Finite Group Databasebeta

← #181520

Subgroups — #181520

Order 512.

3194 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊C8C211yesmaximal, characteristic
2C2×#2280C211yesmaximal, characteristic
3C2×#2280C211yesmaximal, characteristic
4#8024C214yesmaximal, complemented
5#8023C214yesmaximal, complemented
6#8012C214yesmaximal, complemented
7C25×C4C2211yescharacteristic
8C24×C8—21no—
9C22×#117—21no—
10C22×#117C412yescomplemented
11C22×C22⋊C8—24no—
12C22×C22⋊C8—21no—
13C22×C22⋊C8—21no—
14C22×C22⋊C8—21no—
15C22×C22⋊C8C418yescomplemented
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19#2280—24no—
20#2280—21no—
21#2280—21no—
22#2280—21no—
23#2280—21no—
24#2280C418yescomplemented
25#2280C412yescomplemented
26#2280C412yescomplemented
27#2280C2212yes—
28#2280C2212yes—
29C2×#381—21no—
30C2×#381C2212yes—
31C2×#380—21no—
32C2×#380C412yescomplemented
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#379—21no—
36C2×#379—21no—
37C2×C4×C22⋊C4—21no—
38C2×C4×C22⋊C4—21no—
39C2×C4×C22⋊C4—21no—
40C2×C4×C22⋊C4C2212yes—
41#1294C22116yescomplemented
42C26D411yescharacteristic
43C24×C4—48no—
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C414yes—
46C24×C4C2×C411yescharacteristic
47C24×C4C2×C414yes—
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C24×C4D411yescharacteristic
56C24×C4D411yescharacteristic
57C24×C4D414yes—
58C24×C4Q814yes—
59C24×C4C2311yescharacteristic
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C23×C8—28no—
63C23×C8—22no—
64C23×C8—22no—
65C23×C8—22no—
66C23×C8—48no—
67C2×#117—28no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—48no—
71C2×#117—48no—
72C2×#117C2×C414yes—
73C2×#117C2×C414yes—
74C22×C4⋊C4—22no—
75C22×C4⋊C4C2×C414yes—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C42—22no—
83C22×C42—22no—
84C2×C22⋊C8—28no—
85C2×C22⋊C8—28no—
86C2×C22⋊C8—28no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8C2×C418yes—
94C2×C22⋊C8C2×C418yescomplemented
95C2×C22⋊C8C2×C418yescomplemented
96C2×C22⋊C8C2314yes—
97#381—22no—
98#381—22no—
99#381—22no—
100#381—22no—
101#381—24no—
102#381C2×C418yescomplemented
103#380—28no—
104#380—22no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—24no—
109#380C2×C414yes—
110#380C2×C414yes—
111#380C2×C414yes—
112#380C2×C418yes—
113#380C2314yes—
114#379—28no—
115#379—22no—
116#379—22no—
117#379—22no—
118#379—22no—
119C4×C22⋊C4—28no—
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—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—24no—
129C4×C22⋊C4C2×C418yescomplemented
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95C2314yes—
134C25—28no—
135C25—44no—
136C25—44no—
137C25C22⋊C411yescharacteristic
138C25C22⋊C411yescharacteristic
139C25C22⋊C411yescharacteristic
140C25C22⋊C414yes—
141C25C22⋊C411yescharacteristic
142C25C4⋊C414yes—
143C25C2×D411yescharacteristic
144C25C2×D411yescharacteristic
145C25C2×D411yescharacteristic
146C23×C4—28no—
147C23×C4—28no—
148C23×C4—28no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—416no—
152C23×C4—416no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—416no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—416no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4C4214yes—
167C23×C4C4214yes—
168C23×C4C4214yes—
169C23×C4C4214yes—
170C23×C4C22⋊C414yes—
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C414yes—
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C414yes—
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C414yes—
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C414yes—
186C23×C4C22⋊C414yes—
187C23×C4C22⋊C414yes—
188C23×C4C22⋊C411yescharacteristic
189C23×C4C4⋊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×C4C22×C411yescharacteristic
201C23×C4C22×C411yescharacteristic
202C23×C4C22×C411yescharacteristic
203C23×C4C22×C412yes—
204C23×C4C22×C412yes—
205C23×C4C22×C412yes—
206C23×C4C22×C412yes—
207C23×C4C22×C412yes—
208C23×C4C22×C411yescharacteristic
209C23×C4C22×C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C22×C412yes—
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D412yes—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D412yes—
222C23×C4C2×D412yes—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×Q812yes—
225C23×C4C2×Q812yes—
226C23×C4C2×Q812yes—
227C23×C4C2411yescharacteristic
228C22×C8—28no—
229C22×C8—28no—
230C22×C8—28no—
231C22×C8—24no—
232C22×C8—48no—
233C22×C8—432no—
234C22×C8—48no—
235C22×C8—48no—
236#117—432no—
237#117—48no—
238#117—48no—
239#117—48no—
240#117—48no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4C22×C418yes—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—48no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C42—28no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—48no—
265C2×C42—48no—
266C22⋊C8—28no—
267C22⋊C8—28no—
268C22⋊C8—28no—
269C22⋊C8—24no—
270C22⋊C8C22×C418yes—
