Finite Group Databasebeta

← #161448

Subgroups — #161448

Order 512.

1598 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
3C2×#1728C211yesmaximal, characteristic, complemented
4#7685C212yesmaximal, complemented
5#7684C212yesmaximal, complemented
6#7678C214yesmaximal, complemented
7#7565C214yesmaximal, complemented
8C23×M16—21no—
9C2×#532C411yescharacteristic
10C2×#525—21no—
11C23×C42C2211yescharacteristic
12C22×C22⋊C8—21no—
13C22×C4×C8—21no—
14#2857C411yescharacteristic
15#2857C2211yescharacteristic
16#2856C411yescharacteristic
17#2856C2211yescharacteristic
18#2854—21no—
19#2854C2212yes—
20#2850—21no—
21#2850—21no—
22#2847—21no—
23#2847C412yes—
24C4×#121—21no—
25C4×#121—21no—
26C4×#121C412yes—
27C4×#121C2212yes—
28#2297—21no—
29#2285C411yescharacteristic
30C4×C22⋊C8—21no—
31C4×C22⋊C8—21no—
32C4×C22⋊C8C2212yes—
33C4×C22⋊C8C2212yes—
34#1728—21no—
35#1728—21no—
36#1728C2212yes—
37#1728C2212yes—
38C2×#336—21no—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1271C2218yescomplemented
42#1270C2218yescomplemented
43C22×#41—21no—
44C24×C4C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C22×M16—28no—
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—44no—
51C22×M16—44no—
52C23×C8—22no—
53C23×C8—22no—
54#532—44no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#127C2×C411yescharacteristic
57C2×#124—21no—
58#525—44no—
59#525—44no—
60C2×#123—22no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#121C2×C412yes—
69C2×#121C2×C412yes—
70C2×#121C2×C412yes—
71C2×#120—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C412yes—
76C2×#117—21no—
77C22×C42—44no—
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×C22⋊C8—28no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8—44no—
96#391—21no—
97#391—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390—21no—
102#390C2×C412yes—
103#390C2312yes—
104#389—21no—
105#389—21no—
106#389C2×C412yes—
107#389C2312yes—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#384C2×C411yescharacteristic
119#384C2×C411yescharacteristic
120#382—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380C2×C412yes—
125#380C2×C412yes—
126#380C2×C412yes—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2×C412yes—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336C2314yes—
136#336C2314yes—
137C25C22⋊C411yescharacteristic
138C2×#41—44no—
139C2×#41—44no—
140C2×#41—44no—
141C22×Q8—21no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C22×C412yes—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4D814yes—
167C23×C4SD1614yes—
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C2×M16—28no—
178C2×M16—28no—
179C2×M16—44no—
180C2×M16—432no—
181C2×M16—44no—
182C2×M16—44no—
183C2×M16—44no—
184C22×C8—28no—
185C22×C8—28no—
186C22×C8—24no—
187C22×C8—24no—
188C22×C8—44no—
189C22×C8—44no—
190C22×C8—44no—
191C22×C8—44no—
192C4⋊Q8—42no—
193#127—42no—
194#124—44no—
195#123—44no—
196#123—44no—
197C22⋊Q8—44no—
198C22⋊Q8—42no—
199C22⋊Q8—42no—
200#121—28no—
201#121—44no—
202#121—44no—
203#121—44no—
204#121—44no—
205#121—44no—
206#121—42no—
207#121—42no—
208#121—44no—
209#120—48no—
210C4×D4—44no—
211C4×D4—44no—
212C4×D4—48no—
213C4×D4—44no—
214C4×D4—44no—
215#117—44no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2411yescharacteristic
266C22⋊C8—28no—
267C22⋊C8—28no—
268C4×C8—44no—
269C4×C8—44no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—41no—
282C24—41no—
283C24#10214yes—
284C24C2×C22⋊C411yescharacteristic
285C24C2×C22⋊C411yescharacteristic
286C24C2×C22⋊C411yescharacteristic
287#41—816no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×D4—24no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—216no—
315C22×C4—216no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—416no—
359C22×C4—416no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4#9914yes—
364C22×C4#10214yes—
365C22×C4#10214yes—
366C22×C4#10214yes—
367C22×C4#10414yes—
368C22×C4#10414yes—
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×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4#13114yes—
379C22×C4C2×D812yes—
380C22×C4C2×D812yes—
381C22×C4C2×D812yes—
382C22×C4C2×SD1612yes—
383C22×C4C2×SD1612yes—
384C22×C4C2×SD1612yes—
385C22×C4C23×C411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388M16—416no—
389M16—416no—
390C2×C8—28no—
391C2×C8—28no—
392C2×C8—44no—
393C2×C8—416no—
394C2×C8—44no—
395C2×C8—416no—
396C2×C8—44no—
397C2×C8—44no—
398C4⋊C4—24no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—44no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—28no—
453C23—28no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23#40812yes—
467C23C2×#9912yes—
468C23C2×#10212yes—
469C23C2×#10212yes—
470C23C2×#10212yes—
471C23C2×#10412yes—
472C23C2×#10412yes—
473C23C22×C22⋊C411yescharacteristic
474Q8—84no—
475D4—84no—
476D4—84no—
477D4—88no—
478D4—88no—
479D4—88no—
480D4—84no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
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—88no—
533C2×C4#327116yes—
534C2×C4#40812yes—
535C2×C4C2×#9912yes—
536C2×C4C2×#9912yes—
537C2×C4#41312yes—
538C2×C4C2×#10212yes—
539C2×C4C2×#10212yes—
540C2×C4C2×#10212yes—
541C2×C4C2×#10412yes—
542C2×C4C2×#10412yes—
543C2×C4C2×#10412yes—
544C2×C4C2×#10412yes—
545C2×C4C2×#13112yes—
546C2×C4C22×D811yescharacteristic
547C2×C4C22×SD1611yescharacteristic
548C2×C4#57512yes—
549C8—48no—
550C8—48no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—416no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22#149214yes—
579C22C2×#32714yes—
580C22C2×#40811yescharacteristic
581C22C22×#9911yescharacteristic
582C22C2×#41311yescharacteristic
583C22C22×#10211yescharacteristic
584C22#289611yescharacteristic
585C22C22×#10411yescharacteristic
586C22C22×#10411yescharacteristic
587C4—21no—
588C4—21no—
589C4—22no—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—41no—
597C4—41no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4C2×#32714yes—
602C4C2×#32714yes—
603C2—24no—
604C2—24no—
605C2—42no—
606C2—162no—
607C2#1738111yescharacteristic
608C2#1746211yescharacteristic
609C2C2×#149211yescharacteristic
610C2C22×#32711yescharacteristic
611C2#1754211yescharacteristic
612C2#1763811yescharacteristic
613C2#1770411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊C8C2semidirect
3. C2×#1728C2semidirect
4. #7685C2semidirect
5. #7684C2semidirect
6. #7678C2semidirect
7. #7565C2semidirect
41. #1271C22semidirect
42. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#366242
C2×C4×C22⋊C82
C2×#17282
#76852
#76842
#76782
#75652