Finite Group Databasebeta

← #356267

Subgroups — #356267

Order 512.

1900 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2352C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×C4×#99C211yesmaximal, characteristic
4C22⋊C4×C4⋊C4C212yesmaximal, complemented
5#10960C214yesmaximal
6#10854C212yesmaximal, complemented
7#10801C212yesmaximal, complemented
8C4×#342C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10C22×#117C411yescharacteristic, complemented
11C23×C4⋊C4C2211yescharacteristic
12C22×#99C2212yes—
13C22×#99C2212yes—
14#2352—22no—
15#2352—22no—
16#2352—24no—
17C2×#388C412yescomplemented
18C2×#387C412yescomplemented
19C2×#386C2212yes—
20C2×#382C411yescharacteristic, complemented
21C2×#380C411yescharacteristic, complemented
22C2×#380C2212yes—
23C2×#380C2211yescharacteristic
24C2×C4×C4⋊C4—22no—
25C2×C4×C4⋊C4C411yescharacteristic, complemented
26C2×C4×C22⋊C4C2214yes—
27C2×C4×C22⋊C4C2212yes—
28C2×C4×C22⋊C4C2212yes—
29#1799C418yescomplemented
30C4×#99C418yescomplemented
31C2×#342C2214yes—
32C2×#342C2214yes—
33C2×#342C2211yescharacteristic
34C2×#342C2212yes—
35C2×#342C2211yescharacteristic
36#1436—24no—
37C4⋊#95—22no—
38C22⋊#95C2214yes—
39C23×D4C2311yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2312yes—
42C24×C4C2311yescharacteristic
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—24no—
46C2×C4×D4C2×C414yes—
47C2×C4×D4C2×C414yes—
48C2×#117—21no—
49C2×#117—22no—
50C2×#117—21no—
51C2×#117—44no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—44no—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4D414yes—
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2312yes—
67C22×C22⋊C4C2312yes—
68C22×C22⋊C4C2314yes—
69C22×C22⋊C4C2311yescharacteristic
70C22×C22⋊C4C2312yes—
71C22×C42—22no—
72C22×C42—24no—
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C412yes—
75C22×C42C2×C412yes—
76C2×#99—24no—
77C2×#99—24no—
78C2×#99—42no—
79C2×#99—42no—
80C2×#99C2×C418yes—
81C2×#99C2×C418yes—
82#388—24no—
83#388—24no—
84#387—24no—
85#387—24no—
86#386—24no—
87#386—24no—
88#382—22no—
89#382—22no—
90#380—22no—
91#380—22no—
92#380—24no—
93#380—22no—
94#380—22no—
95#380C2×C418yes—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—48no—
100C4×C22⋊C4—24no—
101C4×C22⋊C4—24no—
102C4×C22⋊C4—24no—
103C4×C22⋊C4—28no—
104C4×C22⋊C4—44no—
105C4×C22⋊C4C2×C418yes—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—24no—
109C2×#95—24no—
110C2×#95—24no—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116#342—28no—
117#342—44no—
118#342C2×C418yes—
119#342C2×C414yes—
120#342C2×C414yes—
121C25C22×C411yescharacteristic
122C25C2411yescharacteristic
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4C22×C412yes—
132C22×D4C22×C412yes—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—44no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—82no—
148C23×C4C4214yescomplemented
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C412yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C412yes—
153C23×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C412yes—
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C412yes—
159C23×C4C22×C412yes—
160C23×C4C2×D414yes—
161C23×C4C2×D412yes—
162C23×C4C2×D414yes—
163C23×C4C2×D412yes—
164C23×C4#4114yes—
165C4×D4—28no—
166C4×D4—28no—
167C4×D4—28no—
168C4×D4—28no—
169#117—44no—
170#117—48no—
171#117—44no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4C4214yescomplemented
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C414yes—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C414yes—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4C4214yescomplemented
224C2×C22⋊C4C22×C414yes—
225C2×C22⋊C4C22×C418yes—
226C2×C22⋊C4C22×C412yes—
227C2×C22⋊C4C22×C412yes—
228C2×C22⋊C4C22×C414yes—
229C2×C22⋊C4#4114yes—
230C2×C22⋊C4#4114yes—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—28no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—44no—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42—48no—
243C2×C42—48no—
244C2×C42—84no—
245C2×C42—82no—
246C2×C42C4214yescomplemented
247#99—216no—
248#99—48no—
249#95—24no—
250#95—24no—
251#95—22no—
252#95—22no—
253#95—24no—
254#95—24no—
255#95—28no—
256#95—28no—
257#95—48no—
258#95—48no—
259#95—84no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24C2×C4212yes—
275C24C23×C411yescharacteristic
276C24C23×C411yescharacteristic
277C24C23×C411yescharacteristic
278C24C22×D412yes—
279C24C2×#4112yes—
280C24C2×#4112yes—
281C24C2×#4112yes—
282C24H(5,𝔽2)11yescharacteristic
