Finite Group Databasebeta

← #393486

Subgroups — #393486

Order 512.

1935 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×M16C211yesmaximal, characteristic, complemented
2#36626C211yesmaximal, characteristic
3C2×#1871C212yesmaximal
4C4×#381C211yesmaximal, characteristic, complemented
5#10933C214yesmaximal, complemented
6#10886C214yesmaximal, complemented
7#10732C212yesmaximal, complemented
8C23×M16C2212yes—
9C2×#525C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C22×#104C2214yes—
12C2×C4×M16—22no—
13C2×C4×M16—28no—
14C2×C4×M16C2214yes—
15C22×#97—21no—
16C22×C4×C8—21no—
17#2855C411yescharacteristic, complemented
18#2855C412yescomplemented
19#2855C411yescharacteristic, complemented
20#2854C411yescharacteristic, complemented
21#2854C412yescomplemented
22#2854C411yescharacteristic, complemented
23#2853—22no—
24#2849—21no—
25#2849—21no—
26C4×#123—21no—
27C4×#123—21no—
28C4×#123—22no—
29#2297C2212yes—
30#2296C2212yes—
31#2280C2211yescharacteristic
32C2×#381—21no—
33C2×C4×C22⋊C4C2212yes—
34#1871—28no—
35C2×#339C2214yes—
36C2×#339C2218yes—
37#1434—22no—
38C22⋊#95—21no—
39C4×#95C2214yes—
40C22×#41C2311yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2311yescharacteristic
44C22×M16—24no—
45C22×M16—22no—
46C22×M16—216no—
47C22×M16D418yes—
48C22×M16C2314yes—
49C23×C8—22no—
50#525—24no—
51#525—24no—
52C2×#123—21no—
53C2×#123—22no—
54C2×#123—21no—
55C2×#123—21no—
56C2×C22⋊Q8C2×C411yescharacteristic
57C2×C22⋊Q8C2×C411yescharacteristic
58C2×#121C2×C411yescharacteristic
59C2×#121C2×C411yescharacteristic
60C2×#120—21no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×#117C2311yescharacteristic
64C2×#117C2312yes—
65C22×C22⋊C4—22no—
66C22×C42—24no—
67C22×C42—24no—
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C412yes—
70C22×C42C2×C411yescharacteristic
71C22×C42D412yes—
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D412yes—
75C22×C42C2314yes—
76C22×C42C2312yes—
77C2×#104—28no—
78C2×#104—216no—
79C4×M16—216no—
80C4×M16—216no—
81C2×#97—28no—
82C2×#97—24no—
83C2×C4×C8—28no—
84C2×C4×C8—24no—
85#391C2×C412yes—
86#391C2×C412yes—
87#390C2×C412yes—
88#390C2×C412yes—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#386—21no—
102#386—22no—
103#386—21no—
104#386—22no—
105#386—22no—
106#385—21no—
107#385—22no—
108#385—21no—
109#385—22no—
110#385—22no—
111#381—28no—
112#381—21no—
113#381—22no—
114#381—21no—
115#381—21no—
116#380C2×C412yes—
117#380C2×C411yescharacteristic
118#380C2×C411yescharacteristic
119#379—21no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—24no—
127C4×C22⋊C4—24no—
128C2×#95—24no—
129C2×#95—24no—
130C2×#95—22no—
131#339—28no—
132#339—216no—
133C25C2×D411yescharacteristic
134C2×#41—22no—
135C2×#41—22no—
136C2×#41—28no—
137C22×Q8C22×C411yescharacteristic
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4C22×C411yescharacteristic
142C23×C4—28no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—21no—
148C23×C4—44no—
149C23×C4—42no—
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C412yes—
153C23×C4C22⋊C412yes—
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D414yes—
164C23×C4#4114yes—
165C23×C4C2411yescharacteristic
166C2×M16—232no—
167C2×M16—216no—
168C2×M16—28no—
169C2×M16—24no—
170C2×M16—28no—
171C2×M16—232no—
172C2×M16C2×D418yes—
173C2×M16#41116yes—
174C22×C8—216no—
175C22×C8—28no—
176C22×C8—24no—
177#123—44no—
178#123—48no—
179#123—44no—
180#123—44no—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183#121—24no—
184#121—24no—
185#120—44no—
186C4×D4—42no—
187C4×D4—42no—
188C4×D4—42no—
189C4×D4—42no—
190#117—22no—
191#117—22no—
192#117—28no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C412yes—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—28no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—28no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—28no—
235C2×C42—24no—
236C2×C42—28no—
237C2×C42C22⋊C412yes—
238C2×C42C22⋊C412yes—
239C2×C42C22⋊C412yes—
240C2×C42C22⋊C412yes—
241C2×C42C22×C412yes—
242C2×C42C22×C412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245#104—416no—
246#97—216no—
