Finite Group Databasebeta

← #374605

Subgroups — #374605

Order 512.

1920 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2579C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3C2×C4×#99C211yesmaximal, characteristic
4#13923C212yesmaximal, complemented
5#11222C212yesmaximal, complemented
6#11222C212yesmaximal, complemented
7#10854C212yesmaximal, complemented
8#10745C212yesmaximal
9#10745C212yesmaximal
10C2×#518C2212yes—
11C22×C4×D4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×#99C2212yes—
14#2579—24no—
15#2579C418yescomplemented
16C2×#398—21no—
17C2×#396—21no—
18#2349—24no—
19D4×C4⋊C4—24no—
20#2316—21no—
21#2315—21no—
22#2315C2212yes—
23C2×#387—21no—
24C2×#386C2211yescharacteristic
25C2×#386C2211yescharacteristic
26C2×#385—21no—
27C2×#385C2211yescharacteristic
28C2×#385C2211yescharacteristic
29#2285—21no—
30C2×#380—21no—
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33C2×C4×C22⋊C4C2212yes—
34C4×#99—24no—
35C2×#342C2212yes—
36C2×#342C2214yes—
37C2×#342C2214yes—
38#1441—24no—
39#1436—24no—
40C22⋊#95C2214yes—
41C22⋊#95C2214yes—
42C23×D4C2311yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47#518—24no—
48#518—24no—
49#518—44no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—24no—
53C2×C4×D4—28no—
54C2×C4×D4C2×C414yes—
55C2×C4×D4D414yes—
56C2×#117—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2×C414yescomplemented
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D414yes—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4D411yescharacteristic
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2311yescharacteristic
76C22×C22⋊C4C2311yescharacteristic
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2311yescharacteristic
79C22×C42—22no—
80C22×C42D411yescharacteristic
81C2×#99—24no—
82C2×#99—24no—
83C2×#99Q818yes—
84#398—28no—
85#396—28no—
86#389—28no—
87#387—28no—
88#386—22no—
89#386—22no—
90#386C2×C414yes—
91#386C2×C414yes—
92#385—28no—
93#385—28no—
94#385—28no—
95#385—22no—
96#385—22no—
97#385C2×C414yescomplemented
98#385C2×C414yescomplemented
99#384—22no—
100#382—22no—
101#380—22no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—22no—
110#380C2×C414yescomplemented
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C22⋊C4—24no—
114C4×C22⋊C4—24no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—28no—
122C2×#95—22no—
123C2×#95—28no—
124C2×#95D414yes—
125#342—24no—
126#342—44no—
127#342—44no—
128C25C2×D411yescharacteristic
129C25C2411yescharacteristic
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4C22×C412yes—
135C22×D4C2×D412yes—
136C22×D4C2×Q814yes—
137C22×D4#4114yes—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—44no—
154C23×C4—82no—
155C23×C4—82no—
156C23×C4—82no—
157C23×C4C22⋊C414yes—
158C23×C4C22×C412yes—
159C23×C4C22×C412yes—
160C23×C4C22×C412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D418yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4#4114yes—
171C4×D4—28no—
172C4×D4—28no—
173C4×D4—28no—
174C4×D4—28no—
175C4×D4—416no—
176C4×D4—416no—
177#117—44no—
178#117—44no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—84no—
202C2×C4⋊C4—84no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4C22⋊C414yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4C22⋊C414yes—
235C2×C22⋊C4C22×C412yes—
236C2×C22⋊C4C22×C412yes—
237C2×C22⋊C4C22×C412yes—
238C2×C22⋊C4C2×D412yes—
239C2×C22⋊C4C2×Q818yes—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—44no—
245C2×C42—84no—
246C2×C42C22⋊C414yes—
247#99—216no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—416no—
258#95—44no—
259#95—88no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24C2×C22⋊C412yes—
269C24#12412yes—
270C24C4⋊Q812yes—
271C24C23×C411yescharacteristic
272C24C22×D411yescharacteristic
273C24C22×D412yes—
274C24C22×D411yescharacteristic
275C24C22×Q812yes—
276C24C2×#4112yes—
277C24C2×#4112yes—
278C24H(5,𝔽2)12yes—
