Finite Group Databasebeta

← #381863

Subgroups — #381863

Order 512.

1532 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2001C211yesmaximal, characteristic
2C2×#1969C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C8C211yesmaximal, characteristic
4#14702C212yesmaximal
5#11256C211yesmaximal, characteristic, complemented
6#11190C212yesmaximal, complemented
7#11096C211yesmaximal, characteristic
8#11024C211yesmaximal, characteristic, complemented
9#10970C212yesmaximal
10#10926C212yesmaximal, complemented
11#10820C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C23×C42C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#380C2212yes—
26#2001C418yescomplemented
27#1969—24no—
28C4×C22⋊C8—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1436—22no—
37#1436—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C22⋊#95—22no—
41C22⋊#95—22no—
42C23×Q8C2312yes—
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C22×Q16—22no—
47C22×Q16—22no—
48C22×Q16—42no—
49C22×Q16—42no—
50C22×Q16C2×C414yescomplemented
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—42no—
57C2×C22⋊Q8—42no—
58C2×C22⋊Q8—42no—
59C2×C22⋊Q8—42no—
60C2×C22⋊Q8C2×C418yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C42—22no—
68C22×C42—22no—
69C22×C42—42no—
70C22×C42—42no—
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#107—22no—
76C2×#107—42no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103C2×C418yescomplemented
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8C2×C414yes—
92C2×C4×C8—22no—
93C2×C4×C8—42no—
94C2×C4×C8—42no—
95C2×C4×C8D414yes—
96#391—24no—
97#391—24no—
98#391—42no—
99#391—42no—
100#380—42no—
101#380—42no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111#340—22no—
112#340—42no—
113#340—42no—
114#340—42no—
115#340C2×C414yescomplemented
116#336—22no—
117#336—22no—
118C25C2×D411yescharacteristic
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8C22×C414yes—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4Q1614yescomplemented
155C23×C4C22×C412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4#4112yes—
162C23×C4C2411yescharacteristic
163C2×Q16—28no—
164C2×Q16—48no—
165C2×Q16—44no—
166C2×Q16—44no—
167C2×Q16—48no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8C22×C412yes—
183C22×C8C22×C412yes—
184C22×C8C2×D412yes—
185C22×C8C2×D412yes—
186C22×C8#4112yes—
187C22×C8#4112yes—
188C22⋊Q8—48no—
189C22⋊Q8—48no—
190C22⋊Q8—48no—
191C22⋊Q8—48no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4C22×C414yes—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42—48no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42C2×D412yes—
233C2×C42#4112yes—
234#107—44no—
235#103—48no—
236#103—48no—
237#103—48no—
238C22⋊C8—28no—
239C22⋊C8—28no—
240C4×C8—44no—
241C4×C8—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—44no—
251C24—21no—
252C24—21no—
253C24—22no—
254C24—22no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—22no—
259C24C4×D412yes—
260C24C2×Q1612yes—
261C24#13512yes—
262C24C22×D411yescharacteristic
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—28no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—28no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—48no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—48no—
326C22×C4—44no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4C4×D412yes—
365C22×C4#12112yes—
366C22×C4#12112yes—
367C22×C4C2×Q1612yes—
368C22×C4C2×Q1612yes—
369C22×C4#13512yes—
370C22×C4#13512yes—
371C22×C4#13714yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4H(5,𝔽2)12yes—
379C22×C4H(5,𝔽2)12yes—
380C22×C4H(5,𝔽2)12yes—
381C22×C4H(5,𝔽2)12yes—
382Q16—416no—
383C2×C8—24no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—24no—
387C2×C8—22no—
388C2×C8—24no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—24no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8C4×D414yes—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—416no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C22⋊C4—416no—
417C22⋊C4—88no—
418C42—22no—
419C42—22no—
420C42—24no—
421C42—44no—
422C42—44no—
423C42—41no—
424C42—41no—
425C42—42no—
426C23—24no—
427C23—22no—
428C23—22no—
429C23—24no—
430C23—24no—
431C23—22no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23C4×Q1614yes—
449C23C2×C4×D411yescharacteristic
450C23C2×#12111yescharacteristic
451C23#53211yescharacteristic
452C23#53412yes—
453C23#53512yes—
454C23C22×Q1611yescharacteristic
455C23C2×#13511yescharacteristic
456C23C2×#13511yescharacteristic
457C23C2×#13511yescharacteristic
458C23C2×#13712yes—
459C23#57512yes—
460Q8—44no—
461Q8—44no—
462Q8—44no—
463Q8—44no—
464Q8—82no—
465Q8—82no—
466Q8—82no—
467Q8—82no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—416no—
483C2×C4—416no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4#46714yes—
528C2×C4#47014yes—
529C2×C4C2×C4×D411yescharacteristic
530C2×C4C4×#4111yescharacteristic
531C2×C4#51812yes—
532C2×C4#51812yes—
533C2×C4#52511yescharacteristic
534C2×C4#52911yescharacteristic
535C2×C4#53411yescharacteristic
536C2×C4#53511yescharacteristic
537C2×C4#53511yescharacteristic
538C2×C4#53511yescharacteristic
539C2×C4#53812yes—
540C2×C4#53812yes—
541C2×C4#53812yes—
542C2×C4#53812yes—
543C2×C4#53812yes—
544C2×C4#54012yes—
545C2×C4#54012yes—
546C2×C4#54012yes—
547C8—24no—
548C8—42no—
549C8—42no—
550C8—42no—
551C22—24no—
552C22—28no—
553C22—44no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—42no—
562C22C4×#12111yescharacteristic
563C22#232112yes—
564C22#261711yescharacteristic
565C22#261911yescharacteristic
566C22#266112yes—
567C22C2×C4×Q1611yescharacteristic
568C22C4×#13511yescharacteristic
569C22C4×#13712yes—
570C22C2×#46711yescharacteristic
571C22#305011yescharacteristic
572C22C2×#47011yescharacteristic
573C22#305411yescharacteristic
574C22#316811yescharacteristic
575C22#317011yescharacteristic
576C22#317212yes—
577C22#318912yes—
578C22#319012yes—
579C22#319012yes—
580C22#319112yes—
581C22#320912yes—
582C22#321012yes—
583C22#321112yes—
584C22#321212yes—
585C4—24no—
586C4—24no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—84no—
591C4—84no—
592C4—162no—
593C4—162no—
594C4#232111yescharacteristic
595C4#232811yescharacteristic
596C4#233312yes—
597C4#263111yescharacteristic
598C4#263511yescharacteristic
599C4#264812yes—
600C2—44no—
601C2#1431111yescharacteristic
602C2C4×#46711yescharacteristic
603C2C4×#47011yescharacteristic
604C2#2204212yes—
605C2#2204412yes—
606C2#2973111yescharacteristic
607C2#2973311yescharacteristic
608C2#2980811yescharacteristic
609C2#2980911yescharacteristic
610C2#3031812yes—
611C2#3032112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1969C2semidirect
5. #11256C2semidirect
6. #11190C2semidirect
8. #11024C2semidirect
10. #10926C2semidirect
11. #10820C2semidirect
26. #2001C4semidirect
50. C22×Q16C2×C4semidirect
85. C2×#103C2×C4semidirect
115. #340C2×C4semidirect
154. C23×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#20012
C2×#19692
C2×C4×C22⋊C82
#147022
#112562
#111902
#110962
#110242
#109702
#109262
#108202