Finite Group Databasebeta

← #190426

Subgroups — #190426

Order 512.

959 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34109C211yesmaximal, characteristic
2#17357C211yesmaximal, characteristic
3C2×#1458C211yesmaximal, characteristic
4#8733C212yesmaximal, complemented
5#8732C212yesmaximal, complemented
6#8731C212yesmaximal, complemented
7#8730C212yesmaximal, complemented
8#7869C212yesmaximal, complemented
9#7857C212yesmaximal, complemented
10C22×#117C2211yescharacteristic
11#2920C2212yes—
12C2×#431C2212yes—
13#2347—21no—
14#2346—21no—
15#2345—21no—
16#2344—21no—
17#2343—21no—
18#2343—21no—
19#2342—21no—
20#2342—21no—
21#2298C411yescharacteristic
22#2298C411yescharacteristic
23#2298C2211yescharacteristic
24#2298C2211yescharacteristic
25#2296C411yescharacteristic
26#2296C411yescharacteristic
27#2283C411yescharacteristic
28#2283C2211yescharacteristic
29#2280C411yescharacteristic
30C4×#117C411yescharacteristic
31C4×#117C2211yescharacteristic
32C4×#117C2211yescharacteristic
33C4×#117C2211yescharacteristic
34C2×C4×C22⋊C4C411yescharacteristic
35#1464—21no—
36#1464—21no—
37#1463—21no—
38#1462—21no—
39#1458—21no—
40#1458—21no—
41#1458C2212yes—
42#1458C2212yes—
43#1458C2212yes—
44#1458C2212yes—
45C2×#324—21no—
46C2×#323—21no—
47#1276C2214yescomplemented
48#1276C2214yescomplemented
49#1274C2214yescomplemented
50#1274C2214yescomplemented
51C24×C4C2×C411yescharacteristic
52C2×#117—21no—
53C2×#117—21no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117C2×C411yescharacteristic
59C2×#117C2×C411yescharacteristic
60C2×#117C2×C411yescharacteristic
61C2×#117C2×C411yescharacteristic
62C2×#117D411yescharacteristic
63C2×#117D411yescharacteristic
64C2×#117D411yescharacteristic
65C2×#117D411yescharacteristic
66C2×#117C2311yescharacteristic
67C2×#117C2311yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C22⋊C4—21no—
71C22×C22⋊C4D411yescharacteristic
72C22×C22⋊C4D411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2311yescharacteristic
77#432—22no—
78#432—22no—
79#431—22no—
80#431—22no—
81#431—22no—
82#431—22no—
83#431C2314yes—
84#431C2314yes—
85#423—22no—
86C2×C4⋊C8—22no—
87#407—22no—
88C2×C22⋊C8—22no—
89C4×M16—22no—
90C2×C4×C8—22no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#388—21no—
96#388—21no—
97#388—21no—
98#388—21no—
99#388—21no—
100#388—21no—
101#388—21no—
102#388—21no—
103#387—21no—
104#387—21no—
105#387—21no—
106#387—21no—
107#387—21no—
108#387—21no—
109#387—21no—
110#387—21no—
111#385—21no—
112#385—21no—
113#385—21no—
114#385—21no—
115#383—21no—
116#383—21no—
117#382—21no—
118#382—21no—
119#381—21no—
120#381—21no—
121#381—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126#379—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4C2×C411yescharacteristic
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4C2×C411yescharacteristic
150C4×C22⋊C4C2×C411yescharacteristic
151C4×C22⋊C4C2×C411yescharacteristic
152C4×C22⋊C4C2×C411yescharacteristic
153C4×C22⋊C4C2×C411yescharacteristic
154C4×C22⋊C4C2×C411yescharacteristic
155C4×C22⋊C4C2×C411yescharacteristic
156C4×C22⋊C4C2×C411yescharacteristic
157#376—21no—
158#376—21no—
159#376C2×C411yescharacteristic
160#376C2×C411yescharacteristic
161#376C2311yescharacteristic
162#376C2311yescharacteristic
163C2×#95—21no—
164C43C2×C411yescharacteristic
165C43C2×C411yescharacteristic
166C43C2311yescharacteristic
167C43C2311yescharacteristic
168#324—22no—
169#324—22no—
170#324—22no—
171#324—22no—
172#323—22no—
173#323—22no—
174#323—22no—
175#323—22no—
176C25C22⋊C411yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—24no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—42no—
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C2×M16—42no—
196C22×C8—42no—
197#117—24no—
198#117—22no—
199#117—22no—
200#117—22no—
201#117—22no—
202#117—22no—
203#117—22no—
204#117—22no—
205#117—22no—
206#117—22no—
207#117—22no—
208#117—22no—
209#117—22no—
210#117—22no—
211#117—22no—
212#117—22no—
213#117—22no—
214#117—42no—
