Finite Group Databasebeta

← #363164

Subgroups — #363164

Order 512.

2966 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2C2×#1992C212yesmaximal, complemented
3C2×#1959C212yesmaximal, complemented
4C2×#1749C211yesmaximal, characteristic, complemented
5#14686C211yesmaximal, characteristic, complemented
6#10970C212yesmaximal, complemented
7#10969C212yesmaximal
8#10789C212yesmaximal, complemented
9#10788C212yesmaximal, complemented
10C2×#529—24no—
11C2×#529—28no—
12C2×#529—28no—
13C2×#529C418yescomplemented
14C22×#126—21no—
15C22×#125C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×#123—21no—
18C22×C22⋊Q8C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4—21no—
22C22×C4×D4C2211yescharacteristic
23C22×#117—21no—
24C23×C42C2211yescharacteristic
25C22×#103C2214yes—
26C22×#102C2214yes—
27C22×C22⋊C8C2212yes—
28C22×#97C2212yes—
29C2×#384C2212yes—
30C2×#380—21no—
31C2×#380—21no—
32C2×#380C2212yes—
33#1992—28no—
34#1959—28no—
35#1749—24no—
36C2×#340C2214yes—
37C2×#340C2214yes—
38C2×#336C2212yes—
39C4⋊#95—21no—
40C4⋊#95—21no—
41C4⋊#95C2212yes—
42C23×Q8C2311yescharacteristic
43C23×D4—21no—
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2314yes—
50#529—264no—
51#529—232no—
52#529—232no—
53C2×#126—216no—
54C2×#126—28no—
55C2×#125—24no—
56C2×#125—22no—
57C2×#125C2×C414yes—
58C2×#125D418yes—
59C2×#124—24no—
60C2×#124—22no—
61C2×#124C2×C414yes—
62C2×#124D418yes—
63C2×#123—216no—
64C2×#123—28no—
65C2×C22⋊Q8—24no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8C2×C414yes—
69C2×#121—24no—
70C2×#121—22no—
71C2×#121—24no—
72C2×#121C2×C414yes—
73C2×C4×Q8—24no—
74C2×C4×Q8—24no—
75C2×C4×Q8—22no—
76C2×C4×Q8C2×C414yes—
77C2×C4×D4—216no—
78C2×C4×D4—24no—
79C2×C4×D4—28no—
80C2×C4×D4—24no—
81C2×C4×D4—22no—
82C2×C4×D4C2×C414yes—
83C2×#117—216no—
84C2×#117—28no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4—22no—
100C22×C42—21no—
101C22×C42—24no—
102C22×C42—24no—
103C22×C42C2×C414yes—
104C22×C42D411yescharacteristic
105C22×C42D411yescharacteristic
106C22×C42D414yes—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C2×#103—28no—
110C2×#103—28no—
111C2×#103—44no—
112C2×#102—28no—
113C2×#102—28no—
114C2×#102—44no—
115C2×C22⋊C8—24no—
116C2×C22⋊C8—28no—
117C2×#97—24no—
118C2×#97—24no—
119C2×#97—24no—
120#384—24no—
121#384—24no—
122#380—44no—
123#380—44no—
124#380—44no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128#340—28no—
129#340—28no—
130#336—24no—
131C25—21no—
132C25C2×D411yescharacteristic
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—24no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8C22×C412yes—
139C22×Q8C2×D414yes—
140C22×D4—28no—
141C22×D4—24no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—24no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—44no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4C22×C412yes—
151C22×D4C2×D414yes—
152C23×C4—28no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—42no—
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×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C22×C8—24no—
185C22×C8—24no—
186C22×C8—24no—
187C22×C8—24no—
188C22×C8—24no—
189C22×C8—24no—
190C22×C8#4118yes—
191#126—264no—
192#125—28no—
193#125—216no—
194#125C22⋊C4116yescomplemented
195#124—28no—
196#124—216no—
197#124C22⋊C4116yescomplemented
198#123—264no—
199C22⋊Q8—216no—
200C22⋊Q8—28no—
201C22⋊Q8—28no—
202#121—216no—
203#121—28no—
204#121—28no—
205C4×Q8—216no—
206C4×Q8—28no—
207C4×Q8—28no—
208C4×D4—264no—
209C4×D4—216no—
210C4×D4—28no—
211C4×D4—28no—
212#117—264no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4C2×D414yes—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C42—28no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—28no—
261C2×C42—28no—
262C2×C42—216no—
263C2×C42C22⋊C414yes—
264C2×C42C22⋊C414yes—
265C2×C42C22×C412yes—
266C2×C42C22×C412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D414yes—
269C2×C42C2×D414yes—
270C2×C42C2×D412yes—
271#103—416no—
272#102—416no—
273C22⋊C8—216no—
274#97—48no—
275#95—44no—
276#95—44no—
277#95—44no—
278C24—22no—
279C24—22no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—24no—
288C24—42no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—42no—
296C24—42no—
297C24C2×C22⋊C412yes—
298C24AGL(1,ℤ/8)12yes—
299C24#13712yes—
