Finite Group Databasebeta

← #6497529

Subgroups — #6497529

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3236C212yesmaximal
2C2×#2859C211yesmaximal, characteristic
3C4×#533C211yesmaximal, characteristic
4#33200C214yesmaximal, complemented
5#29871C214yesmaximal, complemented
6#28725C212yesmaximal, complemented
7#28723C212yesmaximal, complemented
8#28693C212yesmaximal, complemented
9#28692C212yesmaximal, complemented
10#27653C214yesmaximal
11#27648C212yesmaximal
12#27647C212yesmaximal
13#21838C212yesmaximal
14#20810C211yesmaximal, characteristic
15C2×#533C2211yescharacteristic
16C22×#125C2212yes—
17#3236—216no—
18C2×#501C2214yes—
19C2×#491C2214yes—
20C2×#484C2214yes—
21C2×#483C2214yes—
22C2×#479C2218yes—
23C2×#432C2212yes—
24#2859—21no—
25#2859—21no—
26#2859—21no—
27#2859—21no—
28#2859C2212yes—
29#2859C2212yes—
30#2859C2212yes—
31#2859C2212yes—
32C4×C4⋊Q8C2211yescharacteristic
33C4×C4⋊Q8C2211yescharacteristic
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2211yescharacteristic
36C4×#125—21no—
37C4×#125—21no—
38C4×C22⋊Q8—21no—
39C4×C22⋊Q8—21no—
40C4×C22⋊Q8C2212yes—
41C4×C22⋊Q8C2212yes—
42#2296—21no—
43#2295—21no—
44#2295C2212yes—
45C2×#389—21no—
46C2×#389—21no—
47C2×#389—21no—
48C2×#389—21no—
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C2×#384C2211yescharacteristic
52C2×#382C2212yes—
53C42×Q8—21no—
54C42×Q8C2212yes—
55C4×#117—21no—
56C2×C43C2211yescharacteristic
57#2085C2218yes—
58#2084C2214yes—
59#2083C2214yes—
60#2074C2218yes—
61#2072C2214yes—
62#2072C2214yes—
63#1992C2214yes—
64#1992C2214yes—
65#1989C2218yes—
66#1949C2218yes—
67#1948C2214yes—
68#1947C2214yes—
69#1842C2212yes—
70#1831C2212yes—
71#1824C2214yes—
72#1823C2214yes—
73#1805C2218yes—
74#1803C2214yes—
75#1803C2214yes—
76#1797C2212yes—
77#1797C2212yes—
78#1767C2212yes—
79#1767C2212yes—
80#533—24no—
81#533D418yes—
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2311yescharacteristic
84C2×#125—21no—
85C2×#125—22no—
86C2×#125—22no—
87C2×#125—28no—
88C2×#125—28no—
89C2×#125C2314yes—
90C2×#125C2314yes—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8C2312yes—
93C2×C4×Q8—21no—
94C2×C4×Q8—21no—
95C2×C4×Q8—21no—
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2312yes—
98C2×C4×Q8C2312yes—
99C2×#117—21no—
100C2×#117—21no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—22no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C42D411yescharacteristic
110C22×C42D411yescharacteristic
111C22×C42D411yescharacteristic
112C22×C42D411yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#501—216no—
117#491—216no—
118#484—216no—
119#483—216no—
120#479—232no—
121#432—28no—
122C2×#107C2318yes—
123C2×#106C2318yes—
124C2×C4⋊C8C2314yes—
125C2×C4⋊C8C2314yes—
126C2×C4⋊C8C2314yes—
127C2×#103C2318yes—
128C2×#103C2318yes—
129C2×C22⋊C8C2314yes—
130C2×#97C2314yes—
131#391—21no—
132#391—21no—
133#391C2312yes—
134#391C2312yes—
135#389—21no—
136#389—21no—
137#389—21no—
138#389—21no—
139#389—21no—
140#389—21no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389C2312yes—
158#389C2312yes—
159#387—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#385—22no—
164#385—22no—
165#384—21no—
166#384—21no—
167#384—21no—
168#384—21no—
169#384C2311yescharacteristic
170#384C2311yescharacteristic
171#384C2312yes—
172#384C2312yes—
173#384C2312yes—
174#384C2312yes—
175#382—21no—
176#382—22no—
177#382—22no—
178#382C2314yes—
179#382C2314yes—
180#380—21no—
181#380C2312yes—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200#376—22no—
201C2×#95—22no—
202C2×#95—22no—
203C43—22no—
204C43D411yescharacteristic
205C43D411yescharacteristic
206C43D411yescharacteristic
207C43D411yescharacteristic
208C43C2311yescharacteristic
209C43C2311yescharacteristic
210C43C2311yescharacteristic
211C43C2311yescharacteristic
212#340C2318yes—
213#340C2318yes—
