Finite Group Databasebeta

← #493972

Subgroups — #493972

Order 512.

1299 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#535C211yesmaximal, characteristic
2#20722C212yesmaximal, complemented
3#12858C212yesmaximal, complemented
4#12798C212yesmaximal, complemented
5#12797C212yesmaximal, complemented
6#12696C212yesmaximal
7#12628C212yesmaximal, complemented
8#12627C212yesmaximal, complemented
9C2×#535C2211yescharacteristic
10C2×C4×#41C411yescharacteristic, complemented
11C22×C4×D4C2211yescharacteristic
12#2328C411yescharacteristic, complemented
13#2328C2211yescharacteristic
14#2326C411yescharacteristic, complemented
15#2326C2211yescharacteristic
16#2324—22no—
17#2321—22no—
18#2317C411yescharacteristic, complemented
19#2311C411yescharacteristic, complemented
20#2309C2211yescharacteristic
21C4×#127C411yescharacteristic, complemented
22C4×#127C2211yescharacteristic
23C4×#124C411yescharacteristic, complemented
24C4×#124C2211yescharacteristic
25C4×#121—22no—
26C4×#120—22no—
27#2280—22no—
28C4×#117C411yescharacteristic, complemented
29C2×#354C2214yes—
30C2×#353C2214yes—
31#1819—22no—
32#1799—22no—
33#1799C2214yes—
34#1781C2214yes—
35C4×#99—22no—
36C4×#99C2214yes—
37#1393C2214yes—
38#1393C2214yes—
39C22×#41C2×C411yescharacteristic
40C23×D4C2311yescharacteristic
41C24×C4D412yes—
42#535—24no—
43#535—24no—
44C2×#127C2×C411yescharacteristic
45C2×#127C2311yescharacteristic
46C2×#124C2×C411yescharacteristic
47C2×#124C2311yescharacteristic
48C2×#121—22no—
49C2×#120—22no—
50C4×#41—42no—
51C4×#41—42no—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×D4—21no—
55C2×C4×D4—21no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—42no—
59C2×C4×D4—42no—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D412yes—
65C2×C4×D4D412yes—
66C2×C4×D4C2311yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×#117—21no—
69C2×#117—42no—
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C22⋊C4D412yes—
75C22×C42—21no—
76C22×C42C2×C411yescharacteristic
77C2×#99—24no—
78C2×#99C2318yes—
79#391C2×C411yescharacteristic
80#391C2×C411yescharacteristic
81#390—22no—
82#390—22no—
83#390—22no—
84#390C2×C411yescharacteristic
85#390C2×C411yescharacteristic
86#390C2×C411yescharacteristic
87#390C2×C411yescharacteristic
88#390C2311yescharacteristic
89#390C2311yescharacteristic
90#389—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#386—22no—
96#386—22no—
97#386—22no—
98#386—22no—
99#386—22no—
100#385—22no—
101#385—22no—
102#385—22no—
103#384C2×C411yescharacteristic
104#384C2×C411yescharacteristic
105#382C2×C411yescharacteristic
106#382C2×C411yescharacteristic
107#381—42no—
108#380—22no—
109#380—24no—
110#380—42no—
111#379—22no—
112#379—24no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—24no—
120C4×C22⋊C4—42no—
121#376C2×C411yescharacteristic
122#376C2×C411yescharacteristic
123C43C2×C411yescharacteristic
124C43C2×C411yescharacteristic
125#354—28no—
126#353—28no—
127#342—24no—
128#342—24no—
129C25C2×D412yes—
130C2×#41—22no—
131C2×#41—22no—
132C2×#41—42no—
133C2×#41—42no—
134C22×Q8C22×C411yescharacteristic
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4C22×C411yescharacteristic
143C22×D4C22×C411yescharacteristic
144C22×D4C2×D412yes—
145C22×D4C2411yescharacteristic
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—84no—
154C23×C4C22⋊C412yes—
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C412yes—
157C23×C4C22⋊C412yes—
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162#127—22no—
163#127—22no—
164#127—22no—
165#127#4114yes—
166#124—22no—
167#124—22no—
168#124—22no—
169#124#4114yes—
170#121—216no—
171#120—216no—
172C4×Q8—42no—
173C4×Q8—41no—
174C4×Q8—41no—
175C4×D4—41no—
176C4×D4—42no—
177C4×D4—41no—
178C4×D4—42no—
179C4×D4—42no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—41no—
184C4×D4—41no—
185C4×D4—41no—
186C4×D4—41no—
187C4×D4—42no—
188C4×D4—44no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—84no—
192C4×D4—84no—
193#117—22no—
194#117—22no—
195#117—44no—
196#117—42no—
197#117—42no—
198#117—88no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4C22⋊C412yes—
231C2×C22⋊C4C22⋊C412yes—
