Finite Group Databasebeta

← #493746

Subgroups — #493746

Order 512.

1427 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#535C211yesmaximal, characteristic
2C2×#2127C212yesmaximal
3C2×#1799C212yesmaximal, complemented
4#12800C212yesmaximal, complemented
5#12799C212yesmaximal, complemented
6#12798C212yesmaximal, complemented
7#12797C212yesmaximal, complemented
8#12582C212yesmaximal, complemented
9C2×#535C2211yescharacteristic
10C2×C4×#41C411yescharacteristic, complemented
11C22×C4×D4C2211yescharacteristic
12C22×#99C2214yes—
13#2328C411yescharacteristic, complemented
14#2328C2211yescharacteristic
15#2326C411yescharacteristic, complemented
16#2326C2211yescharacteristic
17#2324—22no—
18#2321—22no—
19#2317C411yescharacteristic, complemented
20#2311C411yescharacteristic, complemented
21#2309C2211yescharacteristic
22C4×#127C411yescharacteristic, complemented
23C4×#127C2211yescharacteristic
24C4×#124C411yescharacteristic, complemented
25C4×#124C2211yescharacteristic
26C4×#121—22no—
27C4×#120—22no—
28C2×#380—22no—
29C4×#117C411yescharacteristic, complemented
30#2127—28no—
31C2×#354C2214yes—
32C2×#353C2214yes—
33#1799—22no—
34#1799—22no—
35#1799C2214yes—
36#1799C2214yes—
37C2×#342—22no—
38#1393C2214yes—
39#1393C2214yes—
40C22×#41C2×C411yescharacteristic
41C23×D4C2311yescharacteristic
42C24×C4D412yes—
43#535—24no—
44#535D414yes—
45#535Q814yes—
46C2×#127C2×C411yescharacteristic
47C2×#127C2311yescharacteristic
48C2×#124C2×C411yescharacteristic
49C2×#124C2311yescharacteristic
50C2×#121—22no—
51C2×#120—22no—
52C4×#41—42no—
53C4×#41—42no—
54C2×C4×Q8C2×C411yescharacteristic, complemented
55C2×C4×Q8C2×C411yescharacteristic, complemented
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—42no—
61C2×C4×D4—42no—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4C2×C411yescharacteristic, complemented
65C2×C4×D4C2×C411yescharacteristic, complemented
66C2×C4×D4D412yes—
67C2×C4×D4D412yes—
68C2×C4×D4C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117—21no—
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C22×C4⋊C4—42no—
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C22⋊C4—24no—
77C22×C22⋊C4D412yes—
78C22×C42—21no—
79C22×C42C2×C411yescharacteristic
80C2×#99—28no—
81C2×#99—28no—
82C2×#99—24no—
83C2×#99C2318yes—
84#391C2×C411yescharacteristic, complemented
85#391C2×C411yescharacteristic, complemented
86#390—22no—
87#390—22no—
88#390—22no—
89#390C2×C411yescharacteristic
90#390C2×C411yescharacteristic
91#390C2×C411yescharacteristic, complemented
92#390C2×C411yescharacteristic, complemented
93#390C2311yescharacteristic
94#390C2311yescharacteristic
95#389—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#386—22no—
101#386—22no—
102#386—22no—
103#386—22no—
104#386—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#384C2×C411yescharacteristic, complemented
109#384C2×C411yescharacteristic, complemented
110#382C2×C411yescharacteristic, complemented
111#382C2×C411yescharacteristic, complemented
112#380—22no—
113#380—24no—
114#380—24no—
115#380—42no—
116#380—42no—
117#379—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—22no—
124#376C2×C411yescharacteristic, complemented
125#376C2×C411yescharacteristic, complemented
126C2×#95—42no—
127C43C2×C411yescharacteristic, complemented
128C43C2×C411yescharacteristic, complemented
129#354—28no—
130#353—28no—
131#342—24no—
132#342—24no—
133C25C2×D412yes—
134C2×#41—22no—
135C2×#41—42no—
136C2×#41—42no—
137C2×#41C4⋊C414yescomplemented
138C22×Q8C22×C411yescharacteristic
139C22×D4—21no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4C22×C411yescharacteristic
145C22×D4C22×C411yescharacteristic
146C22×D4C2×D412yes—
147C22×D4C2×D412yes—
148C22×D4C2×Q812yes—
149C22×D4#4114yes—
150C22×D4C2411yescharacteristic
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—44no—
159C23×C4—84no—
160C23×C4C22⋊C412yes—
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C412yes—
163C23×C4C22⋊C412yes—
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168#127—22no—
169#127—22no—
170#127C4⋊C414yescomplemented
171#127C2×D412yes—
172#127C2×Q812yes—
173#124—22no—
174#124—22no—
175#124C4⋊C414yescomplemented
176#124C2×D412yes—
177#124C2×Q812yes—
178#121—28no—
179#121—28no—
180#120—28no—
181#120—28no—
182C4×Q8—42no—
183C4×Q8—41no—
184C4×Q8—41no—
185C4×D4—41no—
186C4×D4—42no—
187C4×D4—41no—
188C4×D4—42no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—42no—
192C4×D4—41no—
193C4×D4—42no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4—42no—
198C4×D4—44no—
199C4×D4—42no—
200C4×D4—42no—
201C4×D4—84no—
202C4×D4—84no—
203#117—22no—
204#117—44no—
205#117—42no—
206#117—42no—
207#117C4⋊C414yescomplemented
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—88no—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—48no—
240C2×C22⋊C4—48no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4C22⋊C412yes—
243C2×C22⋊C4C22⋊C412yes—
244C2×C22⋊C4C2×D412yes—
245C2×C22⋊C4C2×D412yes—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265#99—28no—
266#99—28no—
