Finite Group Databasebeta

← #6434408

Subgroups — #6434408

Order 512.

1474 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34638C211yesmaximal, characteristic, complemented
2C2×#2644C211yesmaximal, characteristic
3#30395C214yesmaximal, complemented
4#30392C214yesmaximal, complemented
5#30390C214yesmaximal, complemented
6#30389C214yesmaximal, complemented
7#23747C211yesmaximal, characteristic, complemented
8#23731C211yesmaximal, characteristic
9#23670C212yesmaximal, complemented
10#23657C211yesmaximal, characteristic, complemented
11#23628C211yesmaximal, characteristic
12#23618C212yesmaximal
13#21159C212yesmaximal, complemented
14#21095C212yesmaximal
15#20429C211yesmaximal, characteristic, complemented
16C2×#428C2212yes—
17C2×#428C2212yes—
18C2×#423C2212yes—
19C2×#418C2212yes—
20C2×#418C2212yes—
21C2×C4×M16C2212yes—
22#2647C2212yescomplemented
23#2647C2212yescomplemented
24#2647C2212yescomplemented
25#2647C2212yescomplemented
26#2646—21no—
27#2646—21no—
28#2646—21no—
29#2646—21no—
30#2644—22no—
31#2644—22no—
32#2644C2214yes—
33#2644C2214yes—
34#2402—22no—
35#2402—22no—
36C2×#397—21no—
37C2×#397—21no—
38C2×#397—21no—
39C2×#396—21no—
40#2296C2212yes—
41#2296C2212yes—
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48#2285C2211yescharacteristic
49#2285C2211yescharacteristic
50C2×#376C2211yescharacteristic
51#2101C2214yescomplemented
52#2100C2212yescomplemented
53#2099C2212yescomplemented
54#2094C2212yes—
55#2093C2212yes—
56#2092C2214yes—
57#2088C2214yescomplemented
58#2088C2214yescomplemented
59#2087C2214yescomplemented
60#2086C2214yescomplemented
61#2085C2214yescomplemented
62#2084C2212yescomplemented
63#2083C2212yescomplemented
64#2073C2214yes—
65#2071C2212yes—
66#2071C2212yes—
67#2067C2214yes—
68#2067C2214yes—
69#2065C2214yes—
70#2065C2214yes—
71#1944C2214yescomplemented
72#1943C2214yescomplemented
73#1942C2214yescomplemented
74#1941C2214yescomplemented
75#1936C2214yes—
76#1936C2214yes—
77#1934C2214yes—
78#1934C2214yes—
79#1768C2214yescomplemented
80#1768C2214yescomplemented
81#1736C2212yes—
82#1736C2212yes—
83#1736C2212yes—
84#1736C2212yes—
85#1728C2211yescharacteristic
86#1728C2211yescharacteristic
87C23×D4C2311yescharacteristic
88C22×M16C2312yes—
89C22×M16C2312yes—
90C2×#117C2312yes—
91C2×#117C2312yes—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107#428—28no—
108#428—44no—
109C2×#107C2314yes—
110C2×#107C2314yes—
111C2×#106C2314yes—
112C2×#106C2314yes—
113#423—28no—
114C2×C4⋊C8C2314yes—
115C2×C4⋊C8C2314yes—
116#418—28no—
117#418—44no—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C4×M16—44no—
123C2×#97C2314yes—
124C2×C4×C8C2314yes—
125#400—22no—
126#400—22no—
127#400—22no—
128#400—22no—
129#400—22no—
130#400—22no—
131#400—22no—
132#400—22no—
133#398—22no—
134#398—22no—
135#398—22no—
136#398—22no—
137#398—22no—
138#398—22no—
139#398—22no—
140#398—22no—
141#397—24no—
142#397—24no—
143#397—24no—
144#397—24no—
145#397—24no—
146#397—24no—
147#396—24no—
148#396—24no—
149#390—22no—
150#390—21no—
151#390—21no—
152#390—22no—
153#390C2314yes—
154#390C2312yes—
155#390C2312yes—
156#390C2314yes—
157#389—21no—
158#389—21no—
159#389—22no—
160#389—22no—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2314yes—
168#389C2314yes—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#385—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#382C2312yes—
182#382C2312yes—
183#380—21no—
184#380—21no—
185#380—21no—
186#380—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195#376D412yes—
196#376D412yes—
197#376C2312yes—
198#376C2312yes—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95D411yescharacteristic
203C2×#95D411yescharacteristic
204C2×#95D411yescharacteristic
205C2×#95D411yescharacteristic
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2312yes—
215#336C2312yes—
216#336C2312yes—
217#336C2312yes—
218C25—21no—
219C22×D4—21no—
220C22×D4—24no—
221C22×D4—22no—
222C22×D4—24no—
223C22×D4—22no—
224C22×D4C2412yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2411yescharacteristic
235C2×M16—24no—
236C2×M16—24no—
237C2×M16—24no—
238C2×M16—24no—
239C2×M16—24no—
240C2×M16#4118yes—
241C22×C8C2414yes—
242C22×C8C2414yes—
243#117—24no—
244#117—24no—
245#117—24no—
246#117—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
