Finite Group Databasebeta

← #181500

Subgroups — #181500

Order 512.

3562 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#407C211yesmaximal, characteristic
2C2×#2280C211yesmaximal, characteristic
3C22×C4×C22⋊C4C211yesmaximal, characteristic
4#8020C214yesmaximal, complemented
5#8019C214yesmaximal, complemented
6#8012C214yesmaximal, complemented
7C25×C4C2211yescharacteristic
8C23×M16—21no—
9C22×#117C412yescomplemented
10C23×C22⋊C4—21no—
11C23×C42—21no—
12C2×#407—24no—
13C2×#407—21no—
14C2×#407—21no—
15C2×#407C418yescomplemented
16C2×#407C2212yes—
17C2×#407C2212yes—
18C22×C22⋊C8—21no—
19C22×C22⋊C8C2212yes—
20#2280—21no—
21#2280—21no—
22#2280—21no—
23#2280—21no—
24#2280C412yescomplemented
25#2280C412yescomplemented
26#2280C2212yes—
27#2280C2212yes—
28C2×#381—21no—
29C2×#380C412yescomplemented
30C2×#380C2212yes—
31C2×#379—21no—
32C2×C4×C22⋊C4—24no—
33C2×C4×C22⋊C4—21no—
34C2×C4×C22⋊C4—21no—
35C2×C4×C22⋊C4—21no—
36C2×C4×C22⋊C4—21no—
37C2×C4×C22⋊C4C418yescomplemented
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C22×#95C2212yes—
41#1294C22116yescomplemented
42C26D411yescharacteristic
43C24×C4—22no—
44C24×C4—48no—
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C414yes—
47C24×C4C2×C414yes—
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C24×C4D411yescharacteristic
56C24×C4D414yes—
57C24×C4D411yescharacteristic
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—216no—
75C22×C22⋊C4—28no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C42—28no—
82C22×C42—22no—
83C22×C42—22no—
84C22×C42—22no—
85C22×C42—48no—
86#407—28no—
87#407—28no—
88#407—24no—
89#407—24no—
90#407C2×C418yescomplemented
91#407C2×C418yescomplemented
92C2×C22⋊C8—28no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8C2×C418yescomplemented
98C2×C22⋊C8C2314yes—
99#381—22no—
100#381—22no—
101#381—22no—
102#381—22no—
103#380—22no—
104#380—22no—
105#380—22no—
106#380—22no—
107#380C2×C414yes—
108#380C2×C414yescomplemented
109#380C2×C414yescomplemented
110#380C2314yes—
111#379—22no—
112#379—22no—
113#379—22no—
114#379—22no—
115C4×C22⋊C4—28no—
116C4×C22⋊C4—28no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—24no—
122C4×C22⋊C4—24no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4C2×C418yes—
128C4×C22⋊C4C2×C418yescomplemented
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—24no—
132C2×#95—22no—
133C2×#95C2×C418yescomplemented
134C2×#95C2314yes—
135C25—28no—
136C25—44no—
137C25—44no—
138C25C22⋊C411yescharacteristic
139C25C22⋊C411yescharacteristic
140C25C22⋊C414yes—
141C25C22⋊C411yescharacteristic
142C25C22⋊C411yescharacteristic
143C25C4⋊C414yes—
144C25C2×D411yescharacteristic
145C25C2×D411yescharacteristic
146C25C2×D411yescharacteristic
147C23×C4—28no—
148C23×C4—28no—
149C23×C4—28no—
150C23×C4—216no—
151C23×C4—28no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—416no—
155C23×C4—416no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—416no—
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—44no—
168C23×C4—44no—
169C23×C4—48no—
170C23×C4—48no—
171C23×C4C4214yes—
172C23×C4C4214yes—
173C23×C4C4214yes—
174C23×C4C4214yes—
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C414yes—
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C414yes—
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C414yes—
186C23×C4C22⋊C414yes—
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C414yes—
