Finite Group Databasebeta

← #160709

Subgroups — #160709

Order 512.

1492 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic
3C2×#1728C211yesmaximal, characteristic
4#7617C214yesmaximal, complemented
5#7609C212yesmaximal, complemented
6#7607C212yesmaximal, complemented
7#7566C214yesmaximal, complemented
8C23×M16—21no—
9C22×C22⋊Q8C411yescharacteristic
10C22×C4×Q8C411yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#431—21no—
13C4×C22⋊Q8—21no—
14C4×C22⋊Q8—21no—
15C4×C22⋊Q8—21no—
16C4×C22⋊Q8—21no—
17C4×C22⋊Q8C412yes—
18C4×C22⋊Q8C412yes—
19C4×C22⋊Q8C2212yes—
20C4×C22⋊Q8C2212yes—
21C2×#391C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#387—21no—
24C2×#385—21no—
25C2×#380C411yescharacteristic
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C411yescharacteristic
28C2×C4×C22⋊C4—21no—
29#1817C2212yes—
30#1797—21no—
31#1797C2212yes—
32#1779—21no—
33C4×C22⋊C8—21no—
34C4×C22⋊C8C2212yes—
35#1728—21no—
36#1728—21no—
37#1728C2212yes—
38#1728C2212yes—
39C2×#336—21no—
40C2×#336C2212yes—
41#1272C2218yescomplemented
42#1271C2218yescomplemented
43C23×Q8C2×C411yescharacteristic
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—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—28no—
56C2×C22⋊Q8C2×C412yes—
57C2×C22⋊Q8C2×C412yes—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—44no—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8C2×C412yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42—44no—
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#431—44no—
87C2×C4⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C4×C8—22no—
92#391—21no—
93#391—21no—
94#391C2×C412yes—
95#391C2312yes—
96#389—21no—
97#389—21no—
98#389C2×C412yes—
99#389C2312yes—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#380—21no—
109#380—21no—
110#380C2×C412yes—
111#380C2×C412yes—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C412yes—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336C2314yes—
134#336C2314yes—
135C25C22⋊C411yescharacteristic
136C22×Q8—24no—
137C22×Q8—21no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8C22×C412yes—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—24no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4SD1614yes—
163C23×C4Q1614yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C2×M16—28no—
174C2×M16—28no—
175C2×M16—44no—
176C2×M16—432no—
177C2×M16—44no—
178C2×M16—44no—
179C2×M16—44no—
180C22×C8—28no—
181C22×C8—24no—
182C22×C8—24no—
183C22×C8—44no—
184C22×C8—44no—
185C22⋊Q8—28no—
186C22⋊Q8—28no—
187C22⋊Q8—24no—
188C22⋊Q8—24no—
189C22⋊Q8—24no—
190C22⋊Q8M1618yescomplemented
191C4×Q8—44no—
192C4×Q8—44no—
193C4×Q8—44no—
194C4×Q8—44no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—44no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—44no—
236C2×C42—44no—
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2411yescharacteristic
253C4⋊C8—44no—
254C22⋊C8—44no—
255C22⋊C8—44no—
256C22⋊C8—44no—
257C4×C8—44no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—41no—
273C24—41no—
274C24#10314yes—
275C24C2×C22⋊C411yescharacteristic
276C24C2×C22⋊C411yescharacteristic
277C24C2×C22⋊C411yescharacteristic
278C2×Q8—24no—
279C2×Q8—22no—
280C2×Q8—42no—
281C2×Q8—416no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8C2×M1614yescomplemented
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—216no—
291C22×C4—216no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—416no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—416no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4#9914yes—
352C22×C4#10314yes—
353C22×C4#10314yes—
354C22×C4#10314yes—
355C22×C4#10414yes—
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×M1614yes—
366C22×C4C2×SD1612yes—
367C22×C4C2×SD1612yes—
368C22×C4C2×SD1612yes—
369C22×C4C2×Q1612yes—
370C22×C4C2×Q1612yes—
371C22×C4C2×Q1612yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375M16—416no—
376M16—416no—
377C2×C8—28no—
378C2×C8—44no—
379C2×C8—416no—
380C2×C8—44no—
381C2×C8—44no—
382C2×C8—44no—
383C2×C8—44no—
384C4⋊C4—24no—
385C4⋊C4—28no—
386C4⋊C4—22no—
387C4⋊C4—28no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4C2×M1614yescomplemented
397C22⋊C4—28no—
398C22⋊C4—28no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—44no—
418C42—44no—
419C42—44no—
420C42—44no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—28no—
436C23—28no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23#40712yes—
443C23C2×#9912yes—
444C23C2×#10312yes—
445C23C2×#10312yes—
446C23C2×#10312yes—
447C23C2×#10412yes—
448C23#42112yes—
449C23C22×C22⋊C411yescharacteristic
450Q8—44no—
451Q8—44no—
452Q8—48no—
453C2×C4—28no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—48no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—48no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4#328116yes—
514C2×C4#40712yes—
515C2×C4C2×#9912yes—
516C2×C4C2×#9912yes—
517C2×C4#41312yes—
518C2×C4C2×#10312yes—
519C2×C4C2×#10312yes—
520C2×C4C2×#10312yes—
521C2×C4C2×#10412yes—
522C2×C4C2×#10412yes—
523C2×C4C22×M1612yes—
524C2×C4C22×SD1611yescharacteristic
525C2×C4C22×Q1611yescharacteristic
526C2×C4#57512yes—
527C8—48no—
528C8—84no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—416no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22#148714yes—
550C22C2×#32814yes—
551C22C2×#40711yescharacteristic
552C22C22×#9911yescharacteristic
553C22C2×#41311yescharacteristic
554C22C22×#10311yescharacteristic
555C22#289611yescharacteristic
556C22C22×#10411yescharacteristic
557C22C2×#42111yescharacteristic
558C4—21no—
559C4—21no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—22no—
565C4—44no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—44no—
571C4—82no—
572C4—82no—
573C4C2×#32814yes—
574C4C2×#32814yes—
575C2—24no—
576C2—24no—
577C2—42no—
578C2#1736211yescharacteristic
579C2C2×#148711yescharacteristic
580C2#1744711yescharacteristic
581C2C22×#32811yescharacteristic
582C2#1755911yescharacteristic
583C2#1764211yescharacteristic
584C2#1770811yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC25C22⋊C4
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
4. #7617C2semidirect
5. #7609C2semidirect
6. #7607C2semidirect
7. #7566C2semidirect
41. #1272C22semidirect
42. #1271C22semidirect
190. C22⋊Q8M16semidirect
285. C2×Q8C2×M16semidirect
396. C4⋊C4C2×M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
#207142
C2×#17282
#76172
#76092
#76072
#75662