Finite Group Databasebeta

← #6436719

Subgroups — #6436719

Order 512.

1844 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3123C212yesmaximal
2#35319C211yesmaximal, characteristic, complemented
3#34255C211yesmaximal, characteristic
4#29870C212yesmaximal, complemented
5#29836C214yesmaximal, complemented
6#29798C212yesmaximal, complemented
7#29766C212yesmaximal, complemented
8#29759C212yesmaximal, complemented
9#29723C212yesmaximal
10#29714C212yesmaximal
11#23689C214yesmaximal, complemented
12#23683C212yesmaximal, complemented
13#23677C212yesmaximal, complemented
14#21398C212yesmaximal, complemented
15#20813C211yesmaximal, characteristic, complemented
16C2×#541C2212yes—
17C2×#518C2211yescharacteristic
18#3123—216no—
19C2×#485C2214yes—
20C2×#478C2214yes—
21C2×#461C2214yes—
22C2×#450C2214yes—
23C2×#440C2214yes—
24C2×C4×SD16C2214yes—
25C2×#433C2212yes—
26#2868C2211yescharacteristic, complemented
27#2859C2211yescharacteristic, complemented
28#2857C2212yes—
29#2642C2212yescomplemented
30#2621C2212yes—
31#2506—22no—
32#2506—22no—
33#2496—22no—
34#2496—22no—
35#2357—22no—
36#2354—22no—
37#2342—22no—
38#2342—22no—
39#2327C2212yescomplemented
40#2319C2212yes—
41#2319C2212yes—
42#2317C2211yescharacteristic
43#2309C2211yescharacteristic
44C4×C4⋊Q8C2212yescomplemented
45#2296C2211yescharacteristic
46#2293C2212yes—
47#2293C2212yes—
48C2×#390C2211yescharacteristic
49#2274C2211yescharacteristic
50C4×#117C2211yescharacteristic
51#2085C2214yescomplemented
52#2084C2212yescomplemented
53#2083C2212yescomplemented
54#2073C2214yes—
55#2071C2212yes—
56#2071C2212yes—
57#2068C2214yescomplemented
58#2068C2214yescomplemented
59#2066C2214yescomplemented
60#2066C2214yescomplemented
61#2055C2214yes—
62#2055C2214yes—
63#2055C2214yes—
64#2055C2214yes—
65#2043C2212yescomplemented
66#2043C2212yescomplemented
67#2043C2214yescomplemented
68#2013C2212yes—
69#2013C2212yes—
70#2013C2214yes—
71#1989C2214yescomplemented
72#1981C2214yescomplemented
73#1873C2214yes—
74#1872C2214yes—
75#1843C2212yescomplemented
76#1831C2212yescomplemented
77#1798C2212yes—
78#1797C2212yes—
79#1768C2212yescomplemented
80#1767C2211yescharacteristic, complemented
81C4×C4⋊C8C2211yescharacteristic, complemented
82#1763C2214yescomplemented
83C4×#103C2214yescomplemented
84C23×D4C2311yescharacteristic
85C22×SD16C2314yes—
86C22×SD16C2314yes—
87#541—28no—
88#541—28no—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×#126—24no—
93C2×#125—24no—
94C2×#125C2312yes—
95C2×C22⋊Q8C2314yes—
96#518—24no—
97#518D418yes—
98C2×C4×Q8C2314yes—
99C2×C4×D4—22no—
100C2×C4×D4C2312yes—
101C2×C4×D4C2312yes—
102C2×#117—22no—
103C2×#117—22no—
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4—22no—
110C22×C22⋊C4—22no—
111C22×C42C2311yescharacteristic
112#485—216no—
113#478—216no—
114#461—216no—
115#450—216no—
116#440—216no—
117C4×SD16—216no—
118#433—28no—
119C2×#107C2314yes—
120C2×#106C2314yes—
121C2×C4⋊C8C2314yes—
122C2×C4⋊C8C2314yes—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2312yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2314yes—
134C2×#97C2312yes—
135C2×C4×C8C2312yes—
136#400—24no—
137#400—24no—
138#398—22no—
139#398—22no—
140#397—22no—
141#397—22no—
142#391C2314yes—
143#391C2314yes—
144#390—21no—
145#390—21no—
146#390—22no—
147#390C2312yes—
148#390C2312yes—
149#390C2312yes—
150#390C2312yes—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389C2312yes—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#387—24no—
170#387—24no—
171#385—22no—
172#385—21no—
173#385—21no—
174#384—21no—
175#384—22no—
176#384—22no—
177#384C2312yes—
178#382—21no—
179#382C2312yes—
180#381—22no—
181#381—22no—
182#380—22no—
183#380—22no—
184#380—22no—
185#380—22no—
186#379—21no—
187#379—21no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4D412yes—
193C4×C4⋊C4D412yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4D412yes—
201#376C2311yescharacteristic
202#376C2311yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#336C2312yes—
210#336C2312yes—
211#336C2314yes—
212C25—21no—
213C22×Q8C2414yes—
214C22×D4—21no—
215C22×D4—22no—
216C22×D4—42no—
217C22×D4—42no—
218C22×D4C2×D414yes—
219C22×D4C2412yes—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—21no—
223C23×C4C2×D412yes—
224C23×C4C2411yescharacteristic
225C2×SD16—28no—
226C2×SD16—28no—
227C2×SD16—44no—
228C2×SD16—44no—
229C2×SD16—44no—
230C2×SD16—44no—
231C22×C8C2414yes—
232C22×C8C2414yes—
233C4⋊Q8—24no—
234C4⋊Q8—24no—
235C4⋊Q8—44no—
236C4⋊Q8—42no—
237C4⋊Q8—42no—
238#126—416no—
239#125—24no—
240#125—24no—
241#125—416no—
