Finite Group Databasebeta

← #363888

Subgroups — #363888

Order 512.

2862 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#1901C212yesmaximal, complemented
3#20898C212yesmaximal
4C2×#1821C211yesmaximal, characteristic, complemented
5C22×#343C211yesmaximal, characteristic, complemented
6#11382C212yesmaximal, complemented
7#10968C212yesmaximal, complemented
8#10763C212yesmaximal, complemented
9#10727C212yesmaximal, complemented
10C23×M16C2212yes—
11C2×#554—28no—
12C2×#554—24no—
13C2×#554—24no—
14C2×#554—24no—
15C2×#554—24no—
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125C412yes—
18C22×C22⋊Q8C412yes—
19C22×C22⋊Q8C412yes—
20C22×C22⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C412yes—
24C22×C4×D4C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C22×#101C2214yes—
28C2×#407C2212yes—
29C2×#407C2214yes—
30#1901—28no—
31#1870—24no—
32#1869—24no—
33#1821—24no—
34C2×#343—24no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#343C2214yes—
40#1728C2214yes—
41C23×Q8C2×C412yes—
42C23×Q8C2311yescharacteristic
43C23×D4C2×C411yescharacteristic
44C24×C4C2×C412yes—
45C24×C4C2×C412yes—
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—24no—
50C22×M16—28no—
51C22×M16—24no—
52C22×M16—44no—
53C22×M16C2314yes—
54C22×M16C2314yes—
55C23×C8—22no—
56#554—232no—
57#554—232no—
58#554—232no—
59#554—232no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—24no—
64C2×#125—28no—
65C2×#125—28no—
66C2×#125—28no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—28no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—28no—
75C2×C22⋊Q8—28no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—28no—
79C2×C22⋊Q8—28no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8D418yes—
83C2×C4×Q8—24no—
84C2×C4×Q8—22no—
85C2×C4×Q8—22no—
86C2×C4×Q8—24no—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4—28no—
90C2×C4×D4—24no—
91C2×C4×D4—22no—
92C2×C4×D4—22no—
93C2×C4×D4—28no—
94C2×C4×D4D414yes—
95C2×C4×D4D414yes—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—22no—
103C22×C4⋊C4C2×C412yes—
104C22×C4⋊C4C2×C412yes—
105C22×C4⋊C4C2×C412yes—
106C22×C4⋊C4C2×C412yes—
107C22×C4⋊C4C2×C412yes—
108C22×C4⋊C4C2×C411yescharacteristic
109C22×C4⋊C4C2×C412yes—
110C22×C4⋊C4D414yes—
111C22×C4⋊C4D414yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4C2×C412yes—
116C22×C22⋊C4C2×C412yes—
117C22×C22⋊C4C2×C411yescharacteristic
118C22×C22⋊C4C2×C411yescharacteristic
119C22×C42C2×C412yes—
120C22×C42C2311yescharacteristic
121C2×#101—28no—
122C2×#101—28no—
123C2×#101—44no—
124#407—24no—
125#407—24no—
126#407—28no—
127#407—44no—
128C2×C22⋊C8—22no—
129C2×C22⋊C8—28no—
130C2×C22⋊C8—22no—
131#343—24no—
132#343—24no—
133#343—28no—
134#343—24no—
135#343—24no—
136#336—28no—
137C25C22×C411yescharacteristic
138C25C22×C411yescharacteristic
139C22×Q8—24no—
140C22×Q8—24no—
141C22×Q8—24no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—24no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—84no—
149C22×Q8C2×D414yes—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—28no—
153C22×D4C22⋊C412yes—
154C22×D4C22⋊C412yes—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—84no—
171C23×C4—84no—
172C23×C4C22⋊C418yes—
173C23×C4C22×C412yes—
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C412yes—
179C23×C4C22×C412yes—
180C23×C4C22×C412yes—
181C23×C4C2×D414yes—
182C23×C4C2×D414yes—
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2411yescharacteristic
188C2×M16—24no—
189C2×M16—216no—
190C2×M16—24no—
191C2×M16—24no—
192C2×M16—24no—
193C2×M16—24no—
194C2×M16—24no—
195C2×M16—24no—
196C2×M16—24no—
197C2×M16—24no—
198C2×M16—24no—
199C2×M16—24no—
200C2×M16—48no—
201C2×M16—416no—
202C2×M16—48no—
203C2×M16—44no—
204C2×M16#4118yes—
205C22×C8—24no—
206C22×C8—24no—
207C22×C8—48no—
208C22×C8—44no—
209C4⋊Q8—216no—
210C4⋊Q8—216no—
211#125—264no—
212C22⋊Q8—216no—
213C22⋊Q8—216no—
214C22⋊Q8—264no—
215C22⋊Q8—232no—
216C22⋊Q8—232no—
217C22⋊Q8—28no—
218C22⋊Q8—28no—
219C22⋊Q8—28no—
220C22⋊Q8—28no—
221C4×Q8—216no—
222C4×Q8—216no—
223C4×D4—232no—
224C4×D4—232no—
225C4×D4—28no—
226C4×D4—28no—
227C4×D4—28no—
228C4×D4—28no—
229C2×C4⋊C4—216no—
230C2×C4⋊C4—216no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4C22⋊C418yes—
261C2×C4⋊C4C22⋊C414yes—
