Finite Group Databasebeta

← #214201

Subgroups — #214201

Order 512.

1614 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3#20714C211yesmaximal, characteristic, complemented
4#9076C214yesmaximal, complemented
5#8943C214yesmaximal, complemented
6#8707C212yesmaximal, complemented
7#8310C212yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#432C2212yescomplemented
13C2×#431C2212yescomplemented
14C4×#121—21no—
15C4×#121—21no—
16C4×#121—21no—
17C4×#121—21no—
18C4×#121C412yes—
19C4×#121C412yes—
20C4×#121C2212yes—
21C4×#121C2212yes—
22C2×#390C411yescharacteristic
23C2×#389C411yescharacteristic
24C2×#388—21no—
25C2×#386—21no—
26C2×#380C411yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4—21no—
29#1817—21no—
30#1817—21no—
31#1797—21no—
32#1797—21no—
33#1797C2212yes—
34#1797C2212yescomplemented
35#1779—21no—
36#1779—21no—
37#1749—21no—
38#1749—21no—
39C4×C22⋊C8C2212yes—
40C4×C22⋊C8C2212yescomplemented
41C2×D4⋊C8C2218yescomplemented
42#1279C2218yescomplemented
43C23×D4—21no—
44C23×D4C2×C411yescharacteristic
45C24×C4—21no—
46C24×C4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C2×#121—24no—
50C2×#121—21no—
51C2×#121—21no—
52C2×#121—24no—
53C2×#121C2×C412yes—
54C2×#121C2×C412yes—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4—44no—
62C2×C4×D4—48no—
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2312yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C42—21no—
75C22×C42—44no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#432—44no—
89#431—44no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8C2314yes—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×#97—22no—
99C2×C4×C8C2314yes—
100#390—21no—
101#390—21no—
102#390C2×C412yes—
103#390C2×C412yes—
104#389—21no—
105#389—21no—
106#389C2×C412yes—
107#389C2×C412yes—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#380—21no—
117#380—21no—
118#380C2×C412yes—
119#380C2×C412yes—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2312yes—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135D4⋊C8—48no—
136C25—21no—
137C25C22⋊C411yescharacteristic
138C22×D4—28no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—21no—
142C22×D4—22no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—44no—
146C22×D4—44no—
147C22×D4M1614yes—
148C22×D4C22×C412yes—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
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×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C22×C8—24no—
186C22×C8—24no—
187#121—28no—
188#121—28no—
189#121—24no—
190#121—24no—
191#121—24no—
192#121—24no—
193C4×D4—28no—
194C4×D4—44no—
195C4×D4—44no—
196C4×D4—44no—
197C4×D4—48no—
198C4×D4—48no—
199C4×D4—48no—
200C4×D4—48no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4M1614yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C22×C412yes—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—44no—
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2411yescharacteristic
260C4⋊C8—28no—
261C4⋊C8—44no—
262C22⋊C8—44no—
263C22⋊C8—44no—
264C22⋊C8—44no—
265C22⋊C8—44no—
266C22⋊C8—44no—
267C22⋊C8—44no—
268#97—44no—
269C4×C8—28no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—24no—
281C24—22no—
282C24—22no—
283C24—41no—
284C24—41no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24C2×C22⋊C411yescharacteristic
292C24C2×C22⋊C411yescharacteristic
293C24C2×C22⋊C411yescharacteristic
294C2×D4—28no—
295C2×D4—28no—
296C2×D4—24no—
297C2×D4—22no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—48no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—48no—
304C2×D4—44no—
305C2×D4—432no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4C2×M1614yes—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
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—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
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—416no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—416no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—816no—
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×M1612yes—
396C22×C4C2×M1612yes—
397C22×C4#13514yes—
398C22×C4AGL(1,ℤ/8)14yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C2×C8—44no—
403C2×C8—44no—
404C2×C8—44no—
405C2×C8—44no—
406C2×C8—44no—
407C2×C8—44no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4C2×M1614yes—
417C22⋊C4—28no—
418C22⋊C4—28no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—44no—
443C42—44no—
444C42H(5,𝔽2)14yes—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—28no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—24no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—48no—
469C23—48no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—48no—
478C23—84no—
479C23—84no—
480C23—84no—
481C23—84no—
482C23#40712yes—
483C23#41612yes—
484C23#41812yes—
485C23#42112yes—
486C23#42112yes—
487C23C22×C22⋊C411yescharacteristic
488D4—28no—
489D4—48no—
490D4—416no—
491D4—416no—
492D4—44no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—48no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—88no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—88no—
554C2×C4#40712yes—
555C2×C4#41612yes—
556C2×C4#41812yes—
557C2×C4#51812yes—
558C2×C4#53412yes—
559C2×C4#53412yes—
560C2×C4C22×M1611yescharacteristic
561C2×C4C22×M1611yescharacteristic
562C2×C4#56812yes—
563C2×C4C2×#13512yes—
564C2×C4C2×AGL(1,ℤ/8)12yes—
565C8—84no—
566C8—84no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—416no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—816no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22#148814yes—
597C22#230811yescharacteristic
598C22C2×#40711yescharacteristic
599C22#287911yescharacteristic
600C22C2×#41611yescharacteristic
601C22C2×#41811yescharacteristic
602C22C2×#42111yescharacteristic
603C22C2×#42111yescharacteristic
604C4—22no—
605C4—22no—
606C4—22no—
607C4—22no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—44no—
613C4—41no—
614C4—41no—
615C4—82no—
616C4—82no—
617C4#297312yes—
618C4#297512yes—
619C4#318814yes—
620C2—24no—
621C2—42no—
622C2—42no—
623C2—84no—
624C2C2×#148811yescharacteristic
625C2#1744811yescharacteristic
626C2#1748711yescharacteristic
627C2#1749511yescharacteristic
628C2#2174211yescharacteristic
629C2#2176111yescharacteristic
630C2#2177411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20715C2semidirect
3. #20714C2semidirect
4. #9076C2semidirect
5. #8943C2semidirect
6. #8707C2semidirect
7. #8310C2semidirect
12. C2×#432C22semidirect
13. C2×#431C22semidirect
34. #1797C22semidirect
40. C4×C22⋊C8C22semidirect
41. C2×D4⋊C8C22semidirect
42. #1279C22semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#207152
#207142
#90762
#89432
#87072
#83102