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—82no—
275C2×C4⋊C4—82no—
276C2×C4⋊C4—82no—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—82no—
295C2×C22⋊C4—82no—
296C2×C22⋊C4—82no—
297C2×C22⋊C4—82no—
298C2×C22⋊C4—82no—
299C2×C22⋊C4—82no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42C2×D412yes—
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D412yes—
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2412yes—
313C2×C42C2412yes—
314#107—28no—
315#107—44no—
316#106—28no—
317#106—44no—
318C4⋊C8—28no—
319C4⋊C8—28no—
320#102—28no—
321#102—28no—
322#102—44no—
323#102—44no—
324#97—44no—
325C4×C8—44no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95C2×D412yes—
343#95C2×D412yes—
344#95C2×D412yes—
345#95C2×D412yes—
346#95C2×D412yes—
347#95C2×D412yes—
348#95C2×D412yes—
349#95C2×D412yes—
350C24—22no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24C22×D411yescharacteristic
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—24no—
362C2×D4—22no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—48no—
367C2×D4—48no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D412yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C2×#4114yes—
412C22×C4H(5,𝔽2)12yes—
413C22×C42−1+414yes—
414C22×C42−1+412yes—
415C22×C4C2511yescharacteristic
416M16—216no—
417M16—48no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8C2×#4118yes—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—81no—
440C4⋊C4—81no—
441C4⋊C4—81no—
442C4⋊C4—81no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C4⋊C4—82no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4—84no—
456C4⋊C4—82no—
457C4⋊C4—82no—
458C4⋊C4—84no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—84no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4—84no—
465C22⋊C4—84no—
466C22⋊C4—82no—
467C22⋊C4—82no—
468C22⋊C4—84no—
469C42—24no—
470C42—22no—
471C42—22no—
472C42—24no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—44no—
480C42—44no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—84no—
493C23—84no—
494C23—84no—
495C23—84no—
496C23—84no—
497C23—84no—
498C23—84no—
499C23#53212yes—
500C23#53612yes—
501C23#53712yes—
502C23C23×D411yescharacteristic
503D4—44no—
504D4—44no—
505D4—84no—
506C2×C4—22no—
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—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4#53212yes—
561C2×C4#53612yes—
562C2×C4#53712yes—
563C2×C4#53918yes—
564C2×C4#54118yes—
565C2×C4C22×#4112yes—
566C2×C4C2×H(5,𝔽2)11yescharacteristic
567C2×C4C2×2−1+412yes—
568C2×C4C2×2−1+411yescharacteristic
569C8—28no—
570C8—44no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—168no—
585C22#266414yes—
586C22C2×#53211yescharacteristic
587C22C2×#53611yescharacteristic
588C22C2×#53711yescharacteristic
589C22#358511yescharacteristic
590C22#358511yescharacteristic
591C22#358511yescharacteristic
592C22#358611yescharacteristic
593C4—42no—
594C4—42no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—42no—
600C4—42no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4#266414yes—
609C4C2×#53912yes—
610C4C2×#54112yes—
611C2—24no—
612C2—162no—
613C2C2×#266411yescharacteristic
614C2#6070311yescharacteristic
615C2#6070411yescharacteristic
616C2#6086811yescharacteristic
617C2#6087011yescharacteristic
618C2#6091911yescharacteristic
619C2#6092011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34638C2semidirect
3. #30395C2semidirect
4. #30392C2semidirect
5. #30390C2semidirect
6. #30389C2semidirect
7. #23747C2semidirect
9. #23670C2semidirect
10. #23657C2semidirect
13. #21159C2semidirect
15. #20429C2semidirect
22. #2647C22semidirect
23. #2647C22semidirect
24. #2647C22semidirect
25. #2647C22semidirect
51. #2101C22semidirect
52. #2100C22semidirect
53. #2099C22semidirect
57. #2088C22semidirect
58. #2088C22semidirect
59. #2087C22semidirect
60. #2086C22semidirect
61. #2085C22semidirect
62. #2084C22semidirect
63. #2083C22semidirect
71. #1944C22semidirect
72. #1943C22semidirect
73. #1942C22semidirect
74. #1941C22semidirect
79. #1768C22semidirect
80. #1768C22semidirect

Maximal subgroups

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

subgroupindex
#346382
C2×#26442
#303952
#303922
#303902
#303892
#237472
#237312
#236702
#236572
#236282
#236182
#211592
#210952
#204292