Finite Group Databasebeta

← #481474

Subgroups — #481474

Order 512.

1244 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3050C211yesmaximal, characteristic
2C2×#2112C211yesmaximal, characteristic, complemented
3C22×#341C211yesmaximal, characteristic, complemented
4#11712C212yesmaximal, complemented
5#11711C212yesmaximal, complemented
6#11710C214yesmaximal, complemented
7#11709C214yesmaximal, complemented
8C22×#135C411yescharacteristic
9C23×M16—21no—
10C24×C8C2211yescharacteristic
11C2×#525—21no—
12C22×M32—21no—
13#3050—21no—
14#3050—21no—
15#3050—21no—
16#3050—21no—
17#3050C412yes—
18#3050C412yes—
19#3050C2212yes—
20#3050C2212yes—
21C2×#467C411yescharacteristic
22C2×#467C2211yescharacteristic
23C2×#466C411yescharacteristic
24C2×#466C2211yescharacteristic
25C2×#465—21no—
26C2×#465—21no—
27C2×#427C411yescharacteristic
28C2×#416—21no—
29#2112—21no—
30#2112—21no—
31#2112C2212yes—
32#2112C2212yescomplemented
33C2×C22⋊C16—21no—
34C2×C22⋊C16C2212yescomplemented
35C2×#341—21no—
36C2×#341—21no—
37C2×#341—21no—
38C2×#341C2212yes—
39C2×#341C2212yescomplemented
40C2×#341C2212yescomplemented
41#1350C2218yescomplemented
42#1349C2218yescomplemented
43C22×#41—21no—
44C24×C4D411yescharacteristic
45C2×#135—21no—
46C2×#135—21no—
47C2×#135—44no—
48C2×#135C2×C412yes—
49C2×#135C2×C412yes—
50C22×Q16C2×C411yescharacteristic
51C22×SD16—21no—
52C22×D8C2×C411yescharacteristic
53C22×M16—28no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—44no—
58C22×M16—41no—
59C22×M16—41no—
60C22×M16—41no—
61C23×C8—44no—
62C23×C8—41no—
63C23×C8C2×C411yescharacteristic
64C23×C8C2×C411yescharacteristic
65C23×C8C2×C411yescharacteristic
66C23×C8C2×C411yescharacteristic
67C23×C8D411yescharacteristic
68C23×C8D411yescharacteristic
69C23×C8D411yescharacteristic
70C23×C8D411yescharacteristic
71C23×C8D411yescharacteristic
72C23×C8D411yescharacteristic
73C23×C8D411yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77#525—22no—
78#525—22no—
79#525—22no—
80#525—22no—
81#525—41no—
82#525—41no—
83#525—41no—
84#525—41no—
85C2×#123—41no—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—21no—
88C2×#121—21no—
89C2×#121—21no—
90C2×#120—41no—
91C2×C4×D4—41no—
92C2×C4×D4—41no—
93C2×#117—21no—
94C2×M32—22no—
95C2×M32—22no—
96C2×M32—44no—
97C22×C16—22no—
98#467—21no—
99#467—21no—
100#467—21no—
101#467—21no—
102#467C2×C412yes—
103#467C2×C412yes—
104#467C2×C412yes—
105#467C2312yes—
106#466—21no—
107#466—21no—
108#466—21no—
109#466—21no—
110#466C2×C412yes—
111#466C2×C412yes—
112#466C2×C412yes—
113#466C2312yes—
114#465—22no—
115#465—22no—
116#465—22no—
117#465—22no—
118#465—22no—
119#465—22no—
120#465—22no—
121#465—22no—
122#427—21no—
123#427—21no—
124#427C2×C412yes—
125#427C2×C412yes—
126C2×#107C2×C411yescharacteristic
127C2×#107C2×C411yescharacteristic
128C2×#106—21no—
129#416—22no—
130#416—22no—
131#416—22no—
132#416—22no—
133C2×#103—21no—
134C2×#103—21no—
135C2×#102—21no—
136C2×#102—21no—
137C22⋊C16—22no—
138C22⋊C16—22no—
139C22⋊C16—22no—
140C22⋊C16C2314yes—
141#341—22no—
142#341—22no—
143#341—22no—
144#341C2314yes—
145C25SD1611yescharacteristic
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—41no—
149C2×#41—41no—
150C2×#41—44no—
151C2×#41—44no—
152C22×Q8—21no—
153C22×D4—21no—
154C22×D4—41no—
155C22×D4—41no—
156C22×D4—41no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—41no—
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4D811yescharacteristic
165C23×C4D811yescharacteristic
166C23×C4D811yescharacteristic
167C23×C4D811yescharacteristic
