Finite Group Databasebeta

← #362082

Subgroups — #362082

Order 512.

2006 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2C2×#1799C212yesmaximal
3C2×#1799C211yesmaximal, characteristic
4C2×C4×#99C211yesmaximal, characteristic
5#13907C212yesmaximal, complemented
6#11203C214yesmaximal, complemented
7#10854C212yesmaximal, complemented
8#10761C212yesmaximal
9C22×C4×D4C2212yes—
10C22×C4×D4C2211yescharacteristic
11C23×C4⋊C4C411yescharacteristic, complemented
12C23×C4⋊C4C2211yescharacteristic
13C22×#99C2212yes—
14C22×#99C2211yescharacteristic
15C22×#99C2211yescharacteristic
16D4×C4⋊C4—22no—
17D4×C4⋊C4—22no—
18D4×C4⋊C4—24no—
19C2×#389—22no—
20C2×#389C412yescomplemented
21C2×#385C412yescomplemented
22C2×#385C2212yes—
23C2×#384C411yescharacteristic, complemented
24C2×#380C411yescharacteristic, complemented
25C2×#380C2212yes—
26C2×#380C2214yes—
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4C411yescharacteristic, complemented
29C2×C4×C22⋊C4C2212yes—
30#1799—28no—
31#1799—24no—
32C4×#99—24no—
33C2×#342C2214yes—
34C2×#342C2212yes—
35C2×#342C2212yes—
36C2×#342C2214yes—
37#1441—24no—
38C4⋊#95—22no—
39C22⋊#95C2214yes—
40C23×D4C2311yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2×C412yes—
43C24×C4C2312yes—
44C24×C4C2311yescharacteristic
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—24no—
48C2×C4×D4—24no—
49C2×C4×D4—24no—
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—44no—
53C2×C4×D4D414yes—
54C2×C4×D4Q814yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—44no—
61C22×C4⋊C4—44no—
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C412yes—
71C22×C4⋊C4D412yes—
72C22×C4⋊C4D412yes—
73C22×C4⋊C4D414yes—
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2312yes—
79C22×C22⋊C4C2312yes—
80C22×C22⋊C4C2311yescharacteristic
81C22×C42—22no—
82C22×C42C2×C412yes—
83C22×C42C2×C411yescharacteristic
84C2×#99—22no—
85C2×#99—22no—
86C2×#99—22no—
87C2×#99—22no—
88C2×#99—24no—
89C2×#99—24no—
90C2×#99—24no—
91C2×#99D414yes—
92C2×#99Q814yes—
93#389—24no—
94#389—24no—
95#389—48no—
96#385—24no—
97#385—24no—
98#385—24no—
99#385—24no—
100#384—22no—
101#384—22no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380—22no—
106#380—24no—
107#380—24no—
108#380—28no—
109#380—44no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—21no—
113C4×C22⋊C4—24no—
114C4×C22⋊C4—24no—
115C2×#95—24no—
116C2×#95—24no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—24no—
120C2×#95C2×C412yes—
121C2×#95C2×C412yes—
122C2×#95C2×C412yes—
123#342—28no—
124#342—24no—
125#342—24no—
126#342—44no—
127C25C22×C411yescharacteristic
128C25C2411yescharacteristic
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4C2×D412yes—
134C22×D4C2×D412yes—
135C22×D4C2×Q812yes—
136C22×D4C2×Q812yes—
137C22×D4#4112yes—
138C22×D4#4112yes—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—44no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—44no—
158C23×C4—82no—
159C23×C4—82no—
160C23×C4C4⋊C414yescomplemented
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C412yes—
165C23×C4C22×C412yes—
166C23×C4C22×C412yes—
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C2×D414yes—
171C23×C4C2×D412yes—
172C23×C4C2×D414yes—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×Q812yes—
176C23×C4#4114yes—
177C4×D4—28no—
178C4×D4—28no—
179C4×D4—28no—
180C4×D4—28no—
181C4×D4—48no—
182C4×D4—48no—
183C4×D4—48no—
184C4×D4—48no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—48no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—84no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C414yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C414yes—
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C4⋊C414yescomplemented
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4C4⋊C414yescomplemented
252C2×C22⋊C4C2×D414yes—
253C2×C22⋊C4C2×D412yes—
254C2×C22⋊C4C2×Q814yes—
255C2×C22⋊C4C2×Q812yes—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—42no—
263C2×C42—44no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42C4⋊C414yescomplemented
267#99—28no—
268#99—28no—
269#99—48no—
270#95—22no—
271#95—22no—
272#95—24no—
273#95—24no—
274#95—48no—
275#95—48no—
276#95—44no—
277#95—44no—
278#95—84no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24—41no—
291C24C2×C4⋊C412yes—
292C24C4×D412yes—
293C24C4×Q812yes—
294C24C23×C411yescharacteristic
295C24C22×D412yes—
296C24C22×D411yescharacteristic
297C24C22×D411yescharacteristic
298C24C22×Q811yescharacteristic
299C24C22×Q811yescharacteristic
