Finite Group Databasebeta

← #374238

Subgroups — #374238

Order 512.

1802 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2620C211yesmaximal, characteristic
2C2×#2314C211yesmaximal, characteristic, complemented
3C2×#1959C211yesmaximal, characteristic, complemented
4C2×#1958C211yesmaximal, characteristic, complemented
5C2×#1750C211yesmaximal, characteristic, complemented
6#10970C212yesmaximal, complemented
7#10969C212yesmaximal
8#10932C212yesmaximal, complemented
9#10931C212yesmaximal, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10841C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C22⋊C8C2212yes—
21C22×#97C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2620—24no—
24#2620C418yescomplemented
25C2×#396—22no—
26C2×#394—22no—
27#2314—24no—
28#2314—24no—
29C2×#391C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#388—22no—
32C2×#387—22no—
33C2×#384C2211yescharacteristic
34C2×#384C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2212yes—
38#1959—24no—
39#1958—24no—
40#1750—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2212yes—
46C23×Q8C2311yescharacteristic
47C23×D4—21no—
48C23×D4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#124—22no—
53C2×#124—22no—
54C2×#124—22no—
55C2×#124—22no—
56C2×#124—22no—
57C2×#124C2×C414yes—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—42no—
60C2×C22⋊Q8—42no—
61C2×C22⋊Q8C2×C414yes—
62C2×#121—22no—
63C2×#121—42no—
64C2×#121—42no—
65C2×#121C2×C414yes—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117C2×C414yes—
71C2×#117D414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4D412yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4D412yes—
86C22×C42—22no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—42no—
100C2×#102—42no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—24no—
104C2×#97—22no—
105C2×#97—22no—
106C2×#97—22no—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110#396—216no—
111#394—216no—
112#391—22no—
113#391C2×C414yes—
114#390—22no—
115#390C2×C414yes—
116#388—216no—
117#387—216no—
118#384—22no—
119#384—22no—
120#384C2×C414yes—
121#384D414yescomplemented
122#382—22no—
123#382D414yescomplemented
124#380—24no—
125#380—24no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—24no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132#340—42no—
133#340—42no—
134#340—42no—
135#340—42no—
136#336—24no—
137C25—21no—
138C25C2×D411yescharacteristic
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8C22×C412yes—
149C22×D4—24no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—42no—
160C22×D4—42no—
161C22×D4—82no—
162C22×D4—82no—
163C22×D4C22×C412yes—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4C22×C412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2411yescharacteristic
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8#4114yes—
196C22×C8#4114yes—
197#124—28no—
198#124—28no—
199#124—44no—
200#124—44no—
201C22⋊Q8—48no—
202C22⋊Q8—48no—
203#121—48no—
204#121—48no—
205#117—28no—
206#117—28no—
207#117—28no—
208#117—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4C22⋊C418yes—
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4C2×D414yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—82no—
255C2×C22⋊C4—82no—
256C2×C22⋊C4—82no—
257C2×C22⋊C4—82no—
258C2×C22⋊C4C22⋊C418yes—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—28no—
264C2×C42—82no—
265C2×C42—82no—
266C2×C42C22×C412yes—
267C2×C42C22×C412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42#4112yes—
273C2×C42#4112yes—
274#103—48no—
275#103—48no—
276#102—48no—
277#102—48no—
278C22⋊C8—216no—
279#97—44no—
280C4×C8—44no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24C2×C22⋊C412yes—
307C24C22×D411yescharacteristic
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—48no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—88no—
342C2×D4—84no—
343C2×D4—84no—
344C2×D4—88no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
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×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4C2×C22⋊C414yes—
409C22×C4C2×C22⋊C412yes—
410C22×C4C2×C22⋊C414yes—
411C22×C4#12014yes—
412C22×C4#12314yes—
413C22×C4C23×C411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C4H(5,𝔽2)12yes—
423C2×C8—24no—
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—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C4⋊C4—216no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—48no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—88no—
452C4⋊C4—88no—
453C22⋊C4—216no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—416no—
457C22⋊C4—88no—
458C22⋊C4—88no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—88no—
464C42C4×D414yes—
465C42C4×D414yes—
466C42#12114yes—
467C42#12114yes—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—81no—
503C23—81no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23—81no—
508C23—82no—
509C23—82no—
510C23—81no—
511C23—81no—
512C23—82no—
513C23—82no—
514C23C22×C22⋊C411yescharacteristic
515C23C2×#12011yescharacteristic
516C23C2×#12311yescharacteristic
517C23#53211yescharacteristic
518C23#53211yescharacteristic
519C23#53511yescharacteristic
520C23#53511yescharacteristic
521C23#57612yes—
522C23#57712yes—
523C23#57712yes—
524C23#57812yes—
525Q8—44no—
526Q8—44no—
527Q8—82no—
528Q8—82no—
529D4—44no—
530D4—44no—
531D4—816no—
532D4—82no—
533D4—82no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—82no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4#38618yes—
590C2×C4C2×C4×D411yescharacteristic
591C2×C4C2×C4×D411yescharacteristic
592C2×C4C4×#4112yes—
593C2×C4#51812yes—
594C2×C4C2×#12111yescharacteristic
595C2×C4C2×#12111yescharacteristic
596C2×C4#52512yes—
597C2×C4#53512yes—
598C2×C4#54612yes—
599C2×C4#54612yes—
600C2×C4#55012yes—
601C2×C4#55012yes—
602C2×C4#55312yes—
603C2×C4#55312yes—
604C2×C4#55512yes—
605C2×C4#55512yes—
606C8—44no—
607C8—44no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22—82no—
614C22—82no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—84no—
623C22—84no—
624C22—162no—
625C22—162no—
626C22—161no—
627C22—161no—
628C22—161no—
629C22—161no—
630C22C2×#38611yescharacteristic
631C22#229711yescharacteristic
632C22#230911yescharacteristic
633C22#243111yescharacteristic
634C22#243111yescharacteristic
635C22#250011yescharacteristic
636C22#250011yescharacteristic
637C22#289712yes—
638C22#289812yes—
639C22#301212yes—
640C22#301312yes—
641C22#309812yes—
642C22#309912yes—
643C22#317312yes—
644C22#317512yes—
645C22#317812yes—
646C22#317912yes—
647C22#322011yescharacteristic
648C22#322111yescharacteristic
649C22#322211yescharacteristic
650C22#322311yescharacteristic
651C22#322412yes—
652C22#322512yes—
653C4—28no—
654C4—42no—
655C4—44no—
656C4—84no—
657C4—84no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4#236312yes—
662C4#236312yes—
663C4#241412yes—
664C4#241412yes—
665C2—82no—
666C2—162no—
667C2#1413611yescharacteristic
668C2#2091512yes—
669C2#2141212yes—
670C2#2181311yescharacteristic
671C2#2181411yescharacteristic
672C2#2561012yes—
673C2#2561111yescharacteristic
674C2#2561211yescharacteristic
675C2#2789811yescharacteristic
676C2#2789911yescharacteristic
677C2#2790412yes—
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
2. C2×#2314C2semidirect
3. C2×#1959C2semidirect
4. C2×#1958C2semidirect
5. C2×#1750C2semidirect
6. #10970C2semidirect
8. #10932C2semidirect
9. #10931C2semidirect
10. #10843C2semidirect
11. #10841C2semidirect
24. #2620C4semidirect
121. #384D4semidirect
123. #382D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26202
C2×#23142
C2×#19592
C2×#19582
C2×#17502
#109702
#109692
#109322
#109312
#108432
#108412