Finite Group Databasebeta

← #160808

Subgroups — #160808

Order 512.

1670 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C4×#407C211yesmaximal, characteristic, complemented
4#7633C214yesmaximal, complemented
5#7616C214yesmaximal, complemented
6#7576C212yesmaximal, complemented
7#7572C212yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C411yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#423—21no—
13C2×#407—21no—
14C2×#407—21no—
15C2×C4×M16—21no—
16C4×#121—21no—
17C4×#121—21no—
18C4×#121—21no—
19C4×#121—21no—
20C4×#121C412yes—
21C4×#121C412yes—
22C4×#121C2212yes—
23C4×#121C2212yes—
24C2×#390C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#388—21no—
27C2×#386—21no—
28C2×#380C411yescharacteristic
29C2×C4×C4⋊C4C411yescharacteristic
30C2×C4×C22⋊C4—21no—
31#1797—21no—
32#1797—21no—
33#1797C2212yes—
34#1797C2212yescomplemented
35#1749—21no—
36#1749C2212yescomplemented
37C4×C22⋊C8—21no—
38C4×C22⋊C8C2212yes—
39#1728C2212yescomplemented
40#1728C2212yescomplemented
41#1271C2218yescomplemented
42#1270C2218yescomplemented
43C23×D4—21no—
44C23×D4C2×C411yescharacteristic
45C24×C4—21no—
46C24×C4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C22×M16—22no—
50C22×M16—22no—
51C2×#121—24no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121—24no—
55C2×#121C2×C412yes—
56C2×#121C2×C412yes—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—44no—
64C2×C4×D4—48no—
65C2×C4×D4C2×C412yes—
66C2×C4×D4C2×C412yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C42—21no—
77C22×C42—44no—
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#423—44no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93#407—28no—
94#407—44no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C4×M16—44no—
100C2×#97—22no—
101C2×C4×C8—22no—
102#390—21no—
103#390—21no—
104#390C2×C412yes—
105#390C2312yes—
106#389—21no—
107#389—21no—
108#389C2×C412yes—
109#389C2312yes—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#380—21no—
119#380—21no—
120#380C2×C412yes—
121#380C2×C412yes—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C412yes—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133#336—22no—
134#336—22no—
135#336C2314yes—
136#336C2314yes—
137C25—21no—
138C25C22⋊C411yescharacteristic
139C22×D4—28no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—22no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—44no—
147C22×D4—44no—
148C22×D4M1614yescomplemented
149C22×D4C22×C412yes—
150C23×C4—21no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
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×C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C2×M16—28no—
188C2×M16—44no—
189C2×M16—44no—
190C2×M16—44no—
191C2×M16—44no—
192C2×M16—44no—
193C22×C8—24no—
194C22×C8—24no—
195#121—28no—
196#121—28no—
197#121—24no—
198#121—24no—
199#121—24no—
200#121—24no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4—48no—
205C4×D4—48no—
206C4×D4—48no—
207C4×D4—48no—
208C4×D4—44no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4M1614yescomplemented
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C412yes—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—44no—
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2411yescharacteristic
269C4⋊C8—44no—
270C4⋊C8—44no—
271C22⋊C8—28no—
272C22⋊C8—28no—
273C22⋊C8—44no—
274C22⋊C8—44no—
275#97—44no—
276C4×C8—44no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—24no—
288C24—22no—
289C24—22no—
290C24—41no—
291C24—41no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24C2×C22⋊C411yescharacteristic
299C24C2×C22⋊C411yescharacteristic
300C24C2×C22⋊C411yescharacteristic
301C2×D4—28no—
302C2×D4—28no—
303C2×D4—22no—
304C2×D4—24no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—48no—
308C2×D4—48no—
309C2×D4—44no—
310C2×D4—432no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4C2×M1614yes—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—416no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—816no—
395C22×C4#10414yes—
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×M1612yes—
406C22×C4C2×M1612yes—
407C22×C4AGL(1,ℤ/8)14yes—
408C22×C4AGL(1,ℤ/8)12yes—
409C22×C4#13712yes—
410C22×C4C23×C411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413M16—416no—
414M16—88no—
415C2×C8—28no—
416C2×C8—44no—
417C2×C8—44no—
418C2×C8—44no—
419C2×C8—44no—
420C2×C8—44no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4C2×M1614yes—
430C22⋊C4—28no—
431C22⋊C4—28no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—42no—
437C22⋊C4—44no—
438C22⋊C4—42no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—44no—
457C42—44no—
458C42#9914yes—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—28no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—24no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—48no—
483C23—48no—
484C23—44no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—48no—
492C23—84no—
493C23—84no—
494C23—84no—
495C23—84no—
496C23#40712yes—
497C23#41712yes—
498C23#41812yes—
499C23C2×#10412yes—
500C23#42112yes—
501C23C22×C22⋊C411yescharacteristic
502D4—44no—
503D4—44no—
504D4—416no—
505D4—416no—
506D4—48no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—48no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—88no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—88no—
569C2×C4#40712yes—
570C2×C4C2×#9912yes—
571C2×C4C2×#9912yes—
572C2×C4C2×#9912yes—
573C2×C4#41312yes—
574C2×C4#41712yes—
575C2×C4#41812yes—
576C2×C4C2×#10412yes—
577C2×C4C2×#10412yes—
578C2×C4C22×M1611yescharacteristic
579C2×C4C22×M1611yescharacteristic
580C2×C4C2×AGL(1,ℤ/8)12yes—
581C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
582C2×C4C2×#13711yescharacteristic
583C8—48no—
584C8—84no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—416no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—816no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22#148914yes—
615C22C2×#40711yescharacteristic
616C22C22×#9911yescharacteristic
617C22C2×#41311yescharacteristic
618C22C2×#41811yescharacteristic
619C22C2×#41711yescharacteristic
620C22C22×#10411yescharacteristic
621C22C2×#42111yescharacteristic
622C4—22no—
623C4—22no—
624C4—21no—
625C4—21no—
626C4—22no—
627C4—22no—
628C4—21no—
629C4—21no—
630C4—44no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—82no—
636C4—82no—
637C4#146514yes—
638C4#150714yes—
639C2—24no—
640C2—42no—
641C2—42no—
642C2—84no—
643C2C2×#146511yescharacteristic
644C2C2×#148911yescharacteristic
645C2#1744811yescharacteristic
646C2C2×#150711yescharacteristic
647C2#1759211yescharacteristic
648C2#1764211yescharacteristic
649C2#1774511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20626C2semidirect
3. C4×#407C2semidirect
4. #7633C2semidirect
5. #7616C2semidirect
6. #7576C2semidirect
7. #7572C2semidirect
34. #1797C22semidirect
36. #1749C22semidirect
39. #1728C22semidirect
40. #1728C22semidirect
41. #1271C22semidirect
42. #1270C22semidirect
148. C22×D4M16semidirect
218. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#206262
C4×#4072
#76332
#76162
#75762
#75722