no—
493D4—816no—
494D4—82no—
495D4—82no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—816no—
537C2×C4—82no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4#38814yes—
551C2×C4C2×#11711yescharacteristic
552C2×C4C2×C4×D412yes—
553C2×C4C2×C4×D411yescharacteristic
554C2×C4C4×#4112yes—
555C2×C4C2×#12112yes—
556C2×C4#52512yes—
557C2×C4#52512yes—
558C2×C4#52512yes—
559C2×C4#52512yes—
560C2×C4C2×#12411yescharacteristic
561C2×C4C2×#12611yescharacteristic
562C2×C4#52912yes—
563C2×C4#52912yes—
564C2×C4#52912yes—
565C2×C4#52912yes—
566C2×C4#53212yes—
567C8—44no—
568C8—82no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—44no—
573C22—44no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—44no—
579C22—44no—
580C22—82no—
581C22—82no—
582C22—84no—
583C22—82no—
584C22—84no—
585C22—162no—
586C22—161no—
587C22—161no—
588C22—161no—
589C22—161no—
590C22—162no—
591C22C2×#38811yescharacteristic
592C22C4×#12011yescharacteristic
593C22C4×#12311yescharacteristic
594C22#284711yescharacteristic
595C22#284911yescharacteristic
596C22#285311yescharacteristic
597C22#285411yescharacteristic
598C22#294612yes—
599C22#295112yes—
600C22#300812yes—
601C22#301312yes—
602C22#304111yescharacteristic
603C22#304111yescharacteristic
604C22#304311yescharacteristic
605C22#304411yescharacteristic
606C22#309612yes—
607C22#309612yes—
608C22#309812yes—
609C22#309812yes—
610C22#311912yes—
611C22#312112yes—
612C22#312212yes—
613C22#312412yes—
614C4—24no—
615C4—48no—
616C4—48no—
617C4—44no—
618C4—44no—
619C4—82no—
620C4—161no—
621C4—161no—
622C4#229012yes—
623C4C4×#12112yes—
624C4#285012yes—
625C4#285212yes—
626C2—44no—
627C2—162no—
628C2#1469611yescharacteristic
629C2#2106311yescharacteristic
630C2#2106711yescharacteristic
631C2#2146012yes—
632C2#2159012yes—
633C2#3291111yescharacteristic
634C2#3291811yescharacteristic
635C2#3298612yes—
636C2#3310312yes—
637C2#3311811yescharacteristic
638C2#3312011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2274C2semidirect
2. C2×#1925C2semidirect
5. C2×#1852C2semidirect
7. #10931C2semidirect
8. #10907C2semidirect
9. #10807C2semidirect
10. #10731C2semidirect
11. #10713C2semidirect
30. #1875C4semidirect
94. C2×#102C2×C4semidirect
109. C4×C4⋊C4D4semidirect
120. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22742
C2×#19252
C2×#18752
C2×#18742
C2×#18522
#142572
#109312
#109072
#108072
#107312
#107132