Finite Group Databasebeta

← #365539

Subgroups — #365539

Order 512.

1750 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2367C211yesmaximal, characteristic
2C2×#1983C211yesmaximal, characteristic
3C2×#1959C211yesmaximal, characteristic
4C2×#1948C211yesmaximal, characteristic, complemented
5C2×#1865C212yesmaximal, complemented
6C2×#1864C212yesmaximal
7#13910C211yesmaximal, characteristic, complemented
8#11067C212yesmaximal, complemented
9#11066C212yesmaximal
10#10835C211yesmaximal, characteristic, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#106C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#97C2211yescharacteristic
22#2367—24no—
23#2367—24no—
24C2×#391C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#388—21no—
29C2×#388—21no—
30C2×#385—21no—
31C2×#384—21no—
32C2×#384C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4—21no—
37#1983C418yescomplemented
38#1959—24no—
39#1948—24no—
40#1865—28no—
41#1864—28no—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47#1444—21no—
48#1444—21no—
49C4⋊#95—21no—
50C23×Q8C2311yescharacteristic
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×#124—22no—
56C2×#124—22no—
57C2×#124—22no—
58C2×#124C2×C414yes—
59C2×#124D414yes—
60C2×#124Q814yes—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—24no—
64C2×C4×Q8—42no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—24no—
68C2×C4×D4—42no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C42—21no—
86C22×C42—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—24no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8C2×C414yescomplemented
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—42no—
102C2×#103C2×C418yescomplemented
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102—42no—
108C2×#102C2×C418yes—
109C2×#97—22no—
110C2×#97—22no—
111C2×#97—22no—
112#391—22no—
113#391—22no—
114#390—22no—
115#390—22no—
116#389—24no—
117#389—24no—
118#389—42no—
119#389—42no—
120#388—28no—
121#388—28no—
122#385—28no—
123#384—22no—
124#384—22no—
125#384—44no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4C2×C414yescomplemented
131C4×C4⋊C4D414yes—
132C4×C22⋊C4—28no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95D411yescharacteristic
140C2×#95D411yescharacteristic
141#340—24no—
142#340—24no—
143#340D418yes—
144#336—24no—
145#336—24no—
146C25—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8C22×C412yes—
155C22×Q8C2×D412yes—
156C22×Q8C2×Q812yes—
157C22×D4—22no—
158C22×D4—21no—
159C22×D4—22no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4C22×C412yes—
165C22×D4C2×D412yes—
166C22×D4C2×Q812yes—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—44no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—82no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
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—22no—
196C22×C8C22×C414yes—
197C22×C8C2×D414yes—
198#124—28no—
199#124—28no—
200#124—28no—
201#124C4⋊C4116yescomplemented
202C4×Q8—48no—
203C4×Q8—44no—
204C4×Q8—44no—
205C4×D4—48no—
206C4×D4—44no—
207C4×D4—44no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4—82no—
241C2×C4⋊C4C22×C414yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4#4114yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—82no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—44no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—82no—
275C2×C42C22×C412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×Q812yes—
279C2×C42#4112yes—
280C2×C42#4112yes—
281#106—48no—
282C4⋊C8—28no—
283#103—216no—
284#103—48no—
285#102—216no—
286#102—48no—
287#97—44no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—22no—
293#95—22no—
294#95—44no—
295#95—44no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—42no—
300C24—42no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—42no—
306C24—42no—
307C24C22×D411yescharacteristic
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—42no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8C2×C4⋊C414yes—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—42no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4C2×C4⋊C414yes—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—41no—
