Finite Group Databasebeta

← #6446050

Subgroups — #6446050

Order 512.

1734 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35515C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#25072C214yesmaximal, complemented
4#25068C214yesmaximal, complemented
5#25064C214yesmaximal, complemented
6#25056C214yesmaximal, complemented
7#23735C211yesmaximal, characteristic
8C2×#2093C211yesmaximal, characteristic
9#23660C211yesmaximal, characteristic, complemented
10#23658C211yesmaximal, characteristic, complemented
11#21304C211yesmaximal, characteristic, complemented
12#21173C211yesmaximal, characteristic, complemented
13#21063C211yesmaximal, characteristic
14#21060C211yesmaximal, characteristic
15#21037C211yesmaximal, characteristic, complemented
16#21032C211yesmaximal, characteristic, complemented
17#20803C211yesmaximal, characteristic, complemented
18#20650C211yesmaximal, characteristic
19#20416C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#518C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C2×#405C2211yescharacteristic
34#2580C2214yes—
35#2538C2214yes—
36#2454—22no—
37#2452—22no—
38#2437—24no—
39#2360—24no—
40#2355C2214yescomplemented
41Q8×C4⋊C4C2214yescomplemented
42#2330—22no—
43#2317C2211yescharacteristic
44#2317C2211yescharacteristic
45#2317C2211yescharacteristic
46#2316C2211yescharacteristic
47#2309C2211yescharacteristic
48C4×C4⋊Q8—22no—
49#2296C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#384C2211yescharacteristic
52#2274C2211yescharacteristic
53#2274C2211yescharacteristic
54C4×#117C2211yescharacteristic
55#2094C2212yes—
56#2093C2214yes—
57#2093C2214yes—
58#2093C2212yes—
59#2092C2212yes—
60#2092C2212yes—
61#2085C2214yescomplemented
62#2084C2214yescomplemented
63#2083C2214yescomplemented
64#2083C2214yescomplemented
65#1985C2214yescomplemented
66#1984C2214yescomplemented
67#1949C2214yescomplemented
68#1947C2214yescomplemented
69#1926C2214yes—
70#1926C2214yes—
71#1925C2214yes—
72#1925C2214yes—
73#1921C2212yescomplemented
74#1921C2214yescomplemented
75#1920C2212yescomplemented
76#1919C2212yescomplemented
77#1918C2212yescomplemented
78#1918C2214yescomplemented
79#1847C2212yescomplemented
80#1847C2212yescomplemented
81#1846C2212yescomplemented
82#1845C2212yescomplemented
83#1804C2214yes—
84#1802C2214yes—
85#1762C2214yes—
86C4×#102C2214yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1735C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×D4C2311yescharacteristic
98C22×M16C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C23×C8C2311yescharacteristic
101C23×C8C2311yescharacteristic
102#533—24no—
103#533—24no—
104C2×C4⋊Q8D412yes—
105C2×#125D412yes—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8D414yes—
108#518—24no—
109#518—24no—
110C2×C4×Q8—24no—
111C2×C4×Q8C2314yes—
112C2×C4×D4—22no—
113C2×C4×D4C2314yes—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C22×C4⋊C4—22no—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—22no—
125C22×C22⋊C4—22no—
126C22×C42C2311yescharacteristic
127#428—42no—
128#428—42no—
129#428—42no—
130#427—24no—
131C2×#107—24no—
132C2×#107—42no—
133C2×#107C2311yescharacteristic
134C2×#107C2311yescharacteristic
135C2×#107C2312yes—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#107C2312yes—
140C2×#107C2314yes—
141C2×#106C2311yescharacteristic
142C2×#106C2311yescharacteristic
143C2×#106C2312yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146#424—24no—
147C2×C4⋊C8C2314yes—
148#418—42no—
149#418Q818yescomplemented
150C2×#102—24no—
151C2×#102—42no—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×#102C2314yes—
158#405—24no—
159C2×#97C2312yes—
160C2×C4×C8C2312yes—
161#399—24no—
162#396—24no—
163#396—24no—
164#393—24no—
165#391—24no—
166#390—22no—
167#390—21no—
168#390—21no—
169#390C2312yes—
170#390C2312yes—
171#390C2314yes—
172#389—22no—
173#389—22no—
174#389—24no—
175#389—22no—
176#389—22no—
177#389—24no—
178#389—24no—
179#389C2314yes—
180#389C2312yes—
181#389C2312yes—
182#387—24no—
183#387—24no—
184#385—22no—
185#385—24no—
186#385—24no—
187#385—22no—
188#385—22no—
189#385—22no—
190#384—21no—
191#384—21no—
192#384C2312yes—
193#384C2312yes—
194#384C2312yes—
195#384C2314yes—
196#382C2312yes—
197#382C2312yes—
198#382C2312yes—
199#380—24no—
200#380—22no—
201#379—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—24no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214#376—21no—
215C43—21no—
216#340C2312yes—
217#340C2314yes—
218#340C2312yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#336C2314yes—
226#336C2314yes—
227#336C2312yes—
228#336C2312yes—
229#336C2312yes—
230#336C2312yes—
231C25—21no—
232C22×Q8C2×D414yes—
233C22×D4—21no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4C2×Q814yes—
240C22×D4C2412yes—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—24no—
245C23×C4C2×D412yes—
246C23×C4C2411yescharacteristic
247C2×M16—22no—
248C2×M16—22no—
249C2×M16—22no—
250C2×M16—22no—
251C2×M16—22no—
252C2×M16C2×Q814yes—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8#4114yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2411yescharacteristic