262C2×C4⋊C4C22⋊C418yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4C22⋊C418yes—
279C2×C22⋊C4C22⋊C412yes—
280C2×C22⋊C4C22⋊C412yes—
281C2×C22⋊C4C22⋊C414yes—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42C2×D414yes—
289#101—416no—
290C22⋊C8—416no—
291C22⋊C8—44no—
292C22⋊C8—44no—
293C24—21no—
294C24—21no—
295C24—22no—
296C24—22no—
297C24—24no—
298C24—24no—
299C24—22no—
300C24—22no—
301C24—42no—
302C24—42no—
303C24C2×C22⋊C411yescharacteristic
304C24C2×C22⋊C412yes—
305C24C2×C22⋊C412yes—
306C24C2×C22⋊C412yes—
307C24C2×C22⋊C411yescharacteristic
308C24C23×C411yescharacteristic
309C2×Q8—216no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—28no—
313C2×Q8—28no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—42no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—88no—
322C2×Q8—88no—
323C2×Q8—88no—
324C2×D4—216no—
325C2×D4—216no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4#10118yescomplemented
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—216no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4—88no—
389C22×C4C2×C22⋊C414yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C412yes—
393C22×C4C2×C22⋊C412yes—
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C412yes—
396C22×C4C2×C22⋊C414yes—
397C22×C4C2×C22⋊C414yes—
398C22×C4C2×C22⋊C414yes—
399C22×C4#12014yes—
400C22×C4#12014yes—
401C22×C4#12014yes—
402C22×C4#12014yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C2×#4114yes—
408C22×C4H(5,𝔽2)12yes—
409C22×C42−1+412yes—
410M16—28no—
411M16—28no—
412M16—416no—
413M16—44no—
414M16—44no—
415M16—44no—
416M16—44no—
417M16—44no—
418M16—44no—
419C2×C8—28no—
420C2×C8—416no—
421C2×C8—44no—
422C2×C8—44no—
423C2×C8—44no—
424C2×C8—44no—
425C2×C8—44no—
426C4⋊C4—216no—
427C4⋊C4—216no—
428C4⋊C4—216no—
429C4⋊C4—216no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—216no—
437C4⋊C4—28no—
438C4⋊C4—216no—
439C22⋊C4—216no—
440C22⋊C4—28no—
441C22⋊C4—28no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—28no—
445C42—216no—
446C42—24no—
447C42—24no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—24no—
453C23—24no—
454C23—22no—
455C23—24no—
456C23—22no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—22no—
466C23—42no—
467C23—42no—
468C23—48no—
469C23—48no—
470C23—48no—
471C23—48no—
472C23—42no—
473C23—42no—
474C23—41no—
475C23—42no—
476C23—41no—
477C23#37912yes—
478C23#37912yes—
479C23C2×#10114yes—
480C23C2×#10114yes—
481C23C22×C22⋊C411yescharacteristic
482C23C22×C22⋊C411yescharacteristic
483C23C22×C22⋊C411yescharacteristic
484C23C4×#4112yes—
485C23#52012yes—
486Q8—44no—
487Q8—44no—
488Q8—48no—
489Q8—816no—
490D4—216no—
491D4—216no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—48no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—88no—
536C2×C4—88no—
537C2×C4—816no—
538C2×C4#37914yes—
539C2×C4C2×#10114yes—
540C2×C4C2×#12012yes—
541C2×C4C2×#12012yes—
542C2×C4#52512yes—
543C2×C4#53212yes—
544C2×C4#53511yescharacteristic
545C2×C4#53611yescharacteristic
546C2×C4#53712yes—
547C2×C4#54614yes—
548C2×C4#54614yes—
549C2×C4#54714yes—
550C2×C4#54714yes—
551C8—48no—
552C8—84no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—22no—
558C22—22no—
559C22—24no—
560C22—24no—
561C22—28no—
562C22—22no—
563C22—24no—
564C22—22no—
565C22—22no—
566C22—28no—
567C22—44no—
568C22—48no—
569C22—44no—
570C22—44no—
571C22—48no—
572C22—44no—
573C22—42no—
574C22—42no—
575C22—44no—
576C22—44no—
577C22#178414yes—
578C22C2×#37911yescharacteristic
579C22#229711yescharacteristic
580C22#231111yescharacteristic
581C22#233912yes—
582C22#233912yes—
583C22C22×#10112yes—
584C22#288812yes—
585C4—24no—
586C4—24no—
587C4—42no—
588C4—44no—
589C4—44no—
590C4—44no—
591C4—42no—
592C4—44no—
593C4#240412yes—
594C4#240612yes—
595C4#243014yes—
596C2—21no—
597C2—22no—
598C2—21no—
599C2—24no—
600C2—42no—
601C2—44no—
602C2#1395611yescharacteristic
603C2C2×#178411yescharacteristic
604C2#2059211yescharacteristic
605C2#2248114yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1901C2semidirect
4. C2×#1821C2semidirect
5. C22×#343C2semidirect
6. #11382C2semidirect
7. #10968C2semidirect
8. #10763C2semidirect
9. #10727C2semidirect
331. C2×D4#101semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#19012
#208982
C2×#18212
C22×#3432
#113822
#109682
#107632
#107272