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—88no—
400C22×C4—84no—
401C22×C4—88no—
402C22×C4C4×D412yes—
403C22×C4C4×D412yes—
404C22×C4#12012yes—
405C22×C4#12012yes—
406C22×C4C22⋊Q812yes—
407C22×C4C22⋊Q812yes—
408C22×C4AGL(1,ℤ/8)12yes—
409C22×C4#13712yes—
410C22×C4C23×C411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4C22×Q811yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4112yes—
417C22×C4C2×#4111yescharacteristic
418C22×C4H(5,𝔽2)12yes—
419C22×C42−1+412yes—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—88no—
450C4⋊C4—84no—
451C4⋊C4—84no—
452C4⋊C4—88no—
453C4⋊C4C4×D418yes—
454C4⋊C4#12714yes—
455C22⋊C4—216no—
456C22⋊C4—216no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—84no—
460C42—48no—
461C42—48no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—42no—
467C42—42no—
468C42—84no—
469C42C2×C4⋊C414yes—
470C42C4×D414yes—
471C42C22⋊Q814yes—
472C42#12314yes—
473C23—24no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—44no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23C2×C4×D411yescharacteristic
497C23C2×#12011yescharacteristic
498C23C2×C22⋊Q811yescharacteristic
499C23D4212yes—
500C23#54812yes—
501C23C2×AGL(1,ℤ/8)11yescharacteristic
502C23C2×#13711yescharacteristic
503C23#57511yescharacteristic
504C23#57511yescharacteristic
505C23#57712yes—
506C23#57712yes—
507Q8—28no—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511Q8—44no—
512Q8—42no—
513D4—28no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—44no—
518D4—42no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—816no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4#38514yes—
580C2×C4#38514yes—
581C2×C4C22×C4⋊C411yescharacteristic
582C2×C4C2×C4×D411yescharacteristic
583C2×C4C2×C4×D412yes—
584C2×C4#52012yes—
585C2×C4C2×C22⋊Q811yescharacteristic
586C2×C4C2×#12311yescharacteristic
587C2×C4C2×#12711yescharacteristic
588C2×C4#53211yescharacteristic
589C2×C4#53212yes—
590C2×C4#53511yescharacteristic
591C2×C4#53611yescharacteristic
592C2×C4#54212yes—
593C2×C4D4212yes—
594C2×C4#54712yes—
595C2×C4#54712yes—
596C2×C4D4×Q812yes—
597C2×C4#55312yes—
598C2×C4#55412yes—
599C8—44no—
600C8—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—82no—
605C22—82no—
606C22—84no—
607C22—84no—
608C22—82no—
609C22—82no—
610C22—84no—
611C22—84no—
612C22C2×#38511yescharacteristic
613C22#236112yes—
614C22#243211yescharacteristic
615C22#243311yescharacteristic
616C22#245212yes—
617C22C4×AGL(1,ℤ/8)11yescharacteristic
618C22C4×#13711yescharacteristic
619C22#295212yes—
620C22#300611yescharacteristic
621C22#300711yescharacteristic
622C22#300911yescharacteristic
623C22#301011yescharacteristic
624C22#307911yescharacteristic
625C22#308011yescharacteristic
626C22#308412yes—
627C22#326512yes—
628C22#326612yes—
629C22#326712yes—
630C22#327112yes—
631C22#336912yes—
632C22#337112yes—
633C22#337312yes—
634C22#337612yes—
635C4—24no—
636C4—24no—
637C4—44no—
638C4—42no—
639C4—42no—
640C4—44no—
641C4—48no—
642C4—82no—
643C4—81no—
644C4—81no—
645C4—162no—
646C4C4×#12711yescharacteristic
647C4#232611yescharacteristic
648C4D4×C4⋊C412yes—
649C4#242212yes—
650C4#254211yescharacteristic
651C4#256611yescharacteristic
652C2—84no—
653C2#1397711yescharacteristic
654C2#2087511yescharacteristic
655C2#2088211yescharacteristic
656C2#2300712yes—
657C2#2301312yes—
658C2#2565911yescharacteristic
659C2#2566211yescharacteristic
660C2#2576811yescharacteristic
661C2#2576911yescharacteristic
662C2#2647612yes—
663C2#2647912yes—
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
4. C2×#1948C2semidirect
5. C2×#1865C2semidirect
7. #13910C2semidirect
8. #11067C2semidirect
10. #10835C2semidirect
11. #10703C2semidirect
37. #1983C4semidirect
96. C2×C4⋊C8C2×C4semidirect
102. C2×#103C2×C4semidirect
130. C4×C4⋊C4C2×C4semidirect
201. #124C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23672
C2×#19832
C2×#19592
C2×#19482
C2×#18652
C2×#18642
#139102
#110672
#110662
#108352
#107032