Finite Group Databasebeta

← #6629140

Subgroups — #6629140

Order 512.

6321 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#3092C212yesmaximal, complemented
3#36614C211yesmaximal, characteristic
4#32983C214yesmaximal, complemented
5#32982C214yesmaximal, complemented
6#32979C214yesmaximal, complemented
7#32978C214yesmaximal, complemented
8#20909C214yesmaximal, complemented
9#20844C214yesmaximal, complemented
10#20700C212yesmaximal, complemented
11#20579C211yesmaximal, characteristic, complemented
12C25×C4—21no—
13C25×C4C2211yescharacteristic
14C2×#525C2211yescharacteristic
15C2×#525C2212yes—
16C23×C4⋊C4—21no—
17C23×C4⋊C4—22no—
18C23×C4⋊C4—22no—
19C23×C4⋊C4—22no—
20C23×C4⋊C4—216no—
21C23×C4⋊C4C2214yes—
22C23×C4⋊C4C2214yes—
23C23×C4⋊C4C2214yes—
24C23×C4⋊C4C2212yes—
25#3092—216no—
26C2×#484C2214yescomplemented
27C2×#481C2214yescomplemented
28C2×#483C2214yescomplemented
29C2×#480C2214yescomplemented
30C2×#428C2214yescomplemented
31C2×#417C2214yescomplemented
32C2×#407C2212yes—
33C2×#407C2214yes—
34#2849—21no—
35#2849—22no—
36#2848C2212yes—
37#2848C2212yes—
38#2847C2212yes—
39#2847C2212yes—
40#2846—21no—
41C4×#120—22no—
42C4×#120—22no—
43#2280C2212yes—
44#2280C2211yescharacteristic
45#1874C22116yescomplemented
46#1872C22116yescomplemented
47#1853C22116yescomplemented
48#1852C22116yescomplemented
49#1824C2218yescomplemented
50#1823C2218yescomplemented
51#1779C2214yes—
52#1779C2214yes—
53#1736C2218yescomplemented
54C26D411yescharacteristic
55C22×#41C2311yescharacteristic
56C24×C4—28no—
57C24×C4—216no—
58C24×C4—216no—
59C24×C4—24no—
60C24×C4—24no—
61C24×C4—24no—
62C24×C4—22no—
63C24×C4—416no—
64C24×C4D411yescharacteristic
65C24×C4D412yes—
66C24×C4D412yes—
67C24×C4D412yes—
68C24×C4C2312yes—
69C24×C4C2312yes—
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C22×M16C2314yes—
73#525—28no—
74#525—216no—
75C2×#123—21no—
76C2×#123—22no—
77C2×C22⋊Q8C2311yescharacteristic
78C2×C22⋊Q8C2311yescharacteristic
79C2×C22⋊Q8C2314yes—
80C2×#121C2311yescharacteristic
81C2×#121C2311yescharacteristic
82C2×#121C2314yes—
83C2×#120—22no—
84C2×#120—22no—
85C2×#120—21no—
86C2×#120—22no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×C4×D4—24no—
90C2×#117C2311yescharacteristic
91C2×#117C2312yes—
92C22×C4⋊C4—216no—
93C22×C4⋊C4—232no—
94C22×C4⋊C4—232no—
95C22×C4⋊C4—232no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—22no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4—22no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—232no—
107C22×C4⋊C4—232no—
108C22×C4⋊C4—232no—
109C22×C4⋊C4D4116yescomplemented
110C22×C4⋊C4C23116yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114#484—216no—
115#483—216no—
116#481—216no—
117#480—216no—
118#428—216no—
119C2×#107C2318yescomplemented
120C2×#106C2318yescomplemented
121#417—216no—
122C2×#103C2318yescomplemented
123C2×#102C2318yescomplemented
124#407—28no—
125#407—216no—
126C2×C22⋊C8C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C23116yes—
129#386—24no—
130#386—22no—
131#386—24no—
132#386—22no—
133#385—24no—
134#385—22no—
135#385—24no—
136#385—22no—
137#381—21no—
138#381—22no—
139#380C2312yes—
140#380C2312yes—
141#380C2314yes—
142#380C2314yes—
143#379—22no—
144#379—22no—
145#379—21no—
146#379—22no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4—24no—
150#340C23132yescomplemented
151C25—24no—
152C25—28no—
153C25—28no—
154C25—28no—
155C25C2×D411yescharacteristic
156C25C2×D412yes—
157C25C2×D412yes—
158C25C2×D412yes—
159C2×#41—24no—
160C2×#41—28no—
161C22×Q8C2411yescharacteristic
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—24no—
