Finite Group Databasebeta

← #204654

Subgroups — #204654

Order 512.

1091 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35326C211yesmaximal, characteristic
2#20469C211yesmaximal, characteristic, complemented
3C2×#1521C211yesmaximal, characteristic
4C2×#1515C211yesmaximal, characteristic, complemented
5C2×#1509C211yesmaximal, characteristic
6C2×#1503C211yesmaximal, characteristic, complemented
7#13952C211yesmaximal, characteristic, complemented
8#9683C212yesmaximal, complemented
9#9682C212yesmaximal
10#9681C212yesmaximal
11#9679C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C2×#529C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2857C2212yes—
17#2850C2212yes—
18#2660C411yescharacteristic, complemented
19#2658C411yescharacteristic, complemented
20#2642C411yescharacteristic, complemented
21#2640C411yescharacteristic, complemented
22#2611—22no—
23#2604—21no—
24#2604—21no—
25#2564—21no—
26#2564—21no—
27#2498—22no—
28#2327C412yescomplemented
29#2326C412yescomplemented
30C2×#383C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1768—22no—
34#1749C2212yes—
35C4×C22⋊C8C2212yes—
36#1746—21no—
37#1726—21no—
38#1521—24no—
39#1515—24no—
40#1509—24no—
41#1503—24no—
42C2×#328C2212yes—
43C2×#328C2212yes—
44C2×#328C2212yes—
45C2×#328C2212yes—
46C2×#328C2212yes—
47C2×#328C2211yescharacteristic
48C2×#328C2211yescharacteristic
49C2×#327C2212yes—
50C2×#327C2211yescharacteristic
51C2×#327C2211yescharacteristic
52#1444—21no—
53#1442—21no—
54#1436—22no—
55C4⋊#95C2212yes—
56#1434C2212yes—
57#533—22no—
58#533—22no—
59#533—22no—
60#533—22no—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C412yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64#529—22no—
65#529—22no—
66#529—22no—
67#529—22no—
68C2×#126—21no—
69C2×#125—21no—
70C2×#125C2×C411yescharacteristic
71C2×#124C2×C411yescharacteristic
72C2×#124C2×C412yes—
73C2×#123—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8C2312yes—
76C2×C22⋊Q8C2311yescharacteristic
77C2×#121C2311yescharacteristic
78C2×C4×Q8—21no—
79C2×C4×Q8C2×C412yes—
80C2×C4×Q8C2×C411yescharacteristic
81C2×C4×D4—21no—
82C2×C4×D4C2×C411yescharacteristic
83C2×#117—21no—
84C2×#117—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—22no—
87C22×C42—21no—
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90#432—22no—
91#432D414yes—
92#431—22no—
93#431D414yes—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C22⋊C8C2311yescharacteristic
97C2×C22⋊C8C2311yescharacteristic
98C2×C22⋊C8C2312yes—
99C2×#97—21no—
100C2×#97—22no—
101C2×C4×C8—21no—
102C2×C4×C8—22no—
103#400—21no—
104#400—21no—
105#400—22no—
106#399—21no—
107#399—21no—
108#399—22no—
109#398—22no—
110#397—22no—
111#395—21no—
112#395—21no—
113#393—21no—
114#393—21no—
115#391C2×C412yes—
116#391C2×C412yes—
117#391C2×C412yes—
118#390C2×C412yes—
119#389—22no—
120#389—22no—
121#388—22no—
122#388—22no—
123#388—22no—
124#388—22no—
125#387—22no—
126#387—22no—
127#387—22no—
128#387—22no—
129#386—22no—
130#386—22no—
131#385—22no—
132#385—22no—
133#384C2×C412yes—
134#383—22no—
135#383—22no—
136#382C2×C412yes—
137#380—22no—
138#380—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—22no—
141#376C2×C412yes—
142#376C2×C411yescharacteristic
143#376C2×C411yescharacteristic
144#376D411yescharacteristic
145#376D411yescharacteristic
146#376D412yes—
147C2×#95—22no—
148C2×#95—22no—
149C2×#95—21no—
150C2×#95—21no—
151C2×#95—22no—
152C2×#95D411yescharacteristic
153C2×#95D411yescharacteristic
154C2×#95D412yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158#336—22no—
159#336—22no—
160#328—24no—
161#328—24no—
162#328—24no—
163#328—24no—
164#328—24no—
165#328—22no—
166#328—22no—
167#327—24no—
168#327—22no—
169#327—22no—
170C22×Q8—21no—
171C22×Q8C22×C411yescharacteristic
172C22×Q8C22×C412yes—
173C22×D4—21no—
174C22×D4C22×C411yescharacteristic
175C23×C4—22no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191C4⋊Q8—44no—
192#126—42no—
193#126—42no—
194#125—22no—
195#125—22no—
196#125—42no—
197#125—42no—
198#124—22no—
199#124—22no—
200#124—44no—
201#123—42no—
202#123—42no—
203C22⋊Q8—21no—
204C22⋊Q8—21no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—42no—
212C22⋊Q8—42no—
213#121—21no—
214#121—21no—
215#121—21no—
216#121—21no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—42no—
222C4×Q8—42no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—42no—
