Finite Group Databasebeta

← #160451

Subgroups — #160451

Order 512.

1992 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2297C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
4#7578C214yesmaximal, complemented
5#7575C212yesmaximal, complemented
6#7572C212yesmaximal, complemented
7#7563C214yesmaximal, complemented
8C23×#41C411yescharacteristic
9C22×#117C411yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#431—21no—
12C22×C22⋊C8—21no—
13C22×C4×C8—21no—
14#2297—21no—
15#2297—21no—
16#2297—21no—
17#2297—21no—
18#2297C412yes—
19#2297C412yes—
20#2297C2212yes—
21#2297C2212yes—
22C2×#391C411yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#390C411yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#386—21no—
27C2×#386—21no—
28C2×C4×C22⋊C4—21no—
29C2×C4×C22⋊C4—21no—
30#1817C2212yescomplemented
31#1797—21no—
32#1797C2212yes—
33#1779—21no—
34C4×C22⋊C8—21no—
35C4×C22⋊C8—21no—
36C4×C22⋊C8—21no—
37C4×C22⋊C8C2212yes—
38C4×C22⋊C8C2212yescomplemented
39C4×C22⋊C8C2212yescomplemented
40C2×#336C2212yescomplemented
41#1272C2218yescomplemented
42#1270C2218yescomplemented
43C22×#41—21no—
44C22×#41—21no—
45C22×#41—24no—
46C22×#41—44no—
47C22×#41—44no—
48C22×#41C818yescomplemented
49C22×#41C2×C412yes—
50C22×#41C2×C412yes—
51C23×Q8C2×C411yescharacteristic
52C23×D4—21no—
53C23×D4C2×C411yescharacteristic
54C24×C4—21no—
55C24×C4C2×C411yescharacteristic
56C24×C4D411yescharacteristic
57C24×C4D411yescharacteristic
58C23×C8—22no—
59C2×#117—21no—
60C2×#117—21no—
61C2×#117—44no—
62C2×#117C2×C412yes—
63C2×#117C2×C412yes—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C42—21no—
72C22×C42—44no—
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#431—44no—
86C2×C4⋊C8—22no—
87C2×C22⋊C8—28no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96C2×C4×C8—44no—
97#391—21no—
98#391—21no—
99#391—21no—
100#391—21no—
101#391C2×C412yes—
102#391C2×C412yes—
103#391C2×C412yes—
104#391C2312yes—
105#390—21no—
106#390—21no—
107#390—21no—
108#390—21no—
109#390C2×C412yes—
110#390C2×C412yes—
111#390C2×C412yes—
112#390C2312yes—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133#336—22no—
134#336—22no—
135#336C2314yes—
136#336C2314yes—
137C25—21no—
138C25C22⋊C411yescharacteristic
139C2×#41—28no—
140C2×#41—28no—
141C2×#41—24no—
142C2×#41—24no—
143C2×#41—432no—
144C2×#41—432no—
145C2×#41—44no—
146C2×#41—44no—
147C2×#41—44no—
148C2×#41—44no—
149C2×#41—44no—
150C2×#41—44no—
151C2×#41—44no—
152C2×#41—44no—
153C2×#41C2×C818yes—
154C2×#41C2×C818yescomplemented
155C22×Q8—21no—
156C22×Q8—22no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8C2×C814yescomplemented
160C22×Q8C22×C412yes—
161C22×D4—28no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—21no—
165C22×D4—22no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4C2×C814yescomplemented
171C22×D4C22×C412yes—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—21no—
175C23×C4—28no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22⋊C411yescharacteristic
193C23×C4C22⋊C411yescharacteristic
194C23×C4C22⋊C411yescharacteristic
195C23×C4C22⋊C411yescharacteristic
196C23×C4C22⋊C411yescharacteristic
197C23×C4C2×C814yes—
198C23×C4M1614yes—
199C23×C4C22×C411yescharacteristic
200C23×C4C22×C411yescharacteristic
201C23×C4C22×C411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C22×C8—28no—
209C22×C8—24no—
210C22×C8—24no—
211C22×C8—44no—
212C22×C8—44no—
213#117—44no—
214#117—44no—
215#117—44no—
216#117—44no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
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—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—22no—
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×C42—42no—
251C2×C42—42no—
252C2×C42—44no—
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2411yescharacteristic
269C4⋊C8—44no—
270C22⋊C8—28no—
271C22⋊C8—28no—
272C22⋊C8—44no—
273C22⋊C8—44no—
274C22⋊C8—44no—
275C4×C8—44no—
276C4×C8—44no—
277C4×C8—44no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—24no—
292C24—41no—
293C24—41no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24C22⋊C814yes—
301C24C2×C22⋊C411yescharacteristic
