Finite Group Databasebeta

← #355930

Subgroups — #355930

Order 512.

3150 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#407C211yesmaximal, characteristic
2C2×#2280C211yesmaximal, characteristic
3C2×#2280C211yesmaximal, characteristic
4#10733C214yesmaximal, complemented
5#10732C214yesmaximal, complemented
6#10720C214yesmaximal, complemented
7C25×C4C2211yescharacteristic
8C23×M16C2212yes—
9C22×#117C2212yes—
10C22×#117C2212yes—
11C2×#407—21no—
12C2×#407—24no—
13C2×#407—21no—
14C2×#407C418yescomplemented
15C2×#407C2212yes—
16C2×#407C2212yes—
17C22×C22⋊C8—21no—
18C22×C22⋊C8—21no—
19#2280—21no—
20#2280—21no—
21#2280—21no—
22#2280—21no—
23#2280—24no—
24#2280C412yescomplemented
25#2280C412yescomplemented
26#2280C418yescomplemented
27#2280C2212yes—
28#2280C2212yes—
29C2×#381—21no—
30C2×#381—21no—
31C2×#380—21no—
32C2×#380—21no—
33C2×#380C412yescomplemented
34C2×#380C412yescomplemented
35C2×#379—21no—
36C2×#379—21no—
37C2×C4×C22⋊C4—21no—
38C2×C4×C22⋊C4—21no—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C2×#339C22116yescomplemented
42C26D411yescharacteristic
43C24×C4—48no—
44C24×C4C2×C414yes—
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C414yes—
50C24×C4D411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C24×C4D411yescharacteristic
56C24×C4D414yes—
57C24×C4D411yescharacteristic
58C24×C4Q814yes—
59C24×C4C2311yescharacteristic
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C22×M16—24no—
63C22×M16—22no—
64C22×M16—48no—
65C22×M16C2×C418yescomplemented
66C22×M16C2314yes—
67C23×C8—22no—
68C2×#117—22no—
69C2×#117—24no—
70C2×#117—22no—
71C2×#117—48no—
72C2×#117—48no—
73C2×#117C2×C414yes—
74C2×#117C2×C418yescomplemented
75C2×#117C2314yes—
76C22×C4⋊C4—22no—
77C22×C4⋊C4C2×C414yes—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C42—22no—
85C22×C42C2314yes—
86#407—28no—
87#407—28no—
88#407—24no—
89#407—24no—
90#407C2×C418yes—
91#407C2×C418yescomplemented
92C2×C22⋊C8—28no—
93C2×C22⋊C8—28no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98#381—22no—
99#381—22no—
100#381—28no—
101#381—22no—
102#381—22no—
103#380—28no—
104#380—28no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380C2×C414yes—
110#380C2×C414yes—
111#380C2×C414yes—
112#380C2×C414yes—
113#379—22no—
114#379—22no—
115#379—28no—
116#379—22no—
117#379—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—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—24no—
127C4×C22⋊C4—24no—
128C4×C22⋊C4C2×C418yes—
129C4×C22⋊C4C2×C418yescomplemented
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134#339—416no—
135C25—44no—
136C25—44no—
137C25—44no—
138C25C22⋊C411yescharacteristic
139C25C22⋊C411yescharacteristic
140C25C22⋊C411yescharacteristic
141C25C22⋊C414yes—
142C25C22⋊C411yescharacteristic
143C25C4⋊C414yes—
144C25C2×D411yescharacteristic
145C25C2×D411yescharacteristic
146C25C2×D411yescharacteristic
147C23×C4—28no—
148C23×C4—28no—
149C23×C4—28no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—416no—
157C23×C4—416no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—416no—
163C23×C4—416no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4C4214yes—
168C23×C4C4214yes—
169C23×C4C4214yes—
170C23×C4C4214yes—
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
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⋊C414yes—
186C23×C4C22⋊C414yes—
187C23×C4C22⋊C414yes—
188C23×C4C22⋊C414yes—
189C23×C4C22⋊C414yes—
190C23×C4C4⋊C414yes—
191C23×C4C4⋊C414yes—
192C23×C4C4⋊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×C4C22×C412yes—
202C23×C4C22×C412yes—
203C23×C4C22×C412yes—
204C23×C4C22×C411yescharacteristic
205C23×C4C22×C411yescharacteristic
206C23×C4C22×C412yes—
207C23×C4C22×C412yes—
208C23×C4C22×C412yes—
209C23×C4C22×C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C22×C411yescharacteristic
212C23×C4C22×C411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D412yes—
217C23×C4C2×D412yes—
218C23×C4C2×D412yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×Q812yes—
226C23×C4C2×Q812yes—
227C23×C4C2×Q812yes—
228C23×C4C2411yescharacteristic
229C2×M16—28no—
230C2×M16—24no—
231C2×M16—216no—
232C2×M16—416no—
233C2×M16—416no—
234C2×M16—48no—
235C2×M16C22×C418yes—
236C22×C8—28no—
237C22×C8—24no—
238C22×C8—48no—
239#117—216no—
240#117—416no—
241#117—48no—
242#117—48no—
243#117—416no—
244#117—48no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—48no—
249C2×C4⋊C4C22×C418yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4—48no—
265C2×C42—216no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—48no—
270C2×C42C22×C418yes—
271C22⋊C8—28no—
272C22⋊C8—28no—
273C22⋊C8—28no—
