Finite Group Databasebeta

← #359712

Subgroups — #359712

Order 512.

2554 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2297C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3C2×#1817C211yesmaximal, characteristic, complemented
4C2×#1788C212yesmaximal
5#11050C212yesmaximal, complemented
6#11049C212yesmaximal, complemented
7#10932C212yesmaximal, complemented
8#10931C212yesmaximal, complemented
9#10731C212yesmaximal, complemented
10C23×#41C2211yescharacteristic
11C22×#117—21no—
12C22×#117C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#103C2214yes—
15C22×#102C2214yes—
16C22×C22⋊C8C2212yes—
17#2297—24no—
18#2297C418yescomplemented
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#386—21no—
24C2×#386—21no—
25#2285—28no—
26C2×#384C2212yes—
27C2×#382—21no—
28C2×#380—21no—
29C2×#380C2212yes—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C22⋊C4—21no—
33C2×C4×C22⋊C4—21no—
34#1817—24no—
35#1788—28no—
36C2×#340C2214yes—
37C2×#340C2214yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40C22×#41—24no—
41C22×#41—24no—
42C22×#41—22no—
43C22×#41—24no—
44C22×#41—24no—
45C22×#41—44no—
46C22×#41C2×C414yes—
47C23×Q8C2311yescharacteristic
48C23×D4—21no—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4C2311yescharacteristic
54C23×C8C2314yes—
55C2×#117—28no—
56C2×#117—22no—
57C2×#117—44no—
58C2×#117—44no—
59C2×#117—44no—
60C2×#117C2×C414yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2314yes—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C42—24no—
73C22×C42—24no—
74C22×C42—22no—
75C22×C42—21no—
76C22×C42C2×C414yes—
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D414yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#103—28no—
83C2×#103—28no—
84C2×#103—44no—
85C2×#102—28no—
86C2×#102—28no—
87C2×#102—44no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91#391—22no—
92#391—22no—
93#391C2×C414yes—
94#391C2×C414yes—
95#390—22no—
96#390—22no—
97#390C2×C414yes—
98#390C2×C414yes—
99#386—28no—
100#386—28no—
101#384—28no—
102#382—28no—
103#380—28no—
104#380—24no—
105#380D418yes—
106C4×C4⋊C4—28no—
107C4×C4⋊C4—28no—
108C4×C22⋊C4—28no—
109C4×C22⋊C4—28no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116#340—44no—
117#340—44no—
118#336—24no—
119#336—24no—
120C25—21no—
121C25C2×D411yescharacteristic
122C2×#41—216no—
123C2×#41—28no—
124C2×#41—28no—
125C2×#41—216no—
126C2×#41—48no—
127C2×#41—416no—
128C2×#41—416no—
129C2×#41—416no—
130C2×#41—416no—
131C2×#41—48no—
132C2×#41—48no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—24no—
138C22×Q8C22×C412yes—
139C22×Q8#4114yes—
140C22×D4—28no—
141C22×D4—24no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—24no—
147C22×D4—44no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4C22×C412yes—
151C22×D4#4114yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—28no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—24no—
170C23×C4—42no—
171C23×C4—44no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4C22⋊C414yes—
175C23×C4C22⋊C414yes—
176C23×C4C22×C412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D412yes—
183C23×C4C2×D414yes—
184C23×C4#4114yes—
185C23×C4C2411yescharacteristic
186C22×C8—24no—
187C22×C8—24no—
188C22×C8—24no—
189C22×C8—24no—
190C22×C8—24no—
191C22×C8—24no—
192C22×C8—24no—
193#117—48no—
194#117—432no—
195#117—48no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C2×D418yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C42—216no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—216no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—28no—
240C2×C42—24no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42—44no—
244C2×C42C22⋊C414yes—
245C2×C42C22⋊C414yes—
246C2×C42C22×C412yes—
247C2×C42C22×C412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250#103—416no—
251#102—416no—
252C22⋊C8—48no—
253#95—28no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—28no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—22no—
267C24—22no—
268C24—24no—
269C24—41no—
270C24—41no—
271C24—42no—
272C24—42no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—42no—
278C24C2×C22⋊C412yes—
279C24#12012yes—
280C24#12312yes—
281C24C22×D411yescharacteristic
282#41—432no—
283#41—432no—
284#41—464no—
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—28no—
290C2×Q8—28no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—44no—
294C2×Q8—44no—
295C2×Q8#11714yescomplemented
296C2×D4—216no—
297C2×D4—22no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—28no—
