Finite Group Databasebeta

← #360738

Subgroups — #360738

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2349C211yesmaximal, characteristic
2C2×#1864C212yesmaximal
3C2×#1862C212yesmaximal
4C2×#1846C211yesmaximal, characteristic, complemented
5C2×#1804C211yesmaximal, characteristic
6C2×C4×#102C211yesmaximal, characteristic
7#13954C211yesmaximal, characteristic, complemented
8#11328C212yesmaximal, complemented
9#11327C212yesmaximal, complemented
10#10850C211yesmaximal, characteristic, complemented
11#10834C211yesmaximal, characteristic, complemented
12C22×C4×D4C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C22×#102C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2349—24no—
22#2349—24no—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#385—22no—
27C2×#382C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#380—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C22×#95—21no—
33#1864—28no—
34#1862—28no—
35#1846—24no—
36#1804—24no—
37C4×#102C418yescomplemented
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1444—22no—
46C4⋊#95—21no—
47C4⋊#95—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C24×C4—22no—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4×D4—24no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—42no—
62C2×C4×D4—42no—
63C2×C4×D4C2×C414yes—
64C2×C4×D4D414yes—
65C2×C4×D4Q814yes—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D412yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C42—21no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—42no—
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—42no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102C2×C414yes—
93C2×#102C2×C414yes—
94C2×#102D414yes—
95C2×#102D414yes—
96C2×#102Q814yes—
97C2×#102Q814yes—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8C2×C414yescomplemented
101#389—24no—
102#389—24no—
103#389—24no—
104#389—24no—
105#389—42no—
106#389—42no—
107#385—216no—
108#382—22no—
109#382—22no—
110#382—22no—
111#382D414yes—
112#380—28no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4C2×C414yescomplemented
117C2×#95—28no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95D411yescharacteristic
127C2×#95D411yescharacteristic
128#340—24no—
129#340—24no—
130#340—24no—
131#340C2×C414yescomplemented
132#340C2×C414yescomplemented
133#336—24no—
134#336—24no—
135C25—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4C22×C412yes—
144C22×D4C2×D412yes—
145C22×D4C2×D412yes—
146C22×D4C2×D412yes—
147C22×D4C2×Q812yes—
148C22×D4C2×Q812yes—
149C22×D4C2×Q812yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—28no—
158C23×C4—22no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—82no—
164C23×C4—82no—
165C23×C4—82no—
166C23×C4—82no—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8C22×C412yes—
188C22×C8C22×C412yes—
189C22×C8C2×D412yes—
190C22×C8C2×D412yes—
191C22×C8C2×Q812yes—
192C22×C8C2×Q812yes—
193C4×D4—28no—
194C4×D4—28no—
195C4×D4—28no—
196C4×D4—48no—
197C4×D4—48no—
198C4×D4—48no—
199C4×D4—48no—
200C4×D4C4⋊C4116yescomplemented
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
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—28no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×Q812yes—
241C2×C4⋊C4C2×Q812yes—
242C2×C4⋊C4C2×Q812yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—82no—
257C2×C42C22×C412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×Q812yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263#107—48no—
264#106—48no—
265C4⋊C8—44no—
266#102—48no—
267#102C4⋊C4116yescomplemented
268#102C4⋊C4116yescomplemented
269C4×C8—28no—
270#95—28no—
271#95—24no—
272#95—22no—
273#95—24no—
274#95—24no—
275#95—22no—
276#95—44no—
277#95—44no—
278#95—42no—
279#95—42no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24C22×D411yescharacteristic
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4C2×C4⋊C414yes—
314C2×D4C2×C4⋊C414yes—
315C2×D4C2×C4⋊C414yes—
316C2×D4#12714yes—
317C2×D4C4⋊Q814yes—
318C2×D4C4⋊Q814yes—
319C2×D4C4⋊Q814yes—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—216no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—416no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—416no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—88no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—88no—
373C22×C4—88no—
374C22×C4—88no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4C4×D412yes—
