Finite Group Databasebeta

← #393470

Subgroups — #393470

Order 512.

1376 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2873C211yesmaximal, characteristic
2#20851C211yesmaximal, characteristic
3#20837C211yesmaximal, characteristic
4#20705C211yesmaximal, characteristic
5C2×#1826C211yesmaximal, characteristic
6#10901C214yesmaximal, complemented
7#10894C214yesmaximal, complemented
8#10732C211yesmaximal, characteristic
9#10729C211yesmaximal, characteristic
10C23×M16C2211yescharacteristic
11C23×M16C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C2×#430C2212yes—
14C2×#423C2212yes—
15C2×#407C2211yescharacteristic
16C2×#407C2211yescharacteristic
17#2873—22no—
18#2873—22no—
19#2873—22no—
20#2873C2214yes—
21C2×#405—22no—
22C2×#405—22no—
23C2×C4×M16—22no—
24#2280C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26#1855—22no—
27#1855—22no—
28#1851—22no—
29#1850—22no—
30#1826—22no—
31#1826—22no—
32#1826C412yescomplemented
33#1826C412yescomplemented
34#1826C414yescomplemented
35C2×#344C2214yes—
36#1738C2212yes—
37#1733C2212yes—
38#1733C2212yes—
39C2×#339C2214yes—
40C2×#339C2214yes—
41#1728C2212yes—
42C24×C4C2311yescharacteristic
43C22×M16—21no—
44C22×M16—21no—
45C22×M16—21no—
46C22×M16—21no—
47C22×M16—21no—
48C22×M16—21no—
49C22×M16—21no—
50C22×M16—28no—
51C22×M16—28no—
52C22×M16D412yes—
53C22×M16D412yes—
54C22×M16D412yes—
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2312yes—
58C22×M16C2312yes—
59C22×M16C2312yes—
60C23×C8—21no—
61C23×C8D412yes—
62C2×#117—21no—
63C2×#117—21no—
64C2×#117—21no—
65C2×#117—21no—
66C2×#117—21no—
67C2×#117—21no—
68C2×#117C2×C412yes—
69C2×#117C2×C412yes—
70C2×#117D412yes—
71C2×#117D412yes—
72C2×#117Q812yes—
73C2×#117C2312yes—
74C2×#117C2312yes—
75C22×C4⋊C4Q811yescharacteristic
76C22×C4⋊C4Q811yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C42—21no—
80C22×C42C2312yes—
81#430—28no—
82C2×#108—24no—
83#423—28no—
84C2×C4⋊C8—24no—
85#407—21no—
86#407—21no—
87#407—21no—
88#407—21no—
89#407—21no—
90#407—21no—
91#407C2×C412yes—
92#407C2×C412yes—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97#405—28no—
98#405—28no—
99#405—28no—
100#405—28no—
101C4×M16—28no—
102C4×M16—28no—
103C2×#97—24no—
104C2×C4×C8—24no—
105#381—21no—
106#380—21no—
107#380—21no—
108#379—21no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4C2×C411yescharacteristic
114C4×C22⋊C4C2×C412yes—
115C4×C22⋊C4C2×C411yescharacteristic
116C2×#95—21no—
117#344—28no—
118#341—24no—
119#339—28no—
120#339—24no—
121#339—24no—
122#339C2×C418yes—
123#339C2×C418yes—
124#336—24no—
125C25#4111yescharacteristic
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×Q811yescharacteristic
140C23×C4C2×Q811yescharacteristic
141C23×C4#4111yescharacteristic
142C23×C4#4111yescharacteristic
143C23×C4#4111yescharacteristic
144C23×C4C2411yescharacteristic
145C2×M16—28no—
146C2×M16—28no—
147C2×M16—28no—
148C2×M16—28no—
149C2×M16—28no—
150C2×M16—28no—
151C2×M16—28no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—24no—
160C2×M16—24no—
161C2×M16—28no—
162C2×M16—28no—
163C2×M16—22no—
164C2×M16—28no—
165C2×M16—24no—
166C2×M16—24no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—24no—
172C2×M16—24no—
173C2×M16C22×C414yes—
174C2×M16C2×D414yes—
175C2×M16#4118yes—
176C2×M16#4118yes—
177C22×C8—28no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—28no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8C2×D414yes—
188#117—22no—
189#117—22no—
190#117—22no—
191#117—22no—
192#117—22no—
193#117—22no—
194#117—22no—
195#117—22no—
196#117—22no—
197#117—22no—
198#117—22no—
199#117—22no—
200#117—24no—
201#117—24no—
202#117C4⋊C414yes—
203#117C4⋊C414yes—
204#117C22×C414yes—
205#117C2×D414yes—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4C4⋊C412yes—
214C2×C4⋊C4C4⋊C412yes—
215C2×C4⋊C4C4⋊C412yes—
216C2×C4⋊C4C4⋊C412yes—
217C2×C4⋊C4C2×Q812yes—
218C2×C4⋊C4C2×Q812yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42C22×C412yes—
