Finite Group Databasebeta

← #6457214

Subgroups — #6457214

Order 512.

2322 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3094C211yesmaximal, characteristic
2C22×#481C211yesmaximal, characteristic
3#34204C211yesmaximal, characteristic
4C2×#2317C211yesmaximal, characteristic, complemented
5#27654C218yesmaximal, complemented
6#27651C214yesmaximal, complemented
7#27645C214yesmaximal, complemented
8#23661C212yesmaximal, complemented
9#23632C212yesmaximal, complemented
10#21174C212yesmaximal, complemented
11#20700C211yesmaximal, characteristic, complemented
12#20699C211yesmaximal, characteristic, complemented
13#20650C212yesmaximal, complemented
14#20628C211yesmaximal, characteristic, complemented
15C22×#121C2211yescharacteristic
16C2×#518C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#3094—28no—
21C2×#481—28no—
22C2×#481C2214yes—
23C2×#481C2214yes—
24C2×#481C2212yes—
25C2×#482C2214yes—
26C2×#480C2212yes—
27C2×#428C2212yes—
28C22×#106C2212yes—
29C2×#424C2212yes—
30C2×#418C2212yes—
31C22×#102C2212yes—
32C2×#407C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34#2363—22no—
35#2361—22no—
36#2342—22no—
37D4×C22⋊C4—22no—
38#2324C2214yes—
39#2317—22no—
40#2317—22no—
41#2317C2212yes—
42#2317C2212yes—
43#2317C2214yescomplemented
44#2316C2211yescharacteristic
45#2309C2211yescharacteristic
46C4×#121C2214yes—
47C2×#389—22no—
48C2×#385—22no—
49C2×#384C2212yescomplemented
50C2×#382C2212yescomplemented
51#2280C2211yescharacteristic
52#2274C2211yescharacteristic
53#2085C2218yescomplemented
54#2084C2218yescomplemented
55#2073C2218yescomplemented
56#2071C2218yescomplemented
57#1949C2218yescomplemented
58#1948C2218yescomplemented
59#1825C2214yescomplemented
60#1824C2214yescomplemented
61#1824C2212yescomplemented
62#1823C2212yescomplemented
63#1823C2214yescomplemented
64#1804C2218yescomplemented
65#1802C2218yescomplemented
66#1798C2214yescomplemented
67#1797C2214yescomplemented
68#1736C2212yes—
69#1736C2212yes—
70#1735C2212yes—
71#1735C2212yes—
72#1729C2212yes—
73C23×D4—21no—
74C23×D4C2311yescharacteristic
75C24×C4—21no—
76C24×C4C2311yescharacteristic
77C22×M16C2312yes—
78C23×C8C2312yes—
79C2×#121—21no—
80C2×#121—21no—
81C2×#121—24no—
82C2×#121D418yes—
83C2×#121C2312yes—
84C2×#121C2312yes—
85#518—28no—
86C2×C4×D4—24no—
87C2×C4×D4—22no—
88C2×C4×D4C2314yes—
89C2×#117—28no—
90C2×#117—21no—
91C2×#117—21no—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117D412yes—
95C2×#117D412yes—
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C2×#117C2312yes—
99C2×#117C2312yes—
100C2×#117C2311yescharacteristic
101C2×#117C2312yes—
102C2×#117C2311yescharacteristic
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4D412yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—22no—
119C22×C22⋊C4D412yes—
120C22×C42—21no—
121C22×C42C2312yes—
122#482—216no—
123#481—216no—
124#481—216no—
125#481—28no—
126#480—28no—
127#428—28no—
128C2×#107C2314yes—
129C2×#106—28no—
130C2×#106C2314yes—
131C2×#106C2318yes—
132#424—28no—
133C2×C4⋊C8C2318yes—
134#418—28no—
135C2×#102—28no—
136C2×#102C2314yes—
137C2×#102C2318yes—
138C2×#102C2314yes—
139#407—24no—
140C2×C22⋊C8—24no—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145#390—22no—
146#390—24no—
147#390C2314yes—
148#389—22no—
149#389—24no—
150#389—24no—
151#389—28no—
152#389C2314yes—
153#388—24no—