226C4×D4—42no—
227#117—42no—
228#117—42no—
229#117—42no—
230#117—42no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—82no—
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C412yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42—44no—
265C2×C42—42no—
266C2×C42—81no—
267C2×C42—81no—
268C2×C42C22×C412yes—
269C2×C42C22×C412yes—
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42#4112yes—
279C2×C42#4112yes—
280C4⋊C8—24no—
281C4⋊C8—24no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—22no—
285C22⋊C8C2×D412yes—
286C22⋊C8C2×D412yes—
287C22⋊C8C2×D414yes—
288#97—24no—
289#97—44no—
290C4×C8—24no—
291C4×C8—44no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—21no—
302#95—21no—
303#95—22no—
304#95—21no—
305#95—21no—
306#95—44no—
307#95—44no—
308#95—42no—
309#95—42no—
310#95—82no—
311#95C22⋊C412yes—
312#95C22⋊C412yes—
313#95C22⋊C412yes—
314#95C22⋊C412yes—
315#95C22⋊C411yescharacteristic
316#95C22⋊C411yescharacteristic
317#95C22⋊C411yescharacteristic
318#95C22⋊C411yescharacteristic
319#95C22⋊C412yes—
320#95C22⋊C412yes—
321C24—21no—
322C24C22×D411yescharacteristic
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—82no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—42no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
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—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—82no—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C412yes—
387C22×C4#12412yes—
388C22×C4#12412yes—
389C22×C4#12712yes—
390C22×C4#12712yes—
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—44no—
411C4⋊C4—48no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—82no—
425C4⋊C4—82no—
426C4⋊C4—88no—
427C22⋊C4—44no—
428C22⋊C4—42no—
429C22⋊C4—41no—
430C22⋊C4—42no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—44no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C42—24no—
441C42—24no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—44no—
446C42—84no—
447C42C4×D412yes—
448C42C4×D412yes—
449C42C4×D412yes—
450C42C4×D412yes—
451C42#12112yes—
452C42#12112yes—
453C42#12112yes—
454C42#12112yes—
455C23—22no—
456C23—22no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—42no—
461C23—41no—
462C23C22×C22⋊C411yescharacteristic
463C23C2×#12411yescharacteristic
464C23C2×#12711yescharacteristic
465C23#57711yescharacteristic
466C23#57711yescharacteristic
467C23#57811yescharacteristic
468C23#57811yescharacteristic
469Q8—81no—
470Q8—81no—
471Q8—82no—
472Q8—82no—
473Q8—84no—
474D4—81no—
475D4—81no—
476D4—81no—
477D4—81no—
478D4—81no—
479D4—81no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—84no—
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—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4#39012yes—
545C2×C4#39012yes—
546C2×C4#39012yes—
547C2×C4#39012yes—
548C2×C4#41014yes—
549C2×C4C2×C4×D411yescharacteristic
550C2×C4C2×C4×D411yescharacteristic
551C2×C4C2×#12111yescharacteristic
552C2×C4C2×#12111yescharacteristic
553C8—42no—
554C8—42no—
555C8—44no—
556C22—21no—
557C22—21no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—21no—
563C22—21no—
564C22—81no—
565C22—82no—
566C22—82no—
567C22—81no—
568C22—81no—
569C22C2×#39011yescharacteristic
570C22C2×#41012yes—
571C22#289811yescharacteristic
572C22#289811yescharacteristic
573C22#290112yes—
574C22#313711yescharacteristic
575C22#313711yescharacteristic
576C22#315411yescharacteristic
577C22#315611yescharacteristic
578C4—22no—
579C4—22no—
580C4—24no—
581C4—22no—
582C4—22no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—84no—
591C4—81no—
592C4—82no—
593C4—81no—
594C4—82no—
595C4—81no—
596C4—161no—
597C4—161no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—161no—
603C2#1760912yes—
604C2#1761212yes—
605C2#2120511yescharacteristic
606C2#2121211yescharacteristic
607C2#2125311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20469C2semidirect
4. C2×#1515C2semidirect
6. C2×#1503C2semidirect
7. #13952C2semidirect
8. #9683C2semidirect
11. #9679C2semidirect
18. #2660C4semidirect
19. #2658C4semidirect
20. #2642C4semidirect
21. #2640C4semidirect
28. #2327C4semidirect
29. #2326C4semidirect

Maximal subgroups

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

subgroupindex
#353262
#204692
C2×#15212
C2×#15152
C2×#15092
C2×#15032
#139522
#96832
#96822
#96812
#96792