243C2×C42C22×C412yes—
244#108—48no—
245C4⋊C8—48no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—41no—
251C22⋊C8—41no—
252C22⋊C8—41no—
253C22⋊C8—41no—
254#97—216no—
255C4×C8—216no—
256#95—21no—
257#95—22no—
258#95—21no—
259#95—41no—
260#95—41no—
261C24—24no—
262C24—42no—
263C24—81no—
264C24C4×D411yescharacteristic
265C24C4×D411yescharacteristic
266C24C22⋊Q811yescharacteristic
267C24C22⋊Q811yescharacteristic
268C24#12311yescharacteristic
269C24#12311yescharacteristic
270C24C2×#4111yescharacteristic
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4C2×C4⋊C411yescharacteristic
305C22×C4C2×C4⋊C411yescharacteristic
306C22×C4C2×C4⋊C411yescharacteristic
307C22×C4C2×C4⋊C411yescharacteristic
308C22×C4C4×D411yescharacteristic
309C22×C4C4×D411yescharacteristic
310C22×C4C4×D411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4#12011yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4C22⋊Q811yescharacteristic
319C22×C4C22⋊Q811yescharacteristic
320C22×C4C22⋊Q811yescharacteristic
321C22×C4C22⋊Q811yescharacteristic
322C22×C4C22⋊Q811yescharacteristic
323C22×C4C22⋊Q811yescharacteristic
324C22×C4#12311yescharacteristic
325C22×C4#12311yescharacteristic
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×Q811yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4114yes—
333C22×C4C2×#4114yes—
334M16—28no—
335M16—28no—
336M16—28no—
337M16—28no—
338M16—24no—
339M16—28no—
340M16—28no—
341M16—28no—
342M16—28no—
343M16—44no—
344M16—42no—
345M16—42no—
346M16—42no—
347M16—42no—
348M16#12318yes—
349C2×C8—28no—
350C2×C8—28no—
351C2×C8—28no—
352C2×C8—28no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—28no—
357C2×C8—24no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8#12318yes—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—44no—
372C4⋊C4C2×C4⋊C414yes—
373C4⋊C4C2×C4⋊C414yes—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—44no—
380C22⋊C4—42no—
381C22⋊C4—44no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C42—24no—
385C42—24no—
386C42—24no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42—42no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—88no—
415C23#38511yescharacteristic
416C23#38511yescharacteristic
417C23#38511yescharacteristic
418C23#38511yescharacteristic
419C23C2×C4×D411yescharacteristic
420C23C2×C22⋊Q811yescharacteristic
421C23C2×#12311yescharacteristic
422C23#52914yes—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—88no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4#38511yescharacteristic
468C2×C4#38511yescharacteristic
469C2×C4#38511yescharacteristic
470C2×C4#38511yescharacteristic
471C2×C4C22×C4⋊C411yescharacteristic
472C2×C4C2×C4×D411yescharacteristic
473C2×C4C4×#4114yes—
474C2×C4C2×#12011yescharacteristic
475C2×C4C2×C22⋊Q811yescharacteristic
476C2×C4C2×#12312yes—
477C2×C4C2×#12312yes—
478C2×C4#52512yes—
479C2×C4#52512yes—
480C2×C4#52512yes—
481C2×C4#52512yes—
482C2×C4#52914yes—
483C2×C4#53314yes—
484C8—28no—
485C8—28no—
486C8—44no—
487C8—81no—
488C8—81no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—44no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22#186112yes—
514C22C2×#38511yescharacteristic
515C22C4×#12312yes—
516C22#285312yes—
517C22#285512yes—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—44no—
523C4—44no—
524C4—44no—
525C4—82no—
526C4—162no—
527C4#229612yes—
528C4C4×#12312yes—
529C4#284911yescharacteristic
530C4#284911yescharacteristic
531C4#285512yes—
532C2—22no—
533C2—22no—
534C2—22no—
535C2—22no—
536C2—41no—
537C2—41no—
538C2—41no—
539C2#1470511yescharacteristic
540C2C2×#186111yescharacteristic
541C2#2086711yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #10901C2semidirect
7. #10894C2semidirect
32. #1826C4semidirect
33. #1826C4semidirect
34. #1826C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28732
#208512
#208372
#207052
C2×#18262
#109012
#108942
#107322
#107292