Finite Group Databasebeta

← #399195

Subgroups — #399195

Order 512.

1784 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#106C211yesmaximal, characteristic
2C2×#2096C212yesmaximal, complemented
3C2×#1892C211yesmaximal, characteristic
4C2×#1872C211yesmaximal, characteristic
5C2×#1853C211yesmaximal, characteristic
6#14139C211yesmaximal, characteristic
7#13880C211yesmaximal, characteristic
8#11378C212yesmaximal, complemented
9#11323C212yesmaximal, complemented
10#11122C212yesmaximal, complemented
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#106—28no—
17C22×#106—24no—
18C22×#106C2212yes—
19C22×#106C2212yes—
20C22×#106C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C2×#391C2212yes—
26C2×#385—22no—
27C2×#384C2212yes—
28C2×#380C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#2096—28no—
31#1892—24no—
32#1872—24no—
33#1853—24no—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1444—22no—
41#1441—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C22⋊#95—22no—
45C23×Q8C2311yescharacteristic
46C24×C4—21no—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50C23×C8—22no—
51C23×C8—24no—
52C23×C8—24no—
53C23×C8D414yes—
54C23×C8C2312yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8Q814yes—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4D412yes—
83C22×C42C2312yes—
84C2×#106—232no—
85C2×#106—24no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—42no—
93C2×#106D418yes—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—42no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8D414yes—
104#391—24no—
105#391—42no—
106#385—48no—
107#384—24no—
108#384—42no—
109#380—22no—
110#380—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C2×#95—24no—
114C2×#95—24no—
115C2×#95—21no—
116C2×#95—24no—
117C2×#95D412yes—
118#340—24no—
119#340—24no—
120#340—22no—
121#340—22no—
122#336—24no—
123#336—24no—
124C25C2×D411yescharacteristic
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8C2×Q812yes—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—44no—
146C23×C4—44no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—82no—
152C23×C4—82no—
153C23×C4C2×D414yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×Q812yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4C2411yescharacteristic
162C22×C8—216no—
163C22×C8—216no—
164C22×C8—28no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—28no—
172C22×C8—28no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8C2×D414yes—
180C22×C8C2×D414yes—
181C22×C8C2×D412yes—
182C22×C8C2×D412yes—
183C22×C8#4114yes—
184C22⋊Q8—28no—
185C22⋊Q8—28no—
186C22⋊Q8—44no—
187C22⋊Q8—44no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—48no—
226C2×C4⋊C4—84no—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4C2×Q812yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4SD1618yes—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244#106—216no—
245#106—216no—
246#106—48no—
247#103—48no—
248#103—44no—
249#103—44no—
250C22⋊C8—28no—
251#95—24no—
252#95—48no—
253#95—42no—
254#95—48no—
255#95—84no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24C22⋊Q812yes—
261C24#12312yes—
262C24#12712yes—
263C24C2×SD1614yes—
264C24#13714yes—
265C24C22×D411yescharacteristic
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—42no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—48no—
311C22×C4—48no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—816no—
337C22×C4—816no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4—84no—
341C22×C4—84no—
342C22×C4C22⋊Q814yes—
343C22×C4C22⋊Q814yes—
344C22×C4C22⋊Q812yes—
345C22×C4#12314yes—
346C22×C4#12314yes—
347C22×C4#12312yes—
348C22×C4#12712yes—
349C22×C4C2×SD1618yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D412yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×Q811yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4112yes—
358C22×C4H(5,𝔽2)12yes—
359C22×C42−1+412yes—
360C2×C8—232no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8#12014yes—
377C2×C8#12314yes—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—48no—
385C4⋊C4—48no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—48no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—816no—
400C22⋊C4—216no—
401C22⋊C4—48no—
402C22⋊C4—84no—
403C42—42no—
404C42—42no—
405C42—42no—
406C42—42no—
407C23—28no—
408C23—24no—
409C23—28no—
410C23—24no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—24no—
415C23—24no—
416C23—28no—
417C23—22no—
418C23—28no—
419C23#39914yes—
420C23#47518yes—
421C23#48318yes—
422C23#49214yes—
423C23#49714yes—
424C23C2×C22⋊Q811yescharacteristic
425C23C2×#12311yescharacteristic
426C23C2×#12711yescharacteristic
427C23D4212yes—
428C23#54712yes—
429C23C22×SD1612yes—
430C23C2×#13712yes—
431C23#57712yes—
432C23#57712yes—
433Q8—42no—
434Q8—44no—
435Q8—42no—
436Q8—81no—
437Q8—81no—
438Q8—82no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—28no—
447C2×C4—28no—
448C2×C4—24no—
449C2×C4—28no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—48no—
463C2×C4—48no—
464C2×C4—816no—
465C2×C4—816no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—84no—
476C2×C4—81no—
477C2×C4—84no—
478C2×C4—84no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—84no—
486C2×C4#39914yes—
487C2×C4C2×#12011yescharacteristic
488C2×C4C2×C22⋊Q811yescharacteristic
489C2×C4C2×C22⋊Q811yescharacteristic
490C2×C4C2×#12311yescharacteristic
491C2×C4C2×#12311yescharacteristic
492C2×C4C2×#12311yescharacteristic
493C2×C4#52512yes—
494C2×C4#52512yes—
495C2×C4#53511yescharacteristic
496C2×C4#53611yescharacteristic
497C2×C4#53912yes—
498C2×C4#53912yes—
499C2×C4D4212yes—
500C2×C4#54712yes—
501C2×C4#54812yes—
502C2×C4D4×Q812yes—
503C2×C4#55312yes—
504C2×C4#55412yes—
505C8—24no—
506C8—24no—
507C8—44no—
508C22—216no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—24no—
514C22—24no—
515C22—216no—
516C22#209618yescomplemented
517C22C2×#39911yescharacteristic
518C22#243311yescharacteristic
519C22#244312yes—
520C22#244912yes—
521C22#250111yescharacteristic
522C22C2×#47512yes—
523C22#308412yes—
524C22C2×#48312yes—
525C22#309912yes—
526C22C2×#49211yescharacteristic
527C22C2×#49711yescharacteristic
528C22#315411yescharacteristic
529C22#315411yescharacteristic
530C22#326614yes—
531C22D4×SD1614yes—
532C22#333514yes—
533C22#334414yes—
534C4—24no—
535C4—48no—
536C4—84no—
537C4—84no—
538C4—82no—
539C4—82no—
540C4—161no—
541C4—161no—
542C4#240911yescharacteristic
543C4#244312yes—
544C4#244912yes—
545C4#248711yescharacteristic
546C4#262311yescharacteristic
547C4#284911yescharacteristic
548C2—28no—
549C2#1477711yescharacteristic
550C2C2×#209611yescharacteristic
551C2#2373811yescharacteristic
552C2#2577411yescharacteristic
553C2#2578411yescharacteristic
554C2#2612814yes—
555C2#2634014yes—
556C2#2792411yescharacteristic
557C2#2793311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2096C2semidirect
8. #11378C2semidirect
9. #11323C2semidirect
10. #11122C2semidirect
516. C22#2096semidirect

Maximal subgroups

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

subgroupindex
C23×#1062
C2×#20962
C2×#18922
C2×#18722
C2×#18532
#141392
#138802
#113782
#113232
#111222
#107142