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—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—216no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—216no—
292C24—42no—
293C24—42no—
294C24—48no—
295C24—48no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24—48no—
301C24—48no—
302C24—42no—
303C24—42no—
304C24—464no—
305C24#9514yes—
306C24#9514yes—
307C24#9514yes—
308C24#9514yes—
309C24C2×C22⋊C411yescharacteristic
310C24C2×C22⋊C411yescharacteristic
311C24C2×C22⋊C412yes—
312C24C2×C22⋊C412yes—
313C24C2×C22⋊C411yescharacteristic
314C24C2×C22⋊C411yescharacteristic
315C24C2×C22⋊C411yescharacteristic
316C24C2×C22⋊C411yescharacteristic
317C24C2×C22⋊C412yes—
318C24C2×C4⋊C412yes—
319C24C2×C4⋊C412yes—
320C24C2×C4⋊C412yes—
321C24#13518yes—
322C24C22×D411yescharacteristic
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—216no—
327C22×C4—216no—
328C22×C4—216no—
329C22×C4—216no—
330C22×C4—216no—
331C22×C4—216no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
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—28no—
345C22×C4—28no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—48no—
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—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—48no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4#9514yes—
393C22×C4#9514yes—
394C22×C4#9514yes—
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#9918yes—
405C22×C4C2×C4212yes—
406C22×C4C2×C4212yes—
407C22×C4C2×C4212yes—
408C22×C4C2×C4212yes—
409C22×C4C2×C4212yes—
410C22×C4C2×C4212yes—
411C22×C4C2×C22⋊C412yes—
412C22×C4C2×C22⋊C412yes—
413C22×C4C2×C22⋊C412yes—
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C412yes—
419C22×C4C2×C22⋊C412yes—
420C22×C4C2×C22⋊C412yes—
421C22×C4C2×C22⋊C412yes—
422C22×C4C2×C22⋊C412yes—
423C22×C4C2×C22⋊C412yes—
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C4⋊C412yes—
427C22×C4C2×C4⋊C412yes—
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×C4C23×C411yescharacteristic
442C22×C4C23×C411yescharacteristic
443C22×C4C23×C411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×Q811yescharacteristic
447C2×C8—28no—
448C2×C8—48no—
449C2×C8—416no—
450C2×C8—416no—
451C2×C8—416no—
452C4⋊C4—48no—
453C4⋊C4—416no—
454C4⋊C4—48no—
455C22⋊C4—416no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—416no—
463C22⋊C4—48no—
464C42—48no—
465C42—416no—
466C42—48no—
467C23—232no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—24no—
498C23—28no—
499C23—24no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—416no—
513C23—416no—
514C23—416no—
515C23—416no—
516C23—416no—
517C23—464no—
518C23—464no—
519C23—416no—
520C23C2×#9512yes—
521C23C2×#9512yes—
522C23C2×#9512yes—
523C23C2×#9512yes—
524C23C2×#9512yes—
525C23C2×#9512yes—
526C23C2×#9914yes—
527C23#41614yes—
528C23#41614yes—
529C23#41618yes—
530C23#42114yes—
531C23#42718yes—
532C23C22×C22⋊C411yescharacteristic
533C23C22×C22⋊C411yescharacteristic
534C23C22×C4⋊C411yescharacteristic
535C23C2×#13514yes—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
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—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—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—816no—
594C2×C4—816no—
595C2×C4#342116yes—
596C2×C4C2×#9512yes—
597C2×C4C2×#9512yes—
598C2×C4C2×#9512yes—
599C2×C4C2×#9512yes—
600C2×C4C2×#9512yes—
601C2×C4C2×#9512yes—
602C2×C4C2×#9914yes—
603C2×C4C2×#9914yes—
604C2×C4C22×C4211yescharacteristic
605C2×C4C22×C22⋊C411yescharacteristic
606C2×C4C22×C4⋊C411yescharacteristic
607C2×C4C22×C4⋊C411yescharacteristic
608C8—48no—
609C22—232no—
610C22—21no—
611C22—21no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—21no—
618C22—21no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—28no—
623C22—28no—
624C22—28no—
625C22—28no—
626C22—28no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—48no—
634C22—48no—
635C22—48no—
636C22—48no—
637C22—48no—
638C22—416no—
639C22—48no—
640C22—416no—
641C22—416no—
642C22—44no—
643C22—44no—
644C22#173314yes—
645C22#173514yes—
646C22#173514yes—
647C22C2×#34214yes—
648C22C22×#9511yescharacteristic
649C22C22×#9912yes—
650C22C2×#41612yes—
651C22C2×#41612yes—
652C22C2×#42112yes—
653C22C2×#42712yes—
654C4—21no—
655C4—21no—
656C4—21no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—21no—
662C4—88no—
663C4—88no—
664C4C2×#34214yes—
665C4C2×#34214yes—
666C2—22no—
667C2—24no—
668C2—24no—
669C2—24no—
670C2—22no—
671C2—24no—
672C2—42no—
673C2—42no—
674C2#1757214yes—
675C2C2×#173311yescharacteristic
676C2C2×#173511yescharacteristic
677C2C22×#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. #8024C2semidirect
5. #8023C2semidirect
6. #8012C2semidirect
10. C22×#117C4semidirect
15. C22×C22⋊C8C4semidirect
24. #2280C4semidirect
25. #2280C4semidirect
26. #2280C4semidirect
32. C2×#380C4semidirect
41. #1294C22semidirect
94. C2×C22⋊C8C2×C4semidirect
95. C2×C22⋊C8C2×C4semidirect
102. #381C2×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
C23×C22⋊C82
C2×#22802
#80242
#80232
#80122