Finite Group Databasebeta

← #238267

Subgroups — #238267

Order 512.

1241 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic, complemented
2C2×#1964C211yesmaximal, characteristic
3C2×#1519C211yesmaximal, characteristic
4C2×#1519C211yesmaximal, characteristic
5#10374C218yesmaximal, complemented
6#8023C211yesmaximal, characteristic, complemented
7C2×#1289C211yesmaximal, characteristic, complemented
8C2×#1282C211yesmaximal, characteristic, complemented
9C2×#543—21no—
10C22×#121C2211yescharacteristic
11C22×#99—21no—
12C22×#99—21no—
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15#2857—22no—
16#2659—22no—
17#2549—22no—
18#2549C2214yescomplemented
19#2440—22no—
20#2440—22no—
21#2315—21no—
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26#1964—22no—
27#1964C2214yes—
28#1519—22no—
29#1519—22no—
30#1519—22no—
31#1519C2214yes—
32C2×#327—22no—
33C2×#327C2214yes—
34#1294C2214yescomplemented
35#1289C2214yescomplemented
36#1282C2214yescomplemented
37C23×D4—21no—
38C23×D4Q811yescharacteristic, complemented
39C24×C4C2311yescharacteristic
40C23×C8—21no—
41C23×C8—21no—
42#543—44no—
43#543—44no—
44C2×C4⋊Q8—22no—
45C2×C22⋊Q8—24no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8—24no—
48C2×#121—21no—
49C2×#121—21no—
50C2×#121—21no—
51C2×#121—21no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121—21no—
57C2×#121—21no—
58C2×#121Q812yescomplemented
59C2×#121C2312yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4Q811yescharacteristic, complemented
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4D411yescharacteristic
69C22×C22⋊C4D411yescharacteristic
70C22×C42—21no—
71C2×#99—24no—
72C2×#99—24no—
73C2×#99—44no—
74C2×#99—44no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8C2312yes—
86C2×C22⋊C8C2312yes—
87#398—24no—
88#398—24no—
89#395—24no—
90#395—24no—
91#391—24no—
92#389—22no—
93#389—22no—
94#389—24no—
95#385—24no—
96#385—24no—
97#384—22no—
98#380—24no—
99#380—21no—
100#380—21no—
101#380—21no—
102#380—21no—
103#380—21no—
104#380—21no—
105#380—21no—
106#380D412yes—
107#380D412yes—
108#380D412yes—
109#380C2312yes—
110#380C2312yes—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4D412yes—
114C4×C22⋊C4C2312yes—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95D412yes—
118C2×#95C2312yes—
119#327—28no—
120#327—28no—
121C25—21no—
122C25C2×D411yescharacteristic
123C22×Q8—24no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—41no—
132C22×D4—41no—
133C22×D4—41no—
134C22×D4—41no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—82no—
140C22×D4C2×Q812yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—41no—
147C23×C4—82no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×Q811yescharacteristic
154C23×C4C2×Q811yescharacteristic
155C23×C4#4111yescharacteristic
156C23×C4#4111yescharacteristic
157C23×C4#4111yescharacteristic
158C23×C4#4111yescharacteristic
159C23×C4#4111yescharacteristic
160C23×C4#4111yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—41no—
165C22×C8—41no—
166C22×C8—41no—
167C22×C8—41no—
168C22×C8—82no—
169C22×C8—82no—
170C4⋊Q8—84no—
171C22⋊Q8—48no—
172C22⋊Q8—416no—
173C22⋊Q8—416no—
174C22⋊Q8—48no—
175#121—22no—
176#121—22no—
177#121—22no—
178#121—22no—
179#121—22no—
180#121—22no—
181#121—22no—
182#121—22no—
183#121—41no—
184#121—41no—
185#121—41no—
186#121—41no—
187#121—41no—
188#121—41no—
189#121—41no—
190#121—41no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C2×Q812yes—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
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—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4D814yes—
234C2×C22⋊C4SD1614yes—
235C2×C22⋊C4C2×D412yes—
236C2×C22⋊C4C2×D412yes—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—82no—
240#99—48no—
241#99—48no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C22⋊C8—41no—
247C22⋊C8—41no—
248C22⋊C8—41no—
249C22⋊C8—41no—
250#95—24no—
251#95—24no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—41no—
258#95C2×D414yes—
259#95C2×D414yes—
260C24—22no—
261C24—41no—
262C24—41no—
263C24—42no—
264C24—42no—
265C24—81no—
266C24—81no—
267C24—82no—
268C24—82no—
269C24#12011yescharacteristic
270C24#12011yescharacteristic
271C24C22⋊Q811yescharacteristic
272C24C22⋊Q811yescharacteristic
273C24#12411yescharacteristic
274C24#12411yescharacteristic
275C24C2×D812yes—
276C24C2×SD1612yes—
277C24AGL(1,ℤ/8)12yes—
278C24#13712yes—
279C24C22×D411yescharacteristic
280C2×Q8—48no—
281C2×Q8—84no—
