Finite Group Databasebeta

← #481491

Subgroups — #481491

Order 512.

1124 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34634C211yesmaximal, characteristic
2C2×#1740C211yesmaximal, characteristic
3C2×#1458C211yesmaximal, characteristic
4#11718C212yesmaximal, complemented
5#11717C212yesmaximal, complemented
6#11716C214yesmaximal
7#11715C214yesmaximal, complemented
8C2×#533C411yescharacteristic, complemented
9C22×#117—21no—
10C22×#117C2211yescharacteristic
11C2×#431—21no—
12#2629—21no—
13#2629—21no—
14#2628C411yescharacteristic, complemented
15#2628C411yescharacteristic, complemented
16#2628C2211yescharacteristic
17#2628C2211yescharacteristic
18#2627—21no—
19#2627C412yescomplemented
20#2626—21no—
21#2626C2212yes—
22#2396—21no—
23#2396—21no—
24#2396C412yescomplemented
25#2396C2212yes—
26#2296—21no—
27#2285C411yescharacteristic, complemented
28#2280—21no—
29#1740—21no—
30#1740—21no—
31#1740C2212yes—
32#1740C2212yes—
33C2×#342—21no—
34C2×#342C2212yes—
35#1458—21no—
36#1458—21no—
37#1458C2212yes—
38#1458C2212yes—
39C2×#324—21no—
40C2×#323C2212yes—
41#1352C2218yes—
42#1351C2218yes—
43C24×C4D411yescharacteristic
44#533—41no—
45#533—41no—
46#533—41no—
47#533—41no—
48C2×C4⋊Q8C2×C411yescharacteristic, complemented
49C2×C4⋊Q8C2×C411yescharacteristic, complemented
50C2×#125—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8C2×C412yes—
53C2×C4×Q8—21no—
54C2×C4×Q8C2×C412yescomplemented
55C2×#117—21no—
56C2×#117—21no—
57C2×#117—28no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—41no—
61C2×#117—41no—
62C2×#117—41no—
63C2×#117—41no—
64C2×#117—41no—
65C2×#117—41no—
66C2×#117—44no—
67C2×#117C2×C411yescharacteristic
68C2×#117C2×C411yescharacteristic
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C2×#117D411yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C4⋊C4—41no—
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C22⋊C4—22no—
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4C2311yescharacteristic
81C22×C42—41no—
82C22×C42C2×C411yescharacteristic
83C22×C42D411yescharacteristic
84#431—22no—
85#431—22no—
86#431—41no—
87#431—41no—
88C2×C4⋊C8—41no—
89C2×C22⋊C8—22no—
90C2×C4×C8—41no—
91#400—21no—
92#400—21no—
93#400C2×C412yescomplemented
94#400C2×C412yescomplemented
95#399—21no—
96#399—21no—
97#399C2×C412yes—
98#399C2312yes—
99#398—22no—
100#398—22no—
101#398—22no—
102#398—22no—
103#396—22no—
104#396—22no—
105#396—22no—
106#396—22no—
107#395—21no—
108#395—21no—
109#395C2×C412yescomplemented
110#395C2×C412yescomplemented
111#393—21no—
112#393—21no—
113#393C2×C412yes—
114#393C2312yes—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#385—22no—
120#385—22no—
121#384—21no—
122#382C2×C411yescharacteristic, complemented
123#382C2×C411yescharacteristic, complemented
124#381—22no—
125#380—21no—
126#380—21no—
127#380—21no—
128#380—21no—
129#380—21no—
130#380C2×C412yes—
131#379—22no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4C2×C412yescomplemented
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138#342—22no—
139#342—22no—
140#342—22no—
141#342C2314yes—
142#324—22no—
143#324—22no—
144#323—22no—
145#323C2314yes—
146C25C2×D411yescharacteristic
147C22×Q8—21no—
148C22×Q8C22×C412yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—24no—
154C23×C4—42no—
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4SD1614yes—
162C23×C4Q1614yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C22×C8—42no—
168C4⋊Q8—41no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8—41no—
172#125—42no—
173#125—42no—
174C22⋊Q8—42no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×Q8—42no—
182#117—28no—
183#117—28no—
184#117—41no—
185#117—41no—
186#117—41no—
187#117—41no—
188#117—41no—
189#117—41no—
190#117—41no—
191#117—41no—
192#117—41no—
193#117—41no—
194#117—41no—
195#117—41no—
196#117—41no—
197#117—41no—
198#117—41no—
199#117—41no—
200#117—48no—
201#117—48no—
202#117—41no—
203#117—41no—
204#117—41no—
205#117—41no—
206#117—44no—
207#117—44no—
208#117—44no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—48no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4C22⋊C411yescharacteristic
253C2×C22⋊C4C22⋊C411yescharacteristic
254C2×C22⋊C4C22×C411yescharacteristic
255C2×C22⋊C4C2×D411yescharacteristic
