Finite Group Databasebeta

← #496414

Subgroups — #496414

Order 512.

1175 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2#20726C211yesmaximal, characteristic
3#20722C211yesmaximal, characteristic
4#13295C214yesmaximal, complemented
5#13292C214yesmaximal, complemented
6#12852C214yesmaximal
7C22×C4×D4C2211yescharacteristic
8C23×C4⋊C4C411yescharacteristic, complemented
9#2920—21no—
10#2487—21no—
11#2487—21no—
12#2487—21no—
13#2487—21no—
14#2487C412yescomplemented
15#2487C412yescomplemented
16#2487C2212yes—
17#2487C2212yes—
18C2×#400—21no—
19C2×#399—21no—
20C2×#389—21no—
21C2×#386C2212yes—
22C2×#385C412yescomplemented
23#2280—21no—
24C2×#380—21no—
25C2×#380C411yescharacteristic, complemented
26#1821—21no—
27#1819—21no—
28#1800—21no—
29#1800C2212yes—
30#1799—21no—
31#1799C2212yes—
32#1783C2212yes—
33#1781C2212yes—
34C4×#100—21no—
35C4×#100C2212yes—
36C4×#99—21no—
37C4×#99C2212yes—
38#1390C22116yes—
39C23×D4C2311yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C412yes—
42C24×C4D411yescharacteristic
43C2×C4×D4—41no—
44C2×C4×D4—41no—
45C2×C4×D4—41no—
46C2×C4×D4—41no—
47C2×C4×D4C2×C411yescharacteristic
48C2×C4×D4C2×C411yescharacteristic
49C2×C4×D4D411yescharacteristic
50C2×C4×D4D411yescharacteristic
51C2×C4×D4D411yescharacteristic
52C2×C4×D4D411yescharacteristic
53C2×C4×D4C2311yescharacteristic
54C2×C4×D4C2311yescharacteristic
55C2×#117—41no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—41no—
63C22×C4⋊C4—41no—
64C22×C4⋊C4—41no—
65C22×C4⋊C4—41no—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4D411yescharacteristic
69C22×C22⋊C4—22no—
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4D411yescharacteristic
73C22×C42D411yescharacteristic
74#432—41no—
75#432—41no—
76#431—41no—
77#431—41no—
78#423—22no—
79C2×#100—22no—
80C2×#100C2314yes—
81C2×#99—22no—
82C2×#99C2314yes—
83#407—22no—
84C4×M16—22no—
85#400—22no—
86#400—22no—
87#400—22no—
88#400—22no—
89#399—22no—
90#399—22no—
91#399—22no—
92#399—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#386—22no—
98#386—22no—
99#386C2×C414yes—
100#386C2314yes—
101#385—22no—
102#385—22no—
103#385C2×C414yes—
104#385C2×C414yes—
105#381—41no—
106#380—21no—
107#380—21no—
108#380—22no—
109#380—22no—
110#380—22no—
111#380—22no—
112#380—22no—
113#380—41no—
114#380C2×C412yes—
115#380C2×C412yes—
116#379—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—41no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122#343—22no—
123#343—22no—
124#342—22no—
125#342—22no—
126C25C22×C411yescharacteristic
127C25C2×D411yescharacteristic
128C22×D4—24no—
129C22×D4—41no—
130C22×D4—41no—
131C22×D4—42no—
132C22×D4C22×C411yescharacteristic
133C22×D4C2×D411yescharacteristic
134C22×D4C2×D411yescharacteristic
135C22×D4C2411yescharacteristic
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—84no—
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C414yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C2×M16—24no—
167C4×D4—41no—
168C4×D4—41no—
169C4×D4—41no—
170C4×D4—41no—
171C4×D4—41no—
172C4×D4—41no—
173C4×D4—41no—
174C4×D4—41no—
175C4×D4—41no—
176C4×D4—41no—
177C4×D4—41no—
178C4×D4—41no—
179C4×D4—41no—
180C4×D4—41no—
181C4×D4—41no—
182C4×D4—41no—
183#117—84no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—84no—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—84no—
227C2×C22⋊C4C22⋊C411yescharacteristic
228C2×C22⋊C4C22⋊C411yescharacteristic
229C2×C22⋊C4C22⋊C411yescharacteristic
230C2×C22⋊C4C22⋊C411yescharacteristic
