Finite Group Databasebeta

← #405121

Subgroups — #405121

Order 512.

2610 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#451C211yesmaximal, characteristic
2C2×#2855C211yesmaximal, characteristic
3C22×#391C211yesmaximal, characteristic
4C2×#1872C211yesmaximal, characteristic
5C2×#1817C211yesmaximal, characteristic
6C2×#1787C211yesmaximal, characteristic
7#11067C214yesmaximal, complemented
8#11024C214yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10C24×Q8C2211yescharacteristic
11C23×Q16C2212yes—
12C22×C22⋊Q8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C2×#451—24no—
16C2×#451—24no—
17C2×#451—24no—
18C22×#103C2212yes—
19C22×#103C2214yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23#2855—24no—
24#2855—24no—
25C2×#391—22no—
26C2×#391—24no—
27C2×#391—24no—
28C2×#391C2212yes—
29C2×#391C2214yes—
30C2×#389C2212yes—
31C2×#387—22no—
32C2×#385—21no—
33C2×#381—21no—
34C2×#380C2211yescharacteristic
35C22×#95—21no—
36C22×#95—21no—
37#1872—24no—
38#1817—24no—
39#1787—24no—
40C2×#340C2212yes—
41C2×#340C2214yes—
42C2×#336C2214yes—
43C23×Q8—28no—
44C23×Q8—22no—
45C23×Q8—21no—
46C23×Q8—22no—
47C23×Q8—22no—
48C23×Q8—22no—
49C23×Q8—22no—
50C23×Q8—44no—
51C23×Q8D414yes—
52C23×Q8C2311yescharacteristic
53C23×Q8C2312yes—
54C24×C4—21no—
55C24×C4—21no—
56C24×C4D411yescharacteristic
57C24×C4D411yescharacteristic
58C24×C4C2311yescharacteristic
59C22×Q16—24no—
60C22×Q16—24no—
61C22×Q16—28no—
62C22×Q16—24no—
63C22×Q16—44no—
64C23×C8C2312yes—
65C23×C8C2312yes—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8D414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—44no—
77C22×C4⋊C4D414yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C42—22no—
84C22×C42—22no—
85C22×C42—22no—
86C22×C42—22no—
87C22×C42—22no—
88C22×C42—22no—
89C22×C42D412yes—
90C22×C42C2312yes—
91#451—232no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—28no—
96C2×#103—24no—
97C2×#103—28no—
98C2×#103—44no—
99C2×#103—44no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8D414yes—
106#391—216no—
107#391—24no—
108#391—28no—
109#391—24no—
110#391—28no—
111#389—24no—
112#389—24no—
113#387—216no—
114#385—28no—
115#381—28no—
116#380—22no—
117#380D414yes—
118C2×#95—28no—
119C2×#95—28no—
120C2×#95—24no—
121C2×#95—24no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126#340—24no—
127#340—44no—
128#336—28no—
129C25C2×D411yescharacteristic
130C22×Q8—28no—
131C22×Q8—216no—
132C22×Q8—216no—
133C22×Q8—24no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—24no—
137C22×Q8—28no—
138C22×Q8—28no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—21no—
147C22×Q8—24no—
148C22×Q8—24no—
149C22×Q8—24no—
150C22×Q8—24no—
151C22×Q8—48no—
152C22×Q8—416no—
153C22×Q8—416no—
154C22×Q8—416no—
155C22×Q8—416no—
156C22×Q8—44no—
157C22×Q8—48no—
158C22×Q8—42no—
159C22×Q8—42no—
160C22×Q8—44no—
161C22×Q8—44no—
162C22×Q8C2×D412yes—
163C22×Q8C2×D414yes—
164C22×Q8#4118yes—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—28no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—21no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—22no—
185C23×C4—42no—
186C23×C4—44no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4Q1614yescomplemented
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D412yes—
198C23×C4C2×D412yes—
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4C2×D412yes—
202C23×C4#4112yes—
203C23×C4C2411yescharacteristic
204C2×Q16—216no—
205C2×Q16—416no—
206C2×Q16—416no—
207C2×Q16—48no—
208C2×Q16—48no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—24no—
212C22×C8—24no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—24no—
217C22×C8—24no—
218C22×C8C2×D414yes—
219C22⋊Q8—28no—
220C22⋊Q8—28no—
221C22⋊Q8—44no—
222C22⋊Q8—44no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—48no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D414yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42—44no—
265C2×C42—44no—
266#103—216no—
267#103—416no—
268#103—48no—
269C22⋊C8—28no—
270C22⋊C8—44no—
271#95—216no—
272#95—216no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—24no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24#12012yes—
282C24#12012yes—
283C24#12112yes—
284C24C2×Q1612yes—
285C24#13512yes—
286C24#13712yes—
287C24#13712yes—
288C24C22×D411yescharacteristic
