Finite Group Databasebeta

← #467116

Subgroups — #467116

Order 512.

1404 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2492C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic, complemented
3C2×#1987C211yesmaximal, characteristic, complemented
4#15125C211yesmaximal, characteristic
5#11384C212yesmaximal, complemented
6#11200C212yesmaximal
7#11190C212yesmaximal, complemented
8#11077C211yesmaximal, characteristic, complemented
9#11042C211yesmaximal, characteristic
10#10970C212yesmaximal
11#10807C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2212yes—
19#2492—24no—
20#2492—24no—
21C2×#400—21no—
22C2×#393—21no—
23C2×#393—21no—
24C2×#391—21no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#388—21no—
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#2024—24no—
33#1987—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1446—21no—
43#1442—21no—
44#1439—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×Q8—21no—
49C23×Q8C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—42no—
60C2×C4⋊Q8—42no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—42no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4D411yescharacteristic
77C22×C22⋊C4D411yescharacteristic
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×C4⋊C8—22no—
82C2×C4⋊C8—42no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#103—42no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—42no—
91C2×C22⋊C8—42no—
92#400—28no—
93#393—28no—
94#393—44no—
95#391—28no—
96#391—22no—
97#391—22no—
98#391—22no—
99#391D414yes—
100#388—28no—
101#380—22no—
102#380—22no—
103#380—24no—
104#380—42no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4D414yes—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95D411yescharacteristic
121C2×#95D411yescharacteristic
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—42no—
127#340—42no—
128#336—22no—
129#336—42no—
130C25C2×D411yescharacteristic
131C22×Q8—24no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—44no—
142C22×Q8—44no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—82no—
149C22×Q8—82no—
150C22×Q8C2×D412yes—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C4⋊Q8—28no—
188C4⋊Q8—48no—
189C4⋊Q8—44no—
190C4⋊Q8—48no—
191C4⋊Q8—44no—
192C4⋊Q8—44no—
193C22⋊Q8—48no—
194C22⋊Q8—44no—
195C22⋊Q8—44no—
196C2×C4⋊C4—21no—
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—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4#4114yes—
227C2×C4⋊C4#4114yes—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—82no—
240C2×C22⋊C4Q1614yes—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—82no—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C4⋊C8—44no—
252#103—48no—
253#103—48no—
254C22⋊C8—48no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—22no—
262#95—22no—
263#95—44no—
264#95—42no—
265#95—44no—
266#95—42no—
267#95—82no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—22no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—22no—
277C24—22no—
278C24C2×Q1612yes—
279C24C22×D411yescharacteristic
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—48no—
285C2×Q8—48no—
286C2×Q8—48no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—88no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×Q8—84no—
305C2×Q8—88no—
306C2×Q8—88no—
307C2×Q8—88no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
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#12012yes—
380C22×C4#12012yes—
381C22×C4C2×Q1612yes—
382C22×C4C2×Q1612yes—
383C22×C4AGL(1,ℤ/8)12yes—
384C22×C4#13712yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C2×#4112yes—
388C22×C4C2×#4112yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C4H(5,𝔽2)12yes—
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C42−1+411yescharacteristic
393C22×C42−1+411yescharacteristic
394C22×C42−1+411yescharacteristic
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—41no—
420C4⋊C4—42no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4#12414yes—
430C22⋊C4—28no—
431C22⋊C4—44no—
432C22⋊C4—48no—
433C22⋊C4—84no—
434C22⋊C4—88no—
435C42—22no—
436C42—22no—
437C42—24no—
438C42—41no—
439C42—41no—
440C42—42no—
441C42—84no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—24no—
446C23—24no—
447C23—42no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23—41no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23C2×#12011yescharacteristic
466C23#53611yescharacteristic
467C23#53711yescharacteristic
468C23#54612yes—
469C23#54612yes—
470C23C22×Q1611yescharacteristic
471C23C2×AGL(1,ℤ/8)11yescharacteristic
472C23C2×#13711yescharacteristic
473C23#57511yescharacteristic
474C23#57712yes—
475C23#57712yes—
476Q8—44no—
477Q8—44no—
478Q8—816no—
479Q8—816no—
480Q8—81no—
481Q8—81no—
482Q8—81no—
483Q8—81no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4#45114yes—
549C2×C4#52511yescharacteristic
550C2×C4C2×#12411yescharacteristic
551C2×C4#53511yescharacteristic
552C2×C4#53511yescharacteristic
553C2×C4#53611yescharacteristic
554C2×C4#53611yescharacteristic
555C2×C4#53711yescharacteristic
556C2×C4#53711yescharacteristic
557C2×C4#54212yes—
558C2×C4#54212yes—
559C2×C4#54212yes—
560C2×C4#54212yes—
561C2×C4#54612yes—
562C2×C4#54612yes—
563C2×C4#55612yes—
564C2×C4#55912yes—
565C2×C4H(3,𝔽4)12yes—
566C2×C4#56212yes—
567C8—82no—
568C8—82no—
569C22—44no—
570C22—42no—
571C22—44no—
572C22—42no—
573C22—42no—
574C22—81no—
575C22—82no—
576C22—81no—
577C22—82no—
578C22—81no—
579C22—81no—
580C22#240411yescharacteristic
581C22#243511yescharacteristic
582C22#261611yescharacteristic
583C22#270012yes—
584C22#271212yes—
585C22C2×#45111yescharacteristic
586C22#300611yescharacteristic
587C22#300911yescharacteristic
588C22#301011yescharacteristic
589C22#323011yescharacteristic
590C22#323711yescharacteristic
591C22#324412yes—
592C22#324911yescharacteristic
593C22#325411yescharacteristic
594C22#326312yes—
595C22#329912yes—
596C22#329912yes—
597C22#330612yes—
598C22#331112yes—
599C22#331112yes—
600C22#331612yes—
601C22#332412yes—
602C22#332912yes—
603C4—24no—
604C4—42no—
605C4—42no—
606C4—44no—
607C4—42no—
608C4—82no—
609C4—82no—
610C4—84no—
611C4—81no—
612C4—82no—
613C4—162no—
614C4#259211yescharacteristic
615C4#263211yescharacteristic
616C4#264011yescharacteristic
617C4#270012yes—
618C4#271212yes—
619C4#281911yescharacteristic
620C2—82no—
621C2#1655911yescharacteristic
622C2#2397011yescharacteristic
623C2#2397911yescharacteristic
624C2#2588811yescharacteristic
625C2#2589811yescharacteristic
626C2#2969111yescharacteristic
627C2#2969711yescharacteristic
628C2#3174412yes—
629C2#3176512yes—
630C2#3211412yes—
631C2#3212612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2024C2semidirect
3. C2×#1987C2semidirect
5. #11384C2semidirect
7. #11190C2semidirect
8. #11077C2semidirect
11. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24922
C2×#20242
C2×#19872
#151252
#113842
#112002
#111902
#110772
#110422
#109702
#108072