Finite Group Databasebeta

← #161141

Subgroups — #161141

Order 512.

1890 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2297C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3#20626C211yesmaximal, characteristic, complemented
4#7612C214yesmaximal, complemented
5#7609C212yesmaximal, complemented
6#7606C212yesmaximal, complemented
7#7597C214yesmaximal, complemented
8C23×#41C411yescharacteristic
9C22×#117C411yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#432—21no—
12C2×#423—21no—
13C2×#407—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#1817C2212yes—
31#1797—21no—
32#1797—21no—
33#1797—21no—
34#1797C2212yescomplemented
35#1797C2212yescomplemented
36#1797C2212yescomplemented
37#1779—21no—
38#1749—21no—
39#1749C2212yescomplemented
40#1728C2212yes—
41#1272C2218yescomplemented
42#1270C2218yescomplemented
43C22×#41—28no—
44C22×#41—21no—
45C22×#41—21no—
46C22×#41—44no—
47C22×#41—44no—
48C22×#41C2×C412yes—
49C22×#41C2×C412yes—
50C23×Q8C2×C411yescharacteristic
51C23×D4—21no—
52C23×D4C2×C411yescharacteristic
53C24×C4—21no—
54C24×C4C2×C411yescharacteristic
55C24×C4D411yescharacteristic
56C24×C4D411yescharacteristic
57C22×M16—22no—
58C2×#117—21no—
59C2×#117—21no—
60C2×#117—44no—
61C2×#117C2×C412yes—
62C2×#117C2×C412yes—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C42—21no—
71C22×C42—44no—
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#432—44no—
85#423—44no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89#407—44no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×#97—22no—
96#391—21no—
97#391—21no—
98#391—21no—
99#391—21no—
100#391C2×C412yes—
101#391C2×C412yes—
102#391C2×C412yes—
103#391C2312yes—
104#390—21no—
105#390—21no—
106#390—21no—
107#390—21no—
108#390C2×C412yes—
109#390C2×C412yes—
110#390C2×C412yes—
111#390C2312yes—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120C4×C22⋊C4—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—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132#336—22no—
133#336—22no—
134#336C2314yes—
135#336C2314yes—
136C25—21no—
137C25C22⋊C411yescharacteristic
138C2×#41—28no—
139C2×#41—28no—
140C2×#41—24no—
141C2×#41—24no—
142C2×#41—24no—
143C2×#41—464no—
144C2×#41—44no—
145C2×#41—44no—
146C2×#41—44no—
147C2×#41—44no—
148C2×#41—44no—
149C2×#41—44no—
150C2×#41—44no—
151C2×#41—44no—
152C2×#41M1618yescomplemented
153C22×Q8—24no—
154C22×Q8—21no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8C22×C412yes—
158C22×D4—28no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—24no—
162C22×D4—21no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4C22×C412yes—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—28no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
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×C4C22×C411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C2×M16—44no—
205C2×M16—44no—
206C2×M16—44no—
207C22×C8—24no—
208C22×C8—24no—
209#117—44no—
210#117—44no—
211#117—44no—
212#117—44no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
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—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—42no—
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—44no—
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
266C4⋊C8—44no—
267C4⋊C8—44no—
268C4⋊C8—44no—
269C22⋊C8—44no—
270C22⋊C8—44no—
271C22⋊C8—44no—
272C22⋊C8—44no—
273C22⋊C8—44no—
274#97—44no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—24no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—41no—
291C24—41no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24C2×C22⋊C411yescharacteristic
299C24C2×C22⋊C411yescharacteristic
300C24C2×C22⋊C411yescharacteristic
301#41—416no—
302#41—416no—
303#41—432no—
304#41—432no—
305#41—416no—
306#41—416no—
307C2×Q8—24no—
308C2×Q8—22no—
309C2×Q8—416no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8C2×M1614yes—
315C2×D4—24no—
316C2×D4—28no—
317C2×D4—28no—
318C2×D4—22no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—416no—
322C2×D4—42no—
323C2×D4—432no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—42no—
329C2×D4C2×M1614yes—
330C22×C4—232no—
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—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—28no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—41no—
372C22×C4—41no—
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—41no—
388C22×C4—41no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—44no—
392C22×C4—432no—
393C22×C4—42no—
394C22×C4—416no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4#9914yes—
406C22×C4#10014yes—
407C22×C4C2×C22⋊C411yescharacteristic
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×M1614yes—
417C22×C4AGL(1,ℤ/8)12yes—
418C22×C4#13712yes—
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422M16—88no—
423C2×C8—44no—
424C2×C8—44no—
425C2×C8—44no—
426C2×C8—44no—
427C2×C8—44no—
428C2×C8—44no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—44no—
461C42—44no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—28no—
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—24no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—48no—
486C23—48no—
487C23—48no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—84no—
496C23—84no—
497C23—84no—
498C23—84no—
499C23#40712yes—
500C23C2×#9912yes—
501C23C2×#10012yes—
502C23#41712yes—
503C23#42112yes—
504C23C22×C22⋊C411yescharacteristic
505Q8—44no—
506Q8—44no—
507Q8—48no—
508D4—44no—
509D4—44no—
510D4—48no—
511D4—416no—
512D4—416no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
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—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—416no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—416no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4#40712yes—
575C2×C4C2×#9912yes—
576C2×C4C2×#9912yes—
577C2×C4C2×#10012yes—
578C2×C4C2×#10012yes—
579C2×C4#41712yes—
580C2×C4#42112yes—
581C2×C4C22×M1612yes—
582C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
583C2×C4C2×#13711yescharacteristic
584C2×C4#57512yes—
585C8—84no—
586C8—84no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—416no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—816no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22#151314yes—
617C22C2×#40711yescharacteristic
618C22C22×#9911yescharacteristic
619C22C22×#10011yescharacteristic
620C22C2×#41711yescharacteristic
621C22#289611yescharacteristic
622C22C2×#42111yescharacteristic
623C22C2×#42111yescharacteristic
624C4—21no—
625C4—21no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—44no—
637C4—82no—
638C4—82no—
639C4#146214yes—
640C2—24no—
641C2—42no—
642C2—42no—
643C2—84no—
644C2C2×#146211yescharacteristic
645C2#1745111yescharacteristic
646C2#1745211yescharacteristic
647C2C2×#151311yescharacteristic
648C2#1755911yescharacteristic
649C2#1767111yescharacteristic
650C2#1775011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×#41C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20715C2semidirect
3. #20626C2semidirect
4. #7612C2semidirect
5. #7609C2semidirect
6. #7606C2semidirect
7. #7597C2semidirect
34. #1797C22semidirect
35. #1797C22semidirect
36. #1797C22semidirect
39. #1749C22semidirect
41. #1272C22semidirect
42. #1270C22semidirect
152. C2×#41M16semidirect

Maximal subgroups

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

subgroupindex
C2×#22972
#207152
#206262
#76122
#76092
#76062
#75972