283C24H(5,𝔽2)11yescharacteristic
284C242−1+411yescharacteristic
285C242−1+411yescharacteristic
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—28no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—48no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—84no—
383C22×C4—88no—
384C22×C4—84no—
385C22×C4C2×C4212yes—
386C22×C4C2×C4212yes—
387C22×C4C2×C4212yes—
388C22×C4C2×C4212yes—
389C22×C4C2×C4212yes—
390C22×C4C2×C22⋊C414yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C412yes—
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4#11712yes—
397C22×C4#11712yes—
398C22×C4C4×D414yes—
399C22×C4C4×D414yes—
400C22×C4C4×D414yes—
401C22×C4C4×D414yes—
402C4⋊C4—28no—
403C4⋊C4—28no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—416no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4#9914yescomplemented
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—28no—
421C22⋊C4—24no—
422C22⋊C4—28no—
423C22⋊C4—28no—
424C22⋊C4—24no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—42no—
432C22⋊C4—44no—
433C22⋊C4—84no—
434C22⋊C4#11718yescomplemented
435C42—24no—
436C42—24no—
437C42—416no—
438C42—44no—
439C42—44no—
440C42—82no—
441C42—82no—
442C42—816no—
443C23—22no—
444C23—21no—
445C23—22no—
446C23—21no—
447C23—22no—
448C23—21no—
449C23—22no—
450C23—21no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—81no—
490C23—81no—
491C23—81no—
492C23C22×C4211yescharacteristic
493C23C22×C22⋊C411yescharacteristic
494C23C2×#11711yescharacteristic
495C23C2×#11712yes—
496C23C2×C4×D412yes—
497C23C2×C4×D412yes—
498C23C4×#4111yescharacteristic
499C23C4×#4111yescharacteristic
500C23C4×#4112yes—
501C23#51811yescharacteristic
502C23#51911yescharacteristic
503C23#52011yescharacteristic
504C23#52011yescharacteristic
505C23#52011yescharacteristic
506C23#52011yescharacteristic
507C23#54612yes—
508C23#54712yes—
509C23#54812yes—
510C23#55012yes—
511C23#55212yes—
512C23#55312yes—
513C23#55512yes—
514C23#55612yes—
515D4—44no—
516D4—44no—
517D4—44no—
518D4—44no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—48no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—48no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—84no—
586C2×C4—82no—
587C2×C4—84no—
588C2×C4—82no—
589C2×C4—84no—
590C2×C4—82no—
591C2×C4—84no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4C4×C22⋊C412yes—
598C2×C4C4×C22⋊C414yes—
599C2×C4C4×C22⋊C412yes—
600C2×C4C2×#9914yes—
601C2×C4C2×#9912yes—
602C2×C4#41012yes—
603C22—22no—
604C22—22no—
605C22—21no—
606C22—21no—
607C22—22no—
608C22—22no—
609C22—21no—
610C22—21no—
611C22—22no—
612C22—22no—
613C22—21no—
614C22—21no—
615C22—22no—
616C22—22no—
617C22—21no—
618C22—21no—
619C22—22no—
620C22—22no—
621C22—21no—
622C22—21no—
623C22—22no—
624C22—22no—
625C22—21no—
626C22—21no—
627C22—42no—
628C22—42no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22C2×C4×C22⋊C411yescharacteristic
649C22C4×#11711yescharacteristic
650C22#227411yescharacteristic
651C22#233911yescharacteristic
652C22#234111yescharacteristic
653C22#234611yescharacteristic
654C22#234711yescharacteristic
655C22#235212yes—
656C22#235412yes—
657C22#236212yes—
658C22#236312yes—
659C22C22×#9911yescharacteristic
660C22C2×#41011yescharacteristic
661C22#288311yescharacteristic
662C22#288411yescharacteristic
663C4—22no—
664C4—24no—
665C4—24no—
666C4—48no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—44no—
671C4—42no—
672C4—88no—
673C4—84no—
674C4—82no—
675C4—82no—
676C4C4×#9914yes—
677C2—22no—
678C2—22no—
679C2—22no—
680C2—21no—
681C2—21no—
682C2—21no—
683C2—21no—
684C2—22no—
685C2—82no—
686C2#1380111yescharacteristic
687C2C2×C4×#9911yescharacteristic
688C2#2039311yescharacteristic
689C2#2247611yescharacteristic
690C2#2247811yescharacteristic
691C2#2253411yescharacteristic
692C2#2253511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C23 ▷ 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
4. C22⋊C4×C4⋊C4C2semidirect
6. #10854C2semidirect
7. #10801C2semidirect
8. C4×#342C2semidirect
10. C22×#117C4semidirect
17. C2×#388C4semidirect
18. C2×#387C4semidirect
20. C2×#382C4semidirect
21. C2×#380C4semidirect
25. C2×C4×C4⋊C4C4semidirect
29. #1799C4semidirect
30. C4×#99C4semidirect
148. C23×C4C42semidirect
189. C2×C4⋊C4C42semidirect
223. C2×C22⋊C4C42semidirect
246. C2×C42C42semidirect
414. C4⋊C4#99semidirect
434. C22⋊C4#117semidirect

Maximal subgroups

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

subgroupindex
C2×#23522
C2×#17992
C2×C4×#992
C22⋊C4×C4⋊C42
#109602
#108542
#108012
C4×#3422