Finite Group Databasebeta

← #206925

Subgroups — #206925

Order 512.

1398 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic
3C4×#407C211yesmaximal, characteristic
4#9881C214yesmaximal, complemented
5#9879C212yesmaximal, complemented
6#9878C212yesmaximal, complemented
7#7911C214yesmaximal
8C22×C22⋊Q8C411yescharacteristic
9C22×C4×Q8C411yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#423—21no—
12C2×#407C2212yes—
13C2×#407C2212yes—
14C2×C4×M16—21no—
15C4×C22⋊Q8—21no—
16C4×C22⋊Q8—21no—
17C4×C22⋊Q8—21no—
18C4×C22⋊Q8—21no—
19C4×C22⋊Q8—21no—
20C4×C22⋊Q8—21no—
21C4×C22⋊Q8—21no—
22C4×C22⋊Q8—21no—
23C2×#391C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#387C2212yes—
26C2×#385C412yes—
27C2×#380C411yescharacteristic
28C2×C4×C4⋊C4C412yes—
29C2×C4×C4⋊C4C411yescharacteristic
30C2×C4×C22⋊C4C2212yes—
31#1797—21no—
32#1797—21no—
33#1797C2212yes—
34#1797C2212yes—
35#1749—21no—
36#1749C2212yes—
37C4×C22⋊C8—21no—
38C4×C22⋊C8C2212yes—
39#1728—21no—
40#1728—21no—
41#1280C2218yes—
42#1280C2218yes—
43C23×Q8C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48C22×M16—22no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—24no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—44no—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C412yes—
68C22×C42—44no—
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C412yes—
71C22×C42C2×C412yes—
72C22×C42C2×C411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#423—44no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86#407—24no—
87#407—28no—
88#407Q818yescomplemented
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8C2314yes—
92C2×C22⋊C8C2314yes—
93C4×M16—44no—
94C2×#97—22no—
95C2×C4×C8—22no—
96#391—21no—
97#391—21no—
98#391—21no—
99#391—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#385—22no—
109#385—22no—
110#385—22no—
111#385—22no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C2×#95C2×C412yes—
129C2×#95C2×C412yes—
130C2×#95C2×C412yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133#336—22no—
134#336—22no—
135#336—22no—
136#336—22no—
137C25C22×C411yescharacteristic
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—22no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8M1614yescomplemented
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—42no—
147C23×C4—42no—
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×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C412yes—
166C23×C4C22×C412yes—
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C412yes—
169C23×C4C22×C412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×Q814yes—
175C23×C4#4114yes—
176C23×C4C2411yescharacteristic
177C2×M16—28no—
178C2×M16—44no—
179C2×M16—44no—
180C2×M16—44no—
181C2×M16—44no—
182C2×M16—44no—
183C22×C8—24no—
184C22×C8—24no—
185C22⋊Q8—28no—
186C22⋊Q8—28no—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C4×Q8—44no—
190C4×Q8—44no—
191C4×Q8—44no—
192C4×Q8—44no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4M1614yescomplemented
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—44no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
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—42no—
236C2×C42—44no—
237C2×C42—44no—
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×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C4⋊C8—44no—
255C4⋊C8—44no—
256C22⋊C8—28no—
257C22⋊C8—24no—
258C22⋊C8—28no—
259C22⋊C8—44no—
260C22⋊C8C2×Q818yes—
261#97—44no—
262C4×C8—44no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273C24—21no—
274C24—21no—
275C24—22no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24C2×C22⋊C411yescharacteristic
281C24C2×C22⋊C411yescharacteristic
282C24C4×Q814yes—
283C24C23×C411yescharacteristic
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—48no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—48no—
290C2×Q8—42no—
291C2×Q8—42no—
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—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
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—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—48no—
327C22×C4—48no—
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—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—416no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
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×C4C4×Q814yes—
372C22×C4C22⋊Q814yes—
373C22×C4C22⋊Q814yes—
374C22×C4#12412yes—
375C22×C4#12412yes—
376C22×C4C4⋊Q812yes—
377C22×C4C4⋊Q812yes—
378C22×C4C2×M1612yes—
379C22×C4C2×M1612yes—
380C22×C4C2×M1614yes—
381C22×C4C2×M1614yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×Q812yes—
385C22×C4C2×#4112yes—
386M16—416no—
387M16—88no—
388C2×C8—28no—
389C2×C8—44no—
390C2×C8—44no—
391C2×C8—44no—
392C2×C8—44no—
393C2×C8—44no—
394C4⋊C4—28no—
395C4⋊C4—28no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4—42no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—42no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C22⋊C4—28no—
407C22⋊C4—28no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—44no—
426C42—44no—
427C42—44no—
428C42—44no—
429C42#9914yes—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—24no—
447C23—24no—
448C23—44no—
449C23—44no—
450C23—42no—
451C23—42no—
452C23—44no—
453C23—42no—
454C23—42no—
455C23—44no—
456C23#39112yes—
457C23#39112yes—
458C23C22×C22⋊C411yescharacteristic
459C23C2×C4×Q812yes—
460C23C22×M1611yescharacteristic
461C23C22×M1611yescharacteristic
462C23#56812yes—
463Q8—48no—
464Q8—48no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—28no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4#39112yes—
531C2×C4#39112yes—
532C2×C4#40712yes—
533C2×C4#40712yes—
534C2×C4C2×#9912yes—
535C2×C4C2×#9912yes—
536C2×C4C2×#9912yes—
537C2×C4#41012yes—
538C2×C4#41312yes—
539C2×C4#41312yes—
540C2×C4C2×C22⋊Q812yes—
541C2×C4C2×#12411yescharacteristic
542C2×C4C2×C4⋊Q811yescharacteristic
543C8—48no—
544C8—84no—
545C22—22no—
546C22—22no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—48no—
560C22—48no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22C2×#39111yescharacteristic
570C22C2×#40711yescharacteristic
571C22#287911yescharacteristic
572C22C22×#9911yescharacteristic
573C22C2×#41011yescharacteristic
574C22C2×#41311yescharacteristic
575C22C2×#41311yescharacteristic
576C22Q8×M1612yes—
577C22Q8×M1612yes—
578C4—22no—
579C4—21no—
580C4—21no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—22no—
585C4—41no—
586C4—41no—
587C4—44no—
588C4—44no—
589C4—41no—
590C4—41no—
591C4—82no—
592C4—82no—
593C4#146514yes—
594C4#199614yes—
595C2—22no—
596C2—22no—
597C2—42no—
598C2—42no—
599C2C2×#146511yescharacteristic
600C2#1736611yescharacteristic
601C2#1740011yescharacteristic
602C2#1740111yescharacteristic
603C2#2130311yescharacteristic
604C2C2×#199611yescharacteristic
605C2#2133711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ 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
4. #9881C2semidirect
5. #9879C2semidirect
6. #9878C2semidirect
88. #407Q8semidirect
143. C22×Q8M16semidirect
212. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
#206262
C4×#4072
#98812
#98792
#98782
#79112