263C22×C8C2411yescharacteristic
264C4⋊Q8—28no—
265#125—28no—
266C22⋊Q8—216no—
267C22⋊Q8—88no—
268C4×Q8—28no—
269C4×Q8—28no—
270C4×Q8—88no—
271C4×D4—28no—
272C4×D4—28no—
273C4×D4—48no—
274#117—22no—
275#117—22no—
276#117—22no—
277#117—22no—
278#117—22no—
279#117—22no—
280#117—22no—
281#117—22no—
282#117—22no—
283#117C2×Q814yes—
284#117#4114yes—
285#117#4114yes—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—84no—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—48no—
328C2×C22⋊C4—84no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2412yes—
346#107—24no—
347#107—24no—
348#107—22no—
349#107—22no—
350#107—44no—
351#107—42no—
352#107—42no—
353#107—42no—
354#106—22no—
355#106—22no—
356#106—42no—
357#106—42no—
358#106—42no—
359C4⋊C8—28no—
360#102—24no—
361#102—24no—
362#102—44no—
363#102—42no—
364#102—42no—
365#102C2×Q8116yes—
366#97—24no—
367C4×C8—24no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—24no—
372#95—22no—
373#95—24no—
374#95—24no—
375#95—22no—
376#95—24no—
377#95—24no—
378C24—22no—
379C24—24no—
380C24—42no—
381C24—42no—
382C24—82no—
383C24—82no—
384C24C22×D411yescharacteristic
385C2×Q8—28no—
386C2×Q8—28no—
387C2×Q8—44no—
388C2×D4—28no—
389C2×D4—24no—
390C2×D4—28no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—42no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4—44no—
400C2×D4C22×Q814yes—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—48no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4C22×D414yes—
423C22×C4C22×D412yes—
424C22×C4C22×D414yes—
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×Q812yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C42−1+412yes—
433C22×C4C2511yescharacteristic
434M16—28no—
435M16—44no—
436C2×C8—28no—
437C2×C8—21no—
438C2×C8—22no—
439C2×C8—21no—
440C2×C8—22no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—44no—
450C2×C8C22×Q814yes—
451C2×C8C2×#4112yes—
452C2×C8C2×#4112yes—
453C4⋊C4—24no—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4#12318yes—
486C4⋊C4C22×Q814yes—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4#12318yes—
498C42—22no—
499C42—22no—
500C42—24no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—44no—
506C42—42no—
507C42—42no—
508C42—42no—
509C42—44no—
510C42—84no—
511C42—84no—
512C42C2×#4114yes—
513C42C2×#4114yes—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—48no—
522C23—88no—
523C23—164no—
524C23—162no—
525C23—162no—
526C23C2×#12312yes—
527C23D4×Q812yes—
528C23#55012yes—
529C23C23×D411yescharacteristic
530Q8—416no—
531D4—24no—
532D4—24no—
533D4—48no—
534D4—42no—
535D4—42no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—168no—
568C2×C4—162no—
569C2×C4—162no—
570C2×C4—162no—
571C2×C4C2×#12318yes—
572C2×C4C2×#12312yes—
573C2×C4#53314yes—
574C2×C4#54214yes—
575C2×C4D4×Q812yes—
576C2×C4#55012yes—
577C2×C4#55214yes—
578C2×C4#55414yes—
579C2×C4#57614yes—
580C2×C4C23×Q811yescharacteristic
581C2×C4C22×#4111yescharacteristic
582C2×C4C22×#4111yescharacteristic
583C2×C4C22×#4111yescharacteristic
584C2×C4C2×H(5,𝔽2)11yescharacteristic
585C2×C4C2×2−1+411yescharacteristic
586C8—22no—
587C8—22no—
588C8—24no—
589C8—42no—
590C8—42no—
591C22—24no—
592C22—24no—
593C22—22no—
594C22—22no—
595C22—44no—
596C22—48no—
597C22—84no—
598C22—82no—
599C22—82no—
600C22—168no—
601C22#242314yes—
602C22C22×#12311yescharacteristic
603C22C2×D4×Q811yescharacteristic
604C22C2×#55011yescharacteristic
605C22C2×#57612yes—
606C22#358612yes—
607C4—24no—
608C4—28no—
609C4—42no—
610C4—44no—
611C4—44no—
612C4—82no—
613C4—81no—
614C4—81no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—81no—
619C4—81no—
620C4#242314yes—
621C4C2×#53311yescharacteristic
622C4C2×#54211yescharacteristic
623C4C2×#55211yescharacteristic
624C4C2×#55411yescharacteristic
625C2—24no—
626C2—84no—
627C2C2×#242311yescharacteristic
628C2#6052112yes—
629C2#6138112yes—
630C2#6140911yescharacteristic
631C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34214C2semidirect
3. #25072C2semidirect
4. #25068C2semidirect
5. #25064C2semidirect
6. #25056C2semidirect
9. #23660C2semidirect
10. #23658C2semidirect
11. #21304C2semidirect
12. #21173C2semidirect
15. #21037C2semidirect
16. #21032C2semidirect
17. #20803C2semidirect
40. #2355C22semidirect
41. Q8×C4⋊C4C22semidirect
61. #2085C22semidirect
62. #2084C22semidirect
63. #2083C22semidirect
64. #2083C22semidirect
65. #1985C22semidirect
66. #1984C22semidirect
67. #1949C22semidirect
68. #1947C22semidirect
73. #1921C22semidirect
74. #1921C22semidirect
75. #1920C22semidirect
76. #1919C22semidirect
77. #1918C22semidirect
78. #1918C22semidirect
79. #1847C22semidirect
80. #1847C22semidirect
81. #1846C22semidirect
82. #1845C22semidirect
149. #418Q8semidirect

Maximal subgroups

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

subgroupindex
#355152
#342142
#250722
#250682
#250642
#250562
#237352
C2×#20932
#236602
#236582
#213042
#211732
#210632
#210602
#210372
#210322
#208032
#206502
#204162