165C22×D4—21no—
166C22×D4C2411yescharacteristic
167C23×C4—2128no—
168C23×C4—24no—
169C23×C4—28no—
170C23×C4—28no—
171C23×C4—28no—
172C23×C4—28no—
173C23×C4—216no—
174C23×C4—216no—
175C23×C4—28no—
176C23×C4—28no—
177C23×C4—28no—
178C23×C4—28no—
179C23×C4—28no—
180C23×C4—216no—
181C23×C4—216no—
182C23×C4—28no—
183C23×C4—28no—
184C23×C4—232no—
185C23×C4—232no—
186C23×C4—232no—
187C23×C4—216no—
188C23×C4—216no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—21no—
193C23×C4—4128no—
194C23×C4—432no—
195C23×C4—432no—
196C23×C4—416no—
197C23×C4—416no—
198C23×C4—416no—
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D418yes—
207C23×C4C2×D414yes—
208C23×C4C2×D412yes—
209C23×C4#41116yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2414yes—
213C23×C4C2411yescharacteristic
214C2×M16—28no—
215C2×M16—216no—
216C22×C8C2418yes—
217#123—44no—
218#123—48no—
219C22⋊Q8—24no—
220C22⋊Q8—24no—
221C22⋊Q8—216no—
222#121—24no—
223#121—24no—
224#121—216no—
225#120—44no—
226#120—48no—
227#120—48no—
228#120—48no—
229C4×D4—44no—
230C4×D4—44no—
231C4×D4—416no—
232#117—24no—
233#117—28no—
234C2×C4⋊C4—232no—
235C2×C4⋊C4—232no—
236C2×C4⋊C4—232no—
237C2×C4⋊C4—264no—
238C2×C4⋊C4—264no—
239C2×C4⋊C4—264no—
240C2×C4⋊C4—216no—
241C2×C4⋊C4—216no—
242C2×C4⋊C4—216no—
243C2×C4⋊C4—216no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—264no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4C2×D4116yes—
255C2×C4⋊C4C2×D4116yes—
256C2×C4⋊C4C2×D4116yes—
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C2411yescharacteristic
259C2×C4⋊C4C2411yescharacteristic
260C2×C4⋊C4C2418yes—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—24no—
273C2×C42—21no—
274C2×C42—22no—
275#107—216no—
276#106—216no—
277#103—216no—
278#102—216no—
279C22⋊C8—28no—
280C22⋊C8—28no—
281C22⋊C8—232no—
282#95—22no—
283#95—22no—
284#95—24no—
285#95—24no—
286C24—232no—
287C24—232no—
288C24—2128no—
289C24—232no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—28no—
294C24—28no—
295C24—24no—
296C24—24no—
297C24—28no—
298C24—28no—
299C24—24no—
300C24—24no—
301C24—28no—
302C24—28no—
303C24—24no—
304C24—24no—
305C24—21no—
306C24#12014yes—
307C24#12318yes—
308C24AGL(1,ℤ/8)18yes—
309C24#13718yes—
310C24C22×D411yescharacteristic
311C24C22×D411yescharacteristic
312C24C22×D411yescharacteristic
313C24C22×D414yes—
314#41—416no—
315C2×Q8—22no—
316C2×Q8—24no—
317C2×D4—22no—
318C2×D4—24no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—48no—
323C2×D4—44no—
324C2×D4—48no—
325C2×D4—44no—
326C2×D4—48no—
327C2×D4—44no—
328C22×C4—264no—
329C22×C4—232no—
330C22×C4—264no—
331C22×C4—264no—
332C22×C4—264no—
333C22×C4—264no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—28no—
351C22×C4—232no—
352C22×C4—264no—
353C22×C4—28no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—28no—
357C22×C4—264no—
358C22×C4—216no—
359C22×C4—216no—
360C22×C4—216no—
361C22×C4—28no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—21no—
366C22×C4—4128no—
367C22×C4—4128no—
368C22×C4—464no—
369C22×C4—4128no—
370C22×C4—416no—
371C22×C4—416no—
372C22×C4—432no—
373C22×C4—432no—
374C22×C4—416no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4#12014yes—
378C22×C4#12014yes—
379C22×C4#12014yes—
380C22×C4#123116yes—
381C22×C4#12318yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D414yes—
385C22×C4C22×D414yes—
386C22×C4C22×D414yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D414yes—
389C22×C4C2×#41116yes—
390C22×C4C2×#4118yes—
391C22×C4C2511yescharacteristic
392M16—416no—
393C2×C8—28no—
394C2×C8—216no—
395C4⋊C4—264no—
396C4⋊C4—216no—