191C23×C4C22⋊C414yes—
192C23×C4C22⋊C414yes—
193C23×C4C22⋊C411yescharacteristic
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×C4C4⋊C414yes—
205C23×C4C22×C411yescharacteristic
206C23×C4C22×C411yescharacteristic
207C23×C4C22×C411yescharacteristic
208C23×C4C22×C412yes—
209C23×C4C22×C412yes—
210C23×C4C22×C412yes—
211C23×C4C22×C412yes—
212C23×C4C22×C412yes—
213C23×C4C22×C412yes—
214C23×C4C22×C411yescharacteristic
215C23×C4C22×C411yescharacteristic
216C23×C4C22×C411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D412yes—
225C23×C4C2×D412yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D412yes—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×Q812yes—
230C23×C4C2×Q812yes—
231C23×C4C2×Q812yes—
232C23×C4C2411yescharacteristic
233C2×M16—28no—
234C2×M16—28no—
235C2×M16—48no—
236C2×M16—432no—
237C2×M16—48no—
238C22×C8—28no—
239C22×C8—24no—
240C22×C8—48no—
241#117—48no—
242#117—48no—
243#117—48no—
244#117—48no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4C22×C418yes—
248C2×C22⋊C4—264no—
249C2×C22⋊C4—216no—
250C2×C22⋊C4—216no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—48no—
262C2×C42—24no—
263C2×C42—28no—
264C2×C42—28no—
265C2×C42—24no—
266C2×C42—48no—
267C2×C42—432no—
268C2×C42—48no—
269C2×C42—48no—
270C22⋊C8—28no—
271C22⋊C8—28no—
272C22⋊C8—28no—
273C22⋊C8—24no—
274C22⋊C8C22×C418yes—
275#95—28no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95C22×C418yes—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—216no—
288C24—216no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—42no—
297C24—42no—
298C24—48no—
299C24—48no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—48no—
305C24—42no—
306C24—42no—
307C24—48no—
308C24—464no—
309C24#9514yes—
310C24#9514yes—
311C24#9514yes—
312C24#9514yes—
313C24#10418yes—
314C24C2×C22⋊C411yescharacteristic
315C24C2×C22⋊C411yescharacteristic
316C24C2×C22⋊C412yes—
317C24C2×C22⋊C412yes—
318C24C2×C22⋊C412yes—
319C24C2×C22⋊C411yescharacteristic
320C24C2×C22⋊C411yescharacteristic
321C24C2×C22⋊C411yescharacteristic
322C24C2×C22⋊C411yescharacteristic
323C24C2×C4⋊C412yes—
324C24C2×C4⋊C412yes—
325C24C2×C4⋊C412yes—
326C24C22×D411yescharacteristic
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—216no—
331C22×C4—216no—
332C22×C4—216no—
333C22×C4—216no—
334C22×C4—216no—
335C22×C4—216no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—216no—
349C22×C4—232no—
350C22×C4—28no—
351C22×C4—28no—
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—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—48no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—48no—
393C22×C4—416no—
394C22×C4—464no—
395C22×C4—416no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
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×C4#9514yes—
410C22×C4#9514yes—
411C22×C4#9514yes—
412C22×C4#9918yes—
413C22×C4C2×C4212yes—
414C22×C4C2×C4212yes—
415C22×C4C2×C4212yes—
416C22×C4C2×C4212yes—
417C22×C4C2×C4212yes—
418C22×C4C2×C4212yes—
419C22×C4C2×C22⋊C412yes—
420C22×C4C2×C22⋊C412yes—
421C22×C4C2×C22⋊C412yes—
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C412yes—
427C22×C4C2×C22⋊C412yes—
428C22×C4C2×C22⋊C412yes—
429C22×C4C2×C22⋊C412yes—
430C22×C4C2×C22⋊C412yes—
431C22×C4C2×C22⋊C412yes—
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
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×C4C2×C4⋊C412yes—
446C22×C4C2×C4⋊C412yes—
447C22×C4C2×C4⋊C412yes—
