Finite Group Databasebeta

← #358197

Subgroups — #358197

Order 512.

1940 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#440C211yesmaximal, characteristic
2C2×#2293C211yesmaximal, characteristic
3C2×#1763C211yesmaximal, characteristic, complemented
4C2×#1762C211yesmaximal, characteristic
5C4×#391C211yesmaximal, characteristic, complemented
6#11219C212yesmaximal, complemented
7#11189C212yesmaximal
8#11067C212yesmaximal, complemented
9#11023C212yesmaximal
10#10703C211yesmaximal, characteristic, complemented
11#10693C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#117—21no—
16C2×#440—24no—
17C2×#440—24no—
18C2×#440C418yescomplemented
19C22×#107C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C2×#391C2212yes—
27#2293—24no—
28#2293—24no—
29C2×#390C2212yes—
30C2×#389C2212yes—
31C2×#388—22no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4—21no—
35C2×C43C2211yescharacteristic
36#1763—24no—
37#1762—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1436—21no—
45#1436—21no—
46C4⋊#95C2212yes—
47C4×#95—22no—
48C23×Q8C2311yescharacteristic
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C22×SD16—24no—
52C22×SD16—22no—
53C22×SD16—22no—
54C22×SD16—24no—
55C22×SD16C2×C414yes—
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8C2×C414yes—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4C2×C414yes—
69C2×C4×D4D414yes—
70C2×C4×D4D414yes—
71C2×#117—28no—
72C2×#117—48no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42D411yescharacteristic
86C22×C42D412yes—
87C22×C42D411yescharacteristic
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90#440—232no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#107C2×C414yes—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—24no—
99C2×#103C2×C414yes—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102C2×C414yes—
106C2×#97—22no—
107C2×#97—22no—
108C2×#97—22no—
109C2×#97—22no—
110C2×#97—22no—
111C2×#97C2×C414yes—
112#391—24no—
113#391—24no—
114#390—24no—
115#390—24no—
116#389—24no—
117#389—24no—
118#388—216no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4D414yes—
123C4×C22⋊C4—28no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—21no—
131C43—22no—
132C43—22no—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—24no—
137#336—24no—
138#336—24no—
139C25—21no—
140C22×Q8—21no—
141C22×Q8—22no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8C22×C412yes—
149C22×Q8#4114yes—
150C22×D4—21no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4C22×C412yes—
158C22×D4C2×D412yes—
159C22×D4C2×D412yes—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2411yescharacteristic
175C2×SD16—216no—
176C2×SD16—216no—
177C2×SD16—28no—
178C2×SD16—28no—
179C2×SD16—28no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8C22×C412yes—
194C22×C8C22×C412yes—
195C22×C8#4114yes—
196C22×C8#4114yes—
197C4×Q8—28no—
198C4×Q8—28no—
199C4×Q8—44no—
200C4×Q8—44no—
201C4×D4—28no—
202C4×D4—28no—
203C4×D4—28no—
204C4×D4C22⋊C4116yes—
205#117—432no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4#4114yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—44no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—44no—
264C2×C42—42no—
265C2×C42C22×C412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42#4112yes—
270C2×C42#4112yes—
271#107—28no—
272#103—216no—
273#103—28no—
274#102—216no—
275#102—28no—
276#97—28no—
277#97—44no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—24no—
282#95—44no—
283#95—42no—
284#95—44no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24C22×D411yescharacteristic
295C2×Q8—22no—
296C2×Q8—24no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8#11714yescomplemented
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4C2×C22⋊C414yes—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
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—24no—