168C23×C4SD1611yescharacteristic
169C23×C4SD1611yescharacteristic
170C23×C4SD1611yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174#135—44no—
175#135—44no—
176#135—44no—
177#135—44no—
178C2×Q16—21no—
179C2×Q16—42no—
180C2×Q16C22×C412yes—
181C2×SD16—22no—
182C2×SD16—22no—
183C2×SD16—44no—
184C2×D8—21no—
185C2×D8—42no—
186C2×D8C22×C412yes—
187C2×M16—28no—
188C2×M16—28no—
189C2×M16—28no—
190C2×M16—24no—
191C2×M16—48no—
192C2×M16—48no—
193C2×M16—48no—
194C2×M16—44no—
195C2×M16—44no—
196C2×M16—44no—
197C2×M16—42no—
198C2×M16—42no—
199C2×M16—42no—
200C22×C8—24no—
201C22×C8—24no—
202C22×C8—42no—
203C22×C8—42no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8—42no—
207C22×C8—42no—
208C22×C8—42no—
209C22×C8—42no—
210C22×C8—42no—
211C22×C8—42no—
212C22×C8—42no—
213C22×C8—42no—
214C22×C8—48no—
215C22×C8—42no—
216C22×C8—42no—
217C22×C8—42no—
218C22×C8C22⋊C411yescharacteristic
219C22×C8C22⋊C411yescharacteristic
220C22×C8C22⋊C411yescharacteristic
221C22×C8C22⋊C411yescharacteristic
222C22×C8C22⋊C411yescharacteristic
223C22×C8C22⋊C411yescharacteristic
224C22×C8C22⋊C411yescharacteristic
225C22×C8C22⋊C411yescharacteristic
226C22×C8C22⋊C411yescharacteristic
227C22×C8C22⋊C411yescharacteristic
228C22×C8C22⋊C411yescharacteristic
229C22×C8C22⋊C411yescharacteristic
230C22×C8C22×C411yescharacteristic
231C22×C8C22×C411yescharacteristic
232C22×C8C22×C411yescharacteristic
233C22×C8C22×C411yescharacteristic
234C22×C8C22×C411yescharacteristic
235C22×C8C22×C411yescharacteristic
236C22×C8C2×D411yescharacteristic
237C22×C8C2×D411yescharacteristic
238C22×C8C2×D411yescharacteristic
239C22×C8C2×D411yescharacteristic
240C22×C8C2×D411yescharacteristic
241C22×C8C2×D411yescharacteristic
242C22×C8C2×D411yescharacteristic
243C22×C8C2×D411yescharacteristic
244C22×C8C2×D411yescharacteristic
245C22×C8C2411yescharacteristic
246#123—42no—
247#123—42no—
248#123—42no—
249#123—42no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252C22⋊Q8—22no—
253C22⋊Q8—22no—
254C22⋊Q8—41no—
255C22⋊Q8—41no—
256C22⋊Q8—41no—
257C22⋊Q8—41no—
258#121—22no—
259#121—22no—
260#121—22no—
261#121—22no—
262#121—41no—
263#121—41no—
264#121—41no—
265#121—41no—
266#120—42no—
267#120—42no—
268#120—42no—
269#120—42no—
270C4×D4—42no—
271C4×D4—42no—
272C4×D4—42no—
273C4×D4—42no—
274C4×D4—42no—
275C4×D4—42no—
276C4×D4—42no—
277C4×D4—42no—
278#117—22no—
279#117—22no—
280#117—41no—
281#117—41no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C42—41no—
292M32—44no—
293M32—44no—
294C2×C16—24no—
295C2×C16—44no—
296#107—21no—
297#107—21no—
298#107C22×C412yes—
299#107C22×C412yes—
300#106—22no—
301#106—22no—
302#103—22no—
303#103—22no—
304#103—22no—
305#103—22no—
306#102—22no—
307#102—22no—
308#102—22no—
309#102—22no—
310C24—24no—
311C24—42no—
312C24—42no—
313C24—41no—
314C24#10211yescharacteristic
315C24#10211yescharacteristic
316C24#10211yescharacteristic
317C24#10211yescharacteristic
318C24C2×SD1611yescharacteristic
319C24C2×SD1611yescharacteristic
320C24C2×SD1611yescharacteristic
321#41—44no—
322#41—44no—
323#41—44no—
324#41—44no—
325#41—84no—
326#41—84no—
327C2×Q8—22no—
328C2×Q8—41no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×D4—22no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4—84no—
342C2×D4—84no—
343C2×D4—84no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—84no—
369C22×C4#9914yes—
370C22×C4#10211yescharacteristic
371C22×C4#10211yescharacteristic
372C22×C4#10211yescharacteristic
373C22×C4#10211yescharacteristic
374C22×C4#10211yescharacteristic
375C22×C4#10211yescharacteristic