232C2×C22⋊C4C2×D412yes—
233C2×C22⋊C4C2×D412yes—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—44no—
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254#99—28no—
255#99—28no—
256#99—48no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—44no—
266#95—44no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24—24no—
271C24—42no—
272C24—82no—
273C24—82no—
274C24C2×C22⋊C412yes—
275C24C2×C22⋊C411yescharacteristic
276C24C22×D412yes—
277#41—48no—
278#41—48no—
279C2×Q8—21no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8#11712yescomplemented
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—28no—
286C2×D4—28no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—24no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—48no—
299C2×D4—48no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—82no—
303C2×D4—82no—
304C2×D4—82no—
305C2×D4—82no—
306C2×D4#11712yescomplemented
307C2×D4C2×#4112yes—
308C2×D4C2×#4112yes—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—48no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—48no—
343C22×C4—84no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—84no—
348C22×C4—82no—
349C22×C4—84no—
350C22×C4—81no—
351C22×C4#9912yes—
352C22×C4#9912yes—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C23×C411yescharacteristic
361C4⋊C4—24no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4—88no—
373C22⋊C4—24no—
374C22⋊C4—28no—
375C22⋊C4—28no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—48no—
385C22⋊C4—48no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C22⋊C4—88no—
393C22⋊C4—88no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—22no—
399C42—22no—
400C42—21no—
401C42—21no—
402C42—41no—
403C42—41no—
404C42—42no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—41no—
409C42—41no—
410C42—81no—
411C42—81no—
412C42—88no—
413C42#11712yescomplemented
414C42#11712yescomplemented
415C23—22no—
416C23—21no—
417C23—21no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—41no—
424C23—41no—
425C23—44no—
426C23—44no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—42no—
433C23—44no—
434C23—42no—
435C23—41no—
436C23—41no—
437C23—82no—
438C23—82no—
439C23—88no—
440C23—88no—
441C23C2×#9912yes—
442C23C22×C22⋊C411yescharacteristic
443C23#52514yes—
444Q8—42no—
445Q8—42no—
446D4—42no—
447D4—42no—
448D4—48no—
449D4—42no—
450D4—42no—
451D4—42no—
452D4—42no—
453D4—48no—
454D4—88no—
455D4—88no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—48no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—84no—
507C2×C4—82no—
508C2×C4—88no—
509C2×C4—88no—
510C2×C4—88no—
511C2×C4C2×#9911yescharacteristic
512C2×C4C2×#9912yes—
513C2×C4C2×#9911yescharacteristic
514C2×C4#41012yes—
515C2×C4C2×#11711yescharacteristic
516C2×C4C2×#11711yescharacteristic
517C22—21no—
518C22—21no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—24no—
523C22—44no—
524C22—42no—
525C22—42no—
526C22—44no—
527C22—44no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—88no—
538C22—88no—
539C22—81no—
540C22—81no—
541C22#228011yescharacteristic
542C22C22×#9911yescharacteristic
543C22C2×#41011yescharacteristic
544C4—22no—
545C4—21no—
546C4—22no—
547C4—21no—
548C4—21no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—41no—
554C4—41no—
555C4—42no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4—168no—
561C4#212712yes—
562C2—21no—
563C2—21no—
564C2—44no—
565C2—42no—
566C2—82no—
567C2—82no—
568C2#2072211yescharacteristic
569C2C2×#212711yescharacteristic
570C2#3335811yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20722C2semidirect
3. #12858C2semidirect
4. #12798C2semidirect
5. #12797C2semidirect
7. #12628C2semidirect
8. #12627C2semidirect
10. C2×C4×#41C4semidirect
12. #2328C4semidirect
14. #2326C4semidirect
18. #2317C4semidirect
19. #2311C4semidirect
21. C4×#127C4semidirect
23. C4×#124C4semidirect
28. C4×#117C4semidirect
282. C2×Q8#117semidirect
306. C2×D4#117semidirect
413. C42#117semidirect
414. C42#117semidirect

Maximal subgroups

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

subgroupindex
C4×#5352
#207222
#128582
#127982
#127972
#126962
#126282
#126272