Finite Group Databasebeta

← #181740

Subgroups — #181740

Order 512.

3194 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#407C211yesmaximal, characteristic
2C2×#2280C211yesmaximal, characteristic
3C22×#381C211yesmaximal, characteristic
4#8024C214yesmaximal, complemented
5#8020C214yesmaximal, complemented
6#8015C214yesmaximal, complemented
7C25×C4C2211yescharacteristic
8C23×M16—21no—
9C22×#117C412yescomplemented
10C23×C22⋊C4—21no—
11C2×#407—24no—
12C2×#407—21no—
13C2×#407—21no—
14C2×#407C418yescomplemented
15C2×#407C2212yes—
16C2×#407C2212yes—
17C22×C22⋊C8—21no—
18C22×C22⋊C8C2212yes—
19#2280—21no—
20#2280—21no—
21#2280—21no—
22#2280—21no—
23#2280C412yescomplemented
24#2280C412yescomplemented
25#2280C2212yes—
26#2280C2212yes—
27C2×#381—24no—
28C2×#381—21no—
29C2×#381—21no—
30C2×#381—21no—
31C2×#381C418yescomplemented
32C2×#381C2212yes—
33C2×#381C2212yes—
34C2×#380C412yescomplemented
35C2×#380C2212yes—
36C2×#379—21no—
37C2×C4×C22⋊C4—21no—
38C2×C4×C22⋊C4—21no—
39C22×#95—21no—
40C22×#95C2212yes—
41#1294C22116yescomplemented
42C26D411yescharacteristic
43C24×C4—22no—
44C24×C4—48no—
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C414yes—
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C414yes—
49C24×C4C2×C411yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C24×C4D411yescharacteristic
56C24×C4D411yescharacteristic
57C24×C4D414yes—
58C24×C4D411yescharacteristic
59C24×C4Q814yes—
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C22×M16—28no—
64C22×M16—22no—
65C22×M16—22no—
66C22×M16—48no—
67C23×C8—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—48no—
71C2×#117C2×C414yes—
72C2×#117C2×C414yescomplemented
73C22×C4⋊C4C2×C414yescomplemented
74C22×C22⋊C4—28no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—48no—
81C22×C42—22no—
82#407—28no—
83#407—28no—
84#407—24no—
85#407—24no—
86#407C2×C418yescomplemented
87#407C2×C418yescomplemented
88C2×C22⋊C8—28no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8C2×C418yescomplemented
94C2×C22⋊C8C2314yes—
95#381—28no—
96#381—28no—
97#381—22no—
98#381—22no—
99#381—24no—
100#381—24no—
101#381—22no—
102#381—22no—
103#381C2×C418yes—
104#381C2×C418yescomplemented
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380C2×C414yes—
110#380C2×C414yescomplemented
111#380C2×C414yescomplemented
112#380C2314yes—
113#379—22no—
114#379—22no—
115#379—22no—
116#379—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C2×#95—28no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—24no—
131C2×#95—22no—
132C2×#95C2×C418yescomplemented
133C2×#95C2314yes—
134C25—28no—
135C25—44no—
136C25—44no—
137C25C22⋊C411yescharacteristic
138C25C22⋊C411yescharacteristic
139C25C22⋊C414yes—
140C25C22⋊C411yescharacteristic
141C25C22⋊C411yescharacteristic
142C25C4⋊C414yes—
143C25C2×D411yescharacteristic
144C25C2×D411yescharacteristic
145C25C2×D411yescharacteristic
146C23×C4—28no—
147C23×C4—28no—
148C23×C4—24no—
149C23×C4—28no—
150C23×C4—24no—
151C23×C4—416no—
152C23×C4—416no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—416no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—416no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—48no—
168C23×C4—48no—
169C23×C4—48no—
170C23×C4C4214yes—
171C23×C4C4214yes—
172C23×C4C4214yes—
173C23×C4C4214yes—
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C414yes—