267#99—48no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—44no—
277#95—44no—
278C24—22no—
279C24—22no—
280C24—24no—
281C24—21no—
282C24—82no—
283C24—82no—
284C24C2×C22⋊C412yes—
285C24C2×C22⋊C411yescharacteristic
286C24#12114yes—
287C24C22⋊Q814yes—
288C24C22×D412yes—
289#41—48no—
290#41—48no—
291C2×Q8—21no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8C2×C4⋊C412yescomplemented
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—28no—
299C2×D4—21no—
300C2×D4—41no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—44no—
305C2×D4—42no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—82no—
315C2×D4—82no—
316C2×D4—82no—
317C2×D4—82no—
318C2×D4C2×C4⋊C412yes—
319C2×D4C2×C4⋊C412yescomplemented
320C2×D4#11712yes—
321C2×D4#12114yes—
322C2×D4C22⋊Q814yes—
323C2×D4C22×D411yescharacteristic
324C2×D4C22×Q811yescharacteristic
325C2×D4C2×#4112yes—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—48no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—48no—
358C22×C4—84no—
359C22×C4—82no—
360C22×C4—82no—
361C22×C4—82no—
362C22×C4—84no—
363C22×C4—82no—
364C22×C4—84no—
365C22×C4—81no—
366C22×C4—88no—
367C22×C4—88no—
368C22×C4—88no—
369C22×C4#9912yes—
370C22×C4#9912yes—
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C412yes—
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C4⋊C412yes—
379C22×C4#11712yes—
380C22×C4C23×C411yescharacteristic
381C4⋊C4—24no—
382C4⋊C4—42no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—88no—
393C4⋊C4—88no—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—28no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—48no—
407C22⋊C4—48no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—88no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—22no—
420C42—22no—
421C42—21no—
422C42—21no—
423C42—41no—
424C42—41no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—41no—
430C42—41no—
431C42—81no—
432C42—81no—
433C42C2×C4⋊C412yescomplemented
434C42C2×C4⋊C412yescomplemented
435C23—24no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—24no—
440C23—24no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—41no—
445C23—41no—
446C23—44no—
447C23—44no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23—41no—
456C23—82no—
457C23—82no—
458C23—84no—
459C23—84no—
460C23—88no—
461C23#38014yes—
462C23#38012yes—
463C23C2×#9912yes—
464C23#41012yes—
465C23C22×C22⋊C411yescharacteristic
466C23C2×#12112yes—
467C23C2×C22⋊Q812yes—
468Q8—42no—
469Q8—42no—
470D4—42no—
471D4—42no—
472D4—44no—
473D4—44no—
474D4—41no—
475D4—41no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—48no—
480D4—84no—
481D4—84no—
482D4—88no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—48no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—84no—
534C2×C4—82no—
535C2×C4—88no—
536C2×C4—88no—
537C2×C4—88no—
538C2×C4—1616no—
539C2×C4#38012yes—
540C2×C4C2×#9911yescharacteristic
541C2×C4C2×#9912yes—
542C2×C4C2×#9911yescharacteristic
543C2×C4C22×C4⋊C411yescharacteristic
544C2×C4C2×#11711yescharacteristic
545C22—21no—
546C22—22no—
547C22—21no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—24no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—44no—
557C22—48no—
558C22—48no—
559C22—44no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—82no—
566C22—82no—
567C22—84no—
568C22—84no—
569C22—82no—
570C22—81no—
571C22—81no—
572C22#179914yes—
573C22C2×#38011yescharacteristic
574C22C22×#9911yescharacteristic
575C22C2×#41011yescharacteristic
576C4—22no—
577C4—21no—
578C4—22no—
579C4—21no—
580C4—21no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—42no—
585C4—41no—
586C4—41no—
587C4—42no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4—168no—
593C4#212712yescomplemented
594C2—21no—
595C2—21no—
596C2—24no—
597C2—42no—
598C2—42no—
599C2—44no—
600C2—82no—
601C2C2×#179911yescharacteristic
602C2C2×#212711yescharacteristic
603C2#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
3. C2×#1799C2semidirect
4. #12800C2semidirect
5. #12799C2semidirect
6. #12798C2semidirect
7. #12797C2semidirect
8. #12582C2semidirect
10. C2×C4×#41C4semidirect
13. #2328C4semidirect
15. #2326C4semidirect
19. #2317C4semidirect
20. #2311C4semidirect
22. C4×#127C4semidirect
24. C4×#124C4semidirect
29. C4×#117C4semidirect
54. C2×C4×Q8C2×C4semidirect
55. C2×C4×Q8C2×C4semidirect
64. C2×C4×D4C2×C4semidirect
65. C2×C4×D4C2×C4semidirect
84. #391C2×C4semidirect
85. #391C2×C4semidirect
91. #390C2×C4semidirect
92. #390C2×C4semidirect
108. #384C2×C4semidirect
109. #384C2×C4semidirect
110. #382C2×C4semidirect
111. #382C2×C4semidirect
124. #376C2×C4semidirect
125. #376C2×C4semidirect
127. C43C2×C4semidirect
128. C43C2×C4semidirect
137. C2×#41C4⋊C4semidirect
170. #127C4⋊C4semidirect
175. #124C4⋊C4semidirect
207. #117C4⋊C4semidirect
294. C2×Q8C2×C4⋊C4semidirect
319. C2×D4C2×C4⋊C4semidirect
433. C42C2×C4⋊C4semidirect
434. C42C2×C4⋊C4semidirect
593. C4#2127semidirect

Maximal subgroups

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

subgroupindex
C4×#5352
C2×#21272
C2×#17992
#128002
#127992
#127982
#127972
#125822