Finite Group Databasebeta

← #404930

Subgroups — #404930

Order 512.

1342 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2346C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#1947C211yesmaximal, characteristic, complemented
4C2×#1854C212yesmaximal
5#15150C211yesmaximal, characteristic
6#11228C211yesmaximal, characteristic, complemented
7#11220C212yesmaximal, complemented
8#11200C212yesmaximal
9#11042C211yesmaximal, characteristic
10#10907C212yesmaximal, complemented
11#10806C211yesmaximal, characteristic, complemented
12C22×#117C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2212yes—
17C2×#397—21no—
18C2×#397—21no—
19#2346—24no—
20#2346—24no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#388—21no—
24C2×#388—21no—
25C2×#384C2211yescharacteristic
26C2×#382—21no—
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×#376—21no—
33#2084—24no—
34#1947—24no—
35#1854—28no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1445—21no—
45#1445—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117D414yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4D411yescharacteristic
71C22×C22⋊C4D411yescharacteristic
72C22×C42—21no—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#107—24no—
78C2×#107—24no—
79C2×#107—24no—
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—24no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—22no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88#397—44no—
89#397—44no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389D414yes—
94#388—28no—
95#388—28no—
96#384—22no—
97#384—22no—
98#382—28no—
99#380—22no—
100#380—22no—
101#380—24no—
102#380—24no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4D414yes—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109#376—28no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95D411yescharacteristic
121C2×#95D411yescharacteristic
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—24no—
128#336—24no—
129#336—22no—
130C25C2×D411yescharacteristic
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8#4114yes—
166C22×C8#4114yes—
167#117—28no—
168#117—28no—
169#117—28no—
170#117—28no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
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—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4#4112yes—
212C2×C4⋊C4#4112yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—82no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244#107—48no—
245#106—48no—
246C4⋊C8—28no—
247C22⋊C8—216no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—22no—
254#95—22no—
255#95—42no—
256#95—44no—
257#95—44no—
258#95—42no—
259#95—44no—
260C24—21no—
261C24—21no—
262C24—22no—
263C24—22no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24#12112yes—
271C24C22×D411yescharacteristic
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4#12012yes—
346C22×C4#12012yes—
347C22×C4#12112yes—
348C22×C4#13512yes—
349C22×C4AGL(1,ℤ/8)12yes—
350C22×C4AGL(1,ℤ/8)12yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4112yes—
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4112yes—
359C22×C42−1+412yes—
360C22×C42−1+412yes—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C4⋊C4—84no—
398C4⋊C4—84no—
399C4⋊C4#12114yes—
400C4⋊C4#12114yes—
401C22⋊C4—216no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—84no—
405C22⋊C4—88no—
406C42—216no—
407C42—22no—
408C42—22no—
409C42—24no—
410C42—48no—
411C42—41no—
412C42—41no—
413C42—42no—
414C42—48no—
415C23—22no—
416C23—24no—
417C23—22no—
418C23—22no—
419C23—24no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—41no—
429C23—41no—
430C23—42no—
431C23—41no—
432C23—41no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23C2×#12011yescharacteristic
439C23C2×#12111yescharacteristic
440C23#52511yescharacteristic
441C23#54712yes—
442C23#54712yes—
443C23C2×#13511yescharacteristic
444C23C2×AGL(1,ℤ/8)11yescharacteristic
445C23C2×AGL(1,ℤ/8)11yescharacteristic
446C23#57511yescharacteristic
447C23#57712yes—
448C23#57812yes—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
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#44914yes—
516C2×C4C2×#12111yescharacteristic
517C2×C4C2×#12111yescharacteristic
518C2×C4#52511yescharacteristic
519C2×C4#52511yescharacteristic
520C2×C4#52512yes—
521C2×C4#52511yescharacteristic
522C2×C4#52912yes—
523C2×C4#53211yescharacteristic
524C2×C4#53611yescharacteristic
525C2×C4#53611yescharacteristic
526C2×C4#53612yes—
527C2×C4#54112yes—
528C2×C4#54712yes—
529C2×C4#54812yes—
530C2×C4#55212yes—
531C2×C4#55212yes—
532C2×C4#55212yes—
533C2×C4#55612yes—
534C8—44no—
535C8—44no—
536C22—44no—
537C22—44no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—81no—
542C22—81no—
543C22—82no—
544C22—82no—
545C22—81no—
546C22—81no—
547C22#240611yescharacteristic
548C22#241312yes—
549C22#243211yescharacteristic
550C22#247812yes—
551C22#284711yescharacteristic
552C22C2×#44911yescharacteristic
553C22#300611yescharacteristic
554C22#300911yescharacteristic
555C22#301111yescharacteristic
556C22#303511yescharacteristic
557C22#303811yescharacteristic
558C22#304612yes—
559C22#310611yescharacteristic
560C22#311511yescharacteristic
561C22#312012yes—
562C22#333612yes—
563C22#334012yes—
564C22#334312yes—
565C22#334312yes—
566C22#334812yes—
567C22#334912yes—
568C22#335412yes—
569C22#335412yes—
570C4—24no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—44no—
575C4—44no—
576C4—82no—
577C4—81no—
578C4—82no—
579C4—162no—
580C4—162no—
581C4#239211yescharacteristic
582C4#241512yes—
583C4#247512yes—
584C4#259911yescharacteristic
585C4#261111yescharacteristic
586C4#262111yescharacteristic
587C2—82no—
588C2#1490711yescharacteristic
589C2#2409411yescharacteristic
590C2#2410411yescharacteristic
591C2#2458612yes—
592C2#2459412yes—
593C2#2570011yescharacteristic
594C2#2571611yescharacteristic
595C2#2746712yes—
596C2#2747812yes—
597C2#3290411yescharacteristic
598C2#3291611yescharacteristic
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×#2084C2semidirect
3. C2×#1947C2semidirect
6. #11228C2semidirect
7. #11220C2semidirect
10. #10907C2semidirect
11. #10806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23462
C2×#20842
C2×#19472
C2×#18542
#151502
#112282
#112202
#112002
#110422
#109072
#108062