Finite Group Databasebeta

← #191012

Subgroups — #191012

Order 512.

1456 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic
3C4×#381C211yesmaximal, characteristic
4#8713C212yesmaximal
5#8574C214yesmaximal, complemented
6#8329C212yesmaximal
7#8219C214yesmaximal, complemented
8C22×C22⋊Q8C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic, complemented
10C23×C42C2211yescharacteristic
11C2×#432—21no—
12C4×C22⋊Q8—21no—
13C4×C22⋊Q8—21no—
14C4×C22⋊Q8—21no—
15C4×C22⋊Q8—21no—
16C4×C22⋊Q8C412yescomplemented
17C4×C22⋊Q8C412yescomplemented
18C4×C22⋊Q8C2212yes—
19C4×C22⋊Q8C2212yes—
20C2×#391C411yescharacteristic, complemented
21C2×#389C411yescharacteristic, complemented
22C2×#387—21no—
23C2×#385—21no—
24C2×#381—21no—
25C2×#380C2211yescharacteristic
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C411yescharacteristic, complemented
28C2×C4×C22⋊C4—21no—
29C2×C4×C22⋊C4—21no—
30#1817—21no—
31#1797—21no—
32#1797C2212yes—
33#1779C2212yes—
34#1749—21no—
35#1749C2212yes—
36C2×#328C2218yes—
37#1434—21no—
38#1434C2212yes—
39C22⋊#95C2212yes—
40C4×#95—21no—
41C4×#95C2212yes—
42#1294C2218yes—
43C23×Q8C2×C411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—28no—
50C2×C22⋊Q8C2×C412yes—
51C2×C22⋊Q8C2312yes—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—44no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C412yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—21no—
66C22×C42—21no—
67C22×C42—21no—
68C22×C42—22no—
69C22×C42—44no—
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#432—44no—
83C2×C4⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8C2314yes—
87C2×#97—22no—
88#391—21no—
89#391—21no—
90#391C2×C412yes—
91#391C2×C412yes—
92#389—21no—
93#389—21no—
94#389C2×C412yes—
95#389C2×C412yes—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#385—22no—
101#385—22no—
102#385—22no—
103#385—22no—
104#381—28no—
105#380—21no—
106#380—21no—
107#380C2×C412yes—
108#380C2312yes—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C412yes—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—44no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95C2314yes—
132#336—22no—
133#336—22no—
134#328—216no—
135C25C2×D411yescharacteristic
136C22×Q8—24no—
137C22×Q8—21no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8C22×C412yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—24no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
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×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8—24no—
174C22⋊Q8—28no—
175C22⋊Q8—28no—
176C22⋊Q8—24no—
177C22⋊Q8—24no—
178C22⋊Q8—24no—
179C22⋊Q8#4118yes—
180C4×Q8—44no—
181C4×Q8—44no—
182C4×Q8—44no—
183C4×Q8—44no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42—44no—
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×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C4⋊C8—44no—
249C22⋊C8—28no—
250C22⋊C8—44no—
251C22⋊C8—44no—
252#97—44no—
253#95—28no—
254#95—28no—
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—
264#95—22no—
265#95—28no—
266#95—44no—
267#95—44no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—41no—
273C24—41no—
274C24C2×C22⋊C411yescharacteristic
275C24C2×C22⋊C411yescharacteristic
276C24#13714yes—
277C24C22×D411yescharacteristic
278C2×Q8—24no—
279C2×Q8—22no—
280C2×Q8—42no—
281C2×Q8—416no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8#11714yes—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
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—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—22no—
322C22×C4—22no—
323C22×C4—28no—
324C22×C4—22no—
325C22×C4—22no—
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—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—416no—
345C22×C4—416no—
346C22×C4—416no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4#9914yes—
364C22×C4#10414yes—
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4114yes—
378C2×C8—44no—
379C2×C8—44no—
380C2×C8—44no—
381C4⋊C4—24no—
382C4⋊C4—28no—
383C4⋊C4—22no—
384C4⋊C4—28no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4#11714yes—
394C22⋊C4—28no—
395C22⋊C4—28no—
396C22⋊C4—416no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—44no—
416C42—44no—
417C42—44no—
418C42—44no—
419C42—88no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—28no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—28no—
435C23—21no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23C2×#9912yes—
442C23#41912yes—
443C23#41912yes—
444C23C2×#10412yes—
445C23C22×C22⋊C411yescharacteristic
446C23#52512yes—
447C23C2×#13712yes—
448C23#57512yes—
449Q8—44no—
450Q8—44no—
451Q8—48no—
452C2×C4—28no—
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—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—28no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—21no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
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—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—48no—
500C2×C4—48no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4C2×#9912yes—
522C2×C4C2×#9912yes—
523C2×C4#41012yes—
524C2×C4C2×#10412yes—
525C2×C4C2×#10412yes—
526C2×C4#42112yes—
527C2×C4C2×#11712yes—
528C2×C4#52512yes—
529C8—84no—
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—22no—
545C22—416no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22#150814yes—
551C22#228011yescharacteristic
552C22C22×#9911yescharacteristic
553C22C2×#41011yescharacteristic
554C22C2×#41911yescharacteristic
555C22#289611yescharacteristic
556C22C22×#10411yescharacteristic
557C22C2×#42111yescharacteristic
558C22#311714yes—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—22no—
567C4—44no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—44no—
573C4—88no—
574C4—82no—
575C4—82no—
576C2—24no—
577C2—24no—
578C2—42no—
579C2C2×#150811yescharacteristic
580C2#1753711yescharacteristic
581C2#1758711yescharacteristic
582C2#1759611yescharacteristic
583C2#2072211yescharacteristic
584C2#2074111yescharacteristic
585C2#2074711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8574C2semidirect
7. #8219C2semidirect
9. C22×C4×Q8C4semidirect
16. C4×C22⋊Q8C4semidirect
17. C4×C22⋊Q8C4semidirect
20. C2×#391C4semidirect
21. C2×#389C4semidirect
27. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
#207152
C4×#3812
#87132
#85742
#83292
#82192