Finite Group Databasebeta

← #6433946

Subgroups — #6433946

Order 512.

1671 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34635C211yesmaximal, characteristic
2C2×#2629C211yesmaximal, characteristic, complemented
3#30375C214yesmaximal, complemented
4#30373C214yesmaximal, complemented
5#30371C214yesmaximal, complemented
6#30369C214yesmaximal, complemented
7#23745C211yesmaximal, characteristic, complemented
8#23734C211yesmaximal, characteristic, complemented
9#23642C212yesmaximal, complemented
10#23553C211yesmaximal, characteristic, complemented
11#23482C212yesmaximal
12#23453C211yesmaximal, characteristic, complemented
13#21159C212yesmaximal, complemented
14#21093C212yesmaximal, complemented
15#20377C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#417C2212yes—
22C2×#417C2212yes—
23C2×#407C2212yes—
24C2×C4×M16C2212yes—
25#2629—22no—
26#2629—22no—
27#2629C2214yes—
28#2629C2214yes—
29#2625C2212yes—
30#2625C2212yes—
31#2624C2212yes—
32#2624C2212yes—
33#2623—22no—
34#2622—22no—
35#2395—22no—
36#2395—22no—
37C2×#398—22no—
38C2×#396—22no—
39#2297C2211yescharacteristic
40#2297C2211yescharacteristic
41#2296C2211yescharacteristic
42#2296C2211yescharacteristic
43C2×#389C2212yes—
44C2×#384C2211yescharacteristic
45#2280C2211yescharacteristic
46C2×#380C2212yes—
47#2100C2214yescomplemented
48#2099C2214yescomplemented
49#2096C2212yescomplemented
50#2095C2212yescomplemented
51#2094C2212yescomplemented
52#2093C2212yescomplemented
53#2077C2214yes—
54#2077C2214yes—
55#2076C2214yes—
56#2075C2214yes—
57#2059C2214yescomplemented
58#2059C2214yescomplemented
59#2036C2214yes—
60#2035C2214yes—
61#2034C2214yes—
62#2033C2214yes—
63#2026C2212yescomplemented
64#2026C2212yescomplemented
65#2025C2212yescomplemented
66#2025C2212yescomplemented
67#1944C2214yescomplemented
68#1943C2214yescomplemented
69#1942C2214yescomplemented
70#1941C2214yescomplemented
71#1936C2214yescomplemented
72#1935C2214yescomplemented
73#1934C2214yescomplemented
74#1933C2214yescomplemented
75#1750C2214yes—
76#1750C2214yes—
77#1736C2212yes—
78#1736C2212yes—
79#1736C2212yes—
80#1736C2212yes—
81#1728C2211yescharacteristic
82#1728C2211yescharacteristic
83C22×#41C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C22×M16C2312yes—
86C22×M16C2312yes—
87C2×#125—22no—
88C2×#125—44no—
89C2×#125C2312yes—
90C2×#125C2312yes—
91#525—44no—
92C2×#123—21no—
93C2×C22⋊Q8C2312yes—
94C2×#121C2312yes—
95C2×#120—21no—
96C2×C4×D4—22no—
97C2×#117—44no—
98C2×#117D412yes—
99C2×#117D412yes—
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C2×#117C2312yes—
104C2×#117C2312yes—
105C22×C4⋊C4—22no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4D412yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4—22no—
112C22×C22⋊C4D412yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#428—28no—
116#428—44no—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121#417—28no—
122#417—44no—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127#407—28no—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2314yes—
130C4×M16—44no—
131C2×#97C2314yes—
132C2×C4×C8C2314yes—
133#400—22no—
134#400—22no—
135#399—22no—
136#399—22no—
137#398—22no—
138#398—22no—
139#398—24no—
140#398—24no—
141#398—24no—
142#398—24no—
143#397—22no—
144#397—22no—
145#397—22no—
146#397—22no—
147#397—22no—
148#397—22no—
149#396—24no—
150#396—24no—
151#396—24no—
152#396—24no—
153#394—22no—
154#394—22no—
155#394—22no—
156#394—22no—
157#391C2311yescharacteristic
158#391C2311yescharacteristic
159#391C2312yes—
160#390C2311yescharacteristic
161#390C2311yescharacteristic
162#390C2312yes—
163#389—22no—
164#389—22no—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2314yes—
170#389C2314yes—
171#386—22no—
172#386—22no—
173#385—22no—
174#385—22no—
175#384—22no—
176#384C2312yes—
177#384C2312yes—
178#381—21no—
179#380—22no—
180#380—22no—
181#380C2312yes—
182#380C2314yes—
183#380C2314yes—
184#380C2312yes—
185#379—21no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—21no—
192C4×C22⋊C4—21no—
193C2×#95—22no—
194C2×#95—22no—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#336C2312yes—
204#336C2312yes—
205#336C2312yes—
206#336C2312yes—
207C25C2×D411yescharacteristic
208C2×#41—24no—
209C2×#41—24no—
210C2×#41—22no—
211C2×#41—22no—
212C22×Q8C2411yescharacteristic
213C22×D4—21no—
214C22×D4—22no—
215C22×D4C2411yescharacteristic
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—21no—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D412yes—
225C23×C4C2×D412yes—
226C23×C4C2×D412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C2×M16—24no—
230C2×M16—24no—
231C2×M16—24no—
232C2×M16—24no—
233C2×M16—24no—
234C2×M16#4118yes—
