Finite Group Databasebeta

← #204649

Subgroups — #204649

Order 512.

1318 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35019C211yesmaximal, characteristic
2#20524C211yesmaximal, characteristic, complemented
3C2×#1511C211yesmaximal, characteristic, complemented
4C2×#1510C211yesmaximal, characteristic
5C2×#1508C211yesmaximal, characteristic, complemented
6C2×#1507C211yesmaximal, characteristic
7#13953C211yesmaximal, characteristic, complemented
8#9682C214yesmaximal, complemented
9#9681C214yesmaximal, complemented
10C22×C22⋊Q8C2211yescharacteristic
11C22×#121C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C2×#407C2211yescharacteristic
14#2850C2212yes—
15#2850C2212yes—
16#2639—22no—
17#2591—22no—
18#2561—22no—
19#2504C412yescomplemented
20#2503C412yescomplemented
21#2436C412yescomplemented
22#2435C412yescomplemented
23#2308—22no—
24C2×#381C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27#1753—22no—
28#1753—22no—
29#1749C2212yescomplemented
30C4×C22⋊C8C2212yescomplemented
31#1511—24no—
32#1510—24no—
33#1508—24no—
34#1507—24no—
35C2×#328C2214yescomplemented
36C2×#328C2214yescomplemented
37C2×#327C2214yes—
38C2×#327C2214yes—
39#1439—22no—
40#1436—22no—
41C4⋊#95C2212yescomplemented
42#1434C2212yescomplemented
43C23×Q8C2×C411yescharacteristic
44C23×D4—21no—
45C23×D4C2×C411yescharacteristic
46C24×C4C2311yescharacteristic
47C22×M16—21no—
48C22×M16—21no—
49C2×#124C2×C412yes—
50C2×#124C2×C412yes—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2312yes—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121C2×C412yes—
66C2×#121C2312yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4D412yes—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#407—22no—
80#407—22no—
81#407D414yes—
82#407D414yes—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8C2312yes—
86C2×C22⋊C8C2312yes—
87C2×#97—22no—
88C2×C4×C8—22no—
89#400—24no—
90#397—24no—
91#394—24no—
92#393—24no—
93#391C2×C412yes—
94#391C2×C412yes—
95#390C2×C412yes—
96#390C2×C412yes—
97#388—24no—
98#388—24no—
99#386—24no—
100#386—24no—
101#386—24no—
102#386—24no—
103#382C2×C412yes—
104#382C2×C412yes—
105#381—24no—
106#381—22no—
107#381—22no—
108#379—24no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4C2×C412yes—
112C4×C22⋊C4C2×C412yes—
113C4×C22⋊C4D412yes—
114C4×C22⋊C4D412yes—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95D412yes—
120C2×#95C2312yes—
121C2×#95C2312yes—
122#336—22no—
123#336—22no—
124#328—28no—
125#328—28no—
126#327—28no—
127#327—28no—
128C25—21no—
129C25C2×D411yescharacteristic
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—81no—
134C22×Q8—81no—
135C22×Q8C22×C412yes—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—44no—
142C22×D4—44no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—81no—
146C22×D4—81no—
147C22×D4C22×C412yes—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—81no—
154C23×C4—81no—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4#4112yes—
165C23×C4#4112yes—
166C23×C4C2411yescharacteristic
167C2×M16—24no—
168C2×M16—24no—
169C2×M16—81no—
170C2×M16—81no—
171C2×M16—81no—
172C2×M16—81no—
173C22×C8—22no—
174C22×C8—22no—
175#124—82no—
176#124—82no—
177C22⋊Q8—24no—
178C22⋊Q8—24no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—42no—
184C22⋊Q8—42no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187#121—24no—
188#121—24no—
189#121—22no—
190#121—22no—
191#121—22no—
192#121—22no—
193#121—42no—
194#121—42no—
195#121—42no—
196#121—42no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—81no—
230C2×C22⋊C4—81no—
231C2×C22⋊C4C22⋊C414yes—
232C2×C22⋊C4C22⋊C414yes—
233C2×C42—42no—
234C2×C42—81no—
235C2×C42—81no—
236C2×C42—81no—
237C2×C42—81no—
238C2×C42C22×C412yes—
239C2×C42C22×C412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C22⋊C8—24no—
243C22⋊C8—24no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C22⋊C8C2×D414yes—
247C22⋊C8C2×D414yes—
248#97—82no—
249C4×C8—82no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—44no—
