Finite Group Databasebeta

← #374276

Subgroups — #374276

Order 512.

1726 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×#464C211yesmaximal, characteristic
2C2×#2356C211yesmaximal, characteristic, complemented
3C2×#1958C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic, complemented
5#14153C211yesmaximal, characteristic
6#11024C212yesmaximal, complemented
7#11023C212yesmaximal
8#10932C212yesmaximal, complemented
9#10931C212yesmaximal
10#10850C211yesmaximal, characteristic, complemented
11#10834C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C23×SD16C2211yescharacteristic
14C22×#124C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C2×#464—24no—
17C2×#464—24no—
18C2×#464C418yescomplemented
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2211yescharacteristic
23C22×#102C2212yes—
24C22×#102C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2356—24no—
27#2356—24no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#390C2212yes—
31C2×#387—22no—
32C2×#383—21no—
33C2×#382C2211yescharacteristic
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×#376—21no—
38#1958—24no—
39C4×#103—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1439—22no—
46#1437—21no—
47#1437—21no—
48C23×Q8C2311yescharacteristic
49C23×Q8C2311yescharacteristic
50C23×D4C2311yescharacteristic
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—42no—
54C22×Q16—42no—
55C22×Q16C2×C414yes—
56C22×SD16—22no—
57C22×SD16—22no—
58C22×SD16—42no—
59C22×SD16—42no—
60C22×SD16C2×C414yes—
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2312yes—
64C2×#124—22no—
65C2×#124—22no—
66C2×#124—22no—
67C2×#124—22no—
68C2×#124—22no—
69C2×#124C2×C414yes—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8C2×C414yes—
74C2×C4×Q8D414yes—
75C2×C4×Q8D414yes—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D412yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C42—21no—
86C22×C42—22no—
87C22×C42—21no—
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90#464—232no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8C2×C414yes—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—42no—
102C2×#103C2×C414yes—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102—42no—
108C2×#102C2×C414yes—
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112#391—24no—
113#391—24no—
114#391—24no—
115#391—42no—
116#390—24no—
117#390—42no—
118#387—216no—
119#383—28no—
120#382—22no—
121#382D414yes—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128#376—28no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95D411yescharacteristic
138C2×#95D411yescharacteristic
139#340—24no—
140#340—42no—
141#340—42no—
142#336—42no—
143#336—42no—
144C25—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—21no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8C22×C412yes—
162C22×Q8C22×C412yes—
163C22×Q8C2×D412yes—
164C22×Q8C2×D412yes—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—21no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4C22×C412yes—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
185C2×Q16—28no—
186C2×Q16—48no—
187C2×Q16—48no—
188C2×Q16—44no—
189C2×Q16—44no—
190C2×SD16—28no—
191C2×SD16—48no—
192C2×SD16—48no—
193C2×SD16—44no—
194C2×SD16—44no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8C22×C412yes—
210C22×C8C22×C412yes—
211C22×C8#4114yes—
212#124—28no—
213#124—28no—
214#124—44no—
215#124—44no—
216C4×Q8—28no—
217C4×Q8—28no—
218C4×Q8—28no—
219C4×Q8C22⋊C4116yescomplemented
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4#4114yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—82no—
255C2×C22⋊C4—82no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—28no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—44no—
267C2×C42—44no—
268C2×C42—42no—
269C2×C42—82no—
270C2×C42C22×C412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42#4112yes—
275C2×C42#4112yes—
276C4⋊C8—28no—
277#103—216no—
278#103—28no—
279#103—48no—
280#102—28no—
281#102—48no—
282C4×C8—44no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—24no—
287#95—24no—
288#95—22no—
289#95—22no—
290#95—44no—