282C2×Q8—84no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—82no—
298C2×D4—82no—
299C2×D4—82no—
300C2×D4—82no—
301C2×D4—82no—
302C2×D4—82no—
303C2×D4—82no—
304C2×D4—82no—
305C2×D4—82no—
306C2×D4—82no—
307C2×D4—82no—
308C2×D4—82no—
309C2×D4—82no—
310C2×D4—82no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—81no—
334C22×C4—81no—
335C22×C4—81no—
336C22×C4—81no—
337C22×C4—81no—
338C22×C4—81no—
339C22×C4—81no—
340C22×C4—81no—
341C22×C4—81no—
342C22×C4—81no—
343C22×C4—81no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4#12011yescharacteristic
352C22×C4#12011yescharacteristic
353C22×C4C22⋊Q811yescharacteristic
354C22×C4C22⋊Q811yescharacteristic
355C22×C4C22⋊Q811yescharacteristic
356C22×C4C22⋊Q811yescharacteristic
357C22×C4C22⋊Q811yescharacteristic
358C22×C4C22⋊Q811yescharacteristic
359C22×C4C22⋊Q811yescharacteristic
360C22×C4C22⋊Q811yescharacteristic
361C22×C4C22⋊Q811yescharacteristic
362C22×C4C22⋊Q811yescharacteristic
363C22×C4#12411yescharacteristic
364C22×C4#12411yescharacteristic
365C22×C4#12411yescharacteristic
366C22×C4#12411yescharacteristic
367C22×C4#12411yescharacteristic
368C22×C4#12411yescharacteristic
369C22×C4#12411yescharacteristic
370C22×C4#12411yescharacteristic
371C22×C4#12411yescharacteristic
372C22×C4#12411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×Q811yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—82no—
382C2×C8—82no—
383C2×C8—82no—
384C2×C8—82no—
385C2×C8—82no—
386C2×C8—82no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—48no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—82no—
397C4⋊C4—84no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—84no—
401C4⋊C4—82no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—82no—
418C22⋊C4C2×D814yes—
419C22⋊C4C2×SD1614yes—
420C42—44no—
421C42—82no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—164no—
452C23—164no—
453C23#39311yescharacteristic
454C23#39311yescharacteristic
455C23#39311yescharacteristic
456C23#39311yescharacteristic
457C23#44712yes—
458C23#44712yes—
459C23#44812yes—
460C23#44812yes—
461C23#47412yes—
462C23#47412yes—
463C23#47612yes—
464C23#47612yes—
465C23#48612yes—
466C23#48612yes—
467C23#48912yes—
468C23#48912yes—
469C23C2×#12011yescharacteristic
470C23C2×C22⋊Q811yescharacteristic
471C23C2×#12411yescharacteristic
472C23C22×D811yescharacteristic
473C23C22×SD1611yescharacteristic
474C23C2×AGL(1,ℤ/8)11yescharacteristic
475C23C2×#13711yescharacteristic
476Q8—168no—
477D4—82no—
478D4—82no—
479D4—82no—
480D4—82no—
481D4—82no—
482D4—82no—
483D4—82no—
484D4—84no—
485D4—84no—
486D4—82no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—84no—
531C2×C4—84no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—162no—
538C2×C4—162no—
539C2×C4—164no—
540C2×C4—164no—
541C2×C4#39311yescharacteristic
542C2×C4#39311yescharacteristic
543C2×C4#39311yescharacteristic
544C2×C4#39311yescharacteristic
545C2×C4C2×C22⋊Q811yescharacteristic
546C2×C4C2×C22⋊Q811yescharacteristic
547C2×C4C2×#12411yescharacteristic
548C2×C4C2×#12411yescharacteristic
549C8—82no—
550C8—82no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—42no—
566C22—42no—
567C22—41no—
568C22—41no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—41no—
574C22—41no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—162no—
585C22—162no—
586C22—164no—
587C22—164no—
588C22#202312yes—
589C22#202312yes—
590C22#202312yes—
591C22#202312yes—
592C22C2×#39311yescharacteristic
593C22C2×#44711yescharacteristic
594C22C2×#44811yescharacteristic
595C22C2×#47411yescharacteristic
596C22C2×#47611yescharacteristic
597C22C2×#48611yescharacteristic
598C22C2×#48911yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—44no—
607C4—44no—
608C4—84no—
609C4—84no—
610C4—82no—
611C4—162no—
612C4#203212yes—
613C2—21no—
614C2—21no—
615C2—21no—
616C2—22no—
617C2—22no—
618C2—21no—
619C2—21no—
620C2—21no—
621C2—22no—
622C2—22no—
623C2—162no—
624C2C2×#202311yescharacteristic
625C2#2347211yescharacteristic
626C2C2×#203211yescharacteristic
subgroupG/H
center, socleC22C2×#393
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35002C2semidirect
5. #10374C2semidirect
6. #8023C2semidirect
7. C2×#1289C2semidirect
8. C2×#1282C2semidirect
18. #2549C22semidirect
34. #1294C22semidirect
35. #1289C22semidirect
36. #1282C22semidirect
38. C23×D4Q8semidirect
58. C2×#121Q8semidirect
63. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#19642
C2×#15192
#103742
#80232
C2×#12892
C2×#12822