376C22×C4#10211yescharacteristic
377C22×C4#10211yescharacteristic
378C22×C4#10211yescharacteristic
379C22×C4#10211yescharacteristic
380C22×C4#10211yescharacteristic
381C22×C4#10211yescharacteristic
382C22×C4#10414yes—
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×D811yescharacteristic
390C22×C4C2×D811yescharacteristic
391C22×C4C2×D811yescharacteristic
392C22×C4C2×D811yescharacteristic
393C22×C4C2×D811yescharacteristic
394C22×C4C2×D811yescharacteristic
395C22×C4C2×SD1611yescharacteristic
396C22×C4C2×SD1611yescharacteristic
397C22×C4C2×SD1611yescharacteristic
398C22×C4C22×D411yescharacteristic
399Q16—42no—
400Q16—42no—
401SD16—44no—
402SD16—44no—
403D8—42no—
404D8—42no—
405M16—28no—
406M16—48no—
407M16—48no—
408M16—44no—
409M16—48no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—48no—
439C2×C8—48no—
440C2×C8—48no—
441C2×C8—44no—
442C2×C8C2×C22⋊C411yescharacteristic
443C2×C8C2×C22⋊C411yescharacteristic
444C2×C8C2×C22⋊C411yescharacteristic
445C2×C8C2×C22⋊C411yescharacteristic
446C2×C8C2×C22⋊C411yescharacteristic
447C2×C8C2×C22⋊C411yescharacteristic
448C2×C8C23×C411yescharacteristic
449C2×C8C22×D411yescharacteristic
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C42—42no—
469C42—42no—
470C16—44no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—416no—
492C23—42no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23#32714yes—
497C23C2×#10211yescharacteristic
498C23C2×#10211yescharacteristic
499C23C2×#10211yescharacteristic
500C23C2×#10211yescharacteristic
501C23C2×#10211yescharacteristic
502C23C2×#10211yescharacteristic
503C23C22×SD1611yescharacteristic
504Q8—42no—
505Q8—42no—
506Q8—82no—
507D4—42no—
508D4—42no—
509D4—84no—
510D4—84no—
511D4—84no—
512D4—84no—
513D4—84no—
514D4—84no—
515D4—82no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4#32714yes—
551C2×C4#32714yes—
552C2×C4#32714yes—
553C2×C4C2×#9912yes—
554C2×C4C2×#9912yes—
555C2×C4C2×#9912yes—
556C2×C4C2×#10211yescharacteristic
557C2×C4C2×#10211yescharacteristic
558C2×C4C2×#10211yescharacteristic
559C2×C4C2×#10211yescharacteristic
560C2×C4C2×#10211yescharacteristic
561C2×C4C2×#10211yescharacteristic
562C2×C4C2×#10412yes—
563C2×C4C2×#10412yes—
564C2×C4C2×#10412yes—
565C2×C4C22×C22⋊C411yescharacteristic
566C2×C4C22×D811yescharacteristic
567C8—21no—
568C8—21no—
569C8—21no—
570C8—21no—
571C8—21no—
572C8—21no—
573C8—21no—
574C8—21no—
575C8—48no—
576C22—28no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—164no—
603C22C2×#32712yes—
604C22C2×#32712yes—
605C22C2×#32712yes—
606C22#214312yes—
607C22#214812yes—
608C22C22×#10211yescharacteristic
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4C2×#32712yes—
617C4C2×#32712yes—
618C4C2×#32712yes—
619C4C22×#9911yescharacteristic
620C4C22×#10411yescharacteristic
621C2—22no—
622C2—22no—
623C2—22no—
624C2—22no—
625C2—22no—
626C2—41no—
627C2—41no—
628C2—162no—
629C2C22×#32711yescharacteristic
630C2C2×#214311yescharacteristic
631C2C2×#214811yescharacteristic
subgroupG/H
centerC2×C4C2×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22C22×#102
prime radicalC2×#3050C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2112C2semidirect
3. C22×#341C2semidirect
4. #11712C2semidirect
5. #11711C2semidirect
6. #11710C2semidirect
7. #11709C2semidirect
32. #2112C22semidirect
34. C2×C22⋊C16C22semidirect
39. C2×#341C22semidirect
40. C2×#341C22semidirect
41. #1350C22semidirect
42. #1349C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30502
C2×#21122
C22×#3412
#117122
#117112
#117102
#117092