181C23×C4C22⋊C414yes—
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C414yes—
189C23×C4C22⋊C414yes—
190C23×C4C22⋊C414yes—
191C23×C4C22⋊C414yes—
192C23×C4C22⋊C414yes—
193C23×C4C4⋊C414yes—
194C23×C4C4⋊C414yes—
195C23×C4C4⋊C414yes—
196C23×C4C4⋊C414yes—
197C23×C4C4⋊C414yes—
198C23×C4C4⋊C414yes—
199C23×C4C4⋊C414yes—
200C23×C4C4⋊C414yes—
201C23×C4C4⋊C414yes—
202C23×C4C4⋊C414yes—
203C23×C4C4⋊C414yes—
204C23×C4C22×C411yescharacteristic
205C23×C4C22×C411yescharacteristic
206C23×C4C22×C411yescharacteristic
207C23×C4C22×C412yes—
208C23×C4C22×C412yes—
209C23×C4C22×C412yes—
210C23×C4C22×C412yes—
211C23×C4C22×C411yescharacteristic
212C23×C4C22×C411yescharacteristic
213C23×C4C22×C411yescharacteristic
214C23×C4C22×C412yes—
215C23×C4C22×C412yes—
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D412yes—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D412yes—
227C23×C4C2×D412yes—
228C23×C4C2×Q812yes—
229C23×C4C2×Q812yes—
230C23×C4C2×Q812yes—
231C23×C4C2411yescharacteristic
232C2×M16—28no—
233C2×M16—28no—
234C2×M16—432no—
235C2×M16—48no—
236C2×M16—48no—
237C22×C8—28no—
238C22×C8—24no—
239C22×C8—48no—
240#117—48no—
241#117—48no—
242#117—48no—
243#117—48no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4C22×C418yes—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
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—48no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—432no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—48no—
264C22⋊C8—28no—
265C22⋊C8—28no—
266C22⋊C8—28no—
267C22⋊C8—24no—
268C22⋊C8C22×C418yes—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95C22×C418yes—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—216no—
284C24—216no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—42no—
293C24—42no—
294C24—48no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—48no—
300C24—48no—
301C24—48no—
302C24—42no—
303C24—42no—
304C24—464no—
305C24#9514yes—
306C24#9514yes—
307C24#9514yes—
308C24#9514yes—
309C24#9918yes—
310C24C2×C22⋊C412yes—
311C24C2×C22⋊C411yescharacteristic
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C412yes—
314C24C2×C22⋊C412yes—
315C24C2×C22⋊C411yescharacteristic
316C24C2×C22⋊C411yescharacteristic
317C24C2×C22⋊C411yescharacteristic
318C24C2×C22⋊C411yescharacteristic
319C24C2×C4⋊C412yes—
320C24C2×C4⋊C412yes—
321C24C2×C4⋊C412yes—
322C24C22×D411yescharacteristic
323C22×C4—216no—
324C22×C4—216no—
325C22×C4—216no—
326C22×C4—216no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—48no—
388C22×C4—416no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—416no—
393C22×C4—416no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—832no—
397C22×C4#9514yes—
398C22×C4#9514yes—
399C22×C4#9514yes—
400C22×C4#9514yes—
401C22×C4#9514yes—
402C22×C4#9514yes—
403C22×C4#9514yes—
404C22×C4#9514yes—
405C22×C4#9514yes—
406C22×C4#9514yes—
407C22×C4#9514yes—
408C22×C4#9514yes—
409C22×C4C2×C4212yes—
410C22×C4C2×C4212yes—
411C22×C4C2×C4212yes—
412C22×C4C2×C4212yes—
413C22×C4C2×C4212yes—
414C22×C4C2×C4212yes—
415C22×C4C2×C22⋊C412yes—
416C22×C4C2×C22⋊C412yes—
417C22×C4C2×C22⋊C412yes—
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C412yes—
425C22×C4C2×C22⋊C412yes—
426C22×C4C2×C22⋊C412yes—
427C22×C4C2×C22⋊C412yes—
428C22×C4C2×C22⋊C412yes—
429C22×C4C2×C22⋊C412yes—
430C22×C4C2×C4⋊C412yes—