448C22×C4C2×C4⋊C412yes—
449C22×C4C23×C411yescharacteristic
450C22×C4C23×C411yescharacteristic
451C22×C4C23×C411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×Q811yescharacteristic
455M16—416no—
456M16—416no—
457C2×C8—28no—
458C2×C8—48no—
459C2×C8—416no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C22⋊C4—232no—
463C22⋊C4—232no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C42—416no—
471C42—416no—
472C42—48no—
473C42—48no—
474C23—232no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—28no—
510C23—44no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—416no—
521C23—416no—
522C23—416no—
523C23—416no—
524C23—416no—
525C23—464no—
526C23—464no—
527C23—416no—
528C23#339116yes—
529C23C2×#9512yes—
530C23C2×#9512yes—
531C23C2×#9512yes—
532C23C2×#9512yes—
533C23C2×#9512yes—
534C23C2×#9512yes—
535C23C2×#9914yes—
536C23C2×#10414yes—
537C23C2×#10414yes—
538C23C2×#10414yes—
539C23C22×C22⋊C411yescharacteristic
540C23C22×C22⋊C411yescharacteristic
541C23C22×C4⋊C411yescharacteristic
542C23#57514yes—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
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—216no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
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—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—48no—
596C2×C4—464no—
597C2×C4—432no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—816no—
603C2×C4#342116yes—
604C2×C4C2×#9512yes—
605C2×C4C2×#9512yes—
606C2×C4C2×#9512yes—
607C2×C4C2×#9512yes—
608C2×C4C2×#9512yes—
609C2×C4C2×#9512yes—
610C2×C4C2×#9914yes—
611C2×C4C2×#9914yes—
612C2×C4C22×C4211yescharacteristic
613C2×C4C22×C22⋊C411yescharacteristic
614C2×C4C22×C4⋊C411yescharacteristic
615C2×C4C22×C4⋊C411yescharacteristic
616C8—48no—
617C22—216no—
618C22—216no—
619C22—21no—
620C22—21no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—21no—
627C22—21no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—28no—
632C22—28no—
633C22—28no—
634C22—28no—
635C22—28no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—416no—
643C22—48no—
644C22—48no—
645C22—48no—
646C22—48no—
647C22—416no—
648C22—416no—
649C22—416no—
650C22—44no—
651C22—44no—
652C22C2×#33914yes—
653C22C2×#33914yes—
654C22C2×#33914yes—
655C22C2×#34214yes—
656C22C22×#9511yescharacteristic
657C22C22×#9912yes—
658C22#289612yes—
659C22#289612yes—
660C22C22×#10412yes—
661C22#291112yes—
662C4—21no—
663C4—21no—
664C4—21no—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—21no—
670C4—416no—
671C4—88no—
672C4C2×#34214yes—
673C4C2×#34214yes—
674C2—22no—
675C2—22no—
676C2—24no—
677C2—24no—
678C2—24no—
679C2—24no—
680C2—42no—
681C2—42no—
682C2#1754214yes—
683C2C22×#33911yescharacteristic
684C2#2034711yescharacteristic
685C2C22×#34211yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC26D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8020C2semidirect
5. #8019C2semidirect
6. #8012C2semidirect
9. C22×#117C4semidirect
15. C2×#407C4semidirect
24. #2280C4semidirect
25. #2280C4semidirect
29. C2×#380C4semidirect
37. C2×C4×C22⋊C4C4semidirect
41. #1294C22semidirect
72. C2×#117C2×C4semidirect
73. C22×C4⋊C4C2×C4semidirect
90. #407C2×C4semidirect
91. #407C2×C4semidirect
97. C2×C22⋊C8C2×C4semidirect
108. #380C2×C4semidirect
109. #380C2×C4semidirect
128. C4×C22⋊C4C2×C4semidirect
133. 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×C4×C22⋊C42
#80202
#80192
#80122