302C24C2×C22⋊C411yescharacteristic
303C24C2×C22⋊C411yescharacteristic
304#41—416no—
305#41—432no—
306#41—416no—
307#41—416no—
308#41—416no—
309#41—432no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—48no—
313C2×Q8—48no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8C22×C814yes—
319C2×D4—28no—
320C2×D4—28no—
321C2×D4—22no—
322C2×D4—24no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—432no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4C22×C814yes—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—216no—
341C22×C4—216no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
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—28no—
364C22×C4—28no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—416no—
388C22×C4—416no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—432no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4C22⋊C814yes—
403C22×C4C22⋊C814yes—
404C22×C4C22⋊C814yes—
405C22×C4#9914yes—
406C22×C4#10014yes—
407C22×C4#10414yes—
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C22×C812yes—
418C22×C4C22×C812yes—
419C22×C4C22×C812yes—
420C22×C4C2×M1612yes—
421C22×C4C2×M1612yes—
422C22×C4C2×M1612yes—
423C22×C4#13514yes—
424C22×C4C23×C411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C2×C8—28no—
428C2×C8—44no—
429C2×C8—416no—
430C2×C8—44no—
431C2×C8—44no—
432C2×C8—44no—
433C2×C8—44no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—44no—
465C42—44no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—28no—
481C23—28no—
482C23—24no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—48no—
489C23—48no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—48no—
497C23—84no—
498C23—84no—
499C23—84no—
500C23—84no—
501C23C2×C22⋊C812yes—
502C23C2×C22⋊C812yes—
503C23C2×C22⋊C812yes—
504C23C2×#9912yes—
505C23C2×#10012yes—
506C23#41612yes—
507C23C2×#10412yes—
508C23C22×C22⋊C411yescharacteristic
509Q8—44no—
510Q8—44no—
511Q8—48no—
512D4—44no—
513D4—44no—
514D4—416no—
515D4—416no—
516D4—48no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
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—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—416no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—416no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4#323116yes—
574C2×C4C2×C22⋊C812yes—
575C2×C4C2×C22⋊C812yes—
576C2×C4C2×C22⋊C812yes—
577C2×C4C2×#9912yes—
578C2×C4C2×#9912yes—
579C2×C4C2×#10012yes—
580C2×C4C2×#10012yes—
581C2×C4#41612yes—
582C2×C4C2×#10412yes—
583C2×C4C2×#10412yes—
584C2×C4#42112yes—
585C2×C4C23×C811yescharacteristic
586C2×C4C22×M1611yescharacteristic
587C2×C4C2×#13512yes—
588C2×C4#57512yes—
589C8—48no—
590C8—84no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—416no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—816no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22C2×#32314yes—
620C22#147614yes—
621C22C22×C22⋊C811yescharacteristic
622C22C22×#9911yescharacteristic
623C22C22×#10011yescharacteristic
624C22C2×#41611yescharacteristic
625C22#289611yescharacteristic
626C22C22×#10411yescharacteristic
627C22C2×#42111yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—21no—
636C4—21no—
637C4—41no—
638C4—41no—
639C4—44no—
640C4—82no—
641C4—82no—
642C4C2×#32314yes—
643C4C2×#32314yes—
644C2—24no—
645C2—24no—
646C2—42no—
647C2—84no—
648C2C22×#32311yescharacteristic
649C2C2×#147611yescharacteristic
650C2#1743711yescharacteristic
651C2#1754211yescharacteristic
652C2#1757211yescharacteristic
653C2#1764711yescharacteristic
654C2#1771411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC23×#41C4
LCSG ▷ C2×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. #20714C2semidirect
3. C2×C4×C22⋊C8C2semidirect
4. #7578C2semidirect
5. #7575C2semidirect
6. #7572C2semidirect
7. #7563C2semidirect
30. #1817C22semidirect
38. C4×C22⋊C8C22semidirect
39. C4×C22⋊C8C22semidirect
40. C2×#336C22semidirect
41. #1272C22semidirect
42. #1270C22semidirect
48. C22×#41C8semidirect
154. C2×#41C2×C8semidirect
159. C22×Q8C2×C8semidirect
170. C22×D4C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#22972
#207142
C2×C4×C22⋊C82
#75782
#75752
#75722
#75632