291#95—44no—
292#95—42no—
293C24—22no—
294C24—41no—
295C24—41no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24C22×D411yescharacteristic
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—24no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—42no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8C2×C22⋊C414yes—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—42no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—42no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4C4×D412yes—
404C22×C4C4×D412yes—
405C22×C4#12012yes—
406C22×C4#12012yes—
407C22×C4#12112yes—
408C22×C4#12112yes—
409C22×C4#13514yes—
410C22×C4C23×C411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4111yescharacteristic
418C22×C4H(5,𝔽2)12yes—
419C22×C4H(5,𝔽2)12yes—
420Q16—416no—
421SD16—416no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—42no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—42no—
438C2×C8—42no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—216no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—48no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4C2×C22⋊C414yes—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—88no—
456C22⋊C4—88no—
457C42—216no—
458C42—416no—
459C42—42no—
460C42—42no—
461C42—42no—
462C42—42no—
463C42—84no—
464C42C2×C22⋊C414yes—
465C42C4×D414yes—
466C42#12114yes—
467C42#12314yes—
468C23—44no—
469C23—42no—
470C23—42no—
471C23—44no—
472C23—42no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—81no—
479C23—81no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—81no—
485C23—81no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23C2×C4×D411yescharacteristic
490C23C2×#12011yescharacteristic
491C23C2×#12111yescharacteristic
492C23#54612yes—
493C23#55012yes—
494C23C2×#13512yes—
495C23#57512yes—
496C23#57712yes—
497C23#57812yes—
498Q8—24no—
499Q8—24no—
500Q8—44no—
501Q8—44no—
502Q8—42no—
503Q8—42no—
504Q8—42no—
505Q8—44no—
506Q8—42no—
507Q8—82no—
508Q8—81no—
509Q8—81no—
510D4—42no—
511D4—44no—
512D4—42no—
513D4—82no—
514D4—81no—
515D4—81no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4#38614yes—
568C2×C4#38614yes—
569C2×C4C22×C22⋊C411yescharacteristic
570C2×C4C2×C4×D411yescharacteristic
571C2×C4C4×#4112yes—
572C2×C4#51812yes—
573C2×C4C2×#12111yescharacteristic
574C2×C4C2×#12311yescharacteristic
575C2×C4#52512yes—
576C2×C4#53211yescharacteristic
577C2×C4#53211yescharacteristic
578C2×C4#53512yes—
579C2×C4#53511yescharacteristic
580C2×C4#53511yescharacteristic
581C2×C4#54612yes—
582C2×C4#55012yes—
583C2×C4#55312yes—
584C2×C4#55312yes—
585C2×C4#55512yes—
586C2×C4#55512yes—
587C8—42no—
588C8—42no—
589C8—44no—
590C22—82no—
591C22—84no—
592C22—84no—
593C22—82no—
594C22—82no—
595C22—162no—
596C22—162no—
597C22—162no—
598C22—162no—
599C22C2×#38611yescharacteristic
600C22#236312yes—
601C22#241412yes—
602C22#243111yescharacteristic
603C22#243111yescharacteristic
604C22C4×#13511yescharacteristic
605C22#294311yescharacteristic
606C22#294712yes—
607C22#300212yes—
608C22#301312yes—
609C22#303512yes—
610C22#304311yescharacteristic
611C22#304411yescharacteristic
612C22#329612yes—
613C22#330012yes—
614C22#332312yes—
615C22#332712yes—
616C22#339312yes—
617C22#339412yes—
618C22#340112yes—
619C22#340212yes—
620C4—24no—
621C4—24no—
622C4—44no—
623C4—44no—
624C4—44no—
625C4—84no—
626C4—82no—
627C4—84no—
628C4—82no—
629C4#229711yescharacteristic
630C4#230911yescharacteristic
631C4#236312yes—
632C4#241412yes—
633C4#250011yescharacteristic
634C4#250011yescharacteristic
635C2—162no—
636C2#1413611yescharacteristic
637C2#2092411yescharacteristic
638C2#2093011yescharacteristic
639C2#2309712yes—
640C2#2310312yes—
641C2#2463512yes—
642C2#2463712yes—
643C2#2557612yes—
644C2#2563612yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2356C2semidirect
4. C2×C4×#103C2semidirect
6. #11024C2semidirect
8. #10932C2semidirect
10. #10850C2semidirect
11. #10834C2semidirect
18. C2×#464C4semidirect
219. C4×Q8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4642
C2×#23562
C2×#19582
C2×C4×#1032
#141532
#110242
#110232
#109322
#109312
#108502
#108342