154#388—24no—
155#386—24no—
156#386—24no—
157#385—24no—
158#385—24no—
159#385—24no—
160#385—28no—
161#384—24no—
162#384C2314yes—
163#384C2314yes—
164#382—24no—
165#382C2314yes—
166#382C2314yes—
167#381—21no—
168#380—22no—
169#380C2312yes—
170#380C2312yes—
171#379—21no—
172#379—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4C2314yes—
175C4×C22⋊C4—21no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4—22no—
178C4×C22⋊C4—24no—
179C2×#95—24no—
180#340C2314yes—
181#340C2318yes—
182#340C2314yes—
183#340C2314yes—
184#340C2318yes—
185#340C2314yes—
186#336C2318yes—
187C25—21no—
188C25C2×D411yescharacteristic
189C22×D4—22no—
190C22×D4—28no—
191C22×D4—22no—
192C22×D4—21no—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4—24no—
196C22×D4—84no—
197C22×D4C2×D414yes—
198C22×D4C2412yes—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—44no—
209C23×C4—84no—
210C23×C4C2×D414yes—
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D414yes—
215C23×C4#4114yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C2×M16—24no—
220C2×M16—24no—
221C2×M16—24no—
222C22×C8—24no—
223C22×C8—24no—
224C22×C8—24no—
225C22×C8C2414yes—
226C22×C8C2414yes—
227#121—28no—
228#121—28no—
229#121—24no—
230#121—24no—
231#121C2×D418yes—
232#121C2×D418yes—
233C4×D4—216no—
234C4×D4—48no—
235C4×D4—88no—
236#117—24no—
237#117—28no—
238#117—28no—
239#117—28no—
240#117—24no—
241#117—24no—
242#117—24no—
243#117—24no—
244#117—24no—
245#117—24no—
246#117—44no—
247#117—44no—
248#117#4118yes—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—88no—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2414yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—48no—
299C2×C22⋊C4C2×D412yes—
300C2×C22⋊C4C2×D414yes—
301C2×C22⋊C4C2×D412yes—
302C2×C42—28no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—24no—
308C2×C42—44no—
309C2×C42C2×D414yes—
310C2×C42C2412yes—
311C2×C42C2412yes—
312#107—28no—
313#106—216no—
314#106—28no—
315C4⋊C8—216no—
316#102—216no—
317#102—28no—
318#102—28no—
319C22⋊C8—24no—
320C22⋊C8—24no—
321C22⋊C8—24no—
322C22⋊C8—24no—
323#95—28no—
324#95—28no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95—22no—
329C24—22no—
330C24—22no—
331C24—24no—
332C24—22no—
333C24—24no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24—82no—
338C24—82no—
339C24#12114yes—
340C24C22×D411yescharacteristic
341C24C22×D411yescharacteristic
342C24C22×D411yescharacteristic
343C2×D4—28no—
344C2×D4—28no—
345C2×D4—24no—
346C2×D4—216no—
347C2×D4—24no—
348C2×D4—22no—
349C2×D4—24no—
350C2×D4—48no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—48no—
354C2×D4—84no—
355C2×D4—816no—
356C2×D4C22×D414yes—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—216no—
364C22×C4—216no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—28no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—48no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—48no—
388C22×C4—84no—
389C22×C4—84no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4—82no—
393C22×C4—816no—
394C22×C4#12114yes—
395C22×C4#12114yes—
396C22×C4#12114yes—
397C22×C4C22×D414yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D414yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D412yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4114yes—
409C22×C42−1+414yes—
410C22×C4C2511yescharacteristic