215#117—44no—
216#117—44no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4C22⋊C411yescharacteristic
255C2×C22⋊C4C22⋊C411yescharacteristic
256C2×C22⋊C4C22⋊C411yescharacteristic
257C2×C22⋊C4C22⋊C411yescharacteristic
258C2×C22⋊C4C2×D411yescharacteristic
259C2×C22⋊C4C2×D411yescharacteristic
260C2×C22⋊C4C2×D411yescharacteristic
261C2×C22⋊C4C2×D411yescharacteristic
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—24no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—42no—
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22×C411yescharacteristic
281C2×C42C22×C411yescharacteristic
282C2×C42C22×C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C411yescharacteristic
287C2×C42C22×C411yescharacteristic
288C2×C42C22×C411yescharacteristic
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42#4112yes—
294C2×C42#4112yes—
295C2×C42#4112yes—
296C2×C42#4112yes—
297C2×C42C2411yescharacteristic
298C4⋊C8—24no—
299C4⋊C8—24no—
300C4⋊C8—42no—
301C4⋊C8—42no—
302C22⋊C8—24no—
303C22⋊C8—24no—
304C22⋊C8—42no—
305C22⋊C8—42no—
306#97—42no—
307C4×C8—24no—
308C4×C8—24no—
309C4×C8—42no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334C24—21no—
335C24—21no—
336C24—22no—
337C24—41no—
338C24—42no—
339C24—81no—
340C24C2×C22⋊C411yescharacteristic
341C24C2×C22⋊C411yescharacteristic
342C24C2×C22⋊C411yescharacteristic
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4#11711yescharacteristic
400C22×C4#11711yescharacteristic
401C22×C4#11711yescharacteristic
402C22×C4#11711yescharacteristic
403C22×C4#12312yes—
404C22×C4#12312yes—
405C22×C4#12312yes—
406C22×C4#12312yes—
407C22×C4#13114yes—
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411M16—84no—
412C2×C8—44no—
413C2×C8—44no—
414C2×C8—84no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C22⋊C4—24no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—81no—
450C22⋊C4—81no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—42no—
471C42—42no—
472C42—42no—
473C42—81no—
474C42—81no—
475C42—81no—
476C42—81no—
477C42#11712yes—
478C42#11712yes—
479C42#12312yes—
480C42#12312yes—
481C42#12312yes—
482C42#12312yes—
483C42C2×#4112yes—
484C42C2×#4112yes—
485C23—22no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—81no—
503C23—81no—
504C23—84no—
505C23—84no—
506C23#38111yescharacteristic
507C23#38111yescharacteristic
508C23#40812yes—
509C23C22×C22⋊C411yescharacteristic
510C2×C4—24no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4#38111yescharacteristic
578C2×C4#38111yescharacteristic
579C2×C4#38112yes—
580C2×C4#38112yes—
581C2×C4#40812yes—
582C2×C4C2×#11711yescharacteristic
583C2×C4C2×#11711yescharacteristic
584C2×C4C2×#12312yes—
585C2×C4C2×#12312yes—
586C2×C4C2×#13112yes—
587C2×C4#56812yes—
588C8—84no—
589C8—84no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—22no—
596C22—21no—
597C22—21no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—84no—
608C22—81no—
609C22—81no—
610C22—84no—
611C22#182712yes—
612C22C2×#38111yescharacteristic
613C22C2×#40811yescharacteristic
614C22#287911yescharacteristic
615C4—21no—
616C4—21no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—42no—
623C4—42no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4#292412yes—
636C4#292412yes—
637C2—21no—
638C2—21no—
639C2—22no—
640C2—41no—
641C2—42no—
642C2—81no—
643C2#2068511yescharacteristic
644C2C2×#182711yescharacteristic
645C2#2070811yescharacteristic
subgroupG/H
centerC2×C4C2×#117
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#381
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #2283 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8733C2semidirect
5. #8732C2semidirect
6. #8731C2semidirect
7. #8730C2semidirect
8. #7869C2semidirect
9. #7857C2semidirect
47. #1276C22semidirect
48. #1276C22semidirect
49. #1274C22semidirect
50. #1274C22semidirect

Maximal subgroups

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

subgroupindex
#341092
#173572
C2×#14582
#87332
#87322
#87312
#87302
#78692
#78572