397C4⋊C4—216no—
398C4⋊C4—216no—
399C4⋊C4—28no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—28no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—48no—
406C4⋊C4C22×D4116yes—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—44no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C22⋊C4—44no—
414C22⋊C4—48no—
415C22⋊C4—44no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—48no—
419C42—42no—
420C42—44no—
421C23—216no—
422C23—216no—
423C23—216no—
424C23—216no—
425C23—2256no—
426C23—232no—
427C23—232no—
428C23—232no—
429C23—232no—
430C23—216no—
431C23—216no—
432C23—232no—
433C23—232no—
434C23—232no—
435C23—28no—
436C23—216no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—24no—
441C23—28no—
442C23—216no—
443C23—28no—
444C23—216no—
445C23—22no—
446C23—24no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—22no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23C2×#12012yes—
456C23C2×#12012yes—
457C23C2×#12012yes—
458C23C2×#12318yes—
459C23C2×#12314yes—
460C23C2×AGL(1,ℤ/8)18yes—
461C23C2×AGL(1,ℤ/8)14yes—
462C23C2×#13718yes—
463C23C2×#13714yes—
464C23C23×D411yescharacteristic
465Q8—44no—
466D4—44no—
467D4—816no—
468D4—84no—
469D4—84no—
470D4—84no—
471C2×C4—232no—
472C2×C4—232no—
473C2×C4—216no—
474C2×C4—216no—
475C2×C4—28no—
476C2×C4—216no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—232no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—232no—
487C2×C4—216no—
488C2×C4—432no—
489C2×C4—432no—
490C2×C4—4128no—
491C2×C4—432no—
492C2×C4—416no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—84no—
503C2×C4C2×#12012yes—
504C2×C4C2×#12012yes—
505C2×C4C2×#12012yes—
506C2×C4C2×#12318yes—
507C2×C4C2×#12314yes—
508C2×C4#525116yes—
509C2×C4C23×D412yes—
510C2×C4C22×#4114yes—
511C8—48no—
512C22—264no—
513C22—264no—
514C22—264no—
515C22—232no—
516C22—22no—
517C22—22no—
518C22—28no—
519C22—28no—
520C22—28no—
521C22—28no—
522C22—28no—
523C22—28no—
524C22—28no—
525C22—22no—
526C22—22no—
527C22—28no—
528C22—22no—
529C22—22no—
530C22—28no—
531C22—81no—
532C22—84no—
533C22—81no—
534C22—81no—
535C22#284914yes—
536C22#300314yes—
537C22#300414yes—
538C22#3092116yescomplemented
539C22C22×#12011yescharacteristic
540C22C22×#12312yes—
541C22C22×AGL(1,ℤ/8)12yes—
542C22C22×#13712yes—
543C4—22no—
544C4—22no—
545C4—28no—
546C4—416no—
547C4—81no—
548C4—84no—
549C4—81no—
550C4—81no—
551C4#284914yes—
552C4C2×#52514yes—
553C2—216no—
554C2—24no—
555C2—24no—
556C2—24no—
557C2—161no—
558C2C2×#284911yescharacteristic
559C2C2×#300311yescharacteristic
560C2C2×#300411yescharacteristic
561C2C2×#309214yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36614C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C24×C4⋊C4C2semidirect
2. C2×#3092C2semidirect
4. #32983C2semidirect
5. #32982C2semidirect
6. #32979C2semidirect
7. #32978C2semidirect
8. #20909C2semidirect
9. #20844C2semidirect
10. #20700C2semidirect
11. #20579C2semidirect
26. C2×#484C22semidirect
27. C2×#481C22semidirect
28. C2×#483C22semidirect
29. C2×#480C22semidirect
30. C2×#428C22semidirect
31. C2×#417C22semidirect
45. #1874C22semidirect
46. #1872C22semidirect
47. #1853C22semidirect
48. #1852C22semidirect
49. #1824C22semidirect
50. #1823C22semidirect
53. #1736C22semidirect
109. C22×C4⋊C4D4semidirect
119. C2×#107C23semidirect
120. C2×#106C23semidirect
122. C2×#103C23semidirect
123. C2×#102C23semidirect
150. #340C23semidirect
538. C22#3092semidirect

Maximal subgroups

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

subgroupindex
C24×C4⋊C42
C2×#30922
#366142
#329832
#329822
#329792
#329782
#209092
#208442
#207002
#205792