Finite Group Databasebeta

← #481733

Subgroups — #481733

Order 512.

1137 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34809C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3#17364C211yesmaximal, characteristic
4#11786C212yesmaximal, complemented
5#11773C212yesmaximal, complemented
6#11755C214yesmaximal, complemented
7#11727C214yesmaximal, complemented
8C22×C4×D4C2211yescharacteristic
9C22×#117C411yescharacteristic, complemented
10#2920—21no—
11C22×#99—21no—
12#2487C411yescharacteristic, complemented
13#2487C2211yescharacteristic
14#2486C411yescharacteristic, complemented
15#2486C2211yescharacteristic
16#2485—21no—
17#2485—21no—
18#2484C412yescomplemented
19#2483—21no—
20#2482C2212yes—
21#2405—21no—
22#2395C412yescomplemented
23#2395C2212yes—
24C4×#123—21no—
25C4×#123—21no—
26#2296—21no—
27#2280—21no—
28#2280C411yescharacteristic, complemented
29C2×C4×C22⋊C4—21no—
30C4×#99—21no—
31C4×#99—21no—
32C4×#99C2212yes—
33C4×#99C2212yes—
34C2×#342C2212yes—
35#1466—21no—
36#1466C2212yes—
37#1462—21no—
38#1462C2212yes—
39#1460—21no—
40#1460C2212yes—
41#1352C2218yescomplemented
42#1344C2218yescomplemented
43C23×D4D411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C2×#123—21no—
47C2×#123—21no—
48C2×C4×D4—41no—
49C2×C4×D4—41no—
50C2×C4×D4—41no—
51C2×C4×D4—41no—
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4D411yescharacteristic
55C2×C4×D4D411yescharacteristic
56C2×C4×D4D411yescharacteristic
57C2×C4×D4D411yescharacteristic
58C2×C4×D4C2311yescharacteristic
59C2×C4×D4C2311yescharacteristic
60C2×#117—21no—
61C2×#117—21no—
62C2×#117—21no—
63C2×#117—41no—
64C2×#117—41no—
65C2×#117—41no—
66C2×#117—41no—
67C2×#117C2×C412yes—
68C2×#117C2×C412yescomplemented
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic, complemented
71C22×C4⋊C4C2×C411yescharacteristic, complemented
72C22×C22⋊C4—22no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4D411yescharacteristic
75C22×C22⋊C4C2311yescharacteristic
76C22×C42—21no—
77C22×C42—41no—
78C22×C42D411yescharacteristic
79#432—22no—
80#432—41no—
81#431—22no—
82#431—41no—
83#423—41no—
84C2×#99—28no—
85C2×#99—22no—
86C2×#99—22no—
87#407—22no—
88C4×M16—41no—
89#400C2×C412yescomplemented
90#400C2×C412yescomplemented
91#399C2×C412yes—
92#399C2312yes—
93#398C2×C412yescomplemented
94#398C2×C412yescomplemented
95#397—22no—
96#397—22no—
97#397C2×C412yes—
98#397C2312yes—
99#394—22no—
100#394—22no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#388—22no—
106#388—22no—
107#387—22no—
108#387—22no—
109#386—21no—
110#386—21no—
111#385—22no—
112#385—22no—
113#385—21no—
114#385—21no—
115#381—22no—
116#381—21no—
117#381—21no—
118#381C2×C412yes—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380C2×C411yescharacteristic, complemented
125#380C2×C411yescharacteristic, complemented
126#379—21no—
127#379—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—41no—
136C4×C22⋊C4—41no—
137C4×C22⋊C4C2×C412yescomplemented
138C2×#95—41no—
139#342—22no—
140#342C2314yes—
141#323—22no—
142#323—22no—
143#323—22no—
144#323C2314yes—
145C25C22⋊C411yescharacteristic
146C25C2×D411yescharacteristic
147C22×D4—24no—
148C22×D4—41no—
149C22×D4—41no—
150C22×D4—42no—
151C22×D4C22⋊C411yescharacteristic
152C22×D4C22⋊C411yescharacteristic
153C22×D4C2×D411yescharacteristic
154C22×D4C2×D411yescharacteristic
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—42no—
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C2×M16—42no—
182#123—44no—
183#123—44no—
184C4×D4—41no—
185C4×D4—41no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200#117—44no—
201#117—42no—
202#117—42no—
203#117—41no—
204#117—41no—
205#117—41no—
206#117—41no—
207#117—42no—
208#117—42no—
209#117—42no—
210#117—42no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4C22⋊C411yescharacteristic
252C2×C22⋊C4C22⋊C411yescharacteristic
253C2×C22⋊C4C22×C411yescharacteristic
254C2×C22⋊C4C2×D411yescharacteristic
255C2×C22⋊C4C2×D411yescharacteristic
256C2×C22⋊C4C2×D411yescharacteristic
257C2×C22⋊C4C2×D411yescharacteristic
258C2×C22⋊C4C2411yescharacteristic
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—84no—
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C4⋊C8—42no—