256C2×C22⋊C4C2×D411yescharacteristic
257C2×C22⋊C4C2×D411yescharacteristic
258C2×C22⋊C4C2×D411yescharacteristic
259C2×C22⋊C4C2411yescharacteristic
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—48no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C4⋊C8—42no—
282C4⋊C8—42no—
283C22⋊C8—24no—
284C22⋊C8—42no—
285C4×C8—42no—
286C4×C8—42no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—21no—
292#95—21no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95C22×C412yes—
298#95C22×C412yes—
299C24—21no—
300C24—21no—
301C24—22no—
302C24—81no—
303C24—81no—
304C24C2×C22⋊C411yescharacteristic
305C24C2×C22⋊C411yescharacteristic
306C24C2×SD1612yes—
307C24C2×Q1612yes—
308C24C22×D411yescharacteristic
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—82no—
312C2×Q8—82no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—44no—
349C22×C4—48no—
350C22×C4—81no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4—81no—
354C22×C4—84no—
355C22×C4—84no—
356C22×C4#9911yescharacteristic
357C22×C4#9911yescharacteristic
358C22×C4#9911yescharacteristic
359C22×C4#9911yescharacteristic
360C22×C4#10314yes—
361C22×C4#10314yes—
362C22×C4#10314yes—
363C22×C4#10314yes—
364C22×C4#10414yes—
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×SD1612yes—
376C22×C4C2×SD1612yes—
377C22×C4C2×Q1612yes—
378C22×C4C2×Q1612yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C2×C8—44no—
382C2×C8—84no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C22⋊C4—28no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—48no—
419C22⋊C4—44no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—82no—
424C22⋊C4—82no—
425C22⋊C4—82no—
426C22⋊C4—82no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—48no—
444C42—48no—
445C42—42no—
446C42—42no—
447C42—44no—
448C23—21no—
449C23—21no—
450C23—22no—
451C23—22no—
452C23—21no—
453C23—21no—
454C23—22no—
455C23—22no—
456C23—41no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—81no—
465C23—81no—
466C23—82no—
467C23—82no—
468C23—84no—
469C23C2×#9911yescharacteristic
470C23C2×#9911yescharacteristic
471C23C2×#10312yes—
472C23C2×#10312yes—
473C23C2×#10412yes—
474C23C22×C22⋊C411yescharacteristic
475C23C22×SD1611yescharacteristic
476C23C22×Q1611yescharacteristic
477Q8—84no—
478Q8—84no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—48no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—816no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4#32814yes—
538C2×C4#32814yes—
539C2×C4#32814yes—
540C2×C4#32814yes—
541C2×C4C2×#9911yescharacteristic
542C2×C4C2×#9911yescharacteristic
543C2×C4C2×#9911yescharacteristic
544C2×C4C2×#9911yescharacteristic
545C2×C4C2×#10312yes—
546C2×C4C2×#10312yes—
547C2×C4C2×#10312yes—
548C2×C4C2×#10312yes—
549C2×C4C2×#10412yes—
550C2×C4C2×#10412yes—
551C8—84no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—22no—
561C22—42no—
562C22—42no—
563C22—41no—
564C22—41no—
565C22—42no—
566C22—44no—
567C22—44no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—81no—
573C22—81no—
574C22—82no—
575C22—82no—
576C22C2×#32812yes—
577C22C2×#32812yes—
578C22C22×#9911yescharacteristic
579C22C22×#10311yescharacteristic
580C22C22×#10411yescharacteristic
581C4—22no—
582C4—22no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—82no—
592C4—82no—
593C4—88no—
594C4—82no—
595C4—82no—
596C4C2×#32812yes—
597C4C2×#32812yes—
598C4C2×#32812yes—
599C4C2×#32812yes—
600C2—21no—
601C2—21no—
602C2—22no—
603C2—22no—
604C2—41no—
605C2—41no—
606C2—42no—
607C2C22×#32811yescharacteristic
608C2#3334711yescharacteristic
609C2#3335811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
socleC22C22×#99
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11718C2semidirect
5. #11717C2semidirect
7. #11715C2semidirect
8. C2×#533C4semidirect
14. #2628C4semidirect
15. #2628C4semidirect
19. #2627C4semidirect
24. #2396C4semidirect
27. #2285C4semidirect
48. C2×C4⋊Q8C2×C4semidirect
49. C2×C4⋊Q8C2×C4semidirect
54. C2×C4×Q8C2×C4semidirect
93. #400C2×C4semidirect
94. #400C2×C4semidirect
109. #395C2×C4semidirect
110. #395C2×C4semidirect
122. #382C2×C4semidirect
123. #382C2×C4semidirect
135. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#346342
C2×#17402
C2×#14582
#117182
#117172
#117162
#117152