Finite Group Databasebeta

← #360742

Subgroups — #360742

Order 512.

1866 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2349C211yesmaximal, characteristic
2C2×#1864C212yesmaximal
3C2×#1862C212yesmaximal
4C2×#1847C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic
6C2×#1762C211yesmaximal, characteristic
7#13910C211yesmaximal, characteristic, complemented
8#11328C212yesmaximal, complemented
9#11327C212yesmaximal, complemented
10#10850C211yesmaximal, characteristic, complemented
11#10835C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×#97C2211yescharacteristic
21#2349—24no—
22#2349—24no—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#385—21no—
28C2×#385—21no—
29C2×#384—21no—
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#380—21no—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C22×#95—21no—
36#1864—28no—
37#1862—28no—
38#1847—24no—
39#1802—24no—
40#1762C418yescomplemented
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47#1444—21no—
48#1444—21no—
49C4⋊#95—21no—
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C24×C4—21no—
53C24×C4—21no—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—42no—
66C2×C4×D4—42no—
67C2×C4×D4C2×C414yes—
68C2×C4×D4D414yes—
69C2×C4×D4Q814yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#107—24no—
91C2×#107—24no—
92C2×#107—42no—
93C2×#106—24no—
94C2×#106—24no—
95C2×#106—42no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102C2×C418yes—
103C2×#102D418yes—
104C2×#102Q818yes—
105C2×#97—22no—
106C2×#97—22no—
107C2×#97C2×C414yescomplemented
108#389—22no—
109#389—22no—
110#389—22no—
111#389—22no—
112#389—24no—
113#389—24no—
114#389—42no—
115#389—42no—
116#385—28no—
117#385—28no—
118#384—22no—
119#384—22no—
120#384—44no—
121#382—22no—
122#382D414yes—
123#380—28no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2×C414yescomplemented
128C2×#95—28no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95D411yescharacteristic
137C2×#95D411yescharacteristic
138#340—24no—
139#340—24no—
140#340—24no—
141#340C2×C418yescomplemented
142#336—24no—
143#336—24no—
144C25—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4C22×C412yes—
151C22×D4C2×D412yes—
152C22×D4C2×D414yes—
153C22×D4C2×Q812yes—
154C22×D4C2×Q814yes—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—82no—
172C23×C4—82no—
173C23×C4—82no—
174C23×C4—82no—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2411yescharacteristic
182C22×C8—22no—
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×C8C22×C414yes—
194C22×C8C2×D414yes—
195C22×C8C2×Q814yes—
196C4×D4—28no—
197C4×D4—28no—
198C4×D4—28no—
199C4×D4—48no—
200C4×D4—48no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4C4⋊C4116yescomplemented
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
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—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—82no—
241C2×C4⋊C4—82no—
242C2×C4⋊C4—82no—
243C2×C4⋊C4—82no—
244C2×C4⋊C4—82no—
245C2×C4⋊C4C22×C412yes—
246C2×C4⋊C4C22×C414yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×Q812yes—
252C2×C4⋊C4C2×Q814yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42C22×C412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×Q812yes—
275C2×C42#4112yes—
276C2×C42#4112yes—
277#107—48no—
278#106—48no—
279C4⋊C8—44no—
280#102—48no—
281#102C4⋊C4132yescomplemented
282#97—28no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—22no—
287#95—24no—
288#95—24no—
289#95—22no—
290#95—44no—
291#95—44no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—24no—
296C24—24no—
297C24—41no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24—42no—
302C24—42no—
303C24C22×D411yescharacteristic
304C2×D4—22no—
305C2×D4—24no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4C2×C4⋊C418yes—
318C2×D4C2×C4⋊C414yes—
319C2×D4#12714yes—
320C2×D4C4⋊Q818yes—
321C2×D4C4⋊Q814yes—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—216no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—88no—
381C22×C4—88no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—88no—
