Finite Group Databasebeta

← #204546

Subgroups — #204546

Order 512.

1171 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34972C211yesmaximal, characteristic
2#20546C211yesmaximal, characteristic, complemented
3C4×#424C211yesmaximal, characteristic, complemented
4C2×#1494C211yesmaximal, characteristic
5C2×#1483C211yesmaximal, characteristic
6C2×#1491C211yesmaximal, characteristic, complemented
7C22×Q8⋊C8C211yesmaximal, characteristic, complemented
8#9677C212yesmaximal, complemented
9#9676C212yesmaximal, complemented
10#9675C212yesmaximal, complemented
11#9673C212yesmaximal, complemented
12C2×#529C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#424—21no—
16C22×C4⋊C8C2211yescharacteristic
17C2×C4×M16C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2604—22no—
20#2565C412yes—
21#2545C412yes—
22#2503C411yescharacteristic
23#2501C411yescharacteristic
24#2436C411yescharacteristic
25#2434C411yescharacteristic
26#2413—21no—
27#2413—21no—
28#2396—22no—
29C4×C4⋊Q8C2212yes—
30C4×#124C2212yes—
31C4×#123—21no—
32C4×#123—21no—
33C4×#117C2211yescharacteristic
34#1771—22no—
35#1768—22no—
36#1767C2212yes—
37C4×C4⋊C8C2212yes—
38C4×C4⋊C8C2212yes—
39C4×C4⋊C8C2212yes—
40#1752—21no—
41C8×C22⋊C4—21no—
42#1729—21no—
43#1494—24no—
44#1491—24no—
45#1483—24no—
46C2×Q8⋊C8—24no—
47C2×Q8⋊C8C2212yes—
48C2×Q8⋊C8C2212yes—
49C2×Q8⋊C8C2212yes—
50C2×Q8⋊C8C2212yes—
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊C8C2211yescharacteristic
53C2×Q8⋊C8C2212yes—
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2212yes—
57C23×Q8C2×C411yescharacteristic
58C22×M16—21no—
59C23×C8—21no—
60C23×C8—21no—
61C2×C4⋊Q8C2×C412yes—
62#529—22no—
63#529—22no—
64#529—22no—
65#529—22no—
66C2×#126—21no—
67C2×#125C2×C411yescharacteristic
68C2×#124C2×C411yescharacteristic
69C2×#124C2×C412yes—
70C2×#123—21no—
71C2×#123—21no—
72C2×#123—21no—
73C2×C22⋊Q8C2×C411yescharacteristic
74C2×#121C2×C411yescharacteristic
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—22no—
78C2×C4×Q8—22no—
79C2×C4×Q8—22no—
80C2×C4×Q8—22no—
81C2×C4×Q8C2×C412yes—
82C2×C4×Q8C2311yescharacteristic
83C2×C4×Q8C2312yes—
84C2×C4×D4—21no—
85C2×C4×D4C2311yescharacteristic
86C2×#117—21no—
87C2×#117D411yescharacteristic
88C2×#117D411yescharacteristic
89C22×C4⋊C4—21no—
90C22×C4⋊C4C2×C411yescharacteristic
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93#432—22no—
94#432D414yes—
95#424—28no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8D414yes—
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2312yes—
104C2×C22⋊C8—21no—
105C4×M16—22no—
106C4×M16—22no—
107C2×#97—21no—
108C2×#97—21no—
109C2×#97—22no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8—21no—
113C2×C4×C8—22no—
114C2×C4×C8C2312yes—
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117#400—22no—
118#399—22no—
119#398—22no—
120#398—22no—
121#396—22no—
122#396—22no—
123#395—22no—
124#393—22no—
125#391C2×C412yes—
126#391C2×C412yes—
127#391C2×C412yes—
128#390C2×C412yes—
129#389—22no—
130#389—22no—
131#388—22no—
132#388—22no—
133#388—22no—
134#388—22no—
135#387—22no—
136#387—22no—
137#386—21no—
138#386—21no—
139#385—21no—
140#385—21no—
141#384C2×C412yes—
142#382C2×C412yes—
143#381—21no—
144#381—21no—
145C4×C4⋊C4—22no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4C2×C412yes—
148C4×C4⋊C4D412yes—
149C4×C22⋊C4—22no—
150C4×C22⋊C4—22no—
151C4×C22⋊C4C2×C411yescharacteristic
152C4×C22⋊C4C2×C411yescharacteristic
153C4×C22⋊C4D411yescharacteristic
154C4×C22⋊C4D411yescharacteristic
155#376D412yes—
156C43C2312yes—
157#336—22no—
158#336—22no—
159#336—22no—
160#336—22no—
161Q8⋊C8—24no—
162Q8⋊C8—24no—
163Q8⋊C8—22no—
164Q8⋊C8—22no—
165Q8⋊C8—24no—
166Q8⋊C8—24no—
167Q8⋊C8—24no—
168D4⋊C8—22no—
169D4⋊C8—22no—
170D4⋊C8—24no—
171C22×Q8—21no—
172C22×Q8—24no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—42no—
176C22×Q8C22×C411yescharacteristic
177C22×Q8C22×C412yes—
178C22×D4—21no—
179C22×D4C22×C411yescharacteristic
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4C22×C411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C2×M16—24no—
187C2×M16—42no—
188C2×M16—42no—
189C22×C8—24no—
190C22×C8—22no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—24no—
197C22×C8—42no—
198C22×C8—42no—
199C22×C8—82no—
200C4⋊Q8—24no—
201C4⋊Q8—42no—
202#126—42no—
203#126—42no—
204#125—22no—
205#125—22no—
206#124—24no—
207#124—22no—
208#124—22no—
209#124—42no—
210#123—44no—
211#123—42no—
212#123—42no—
213#123—44no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216#121—22no—
217#121—22no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—42no—