274C22⋊C8—28no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—28no—
280#95—28no—
281C24—24no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—48no—
294C24—42no—
295C24—48no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24—48no—
301C24—48no—
302C24—48no—
303C24—48no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—464no—
309C24#9514yes—
310C24#9514yes—
311C24#9514yes—
312C24#9514yes—
313C24C2×C22⋊C411yescharacteristic
314C24C2×C22⋊C411yescharacteristic
315C24C2×C22⋊C411yescharacteristic
316C24C2×C22⋊C411yescharacteristic
317C24C2×C22⋊C412yes—
318C24C2×C22⋊C412yes—
319C24C2×C22⋊C411yescharacteristic
320C24C2×C22⋊C411yescharacteristic
321C24C2×C22⋊C412yes—
322C24C2×C4⋊C412yes—
323C24C2×C4⋊C412yes—
324C24C2×C4⋊C412yes—
325C24C22×D411yescharacteristic
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—28no—
344C22×C4—24no—
345C22×C4—28no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
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—48no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—48no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4#9514yes—
396C22×C4#9514yes—
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×C4C2×C4212yes—
408C22×C4C2×C4212yes—
409C22×C4C2×C4212yes—
410C22×C4C2×C4212yes—
411C22×C4C2×C4212yes—
412C22×C4C2×C4212yes—
413C22×C4C2×C22⋊C412yes—
414C22×C4C2×C22⋊C412yes—
415C22×C4C2×C22⋊C412yes—
416C22×C4C2×C22⋊C412yes—
417C22×C4C2×C22⋊C412yes—
418C22×C4C2×C22⋊C412yes—
419C22×C4C2×C22⋊C412yes—
420C22×C4C2×C22⋊C412yes—
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C412yes—
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C4⋊C412yes—
429C22×C4C2×C4⋊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×C4C23×C411yescharacteristic
444C22×C4C23×C411yescharacteristic
445C22×C4C23×C411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×Q811yescharacteristic
449C22×C4H(5,𝔽2)18yes—
450M16—216no—
451M16—416no—
452M16—48no—
453C2×C8—28no—
454C2×C8—416no—
455C2×C8—48no—
456C4⋊C4—216no—
457C4⋊C4—416no—
458C4⋊C4—48no—
459C22⋊C4—416no—
460C22⋊C4—416no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—416no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C42—216no—
469C42—48no—
470C42—48no—
471C42—48no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—24no—
503C23—24no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—416no—
521C23—416no—
522C23—416no—
523C23—416no—
524C23—416no—
525C23—416no—
526C23—464no—
527C23—464no—
528C23—416no—
529C23C2×#9512yes—
530C23C2×#9512yes—
531C23C2×#9512yes—
532C23C2×#9512yes—
533C23C2×#9512yes—
534C23C2×#9512yes—
535C23#42114yes—
536C23#42114yes—
537C23C22×C22⋊C411yescharacteristic
538C23C22×C22⋊C411yescharacteristic
539C23C22×C4⋊C411yescharacteristic
540C23#53414yes—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
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—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—816no—
596C2×C4—816no—
597C2×C4C2×#9512yes—
598C2×C4C2×#9512yes—
599C2×C4C2×#9512yes—
600C2×C4C2×#9512yes—
601C2×C4C2×#9512yes—
602C2×C4C2×#9512yes—
603C2×C4C22×C4211yescharacteristic
604C2×C4C22×C22⋊C411yescharacteristic
605C2×C4C22×C4⋊C411yescharacteristic
606C2×C4C22×C4⋊C411yescharacteristic
607C2×C4#51814yes—
608C2×C4#51814yes—
609C2×C4#51814yes—
610C2×C4#53414yes—
611C2×C4#53414yes—
612C2×C4#54314yes—
613C8—48no—
614C22—28no—
615C22—28no—
616C22—28no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—48no—
638C22—48no—
639C22—48no—
640C22—48no—
641C22—416no—
642C22—48no—
643C22—48no—
644C22—416no—
645C22—416no—
646C22—416no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22#173314yes—
652C22#173314yes—
653C22C22×#9511yescharacteristic
654C22#230812yes—
655C22#230812yes—
656C22#231512yes—
657C22C2×#42112yes—
658C22C2×#42112yes—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—88no—
666C4—88no—
667C4#227312yes—
668C4#230812yes—
669C4#231512yes—
670C4#231512yes—
671C2—22no—
672C2—22no—
673C2—24no—
674C2—24no—
675C2—42no—
676C2—42no—
677C2—42no—
678C2—42no—
679C2#1379411yescharacteristic
680C2C2×#173311yescharacteristic
681C2C2×#173311yescharacteristic
682C2#2177414yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC26D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10733C2semidirect
5. #10732C2semidirect
6. #10720C2semidirect
14. C2×#407C4semidirect
24. #2280C4semidirect
25. #2280C4semidirect
26. #2280C4semidirect
33. C2×#380C4semidirect
34. C2×#380C4semidirect
41. C2×#339C22semidirect
65. C22×M16C2×C4semidirect
74. C2×#117C2×C4semidirect
91. #407C2×C4semidirect
129. C4×C22⋊C4C2×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
#107332
#107322
#107202