276C4⋊C8—42no—
277#99—28no—
278#99—28no—
279C22⋊C8—24no—
280C22⋊C8—42no—
281#97—42no—
282C4×C8—42no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—42no—
293#95—42no—
294#95C22×C412yes—
295#95C22×C412yes—
296C24—21no—
297C24—22no—
298C24—21no—
299C24—22no—
300C24—21no—
301C24—21no—
302C24—22no—
303C24—41no—
304C24—42no—
305C24—42no—
306C24—81no—
307C24—81no—
308C24C2×C22⋊C411yescharacteristic
309C24C2×C22⋊C411yescharacteristic
310C24C2×C22⋊C411yescharacteristic
311C24C2×C22⋊C411yescharacteristic
312C24C22×D411yescharacteristic
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—48no—
325C2×D4—48no—
326C2×D4#10414yes—
327C22×C4—24no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
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—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—41no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
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—44no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—48no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—81no—
383C22×C4—82no—
384C22×C4—84no—
385C22×C4—84no—
386C22×C4#9911yescharacteristic
387C22×C4#9911yescharacteristic
388C22×C4#9911yescharacteristic
389C22×C4#9911yescharacteristic
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×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400M16—84no—
401C2×C8—44no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—44no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—44no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C22⋊C4—28no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—48no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—42no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—44no—
437C22⋊C4—82no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—44no—
449C42—44no—
450C42—42no—
451C42—42no—
452C42—84no—
453C42—84no—
454C23—24no—
455C23—24no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—84no—
488C23—84no—
489C23—84no—
490C23—84no—
491C23C2×#9911yescharacteristic
492C23C2×#9911yescharacteristic
493C23C2×#10412yes—
494C23C2×#10412yes—
495C23C22×C22⋊C411yescharacteristic
496C23#57512yes—
497D4—44no—
498D4—44no—
499D4—44no—
500D4—44no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
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—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—48no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—816no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4#35214yes—
566C2×C4C2×#9911yescharacteristic
567C2×C4C2×#9911yescharacteristic
568C2×C4C2×#9911yescharacteristic
569C2×C4C2×#9911yescharacteristic
570C2×C4C2×#10412yes—
571C8—84no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—22no—
579C22—21no—
580C22—22no—
581C22—22no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—48no—
588C22—42no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—81no—
599C22—81no—
600C22—84no—
601C22—81no—
602C22—81no—
603C22—84no—
604C22—84no—
605C22C2×#35212yes—
606C22C22×#9911yescharacteristic
607C22#289611yescharacteristic
608C22C22×#10411yescharacteristic
609C4—21no—
610C4—21no—
611C4—22no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—82no—
620C4—82no—
621C4—88no—
622C4—82no—
623C4—82no—
624C4C2×#35212yes—
625C4C2×#35212yes—
626C2—21no—
627C2—21no—
628C2—22no—
629C2—42no—
630C2—41no—
631C2—42no—
632C2—42no—
633C2—81no—
634C2#1754211yescharacteristic
635C2C22×#35211yescharacteristic
636C2#3335811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
socleC22C22×#99
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11786C2semidirect
5. #11773C2semidirect
6. #11755C2semidirect
7. #11727C2semidirect
9. C22×#117C4semidirect
12. #2487C4semidirect
14. #2486C4semidirect
18. #2484C4semidirect
22. #2395C4semidirect
28. #2280C4semidirect
41. #1352C22semidirect
42. #1344C22semidirect
68. C2×#117C2×C4semidirect
70. C22×C4⋊C4C2×C4semidirect
71. C22×C4⋊C4C2×C4semidirect
89. #400C2×C4semidirect
90. #400C2×C4semidirect
93. #398C2×C4semidirect
94. #398C2×C4semidirect
124. #380C2×C4semidirect
125. #380C2×C4semidirect
137. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#348092
C2×C4×#992
#173642
#117862
#117732
#117552
#117272