259#95—44no—
260#95—42no—
261#95—82no—
262#95C22⋊C414yes—
263#95C22⋊C414yes—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—42no—
268C24—41no—
269C24—41no—
270C24—44no—
271C24—42no—
272C24—81no—
273C24—81no—
274C24—82no—
275C24C2×C22⋊C411yescharacteristic
276C24C2×C22⋊C411yescharacteristic
277C24#12412yes—
278C24#12712yes—
279C24C22×D411yescharacteristic
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—81no—
286C2×Q8—81no—
287C2×Q8—84no—
288C2×Q8—84no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—48no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—42no—
300C2×D4—81no—
301C2×D4—81no—
302C2×D4—84no—
303C2×D4—84no—
304C2×D4—84no—
305C2×D4—84no—
306C2×D4—88no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
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—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
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—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—81no—
354C22×C4—81no—
355C22×C4C2×C22⋊C414yes—
356C22×C4C2×C22⋊C414yes—
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C4×D412yes—
360C22×C4C4×D412yes—
361C22×C4C4×D412yes—
362C22×C4C4×D412yes—
363C22×C4#12112yes—
364C22×C4#12112yes—
365C22×C4#12112yes—
366C22×C4#12112yes—
367C22×C4#12412yes—
368C22×C4#12712yes—
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375M16—84no—
376M16—84no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—88no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—48no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—44no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—44no—
402C22⋊C4—84no—
403C22⋊C4—84no—
404C22⋊C4—84no—
405C22⋊C4—84no—
406C22⋊C4—84no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42—84no—
412C23—21no—
413C23—21no—
414C23—22no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—24no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—84no—
440C23—84no—
441C23—84no—
442C23—164no—
443C23—164no—
444C23#39012yes—
445C23#39012yes—
446C23C22×C22⋊C411yescharacteristic
447C23C2×#12411yescharacteristic
448C23C2×#12711yescharacteristic
449C23#57512yes—
450C23#57512yes—
451Q8—82no—
452Q8—82no—
453Q8—84no—
454D4—82no—
455D4—82no—
456D4—84no—
457D4—88no—
458D4—82no—
459D4—82no—
460D4—84no—
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—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4—84no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4#39012yes—
518C2×C4#39012yes—
519C2×C4C2×C4×D411yescharacteristic
520C2×C4C2×C4×D411yescharacteristic
521C2×C4C2×#12111yescharacteristic
522C2×C4C2×#12111yescharacteristic
523C8—44no—
524C8—44no—
525C22—21no—
526C22—21no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—84no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—84no—
549C22—84no—
550C22—82no—
551C22—168no—
552C22C2×#39011yescharacteristic
553C22#288311yescharacteristic
554C22#288311yescharacteristic
555C22#289612yes—
556C22#290112yes—
557C22#314011yescharacteristic
558C22#314011yescharacteristic
559C22#315312yes—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—22no—
565C4—22no—
566C4—44no—
567C4—44no—
568C4—84no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4—162no—
573C2—22no—
574C2—21no—
575C2—21no—
576C2—42no—
577C2—42no—
578C2—162no—
579C2#1762312yes—
580C2#1762512yes—
581C2#2120411yescharacteristic
582C2#2121311yescharacteristic
583C2#2125311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20524C2semidirect
3. C2×#1511C2semidirect
5. C2×#1508C2semidirect
7. #13953C2semidirect
8. #9682C2semidirect
9. #9681C2semidirect
19. #2504C4semidirect
20. #2503C4semidirect
21. #2436C4semidirect
22. #2435C4semidirect
29. #1749C22semidirect
30. C4×C22⋊C8C22semidirect
35. C2×#328C22semidirect
36. C2×#328C22semidirect
41. C4⋊#95C22semidirect
42. #1434C22semidirect

Maximal subgroups

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

subgroupindex
#350192
#205242
C2×#15112
C2×#15102
C2×#15082
C2×#15072
#139532
#96822
#96812