300C24C22×D411yescharacteristic
301C2×Q8—22no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—28no—
306C2×Q8—28no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8C2×C22⋊C414yes—
312C2×D4—216no—
313C2×D4—22no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—28no—
318C2×D4—28no—
319C2×D4—48no—
320C2×D4—416no—
321C2×D4—416no—
322C2×D4—48no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4C2×C22⋊C414yes—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—216no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—416no—
370C22×C4—416no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—48no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4C2×C22⋊C412yes—
393C22×C4C2×C22⋊C412yes—
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C412yes—
396C22×C4C2×C22⋊C412yes—
397C22×C4#12014yes—
398C22×C4#12014yes—
399C22×C4AGL(1,ℤ/8)12yes—
400C22×C4#13712yes—
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C2×#4114yes—
406C22×C4H(5,𝔽2)12yes—
407C22×C42−1+412yes—
408C2×C8—28no—
409C2×C8—28no—
410C2×C8—28no—
411C2×C8—44no—
412C2×C8—44no—
413C2×C8—44no—
414C2×C8—44no—
415C4⋊C4—216no—
416C4⋊C4—216no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—216no—
422C4⋊C4—216no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—88no—
428C4⋊C4—88no—
429C4⋊C4C2×C22⋊C414yes—
430C4⋊C4C2×C22⋊C414yes—
431C22⋊C4—216no—
432C22⋊C4—216no—
433C22⋊C4—216no—
434C22⋊C4—216no—
435C22⋊C4—216no—
436C22⋊C4—88no—
437C42—216no—
438C42—22no—
439C42—22no—
440C42—24no—
441C42—28no—
442C42—28no—
443C42C2×C22⋊C414yes—
444C42C2×C22⋊C414yes—
445C42#12018yes—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—24no—
451C23—24no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—81no—
486C23—81no—
487C23—82no—
488C23—81no—
489C23—81no—
490C23—82no—
491C23#41714yes—
492C23C22×C22⋊C411yescharacteristic
493C23C2×#12012yes—
494C23#53212yes—
495C23#53511yescharacteristic
496C23#53611yescharacteristic
497C23C2×AGL(1,ℤ/8)11yescharacteristic
498C23C2×AGL(1,ℤ/8)11yescharacteristic
499C23C2×#13711yescharacteristic
500C23C2×#13711yescharacteristic
501C23#57512yes—
502C23#57512yes—
503Q8—28no—
504Q8—44no—
505Q8—48no—
506D4—28no—
507D4—432no—
508D4—44no—
509D4—48no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—416no—
532C2×C4—416no—
533C2×C4—432no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4#37914yes—
561C2×C4#37914yes—
562C2×C4#41714yes—
563C2×C4C22×C22⋊C411yescharacteristic
564C2×C4C22×C22⋊C411yescharacteristic
565C2×C4C4×#4112yes—
566C2×C4#52012yes—
567C2×C4C2×#12012yes—
568C2×C4#52512yes—
569C2×C4#53712yes—
570C2×C4#54614yes—
571C2×C4#54714yes—
572C2×C4#54814yes—
573C2×C4#55014yes—
574C8—48no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—84no—
590C22—84no—
591C22—84no—
592C22—84no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22C2×#37911yescharacteristic
598C22#229711yescharacteristic
599C22#231111yescharacteristic
600C22#243112yes—
601C22#243212yes—
602C22C2×#41711yescharacteristic
603C22C2×#41711yescharacteristic
604C22#289611yescharacteristic
605C22#289611yescharacteristic
606C22#300612yes—
607C22#300612yes—
608C22#300712yes—
609C22#300712yes—
610C22#317112yes—
611C22#317212yes—
612C22#318012yes—
613C22#318112yes—
614C22#321711yescharacteristic
615C22#321811yescharacteristic
616C22#321912yes—
617C22#323811yescharacteristic
618C22#323911yescharacteristic
619C22#324012yes—
620C4—24no—
621C4—24no—
622C4—24no—
623C4—42no—
624C4—44no—
625C4—42no—
626C4—44no—
627C4—44no—
628C4—164no—
629C4#233912yes—
630C4#234112yes—
631C4#240714yes—
632C2—44no—
633C2—84no—
634C2#1393711yescharacteristic
635C2#2060111yescharacteristic
636C2#2060111yescharacteristic
637C2#2140311yescharacteristic
638C2#2140611yescharacteristic
639C2#2186711yescharacteristic
640C2#2186811yescharacteristic
641C2#2560412yes—
642C2#2560512yes—
643C2#2567812yes—
644C2#2567912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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×#1992C2semidirect
3. C2×#1959C2semidirect
4. C2×#1749C2semidirect
5. #14686C2semidirect
6. #10970C2semidirect
8. #10789C2semidirect
9. #10788C2semidirect
13. C2×#529C4semidirect
194. #125C22⋊C4semidirect
197. #124C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5292
C2×#19922
C2×#19592
C2×#17492
#146862
#109702
#109692
#107892
#107882