389C22×C4—88no—
390C22×C4C4×D412yes—
391C22×C4C4×D412yes—
392C22×C4#12012yes—
393C22×C4#12012yes—
394C22×C4C22⋊Q812yes—
395C22×C4C22⋊Q812yes—
396C22×C4AGL(1,ℤ/8)12yes—
397C22×C4#13712yes—
398C22×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D412yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×Q811yescharacteristic
403C22×C4C22×Q812yes—
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4H(5,𝔽2)12yes—
407C22×C42−1+412yes—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8C2×C4⋊C418yes—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—88no—
441C4⋊C4—84no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4—84no—
445C4⋊C4—88no—
446C4⋊C4C2×C4⋊C418yes—
447C4⋊C4C2×C4⋊C414yes—
448C22⋊C4—216no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C42—48no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—42no—
457C42—42no—
458C42C2×C4⋊C414yes—
459C42C4×D414yes—
460C42C22⋊Q814yes—
461C42#12314yes—
462C23—28no—
463C23—24no—
464C23—24no—
465C23—28no—
466C23—24no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23C2×C4×D411yescharacteristic
486C23C2×#12011yescharacteristic
487C23C2×C22⋊Q811yescharacteristic
488C23D4212yes—
489C23D4×Q812yes—
490C23C2×AGL(1,ℤ/8)11yescharacteristic
491C23C2×#13711yescharacteristic
492C23#57511yescharacteristic
493C23#57511yescharacteristic
494C23#57612yes—
495C23#57712yes—
496D4—24no—
497D4—28no—
498D4—28no—
499D4—24no—
500D4—24no—
501D4#38418yescomplemented
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—816no—
551C2×C4—816no—
552C2×C4—816no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4#38514yes—
565C2×C4#38514yes—
566C2×C4C22×C4⋊C411yescharacteristic
567C2×C4C22×C4⋊C412yes—
568C2×C4C2×C4×D411yescharacteristic
569C2×C4#52012yes—
570C2×C4C2×C22⋊Q811yescharacteristic
571C2×C4C2×#12311yescharacteristic
572C2×C4C2×#12711yescharacteristic
573C2×C4C2×C4⋊Q812yes—
574C2×C4C2×C4⋊Q811yescharacteristic
575C2×C4#53511yescharacteristic
576C2×C4#53611yescharacteristic
577C2×C4#54412yes—
578C2×C4D4212yes—
579C2×C4#54712yes—
580C2×C4#54712yes—
581C2×C4D4×Q812yes—
582C2×C4#55412yes—
583C2×C4#55412yes—
584C8—28no—
585C8—82no—
586C22—28no—
587C22—44no—
588C22—44no—
589C22—48no—
590C22—48no—
591C22—44no—
592C22—44no—
593C22—48no—
594C22—48no—
595C22—44no—
596C22—44no—
597C22C2×#38511yescharacteristic
598C22D4×C4⋊C412yes—
599C22#243311yescharacteristic
600C22#243411yescharacteristic
601C22#244012yes—
602C22C4×AGL(1,ℤ/8)11yescharacteristic
603C22C4×#13711yescharacteristic
604C22#294612yes—
605C22#300611yescharacteristic
606C22#300711yescharacteristic
607C22#300811yescharacteristic
608C22#300911yescharacteristic
609C22#307911yescharacteristic
610C22#308011yescharacteristic
611C22#308112yes—
612C22#326412yes—
613C22#326612yes—
614C22#326712yes—
615C22#326812yes—
616C22#338412yes—
617C22#338512yes—
618C22#338812yes—
619C22#338912yes—
620C4—24no—
621C4—24no—
622C4—44no—
623C4—48no—
624C4—42no—
625C4—42no—
626C4—81no—
627C4—82no—
628C4—81no—
629C4—82no—
630C4—162no—
631C4C2×#38411yescharacteristic
632C4#231711yescharacteristic
633C4D4×C4⋊C412yes—
634C4#244012yes—
635C4#250111yescharacteristic
636C4#250211yescharacteristic
637C2—48no—
638C2#1388011yescharacteristic
639C2#2087511yescharacteristic
640C2#2087611yescharacteristic
641C2#2267612yes—
642C2#2267812yes—
643C2#2576911yescharacteristic
644C2#2577011yescharacteristic
645C2#2585011yescharacteristic
646C2#2585111yescharacteristic
647C2#2605812yes—
648C2#2605912yes—
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×#1847C2semidirect
7. #13910C2semidirect
8. #11328C2semidirect
9. #11327C2semidirect
10. #10850C2semidirect
11. #10835C2semidirect
40. #1762C4semidirect
107. C2×#97C2×C4semidirect
127. C4×C4⋊C4C2×C4semidirect
141. #340C2×C4semidirect
205. C4×D4C4⋊C4semidirect
281. #102C4⋊C4semidirect
501. D4#384semidirect

Maximal subgroups

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

subgroupindex
C2×#23492
C2×#18642
C2×#18622
C2×#18472
C2×#18022
C2×#17622
#139102
#113282
#113272
#108502
#108352