Finite Group Databasebeta

← #404968

Subgroups — #404968

Order 512.

1538 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2C2×#2071C211yesmaximal, characteristic
3C2×#1948C211yesmaximal, characteristic, complemented
4C2×#1864C212yesmaximal
5#15196C211yesmaximal, characteristic, complemented
6#11207C212yesmaximal, complemented
7#11202C211yesmaximal, characteristic, complemented
8#11189C212yesmaximal
9#11076C211yesmaximal, characteristic
10#10908C212yesmaximal, complemented
11#10806C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117—21no—
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2212yes—
18#2354—24no—
19#2354—24no—
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389C2212yes—
23C2×#388—21no—
24C2×#388—21no—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#380—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#2071—24no—
32#1948—24no—
33#1864—28no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1444—21no—
43#1444—21no—
44#1441—21no—
45#1441—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4D414yes—
58C2×#117—28no—
59C2×#117—44no—
60C2×#117—44no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
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—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C42—21no—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—24no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—22no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94#390—22no—
95#390—22no—
96#390—22no—
97#390D414yes—
98#389—24no—
99#389—24no—
100#388—28no—
101#388—28no—
102#384—22no—
103#384—22no—
104#384—22no—
105#384—22no—
106#380—28no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4D414yes—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95D411yescharacteristic
121C2×#95D411yescharacteristic
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—24no—
127#336—24no—
128#336—22no—
129#336—24no—
130C25—21no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4C2×D412yes—
139C22×D4C2×D412yes—
140C22×D4#4112yes—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8#4114yes—
174C4×D4—28no—
175C4×D4—28no—
176C4×D4—28no—
177C4×D4—28no—
178#117—432no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
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—24no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4SD1614yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4#4114yes—
222C2×C4⋊C4#4112yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42#4112yes—
256#106—48no—
257C4⋊C8—28no—
258#102—216no—
259#102—48no—
260#95—24no—
261#95—24no—
262#95—22no—
263#95—22no—
264#95—42no—
265#95—44no—
266#95—42no—
267#95—82no—
268#95—82no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—42no—
273C24—42no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—42no—
279C24—42no—
280C24C22×D411yescharacteristic
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—41no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4#12114yes—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—216no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—48no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—48no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—48no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
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—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4#12012yes—
361C22×C4#12012yes—
362C22×C4#12112yes—
363C22×C4#12112yes—
364C22×C4C2×SD1612yes—
365C22×C4C2×SD1612yes—
366C22×C4C2×SD1612yes—
367C22×C4AGL(1,ℤ/8)12yes—
368C22×C4#13712yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4112yes—
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4112yes—
377C22×C42−1+412yes—
378C22×C42−1+412yes—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—28no—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—44no—
403C4⋊C4—48no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4—88no—
416C4⋊C4—84no—
417C4⋊C4#12114yes—
418C22⋊C4—216no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—84no—
422C22⋊C4—84no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—24no—
430C42—48no—
431C42—41no—
432C42—41no—
433C42—42no—
434C42—48no—
435C23—24no—
436C23—42no—
437C23—42no—
438C23—44no—
439C23—42no—
440C23—44no—
441C23—42no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—44no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23C2×#12011yescharacteristic
459C23C2×#12111yescharacteristic
460C23#52511yescharacteristic
461C23#54712yes—
462C23#54812yes—
463C23C22×SD1611yescharacteristic
464C23C2×AGL(1,ℤ/8)11yescharacteristic
465C23C2×#13711yescharacteristic
466C23#57511yescharacteristic
467C23#57612yes—
468C23#57712yes—
469D4—28no—
470D4—42no—
471D4—42no—
472D4—42no—
473D4—42no—
474D4—44no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—432no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
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—82no—
536C2×C4—82no—
537C2×C4#45014yes—
538C2×C4#46114yes—
539C2×C4C2×#12111yescharacteristic
540C2×C4C2×#12111yescharacteristic
541C2×C4#52511yescharacteristic
542C2×C4#52511yescharacteristic
543C2×C4#52512yes—
544C2×C4#52511yescharacteristic
545C2×C4#52912yes—
546C2×C4#53211yescharacteristic
547C2×C4#53611yescharacteristic
548C2×C4#53611yescharacteristic
549C2×C4#53612yes—
550C2×C4#54112yes—
551C2×C4#54712yes—
552C2×C4#54712yes—
553C2×C4#55212yes—
554C2×C4#55212yes—
555C2×C4#55212yes—
556C2×C4#55612yes—
557C8—44no—
558C8—82no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—84no—
567C22—84no—
568C22—84no—
569C22—84no—
570C22#240611yescharacteristic
571C22#241512yes—
572C22#243211yescharacteristic
573C22#247512yes—
574C22#284711yescharacteristic
575C22C2×#45011yescharacteristic
576C22#300711yescharacteristic
577C22#300811yescharacteristic
578C22#301011yescharacteristic
579C22C2×#46111yescharacteristic
580C22#303611yescharacteristic
581C22#304512yes—
582C22#310111yescharacteristic
583C22#311011yescharacteristic
584C22#311912yes—
585C22#333512yes—
586C22#334512yes—
587C22#334712yes—
588C22#335712yes—
589C22#336212yes—
590C22#336612yes—
591C22#336912yes—
592C22#337612yes—
593C4—24no—
594C4—24no—
595C4—42no—
596C4—42no—
597C4—44no—
598C4—42no—
599C4—81no—
600C4—82no—
601C4—82no—
602C4—84no—
603C4—162no—
604C4#239211yescharacteristic
605C4#241312yes—
606C4#247812yes—
607C4#259911yescharacteristic
608C4#261111yescharacteristic
609C4#262111yescharacteristic
610C2—84no—
611C2#1490711yescharacteristic
612C2#2408911yescharacteristic
613C2#2410511yescharacteristic
614C2#2468012yes—
615C2#2468812yes—
616C2#2564911yescharacteristic
617C2#2566211yescharacteristic
618C2#2736012yes—
619C2#2738112yes—
620C2#3290511yescharacteristic
621C2#3291311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1948C2semidirect
5. #15196C2semidirect
6. #11207C2semidirect
7. #11202C2semidirect
10. #10908C2semidirect
11. #10806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#20712
C2×#19482
C2×#18642
#151962
#112072
#112022
#111892
#110762
#109082
#108062