231C2×C22⋊C4C22×C414yes—
232C2×C22⋊C4C22×C411yescharacteristic
233C2×C22⋊C4C22×C411yescharacteristic
234C2×C22⋊C4C2×D411yescharacteristic
235C2×C22⋊C4C2×D411yescharacteristic
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—42no—
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C4⋊C8—42no—
246C4⋊C8—42no—
247#100—28no—
248#100—44no—
249#99—28no—
250#99—44no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253#97—42no—
254C4×C8—42no—
255#95—24no—
256#95—24no—
257#95—22no—
258#95—22no—
259#95—24no—
260#95—24no—
261#95—42no—
262#95—42no—
263C24—22no—
264C24—21no—
265C24—21no—
266C24—22no—
267C24—22no—
268C24—21no—
269C24—21no—
270C24—41no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24C2×C22⋊C411yescharacteristic
276C24C2×C22⋊C411yescharacteristic
277C24C2×C22⋊C411yescharacteristic
278C24C23×C411yescharacteristic
279C24C22×D411yescharacteristic
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—48no—
292C2×D4—48no—
293C2×D42−1+414yes—
294C22×C4—24no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—41no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4—88no—
355C22×C4—88no—
356C22×C4#9911yescharacteristic
357C22×C4#9911yescharacteristic
358C22×C4#9911yescharacteristic
359C22×C4#9911yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370M16—44no—
371M16—44no—
372C2×C8—44no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—84no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—84no—
401C22⋊C4—84no—
402C22⋊C4—84no—
403C22⋊C4—84no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—84no—
413C23—21no—
414C23—21no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—24no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23—48no—
437C23—48no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—48no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—81no—
448C23—81no—
449C23—84no—
450C23C2×#9911yescharacteristic
451C23C2×#9911yescharacteristic
452C23#41012yes—
453C23C22×C22⋊C411yescharacteristic
454C23#51912yes—
455C23#53612yes—
456D4—44no—
457D4—44no—
458D4—44no—
459D4—44no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—88no—
513C2×C4—84no—
514C2×C4—88no—
515C2×C4—84no—
516C2×C4—88no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4#35214yes—
521C2×C4C2×#9911yescharacteristic
522C2×C4C2×#9911yescharacteristic
523C2×C4C2×#9911yescharacteristic
524C2×C4C2×#9911yescharacteristic
525C2×C4#53612yes—
526C8—84no—
527C22—24no—
528C22—24no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—48no—
544C22—48no—
545C22—42no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—81no—
556C22—84no—
557C22—84no—
558C22—81no—
559C22C2×#35212yes—
560C22#212212yes—
561C22#231011yescharacteristic
562C22C22×#9911yescharacteristic
563C22C2×#41011yescharacteristic
564C4—21no—
565C4—21no—
566C4—22no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—84no—
575C4—84no—
576C4—164no—
577C4C2×#35212yes—
578C4C2×#35212yes—
579C2—21no—
580C2—21no—
581C2—22no—
582C2—22no—
583C2—42no—
584C2—42no—
585C2—42no—
586C2—81no—
587C2#2182511yescharacteristic
588C2C22×#35211yescharacteristic
589C2C2×#212211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ 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
4. #13295C2semidirect
5. #13292C2semidirect
8. C23×C4⋊C4C4semidirect
14. #2487C4semidirect
15. #2487C4semidirect
22. C2×#385C4semidirect
25. C2×#380C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
#207262
#207222
#132952
#132922
#128522