235C22×C8C2414yes—
236C22×C8C2414yes—
237#125—44no—
238#125—44no—
239#125—48no—
240#123—44no—
241C22⋊Q8—42no—
242C22⋊Q8—42no—
243#121—42no—
244#121—42no—
245#120—44no—
246C4×D4—48no—
247#117—24no—
248#117—24no—
249#117—22no—
250#117—22no—
251#117—44no—
252#117—44no—
253#117—42no—
254#117—42no—
255#117—42no—
256#117—42no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4—84no—
277C2×C4⋊C4—84no—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2414yes—
283C2×C4⋊C4C2414yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2411yescharacteristic
287C2×C4⋊C4C2411yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—84no—
303C2×C22⋊C4C2×D412yes—
304C2×C22⋊C4C2×D412yes—
305C2×C22⋊C4C2×D412yes—
306C2×C22⋊C4C2×D412yes—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42C2×D412yes—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317C2×C42C2412yes—
318#107—28no—
319#107—44no—
320#106—28no—
321#106—44no—
322#103—28no—
323#103—44no—
324#102—28no—
325#102—44no—
326C22⋊C8—28no—
327C22⋊C8—28no—
328#97—44no—
329C4×C8—44no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
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#95—22no—
343#95—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—21no—
348C24—42no—
349C24—42no—
350C24C22×D412yes—
351C24C22×D411yescharacteristic
352#41—48no—
353#41—48no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—44no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—82no—
367C2×D4—82no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C2×#4114yes—
411C22×C4H(5,𝔽2)14yes—
412C22×C42−1+412yes—
413C22×C42−1+412yes—
414C22×C4C2511yescharacteristic
415M16—216no—
416M16—48no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8C2×#4118yes—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C4⋊C4—84no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—82no—
460C22⋊C4—82no—
461C22⋊C4—82no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4—82no—
465C22⋊C4—84no—
466C22⋊C4—84no—
467C42—22no—
468C42—22no—
469C42—24no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—42no—
475C42—42no—
476C42—42no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—48no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—82no—
494C23—82no—
495C23—84no—
496C23#53212yes—
497C23#53412yes—
498C23#53612yes—
499C23C23×D411yescharacteristic
500Q8—42no—
501Q8—42no—
502D4—42no—
503D4—42no—
504D4—88no—
505D4—84no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—48no—
522C2×C4—44no—
523C2×C4—48no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—44no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—84no—
548C2×C4—88no—
549C2×C4—88no—
550C2×C4—84no—
551C2×C4#53212yes—
552C2×C4#53412yes—
553C2×C4#53612yes—
554C2×C4#53918yes—
555C2×C4#54218yes—
556C2×C4C22×#4112yes—
557C2×C4C2×H(5,𝔽2)12yes—
558C2×C4C2×2−1+411yescharacteristic
559C2×C4C2×2−1+411yescharacteristic
560C8—28no—
561C8—44no—
562C22—22no—
563C22—22no—
564C22—24no—
565C22—24no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—88no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—82no—
578C22—82no—
579C22#266314yes—
580C22C2×#53211yescharacteristic
581C22C2×#53411yescharacteristic
582C22C2×#53611yescharacteristic
583C22#358512yes—
584C22#358612yes—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—81no—
591C4—81no—
592C4—84no—
593C4—81no—
594C4—84no—
595C4—82no—
596C4—82no—
597C4#266314yes—
598C4C2×#53912yes—
599C4C2×#54212yes—
600C2—24no—
601C2—82no—
602C2—161no—
603C2C2×#266311yescharacteristic
604C2#6070411yescharacteristic
605C2#6070511yescharacteristic
606C2#6075512yes—
607C2#6086712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
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
2. C2×#2629C2semidirect
3. #30375C2semidirect
4. #30373C2semidirect
5. #30371C2semidirect
6. #30369C2semidirect
7. #23745C2semidirect
8. #23734C2semidirect
9. #23642C2semidirect
10. #23553C2semidirect
12. #23453C2semidirect
13. #21159C2semidirect
14. #21093C2semidirect
15. #20377C2semidirect
47. #2100C22semidirect
48. #2099C22semidirect
49. #2096C22semidirect
50. #2095C22semidirect
51. #2094C22semidirect
52. #2093C22semidirect
57. #2059C22semidirect
58. #2059C22semidirect
63. #2026C22semidirect
64. #2026C22semidirect
65. #2025C22semidirect
66. #2025C22semidirect
67. #1944C22semidirect
68. #1943C22semidirect
69. #1942C22semidirect
70. #1941C22semidirect
71. #1936C22semidirect
72. #1935C22semidirect
73. #1934C22semidirect
74. #1933C22semidirect

Maximal subgroups

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

subgroupindex
#346352
C2×#26292
#303752
#303732
#303712
#303692
#237452
#237342
#236422
#235532
#234822
#234532
#211592
#210932
#203772