379C22×C4C4×D412yes—
380C22×C4#12012yes—
381C22×C4#12012yes—
382C22×C4C22⋊Q812yes—
383C22×C4C22⋊Q812yes—
384C22×C4#13514yes—
385C22×C4C23×C411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×Q811yescharacteristic
391C22×C4C22×Q811yescharacteristic
392C22×C4C22×Q811yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4H(5,𝔽2)11yescharacteristic
396C22×C4H(5,𝔽2)11yescharacteristic
397C22×C42−1+411yescharacteristic
398C22×C42−1+411yescharacteristic
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8C2×C4⋊C414yes—
419C2×C8C2×C4⋊C414yes—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—48no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—88no—
439C4⋊C4—88no—
440C4⋊C4—88no—
441C4⋊C4C2×C4⋊C414yes—
442C4⋊C4C2×C4⋊C414yes—
443C4⋊C4C2×C4⋊C414yes—
444C22⋊C4—216no—
445C22⋊C4—416no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—84no—
451C42C2×C4⋊C414yes—
452C42C4×D414yes—
453C42C22⋊Q814yes—
454C42#12314yes—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23C2×C4×D411yescharacteristic
479C23C2×#12011yescharacteristic
480C23C2×C22⋊Q811yescharacteristic
481C23D4211yescharacteristic
482C23D4211yescharacteristic
483C23D4×Q811yescharacteristic
484C23D4×Q811yescharacteristic
485C23C2×#13512yes—
486C23#57512yes—
487C23#57612yes—
488C23#57712yes—
489D4—24no—
490D4—24no—
491D4—24no—
492D4—24no—
493D4—24no—
494D4—24no—
495D4—24no—
496D4#38418yes—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—816no—
538C2×C4—832no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4#38514yes—
551C2×C4#38514yes—
552C2×C4C22×C4⋊C411yescharacteristic
553C2×C4C22×C4⋊C411yescharacteristic
554C2×C4C22×C4⋊C411yescharacteristic
555C2×C4C2×C4×D411yescharacteristic
556C2×C4#52011yescharacteristic
557C2×C4#52011yescharacteristic
558C2×C4C2×C22⋊Q811yescharacteristic
559C2×C4C2×#12311yescharacteristic
560C2×C4C2×#12711yescharacteristic
561C2×C4C2×C4⋊Q811yescharacteristic
562C2×C4C2×C4⋊Q811yescharacteristic
563C2×C4C2×C4⋊Q811yescharacteristic
564C2×C4#53511yescharacteristic
565C2×C4#53611yescharacteristic
566C2×C4#54411yescharacteristic
567C2×C4#54411yescharacteristic
568C2×C4D4211yescharacteristic
569C2×C4D4211yescharacteristic
570C2×C4#54711yescharacteristic
571C2×C4#54711yescharacteristic
572C2×C4#54711yescharacteristic
573C2×C4#54711yescharacteristic
574C2×C4D4×Q811yescharacteristic
575C2×C4D4×Q811yescharacteristic
576C2×C4#55411yescharacteristic
577C2×C4#55411yescharacteristic
578C2×C4#55411yescharacteristic
579C2×C4#55411yescharacteristic
580C8—24no—
581C8—24no—
582C8—82no—
583C22—28no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—48no—
591C22—48no—
592C22—48no—
593C22—48no—
594C22C2×#38511yescharacteristic
595C22D4×C4⋊C411yescharacteristic
596C22D4×C4⋊C411yescharacteristic
597C22#243311yescharacteristic
598C22#243411yescharacteristic
599C22#244011yescharacteristic
600C22#244011yescharacteristic
601C22C4×#13511yescharacteristic
602C22#294311yescharacteristic
603C22#294811yescharacteristic
604C22#294911yescharacteristic
605C22#300212yes—
606C22#301212yes—
607C22#307612yes—
608C22#308112yes—
609C22#327312yes—
610C22#327512yes—
611C22#328012yes—
612C22#328212yes—
613C22#338012yes—
614C22#338112yes—
615C22#338812yes—
616C22#338912yes—
617C4—24no—
618C4—24no—
619C4—44no—
620C4—44no—
621C4—44no—
622C4—44no—
623C4—82no—
624C4—84no—
625C4—162no—
626C4C2×#38411yescharacteristic
627C4#231711yescharacteristic
628C4D4×C4⋊C411yescharacteristic
629C4D4×C4⋊C411yescharacteristic
630C4#244011yescharacteristic
631C4#244011yescharacteristic
632C4#250111yescharacteristic
633C4#250211yescharacteristic
634C2—48no—
635C2#1388011yescharacteristic
636C2#2087211yescharacteristic
637C2#2087911yescharacteristic
638C2#2267211yescharacteristic
639C2#2267311yescharacteristic
640C2#2268211yescharacteristic
641C2#2268311yescharacteristic
642C2#2576212yes—
643C2#2584312yes—
644C2#2604912yes—
645C2#2605112yes—
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×#1846C2semidirect
7. #13954C2semidirect
8. #11328C2semidirect
9. #11327C2semidirect
10. #10850C2semidirect
11. #10834C2semidirect
37. C4×#102C4semidirect
100. C2×C4×C8C2×C4semidirect
116. C4×C4⋊C4C2×C4semidirect
131. #340C2×C4semidirect
132. #340C2×C4semidirect
200. C4×D4C4⋊C4semidirect
267. #102C4⋊C4semidirect
268. #102C4⋊C4semidirect

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×#18462
C2×#18042
C2×C4×#1022
#139542
#113282
#113272
#108502
#108342