302C2×D4—28no—
303C2×D4—48no—
304C2×D4—416no—
305C2×D4—416no—
306C2×D4—48no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4#11714yescomplemented
312C22×C4—28no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—28no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—216no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—28no—
353C22×C4—44no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—416no—
358C22×C4—416no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—48no—
363C22×C4—44no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C4×D418yes—
383C22×C4#12012yes—
384C22×C4#12118yes—
385C22×C4#12312yes—
386C22×C4#13514yes—
387C22×C4AGL(1,ℤ/8)12yes—
388C22×C4AGL(1,ℤ/8)12yes—
389C22×C4#13712yes—
390C22×C4#13712yes—
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C2×#4112yes—
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4H(5,𝔽2)14yes—
398C2×C8—44no—
399C2×C8—44no—
400C2×C8—44no—
401C2×C8—44no—
402C2×C8—44no—
403C2×C8—44no—
404C2×C8—44no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—28no—
408C4⋊C4—28no—
409C4⋊C4—416no—
410C4⋊C4—416no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—416no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C42—24no—
422C42—24no—
423C42—24no—
424C42—24no—
425C42—28no—
426C42—28no—
427C42—48no—
428C42—416no—
429C23—22no—
430C23—24no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—24no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—44no—
463C23—42no—
464C23—82no—
465C23—81no—
466C23—81no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—81no—
473C23—81no—
474C23#38614yes—
475C23C22×C22⋊C411yescharacteristic
476C23C2×#12011yescharacteristic
477C23C2×#12311yescharacteristic
478C23#52512yes—
479C23#53512yes—
480C23C2×#13512yes—
481C23C2×AGL(1,ℤ/8)11yescharacteristic
482C23C2×AGL(1,ℤ/8)11yescharacteristic
483C23C2×#13711yescharacteristic
484C23C2×#13711yescharacteristic
485C23#57512yes—
486Q8—28no—
487Q8—44no—
488Q8—48no—
489D4—28no—
490D4—432no—
491D4—44no—
492D4—48no—
493C2×C4—232no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—216no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—416no—
522C2×C4—432no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—48no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—82no—
541C2×C4—84no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—84no—
545C2×C4—82no—
546C2×C4#38614yes—
547C2×C4#41614yes—
548C2×C4#41714yes—
549C2×C4#41714yes—
550C2×C4C2×#11711yescharacteristic
551C2×C4C2×#11711yescharacteristic
552C2×C4C2×C4×D412yes—
553C2×C4#51812yes—
554C2×C4C2×#12112yes—
555C2×C4#52512yes—
556C2×C4#53512yes—
557C2×C4#54614yes—
558C2×C4#54614yes—
559C2×C4#55114yes—
560C2×C4#55114yes—
561C8—84no—
562C22—24no—
563C22—24no—
564C22—24no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—82no—
577C22—82no—
578C22—84no—
579C22—84no—
580C22—82no—
581C22—82no—
582C22—84no—
583C22—84no—
584C22#228011yescharacteristic
585C22C2×#38611yescharacteristic
586C22#230911yescharacteristic
587C22#240412yes—
588C22#248212yes—
589C22C2×#41611yescharacteristic
590C22C2×#41711yescharacteristic
591C22C2×#41711yescharacteristic
592C22#289611yescharacteristic
593C22#300212yes—
594C22#300311yescharacteristic
595C22#300411yescharacteristic
596C22#309112yes—
597C22#309211yescharacteristic
598C22#309211yescharacteristic
599C22#310612yes—
600C22#310712yes—
601C22#311512yes—
602C22#311612yes—
603C22#320712yes—
604C22#321012yes—
605C22#321412yes—
606C22#321612yes—
607C4—24no—
608C4—24no—
609C4—24no—
610C4—24no—
611C4—44no—
612C4—42no—
613C4—82no—
614C4—82no—
615C4—88no—
616C4#234212yes—
617C4#234212yes—
618C4#241114yes—
619C2—44no—
620C2—84no—
621C2#1386811yescharacteristic
622C2#2073711yescharacteristic
623C2#2074311yescharacteristic
624C2#2090811yescharacteristic
625C2#2090911yescharacteristic
626C2#2180511yescharacteristic
627C2#2180911yescharacteristic
628C2#2393212yes—
629C2#2393512yes—
630C2#2758412yes—
631C2#2758712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
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
2. C2×#2285C2semidirect
3. C2×#1817C2semidirect
5. #11050C2semidirect
6. #11049C2semidirect
7. #10932C2semidirect
8. #10931C2semidirect
9. #10731C2semidirect
18. #2297C4semidirect
295. C2×Q8#117semidirect
311. C2×D4#117semidirect

Maximal subgroups

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

subgroupindex
C2×#22972
C2×#22852
C2×#18172
C2×#17882
#110502
#110492
#109322
#109312
#107312