289C2×Q8—216no—
290C2×Q8—24no—
291C2×Q8—28no—
292C2×Q8—24no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—22no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—28no—
299C2×Q8—28no—
300C2×Q8—28no—
301C2×Q8—28no—
302C2×Q8—28no—
303C2×Q8—432no—
304C2×Q8—432no—
305C2×Q8—48no—
306C2×Q8—416no—
307C2×Q8—416no—
308C2×Q8—432no—
309C2×Q8—432no—
310C2×Q8—42no—
311C2×Q8—41no—
312C2×Q8—42no—
313C2×Q8—41no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—48no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×Q8#12318yes—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—216no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—28no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—44no—
364C22×C4—48no—
365C22×C4—416no—
366C22×C4—416no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—48no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—48no—
397C22×C4—88no—
398C22×C4—88no—
399C22×C4#12014yes—
400C22×C4#12012yes—
401C22×C4#12012yes—
402C22×C4#12114yes—
403C22×C4#12114yes—
404C22×C4#12112yes—
405C22×C4#12714yes—
406C22×C4C2×Q1614yes—
407C22×C4#13714yes—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4114yes—
415C22×C4H(5,𝔽2)14yes—
416Q16—432no—
417C2×C8—24no—
418C2×C8—28no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—44no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—44no—
426C2×C8—44no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—28no—
432C4⋊C4—416no—
433C4⋊C4—48no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—48no—
446C23—24no—
447C23—22no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—22no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—24no—
462C23#39214yes—
463C23#45114yes—
464C23#45114yes—
465C23#45214yes—
466C23#45214yes—
467C23#46714yes—
468C23#47014yes—
469C23C2×#12011yescharacteristic
470C23C2×#12011yescharacteristic
471C23C2×#12111yescharacteristic
472C23#52512yes—
473C23#53512yes—
474C23C22×Q1611yescharacteristic
475C23C2×#13511yescharacteristic
476C23C2×#13712yes—
477C23C2×#13711yescharacteristic
478C23C2×#13711yescharacteristic
479C23#57512yes—
480Q8—28no—
481Q8—28no—
482Q8—432no—
483Q8—44no—
484Q8—44no—
485Q8—48no—
486Q8—48no—
487Q8—44no—
488Q8—44no—
489Q8—82no—
490Q8—82no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—416no—
507C2×C4—432no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—416no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—816no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—81no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4#39214yes—
545C2×C4C2×#12011yescharacteristic
546C2×C4C2×#12111yescharacteristic
547C2×C4C2×#12111yescharacteristic
548C2×C4C2×#12312yes—
549C2×C4#52512yes—
550C2×C4C2×#12711yescharacteristic
551C2×C4#53412yes—
552C2×C4#53412yes—
553C2×C4#54614yes—
554C2×C4#54614yes—
555C2×C4#54614yes—
556C2×C4#55114yes—
557C8—44no—
558C8—82no—
559C22—24no—
560C22—24no—
561C22—28no—
562C22—28no—
563C22—28no—
564C22—24no—
565C22—24no—
566C22—28no—
567C22—24no—
568C22—24no—
569C22#200118yes—
570C22C2×#39211yescharacteristic
571C22#240412yes—
572C22#241112yes—
573C22#261311yescharacteristic
574C22#284911yescharacteristic
575C22C2×#45111yescharacteristic
576C22C2×#45111yescharacteristic
577C22C2×#45211yescharacteristic
578C22C2×#45211yescharacteristic
579C22#300411yescharacteristic
580C22#300511yescharacteristic
581C22#300712yes—
582C22C2×#46711yescharacteristic
583C22C2×#47011yescharacteristic
584C22#305612yes—
585C22#310412yes—
586C22#310812yes—
587C22#311312yes—
588C22#311712yes—
589C22#320912yes—
590C22#321112yes—
591C22#321512yes—
592C22#321812yes—
593C4—24no—
594C4—48no—
595C4—44no—
596C4—44no—
597C4—82no—
598C4—82no—
599C4—84no—
600C4—84no—
601C4#240512yes—
602C4#241212yes—
603C4#253412yes—
604C4#253912yes—
605C2—28no—
606C2#1491111yescharacteristic
607C2C2×#200111yescharacteristic
608C2#2339211yescharacteristic
609C2#2393412yes—
610C2#2393712yes—
611C2#2450212yes—
612C2#2450812yes—
613C2#2962611yescharacteristic
614C2#2962711yescharacteristic
615C2#3300811yescharacteristic
616C2#3301011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
7. #11067C2semidirect
8. #11024C2semidirect
193. C23×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C22×#4512
C2×#28552
C22×#3912
C2×#18722
C2×#18172
C2×#17872
#110672
#110242
#107302