411M16—48no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—48no—
419C4⋊C4—24no—
420C4⋊C4—28no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—88no—
437C4⋊C4—88no—
438C4⋊C4—84no—
439C4⋊C4#12318yes—
440C4⋊C4C22×D414yes—
441C22⋊C4—24no—
442C22⋊C4—28no—
443C22⋊C4—28no—
444C22⋊C4—28no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—44no—
450C22⋊C4—48no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C22⋊C4—82no—
454C22⋊C4#12318yes—
455C42—28no—
456C42—24no—
457C42—44no—
458C42—44no—
459C42—42no—
460C42—42no—
461C42—84no—
462C42C2×#4118yes—
463C23—22no—
464C23—22no—
465C23—21no—
466C23—21no—
467C23—22no—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—28no—
472C23—28no—
473C23—22no—
474C23—24no—
475C23—44no—
476C23—48no—
477C23—48no—
478C23—44no—
479C23—42no—
480C23—42no—
481C23—44no—
482C23—48no—
483C23—84no—
484C23—84no—
485C23—164no—
486C23—164no—
487C23C2×#12112yes—
488C23C2×#12112yes—
489C23C2×#12112yes—
490C23C2×#12312yes—
491C23#53612yes—
492C23#57714yes—
493C23C23×D411yescharacteristic
494D4—28no—
495D4—416no—
496D4—44no—
497D4—48no—
498D4—88no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—416no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—48no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—84no—
531C2×C4—84no—
532C2×C4—88no—
533C2×C4—88no—
534C2×C4—88no—
535C2×C4—84no—
536C2×C4—168no—
537C2×C4C2×#12112yes—
538C2×C4C2×#12112yes—
539C2×C4C2×#12112yes—
540C2×C4C2×#12318yes—
541C2×C4C2×#12312yes—
542C2×C4#53612yes—
543C2×C4#54718yes—
544C2×C4#55218yes—
545C2×C4C23×D411yescharacteristic
546C2×C4C22×#4112yes—
547C2×C4C22×#4111yescharacteristic
548C2×C4C2×2−1+412yes—
549C8—44no—
550C8—44no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—24no—
555C22—24no—
556C22—22no—
557C22—24no—
558C22—24no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—416no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—84no—
569C22—84no—
570C22—168no—
571C22#248614yes—
572C22#304214yes—
573C22C22×#12111yescharacteristic
574C22C22×#12311yescharacteristic
575C22C2×#53611yescharacteristic
576C22C2×#57712yes—
577C22#358512yes—
578C4—28no—
579C4—22no—
580C4—22no—
581C4—44no—
582C4—44no—
583C4—82no—
584C4—82no—
585C4—84no—
586C4—82no—
587C4—82no—
588C4#248614yes—
589C4C2×#54712yes—
590C4C2×#55212yes—
591C2—24no—
592C2—24no—
593C2—42no—
594C2—84no—
595C2C2×#248611yescharacteristic
596C2C2×#304211yescharacteristic
597C2#6042611yescharacteristic
598C2#6052512yes—
599C2#6087312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#2317C2semidirect
5. #27654C2semidirect
6. #27651C2semidirect
7. #27645C2semidirect
8. #23661C2semidirect
9. #23632C2semidirect
10. #21174C2semidirect
11. #20700C2semidirect
12. #20699C2semidirect
13. #20650C2semidirect
14. #20628C2semidirect
43. #2317C22semidirect
49. C2×#384C22semidirect
50. C2×#382C22semidirect
53. #2085C22semidirect
54. #2084C22semidirect
55. #2073C22semidirect
56. #2071C22semidirect
57. #1949C22semidirect
58. #1948C22semidirect
59. #1825C22semidirect
60. #1824C22semidirect
61. #1824C22semidirect
62. #1823C22semidirect
63. #1823C22semidirect
64. #1804C22semidirect
65. #1802C22semidirect
66. #1798C22semidirect
67. #1797C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30942
C22×#4812
#342042
C2×#23172
#276542
#276512
#276452
#236612
#236322
#211742
#207002
#206992
#206502
#206282