214#340C2318yes—
215#340C2318yes—
216#336C2314yes—
217#336C2314yes—
218C22×Q8—21no—
219C22×Q8C2412yes—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2411yescharacteristic
229C22×C8C2418yes—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8C2×D414yes—
233C4⋊Q8C2×D414yes—
234#125—28no—
235#125—28no—
236#125—28no—
237#125—28no—
238#125—28no—
239#125—48no—
240#125—48no—
241C22⋊Q8—28no—
242C22⋊Q8—24no—
243C22⋊Q8C2×D418yes—
244C4×Q8—28no—
245C4×Q8—24no—
246C4×Q8—28no—
247C4×Q8—28no—
248C4×Q8—44no—
249C4×Q8—44no—
250C4×Q8C2×D418yes—
251#117—28no—
252#117—44no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—48no—
288C2×C4⋊C4—48no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4C2414yes—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D414yes—
332C2×C42#4114yes—
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339#107—216no—
340#106—216no—
341C4⋊C8—28no—
342C4⋊C8—28no—
343C4⋊C8—28no—
344#103—216no—
345#103—216no—
346C22⋊C8—28no—
347#97—28no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362C24C22×D411yescharacteristic
363C2×Q8—24no—
364C2×Q8—22no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8C22×D414yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4#12114yes—
400C22×C4#12114yes—
401C22×C4#13714yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D412yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C2×#4112yes—
417C22×C4C2511yescharacteristic
418C2×C8—28no—
419C2×C8—28no—
420C2×C8—28no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—28no—
425C4⋊C4—28no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4C22×D414yes—
456C22⋊C4—28no—
457C22⋊C4—44no—
458C42—28no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—44no—
482C42—44no—
483C42—44no—
484C42—44no—
485C42—44no—
486C42#12114yes—
487C42#12114yes—
488C42C22×D412yes—
489C42C22×D412yes—
490C42C2×#4112yes—
491C42C2×#4112yes—
492C422−1+414yes—
493C422−1+414yes—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23C2×#12112yes—
499C23C2×#13712yes—
500C23#57512yes—
501C23C23×D411yescharacteristic
502Q8—28no—
503Q8—48no—
504Q8—44no—
505C2×C4—28no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—28no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—88no—
558C2×C4—88no—
559C2×C4C2×#12112yes—
560C2×C4C2×#12112yes—
561C2×C4C2×#12112yes—
562C2×C4C2×#12112yes—
563C2×C4C2×#12112yes—
564C2×C4#53612yes—
565C2×C4#53612yes—
566C2×C4#53612yes—
567C2×C4#53612yes—
568C2×C4C2×#13712yes—
569C2×C4C2×#13712yes—
570C2×C4#57512yes—
571C2×C4#57512yes—
572C2×C4C23×D411yescharacteristic
573C2×C4C22×#4111yescharacteristic
574C2×C4C2×2−1+412yes—
575C2×C4C2×2−1+412yes—
576C8—48no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22#303714yes—
584C22C22×#12111yescharacteristic
585C22C2×#53611yescharacteristic
586C22C2×#53611yescharacteristic
587C22C22×#13711yescharacteristic
588C22C2×#57511yescharacteristic
589C22C2×#57511yescharacteristic
590C22C2×#57511yescharacteristic
591C4—22no—
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—22no—
601C4—41no—
602C4—41no—
603C4—44no—
604C4—44no—
605C4—82no—
606C4—82no—
607C4—84no—
608C4—84no—
609C4#323614yescomplemented
610C4#349912yes—
611C4#350912yes—
612C2—24no—
613C2#3531611yescharacteristic
614C2C2×#303711yescharacteristic
615C2#6041511yescharacteristic
616C2C2×#323611yescharacteristic
617C2#6085411yescharacteristic
618C2#6086111yescharacteristic
619C2#6086511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #33200C2semidirect
5. #29871C2semidirect
6. #28725C2semidirect
7. #28723C2semidirect
8. #28693C2semidirect
9. #28692C2semidirect
609. C4#3236semidirect

Maximal subgroups

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

subgroupindex
C2×#32362
C2×#28592
C4×#5332
#332002
#298712
#287252
#287232
#286932
#286922
#276532
#276482
#276472
#218382
#208102