247C4×C8—216no—
248#95—216no—
249#95—21no—
250#95—22no—
251#95—21no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—48no—
261#95—44no—
262C24—24no—
263C24—22no—
264C24—22no—
265C24—21no—
266C24—42no—
267C24C2×C22⋊C411yescharacteristic
268C24C2×C22⋊C411yescharacteristic
269C24#12114yes—
270C24C22×D411yescharacteristic
271#41—44no—
272#41—44no—
273#41—48no—
274C2×Q8—22no—
275C2×Q8—24no—
276C2×D4—22no—
277C2×D4—24no—
278C2×D4—44no—
279C2×D4—42no—
280C2×D4—44no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—28no—
292C22×C4—24no—
293C22×C4—28no—
294C22×C4—24no—
295C22×C4—216no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—28no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—416no—
321C22×C4—416no—
322C22×C4—48no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—48no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—44no—
329C22×C4#10414yescomplemented
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4C4×D414yes—
337C22×C4C4×D414yes—
338C22×C4#12012yes—
339C22×C4#12012yes—
340C22×C4#12114yes—
341C22×C4#12312yes—
342C22×C4#12312yes—
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D412yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C2×#4112yes—
347C22×C4C2×#4118yes—
348M16—232no—
349M16—28no—
350M16—216no—
351M16#123116yescomplemented
352C2×C8—232no—
353C2×C8—28no—
354C2×C8—216no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—416no—
368C22⋊C4—44no—
369C22⋊C4—42no—
370C22⋊C4—44no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C42—24no—
380C42—24no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—24no—
386C42—24no—
387C42—24no—
388C42—24no—
389C42—41no—
390C42—41no—
391C42—44no—
392C42—44no—
393C42—44no—
394C42—48no—
395C23—21no—
396C23—22no—
397C23—22no—
398C23—21no—
399C23—22no—
400C23—28no—
401C23—28no—
402C23—21no—
403C23—22no—
404C23—21no—
405C23—22no—
406C23—44no—
407C23—48no—
408C23—48no—
409C23—41no—
410C23—42no—
411C23—41no—
412C23—41no—
413C23—42no—
414C23#38612yes—
415C23#38612yes—
416C23C2×#10412yes—
417C23#42112yes—
418C23C22×C22⋊C411yescharacteristic
419C23C2×#12112yes—
420C23#53214yes—
421Q8—44no—
422D4—44no—
423D4—84no—
424D4—84no—
425D4—84no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—28no—
440C2×C4—28no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—48no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—48no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—48no—
454C2×C4—416no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—88no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4#38612yes—
474C2×C4#38612yes—
475C2×C4C2×#10414yes—
476C2×C4#42114yes—
477C2×C4C2×C4×D412yes—
478C2×C4C4×#4114yes—
479C2×C4C2×#12011yescharacteristic
480C2×C4C2×#12314yes—
481C2×C4C2×#12311yescharacteristic
482C2×C4#52514yes—
483C2×C4#52514yes—
484C2×C4#52918yes—
485C8—216no—
486C22—22no—
487C22—24no—
488C22—24no—
489C22—24no—
490C22—24no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—416no—
498C22—42no—
499C22—42no—
500C22—81no—
501C22—82no—
502C22—82no—
503C22—81no—
504C22—81no—
505C22#187114yes—
506C22C2×#38611yescharacteristic
507C22#229712yes—
508C22#285414yes—
509C22C22×#10411yescharacteristic
510C22C2×#42112yes—
511C22C2×#42111yescharacteristic
512C4—22no—
513C4—22no—
514C4—24no—
515C4—24no—
516C4—42no—
517C4—42no—
518C4—88no—
519C4—81no—
520C4—82no—
521C4—82no—
522C4—81no—
523C4—81no—
524C4C4×#12314yes—
525C4#284912yes—
526C4#285312yes—
527C2—24no—
528C2—24no—
529C2—42no—
530C2—161no—
531C2#1470611yescharacteristic
532C2C2×#187111yescharacteristic
533C2#2090211yescharacteristic
534C2#2138212yes—
535C2#2138312yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C4×M16C2semidirect
4. C4×#381C2semidirect
5. #10933C2semidirect
6. #10886C2semidirect
7. #10732C2semidirect
17. #2855C4semidirect
18. #2855C4semidirect
19. #2855C4semidirect
20. #2854C4semidirect
21. #2854C4semidirect
22. #2854C4semidirect
329. C22×C4#104semidirect
351. M16#123semidirect

Maximal subgroups

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

subgroupindex
C22×C4×M162
#366262
C2×#18712
C4×#3812
#109332
#108862
#107322