279C242−1+412yes—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—28no—
285C2×D4—28no—
286C2×D4—28no—
287C2×D4—28no—
288C2×D4C4×Q818yes—
289C2×D4C22⋊Q818yes—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—28no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—28no—
311C22×C4—28no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—48no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—44no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4—82no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C412yes—
364C22×C4C4×D414yes—
365C22×C4C4×D414yes—
366C22×C4#12011yescharacteristic
367C22×C4#12012yes—
368C22×C4#12011yescharacteristic
369C22×C4#12114yes—
370C22×C4#12114yes—
371C22×C4#12312yes—
372C22×C4#12312yes—
373C22×C4#12412yes—
374C22×C4C4⋊Q812yes—
375C4⋊C4—28no—
376C4⋊C4—48no—
377C4⋊C4—48no—
378C4⋊C4—48no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—48no—
382C4⋊C4—84no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—28no—
390C22⋊C4—24no—
391C22⋊C4—24no—
392C22⋊C4—28no—
393C22⋊C4—28no—
394C22⋊C4—48no—
395C22⋊C4—48no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—84no—
405C42—24no—
406C42—24no—
407C42—48no—
408C42—88no—
409C23—24no—
410C23—22no—
411C23—24no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—41no—
420C23—41no—
421C23—42no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—44no—
433C23—42no—
434C23—42no—
435C23—81no—
436C23—82no—
437C23—81no—
438C23—82no—
439C23#39114yes—
440C23C22×C22⋊C411yescharacteristic
441C23C2×C4×D412yes—
442C23C2×C4×Q812yes—
443C23#52012yes—
444C23C2×#12011yescharacteristic
445C23C2×#12112yes—
446C23C2×C22⋊Q812yes—
447C23C2×#12311yescharacteristic
448C23C2×#12411yescharacteristic
449C23C2×C4⋊Q811yescharacteristic
450C23#53511yescharacteristic
451C23#53611yescharacteristic
452C23#53712yes—
453C23#54614yes—
454C23D4×Q814yes—
455C23#55214yes—
456C23#55414yes—
457D4—216no—
458D4—216no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—48no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—48no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—416no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—48no—
502C2×C4—48no—
503C2×C4—48no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—84no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—84no—
510C2×C4—88no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—84no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—84no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—88no—
520C2×C4—88no—
521C2×C4—168no—
522C2×C4#38612yes—
523C2×C4#38612yes—
524C2×C4#38614yes—
525C2×C4#39114yes—
526C22—28no—
527C22—22no—
528C22—22no—
529C22—21no—
530C22—22no—
531C22—22no—
532C22—21no—
533C22—21no—
534C22—22no—
535C22—22no—
536C22—21no—
537C22—28no—
538C22—22no—
539C22—22no—
540C22—24no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—42no—
550C22—42no—
551C22—44no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22C2×#38611yescharacteristic
556C22C2×#39111yescharacteristic
557C22#231111yescharacteristic
558C22#236014yes—
559C22#241714yes—
560C22#243411yescharacteristic
561C22#243511yescharacteristic
562C22#250211yescharacteristic
563C22#250311yescharacteristic
564C22#302512yes—
565C22#302612yes—
566C4—24no—
567C4—22no—
568C4—22no—
569C4—24no—
570C4—44no—
571C4—42no—
572C4—42no—
573C4—82no—
574C4—82no—
575C4—88no—
576C4—88no—
577C4—82no—
578C2—21no—
579C2—24no—
580C2—21no—
581C2—21no—
582C2—21no—
583C2—24no—
584C2—44no—
585C2#1413911yescharacteristic
586C2#2095911yescharacteristic
587C2#2096011yescharacteristic
588C2#2585412yes—
589C2#2590112yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13923C2semidirect
5. #11222C2semidirect
6. #11222C2semidirect
7. #10854C2semidirect
15. #2579C4semidirect
67. C22×C4⋊C4C2×C4semidirect
97. #385C2×C4semidirect
98. #385C2×C4semidirect
110. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25792
#341812
C2×C4×#992
#139232
#112222
#108542
#107452