229C4×Q8—42no—
230C4×Q8—42no—
231C4×Q8—42no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—42no—
237C4×D4—42no—
238#117—24no—
239#117—42no—
240#117—42no—
241#117—42no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4C22⋊C412yes—
258C2×C4⋊C4C22⋊C412yes—
259C2×C4⋊C4C22×C412yes—
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C42—22no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42C22×C412yes—
281C2×C42C22×C412yes—
282C2×C42C22×C411yescharacteristic
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42#4112yes—
292C2×C42#4112yes—
293C2×C42C2411yescharacteristic
294C4⋊C8—28no—
295C4⋊C8—28no—
296C4⋊C8—24no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—22no—
300C4⋊C8C2×D412yes—
301C4⋊C8C2×D412yes—
302C4⋊C8C2×D414yes—
303C22⋊C8—24no—
304#97—24no—
305#97—42no—
306#97—82no—
307C4×C8—22no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—22no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—42no—
314C4×C8—82no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95C22⋊C412yes—
324#95C22⋊C412yes—
325#95C22⋊C412yes—
326#95C22⋊C412yes—
327C24—21no—
328C24C2×C22⋊C411yescharacteristic
329C2×Q8—22no—
330C2×Q8—28no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—44no—
340C2×Q8—48no—
341C2×Q8—44no—
342C2×Q8—42no—
343C2×Q8#13114yes—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×C22⋊C412yes—
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C412yes—
378C22×C4C4×D412yes—
379C22×C4C4×D412yes—
380C22×C4#12112yes—
381C22×C4#12112yes—
382C22×C4#12412yes—
383C22×C4#12712yes—
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387M16—48no—
388C2×C8—24no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—24no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—42no—
395C2×C8—48no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—42no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—84no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4—41no—
419C4⋊C4—42no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—42no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—44no—
440C22⋊C4#10414yes—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—24no—
450C42—42no—
451C42—42no—
452C42—42no—
453C42—41no—
454C42—41no—
455C42—44no—
456C42C4×D412yes—
457C42C4×D412yes—
458C42#12112yes—
459C42#12112yes—
460C42#12412yes—
461C42#12712yes—
462C42C22×D411yescharacteristic
463C42C22×D411yescharacteristic
464C42C2×#4111yescharacteristic
465C42C2×#4111yescharacteristic
466C23—22no—
467C23—22no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23#39012yes—
474C23C2×#10412yes—
475C23#42112yes—
476C23C22×C22⋊C411yescharacteristic
477Q8—24no—
478Q8—24no—
479Q8—48no—
480Q8—42no—
481Q8—42no—
482Q8—41no—
483Q8—41no—
484Q8—42no—
485D4—41no—
486D4—41no—
487D4—42no—
488D4—84no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—48no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—84no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4#39012yes—
533C2×C4#39012yes—
534C2×C4#39012yes—
535C2×C4C2×#10414yes—
536C2×C4C2×C4×D411yescharacteristic
537C2×C4C2×C4×D411yescharacteristic
538C2×C4C2×#12111yescharacteristic
539C2×C4C2×#12111yescharacteristic
540C2×C4C2×#12411yescharacteristic
541C2×C4C2×#12711yescharacteristic
542C2×C4C2×#13112yes—
543C2×C4#56812yes—
544C2×C4#57712yes—
545C2×C4#57812yes—
546C8—44no—
547C8—42no—
548C8—42no—
549C8—42no—
550C8—42no—
551C8—44no—
552C22—22no—
553C22—24no—
554C22—24no—
555C22—22no—
556C22—81no—
557C22—81no—
558C22—82no—
559C22—82no—
560C22—81no—
561C22#198014yes—
562C22C2×#39011yescharacteristic
563C22#288012yes—
564C22#289812yes—
565C22C22×#10411yescharacteristic
566C22C2×#42111yescharacteristic
567C4—21no—
568C4—21no—
569C4—22no—
570C4—22no—
571C4—24no—
572C4—22no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—44no—
577C4—82no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4#292911yescharacteristic
582C4#293011yescharacteristic
583C4#293312yes—
584C4#304211yescharacteristic
585C4#304411yescharacteristic
586C4#304612yes—
587C2—24no—
588C2—161no—
589C2#1751912yes—
590C2#1752412yes—
591C2#2119111yescharacteristic
592C2C2×#198011yescharacteristic
593C2#2126211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20546C2semidirect
3. C4×#424C2semidirect
6. C2×#1491C2semidirect
7. C22×Q8⋊C8C2semidirect
8. #9677C2semidirect
9. #9676C2semidirect
10. #9675C2semidirect
11. #9673C2semidirect

Maximal subgroups

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

subgroupindex
#349722
#205462
C4×#4242
C2×#14942
C2×#14832
C2×#14912
C22×Q8⋊C82
#96772
#96762
#96752
#96732