Finite Group Databasebeta

← #161049

Subgroups — #161049

Order 512.

1344 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×#1817C211yesmaximal, characteristic
3#20626C211yesmaximal, characteristic
4#7648C214yesmaximal, complemented
5#7609C212yesmaximal, complemented
6#7607C212yesmaximal, complemented
7#7589C214yesmaximal, complemented
8C22×C22⋊Q8C411yescharacteristic
9C22×C4×Q8C411yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#423—21no—
12C2×#407—21no—
13C22×C22⋊C8—21no—
14C4×C22⋊Q8—21no—
15C4×C22⋊Q8—21no—
16C4×C22⋊Q8—21no—
17C4×C22⋊Q8—21no—
18C4×C22⋊Q8C412yes—
19C4×C22⋊Q8C412yes—
20C4×C22⋊Q8C2212yes—
21C4×C22⋊Q8C2212yes—
22C2×#391C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#387—21no—
25C2×#385—21no—
26C2×#380C411yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C411yescharacteristic
29C2×C4×C22⋊C4—21no—
30#1817—21no—
31#1817—21no—
32#1817C2212yes—
33#1817C2212yes—
34#1797—21no—
35#1797—21no—
36#1797C2212yes—
37#1797C2212yes—
38#1728C2212yes—
39C2×#336—21no—
40C2×#336C2212yes—
41#1272C2218yescomplemented
42#1271C2218yescomplemented
43C23×Q8C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C22×M16—22no—
48C23×C8—22no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—28no—
52C2×C22⋊Q8C2×C412yes—
53C2×C22⋊Q8C2×C412yes—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—44no—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C412yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C42—21no—
68C22×C42—21no—
69C22×C42—44no—
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#423—44no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85#407—44no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—44no—
91#391—21no—
92#391—21no—
93#391C2×C412yes—
94#391C2312yes—
95#389—21no—
96#389—21no—
97#389C2×C412yes—
98#389C2312yes—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#380—21no—
108#380—21no—
109#380C2×C412yes—
110#380C2×C412yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C412yes—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336C2314yes—
133#336C2314yes—
134C25C22⋊C411yescharacteristic
135C22×Q8—24no—
136C22×Q8—21no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8C22×C412yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4C22⋊C411yescharacteristic
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×C4SD1614yes—
162C23×C4Q1614yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C2×M16—44no—
173C2×M16—44no—
174C2×M16—44no—
175C22×C8—24no—
176C22×C8—24no—
177C22×C8—44no—
178C22×C8—44no—
179C22×C8—44no—
180C22⋊Q8—28no—
181C22⋊Q8—28no—
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C22⋊Q8M1618yescomplemented
186C4×Q8—44no—
187C4×Q8—44no—
188C4×Q8—44no—
189C4×Q8—44no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—44no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—44no—
231C2×C42—44no—
232C2×C42C22⋊C411yescharacteristic
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22⋊C411yescharacteristic
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2411yescharacteristic
248C4⋊C8—44no—
249C4⋊C8—44no—
250C22⋊C8—44no—
251C22⋊C8—44no—
252C22⋊C8—44no—
253C22⋊C8—44no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—41no—
269C24—41no—
270C24#10314yes—
271C24C2×C22⋊C411yescharacteristic
272C24C2×C22⋊C411yescharacteristic
273C24C2×C22⋊C411yescharacteristic
274C2×Q8—24no—
275C2×Q8—22no—
276C2×Q8—42no—
277C2×Q8—416no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8C2×M1614yes—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
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—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—416no—
337C22×C4—416no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4#9914yes—
349C22×C4#10314yes—
350C22×C4#10314yes—
351C22×C4#10314yes—
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×M1614yes—
362C22×C4C2×SD1612yes—
363C22×C4C2×SD1612yes—
364C22×C4C2×SD1612yes—
365C22×C4C2×Q1612yes—
366C22×C4C2×Q1612yes—
367C22×C4C2×Q1612yes—
368C22×C4C23×C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371M16—88no—
372C2×C8—44no—
373C2×C8—44no—
374C2×C8—44no—
375C2×C8—44no—
376C2×C8—44no—
377C2×C8—44no—
378C2×C8—88no—
379C4⋊C4—24no—
380C4⋊C4—28no—
381C4⋊C4—22no—
382C4⋊C4—28no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—42no—
391C4⋊C4C2×M1614yes—
392C22⋊C4—28no—
393C22⋊C4—28no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—44no—
413C42—44no—
414C42—44no—
415C42—44no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—28no—
431C23—28no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23#40712yes—
438C23C2×#9912yes—
439C23C2×#10312yes—
440C23C2×#10312yes—
441C23C2×#10312yes—
442C23#42112yes—
443C23#42112yes—
444C23C22×C22⋊C411yescharacteristic
445Q8—44no—
446Q8—44no—
447Q8—48no—
448C2×C4—28no—
449C2×C4—28no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
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—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—48no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
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—84no—
510C2×C4—84no—
511C2×C4#40712yes—
512C2×C4C2×#9912yes—
513C2×C4C2×#9912yes—
514C2×C4#41312yes—
515C2×C4C2×#10312yes—
516C2×C4C2×#10312yes—
517C2×C4C2×#10312yes—
518C2×C4C22×M1612yes—
519C2×C4C22×SD1611yescharacteristic
520C2×C4C22×Q1611yescharacteristic
521C2×C4#57512yes—
522C8—84no—
523C8—84no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—416no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22#148714yes—
545C22#152014yes—
546C22C2×#40711yescharacteristic
547C22C22×#9911yescharacteristic
548C22C2×#41311yescharacteristic
549C22C22×#10311yescharacteristic
550C22#289611yescharacteristic
551C22C2×#42111yescharacteristic
552C22C2×#42111yescharacteristic
553C4—21no—
554C4—21no—
555C4—22no—
556C4—22no—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—44no—
565C4—41no—
566C4—41no—
567C4—44no—
568C4—82no—
569C4—82no—
570C2—24no—
571C2—24no—
572C2—42no—
573C2#1736211yescharacteristic
574C2C2×#148711yescharacteristic
575C2#1745211yescharacteristic
576C2C2×#152011yescharacteristic
577C2#1758711yescharacteristic
578C2#1766811yescharacteristic
579C2#1770811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
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
4. #7648C2semidirect
5. #7609C2semidirect
6. #7607C2semidirect
7. #7589C2semidirect
41. #1272C22semidirect
42. #1271C22semidirect
185. C22⋊Q8M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C2×#18172
#206262
#76482
#76092
#76072
#75892