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—216no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—48no—
357C22×C4—416no—
358C22×C4—48no—
359C22×C4—416no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4C4×D412yes—
382C22×C4C4×D412yes—
383C22×C4#12012yes—
384C22×C4#12012yes—
385C22×C4#12112yes—
386C22×C4#12112yes—
387C22×C4AGL(1,ℤ/8)14yes—
388C22×C4#13714yes—
389C22×C4C23×C411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C2×#4112yes—
394C22×C4C2×#4112yes—
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4112yes—
397C22×C4C2×#4112yes—
398C22×C4C2×#4111yescharacteristic
399SD16—232no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8#11714yescomplemented
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—416no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—48no—
434C4⋊C4C2×C22⋊C414yes—
435C22⋊C4—216no—
436C22⋊C4—416no—
437C22⋊C4—416no—
438C42—28no—
439C42—48no—
440C42—44no—
441C42—48no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—42no—
446C42—48no—
447C42—84no—
448C42C2×C22⋊C414yes—
449C42C4×D414yes—
450C42#12114yes—
451C42#12314yes—
452C23—24no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—42no—
458C23—42no—
459C23—44no—
460C23—44no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23C2×C4×D411yescharacteristic
474C23C2×#12011yescharacteristic
475C23C2×#12111yescharacteristic
476C23#52512yes—
477C23#53212yes—
478C23C2×AGL(1,ℤ/8)12yes—
479C23C2×#13712yes—
480C23#57512yes—
481C23#57512yes—
482Q8—24no—
483Q8—24no—
484Q8—44no—
485Q8—44no—
486Q8—42no—
487Q8—42no—
488D4—24no—
489D4—24no—
490D4—44no—
491D4—44no—
492D4—42no—
493D4—42no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—432no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4#38614yes—
547C2×C4#38614yes—
548C2×C4C22×C22⋊C411yescharacteristic
549C2×C4C2×#11711yescharacteristic
550C2×C4C2×#11711yescharacteristic
551C2×C4C2×C4×D411yescharacteristic
552C2×C4C4×#4111yescharacteristic
553C2×C4C4×#4111yescharacteristic
554C2×C4C2×#12111yescharacteristic
555C2×C4C2×#12311yescharacteristic
556C2×C4#52512yes—
557C2×C4#52512yes—
558C2×C4#52512yes—
559C2×C4#52512yes—
560C2×C4#52912yes—
561C2×C4#52912yes—
562C2×C4#52912yes—
563C2×C4#52912yes—
564C2×C4#53212yes—
565C2×C4#53212yes—
566C8—24no—
567C8—44no—
568C8—42no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—84no—
573C22—84no—
574C22—84no—
575C22—84no—
576C22—84no—
577C22—84no—
578C22C2×#38611yescharacteristic
579C22#229011yescharacteristic
580C22#229311yescharacteristic
581C22#238812yes—
582C22#239212yes—
583C22C4×AGL(1,ℤ/8)12yes—
584C22C4×#13712yes—
585C22#300311yescharacteristic
586C22#300511yescharacteristic
587C22#300712yes—
588C22#303612yes—
589C22#303911yescharacteristic
590C22#304011yescharacteristic
591C22#311112yes—
592C22#311212yes—
593C22#311512yes—
594C22#311712yes—
595C22#318012yes—
596C22#318112yes—
597C22#318212yes—
598C22#318312yes—
599C4—24no—
600C4—24no—
601C4—44no—
602C4—44no—
603C4—44no—
604C4—44no—
605C4—84no—
606C4—84no—
607C4—82no—
608C4#228011yescharacteristic
609C4#229011yescharacteristic
610C4#229311yescharacteristic
611C4#229711yescharacteristic
612C4#239212yes—
613C4#239512yes—
614C2—84no—
615C2#1383911yescharacteristic
616C2#2092511yescharacteristic
617C2#2092711yescharacteristic
618C2#2112412yes—
619C2#2128812yes—
620C2#2357612yes—
621C2#2357812yes—
622C2#2368312yes—
623C2#2368612yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1763C2semidirect
5. C4×#391C2semidirect
6. #11219C2semidirect
8. #11067C2semidirect
10. #10703C2semidirect
11. #10693C2semidirect
18. C2×#440C4semidirect
312. C2×Q8#117semidirect
416. C2×C8#117semidirect

Maximal subgroups

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

subgroupindex
C22×#4402
C2×#22932
C2×#17632
C2×#17622
C4×#3912
#112192
#111892
#110672
#110232
#107032
#106932