242C22⋊Q8—28no—
243C22⋊Q8—28no—
244C4×Q8—28no—
245C4×Q8—28no—
246C4×D4—216no—
247C4×D4—24no—
248C4×D4—24no—
249C4×D4C2×D418yes—
250C4×D4C2×D418yes—
251#117—22no—
252#117—42no—
253#117—42no—
254#117—84no—
255#117—84no—
256#117C2×D414yes—
257#117C2×D414yes—
258#117#4114yes—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4C2×D411yescharacteristic
276C2×C4⋊C4C2×D411yescharacteristic
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2414yes—
281C2×C4⋊C4C2412yes—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—216no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—84no—
297C2×C22⋊C4C2×D411yescharacteristic
298C2×C22⋊C4C2×D411yescharacteristic
299C2×C22⋊C4C2×D412yes—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—42no—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314C2×C42C2411yescharacteristic
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317#107—28no—
318#106—28no—
319C4⋊C8—28no—
320C4⋊C8—44no—
321C4⋊C8—42no—
322C4⋊C8—42no—
323#103—28no—
324#103—28no—
325#103—44no—
326#103—44no—
327#103—44no—
328#103—44no—
329#102—28no—
330#102—28no—
331C22⋊C8—28no—
332#97—24no—
333C4×C8—24no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—22no—
345#95—22no—
346#95C2×D412yes—
347#95C2×D412yes—
348#95C2×D412yes—
349#95C2×D412yes—
350C24—22no—
351C24—24no—
352C24—42no—
353C24—42no—
354C24—82no—
355C24—82no—
356C24C22×D411yescharacteristic
357C2×Q8—24no—
358C2×Q8—24no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—48no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—48no—
371C2×D4C22×D414yes—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D412yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C2×#4112yes—
401C22×C4C2511yescharacteristic
402SD16—416no—
403SD16—416no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4#12118yes—
433C4⋊C4C22×D414yes—
434C22⋊C4—24no—
435C22⋊C4—28no—
436C22⋊C4—28no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—42no—
440C22⋊C4—48no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C22⋊C4—82no—
449C22⋊C4#12118yes—
450C42—24no—
451C42—24no—
452C42—24no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—42no—
458C42—42no—
459C42—42no—
460C42—42no—
461C42—48no—
462C42—82no—
463C42—82no—
464C42C22×D412yes—
465C42C22×D412yes—
466C42C2×#4112yes—
467C42C2×#4112yes—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—24no—
473C23—48no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—88no—
481C23—164no—
482C23—162no—
483C23—162no—
484C23C2×#12112yes—
485C23C23×D411yescharacteristic
486Q8—44no—
487Q8—44no—
488Q8—44no—
489Q8—44no—
490D4—28no—
491D4—48no—
492D4—44no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—84no—
530C2×C4—88no—
531C2×C4—88no—
532C2×C4—162no—
533C2×C4C2×#12112yes—
534C2×C4C2×#12114yes—
535C2×C4C2×#12114yes—
536C2×C4#57714yes—
537C2×C4C23×D411yescharacteristic
538C2×C4C22×#4111yescharacteristic
539C2×C4#58512yes—
540C2×C4#58512yes—
541C8—44no—
542C8—44no—
543C22—24no—
544C22—22no—
545C22—42no—
546C22—41no—
547C22—41no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—168no—
556C22C22×#12111yescharacteristic
557C22#344911yescharacteristic
558C22#344911yescharacteristic
559C22C2×#57712yes—
560C22#358612yes—
561C4—24no—
562C4—24no—
563C4—24no—
564C4—22no—
565C4—44no—
566C4—42no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—84no—
572C4—82no—
573C4—84no—
574C4—82no—
575C4#344912yes—
576C4#345512yes—
577C2—42no—
578C2—84no—
579C2#3463311yescharacteristic
580C2#6042912yes—
581C2#6098212yes—
582C2#6101211yescharacteristic
583C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35319C2semidirect
4. #29870C2semidirect
5. #29836C2semidirect
6. #29798C2semidirect
7. #29766C2semidirect
8. #29759C2semidirect
11. #23689C2semidirect
12. #23683C2semidirect
13. #23677C2semidirect
14. #21398C2semidirect
15. #20813C2semidirect
26. #2868C22semidirect
27. #2859C22semidirect
29. #2642C22semidirect
39. #2327C22semidirect
44. C4×C4⋊Q8C22semidirect
51. #2085C22semidirect
52. #2084C22semidirect
53. #2083C22semidirect
57. #2068C22semidirect
58. #2068C22semidirect
59. #2066C22semidirect
60. #2066C22semidirect
65. #2043C22semidirect
66. #2043C22semidirect
67. #2043C22semidirect
71. #1989C22semidirect
72. #1981C22semidirect
75. #1843C22semidirect
76. #1831C22semidirect
79. #1768C22semidirect
80. #1767C22semidirect
81. C4×C4⋊C8C22semidirect
82. #1763C22semidirect
83. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31232
#353192
#342552
#298702
#298362
#297982
#297662
#297592
#297232
#297142
#236892
#236832
#236772
#213982
#208132