431C22×C4C2×C4⋊C412yes—
432C22×C4C2×C4⋊C412yes—
433C22×C4C2×C4⋊C412yes—
434C22×C4C2×C4⋊C412yes—
435C22×C4C2×C4⋊C412yes—
436C22×C4C2×C4⋊C412yes—
437C22×C4C2×C4⋊C412yes—
438C22×C4C2×C4⋊C412yes—
439C22×C4C2×C4⋊C412yes—
440C22×C4C2×C4⋊C412yes—
441C22×C4C2×C4⋊C412yes—
442C22×C4C2×C4⋊C412yes—
443C22×C4C2×C4⋊C412yes—
444C22×C4C2×C4⋊C412yes—
445C22×C4C23×C411yescharacteristic
446C22×C4C23×C411yescharacteristic
447C22×C4C23×C411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×Q811yescharacteristic
451M16—416no—
452M16—416no—
453C2×C8—28no—
454C2×C8—48no—
455C2×C8—416no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—416no—
461C22⋊C4—416no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C42—48no—
467C42—48no—
468C23—232no—
469C23—22no—
470C23—22no—
471C23—24no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—28no—
496C23—28no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—24no—
503C23—24no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—416no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—416no—
516C23—416no—
517C23—416no—
518C23—416no—
519C23—464no—
520C23—464no—
521C23—416no—
522C23#342116yes—
523C23C2×#9512yes—
524C23C2×#9512yes—
525C23C2×#9512yes—
526C23C2×#9512yes—
527C23C2×#9512yes—
528C23C2×#9512yes—
529C23C2×#9914yes—
530C23C2×#9914yes—
531C23C2×#9914yes—
532C23#42114yes—
533C23C22×C22⋊C411yescharacteristic
534C23C22×C22⋊C411yescharacteristic
535C23C22×C4⋊C411yescharacteristic
536C23#57514yes—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—816no—
598C2×C4—832no—
599C2×C4—816no—
600C2×C4C2×#9512yes—
601C2×C4C2×#9512yes—
602C2×C4C2×#9512yes—
603C2×C4C2×#9512yes—
604C2×C4C2×#9512yes—
605C2×C4C2×#9512yes—
606C2×C4C22×C4211yescharacteristic
607C2×C4C22×C22⋊C411yescharacteristic
608C2×C4C22×C4⋊C411yescharacteristic
609C2×C4C22×C4⋊C411yescharacteristic
610C8—48no—
611C22—216no—
612C22—216no—
613C22—21no—
614C22—21no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—21no—
623C22—21no—
624C22—22no—
625C22—28no—
626C22—28no—
627C22—28no—
628C22—28no—
629C22—28no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—416no—
637C22—48no—
638C22—48no—
639C22—48no—
640C22—48no—
641C22—416no—
642C22—416no—
643C22—416no—
644C22—44no—
645C22—44no—
646C22#173314yes—
647C22C2×#34214yes—
648C22C2×#34214yes—
649C22C2×#34214yes—
650C22C22×#9511yescharacteristic
651C22C22×#9912yes—
652C22#289612yes—
653C22#289612yes—
654C22C2×#42112yes—
655C22#291112yes—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—21no—
666C4—88no—
667C4—88no—
668C2—22no—
669C2—22no—
670C2—24no—
671C2—24no—
672C2—24no—
673C2—24no—
674C2—42no—
675C2—42no—
676C2#1758714yes—
677C2C2×#173311yescharacteristic
678C2#2034711yescharacteristic
679C2C22×#34211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8024C2semidirect
5. #8020C2semidirect
6. #8015C2semidirect
9. C22×#117C4semidirect
14. C2×#407C4semidirect
23. #2280C4semidirect
24. #2280C4semidirect
31. C2×#381C4semidirect
34. C2×#380C4semidirect
41. #1294C22semidirect
72. C2×#117C2×C4semidirect
73. C22×C4⋊C4C2×C4semidirect
86. #407C2×C4semidirect
87. #407C2×C4semidirect
93. C2×C22⋊C8C2×C4semidirect
104. #381C2×C4semidirect
110. #380C2×C4semidirect
111. #380C2×C4semidirect
132. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4072
C2×#22802
C22×#3812
#80242
#80202
#80152