300C24C2×#4112yes—
301C24C2×#4111yescharacteristic
302C24C2×#4111yescharacteristic
303C24H(5,𝔽2)12yes—
304C242−1+412yes—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4#12114yes—
318C2×D4C22⋊Q814yes—
319C2×D4#12514yes—
320C2×D4C4⋊Q814yes—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—28no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—48no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4—84no—
406C22×C4—88no—
407C22×C4—84no—
408C22×C4—88no—
409C22×C4—88no—
410C22×C4C2×C22⋊C412yes—
411C22×C4C2×C22⋊C414yes—
412C22×C4C2×C22⋊C412yes—
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C412yes—
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C4⋊C412yes—
417C22×C4C2×C4⋊C412yes—
418C22×C4C2×C4⋊C412yes—
419C22×C4C2×C4⋊C412yes—
420C22×C4C2×C4⋊C412yes—
421C22×C4#11712yes—
422C22×C4#11712yes—
423C22×C4C4×D412yes—
424C22×C4C4×Q812yes—
425C22×C4#12114yes—
426C22×C4#12114yes—
427C22×C4C22⋊Q814yes—
428C22×C4C22⋊Q814yes—
429C4⋊C4—28no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—28no—
433C4⋊C4—24no—
434C4⋊C4—42no—
435C4⋊C4—44no—
436C4⋊C4—48no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—84no—
444C4⋊C4—88no—
445C4⋊C4—816no—
446C4⋊C4#9914yescomplemented
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—28no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—48no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C42—24no—
466C42—24no—
467C42—44no—
468C42—44no—
469C42—88no—
470C23—22no—
471C23—21no—
472C23—22no—
473C23—21no—
474C23—22no—
475C23—21no—
476C23—22no—
477C23—21no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23#38914yescomplemented
519C23C22×C22⋊C411yescharacteristic
520C23C22×C4⋊C411yescharacteristic
521C23C2×#11711yescharacteristic
522C23C2×C4×D411yescharacteristic
523C23C2×C4×Q811yescharacteristic
524C23#51811yescharacteristic
525C23#51911yescharacteristic
526C23C2×#12112yes—
527C23C2×#12111yescharacteristic
528C23C2×C22⋊Q812yes—
529C23C2×C22⋊Q811yescharacteristic
530C23C2×#12511yescharacteristic
531C23C2×C4⋊Q811yescharacteristic
532C23#53712yes—
533C23#54412yes—
534C23D4212yes—
535C23#54612yes—
536C23#54712yes—
537C23#54812yes—
538C23D4×Q812yes—
539C23#55212yes—
540C23#55312yes—
541C23#55412yes—
542D4—28no—
543D4—28no—
544D4—28no—
545D4—28no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—88no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—88no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4#38012yes—
626C2×C4#38014yes—
627C2×C4#38012yes—
628C2×C4#38914yescomplemented
629C2×C4C2×#9914yes—
630C2×C4C2×#9912yes—
631C2×C4#41012yes—
632C22—22no—
633C22—21no—
634C22—21no—
635C22—22no—
636C22—22no—
637C22—21no—
638C22—21no—
639C22—22no—
640C22—21no—
641C22—21no—
642C22—22no—
643C22—22no—
644C22—21no—
645C22—21no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—21no—
651C22—21no—
652C22—22no—
653C22—22no—
654C22—21no—
655C22—21no—
656C22—24no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22C2×#38011yescharacteristic
678C22C2×#38911yescharacteristic
679C22#231611yescharacteristic
680C22D4×C22⋊C411yescharacteristic
681C22Q8×C22⋊C411yescharacteristic
682C22#234411yescharacteristic
683C22#234511yescharacteristic
684C22#241612yes—
685C22#241712yes—
686C22#244012yes—
687C22#244212yes—
688C22C22×#9911yescharacteristic
689C22C2×#41011yescharacteristic
690C22#288311yescharacteristic
691C22#288411yescharacteristic
692C4—24no—
693C4—24no—
694C4—22no—
695C4—24no—
696C4—44no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—84no—
701C4—84no—
702C4—82no—
703C4—82no—
704C4—164no—
705C4#179914yescomplemented
706C2—22no—
707C2—22no—
708C2—21no—
709C2—21no—
710C2—21no—
711C2—21no—
712C2—22no—
713C2—22no—
714C2—44no—
715C2#1390711yescharacteristic
716C2C2×#179911yescharacteristic
717C2#2063411yescharacteristic
718C2D4×#9911yescharacteristic
719C2#2240611yescharacteristic
720C2Q8×#9911yescharacteristic
721C2#2251311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #13907C2semidirect
6. #11203C2semidirect
7. #10854C2semidirect
11. C23×C4⋊C4C4semidirect
20. C2×#389C4semidirect
21. C2×#385C4semidirect
23. C2×#384C4semidirect
24. C2×#380C4semidirect
28. C2×C4×C4⋊C4C4semidirect
160. C23×C4C4⋊C4semidirect
223. C2×C4⋊C4C4⋊C4semidirect
251. C2×C22⋊C4C4⋊C4semidirect
266. C2×C42C4⋊C4semidirect
446. C4⋊C4#99semidirect
518. C23#389semidirect
628. C2×C4#389semidirect
705. C4#1799semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
C2×#17992
C2×C4×#992
#139072
#112032
#108542
#107612