Finite Group Databasebeta

← #6435217

Subgroups — #6435217

Order 512.

1668 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35330C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic
3#29991C214yesmaximal, complemented
4#29990C214yesmaximal, complemented
5#29986C214yesmaximal, complemented
6#29982C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8#23744C211yesmaximal, characteristic, complemented
9#23731C211yesmaximal, characteristic
10C2×#2092C211yesmaximal, characteristic
11#23556C211yesmaximal, characteristic, complemented
12#23555C211yesmaximal, characteristic, complemented
13#23456C211yesmaximal, characteristic
14#23456C211yesmaximal, characteristic
15#21174C211yesmaximal, characteristic, complemented
16#21173C211yesmaximal, characteristic, complemented
17#20754C211yesmaximal, characteristic, complemented
18#20650C211yesmaximal, characteristic
19#20650C211yesmaximal, characteristic
20C2×#544C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C2×#518C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33#2862C2212yescomplemented
34#2859C2212yescomplemented
35#2660C2212yescomplemented
36#2660C2212yescomplemented
37#2550—22no—
38#2550—22no—
39#2549—24no—
40#2357—22no—
41#2354—22no—
42#2342—24no—
43#2319C2214yes—
44#2317C2211yescharacteristic
45#2317C2211yescharacteristic
46#2317C2211yescharacteristic
47#2309C2211yescharacteristic
48#2296C2211yescharacteristic
49#2293C2214yes—
50C2×#390C2211yescharacteristic
51#2285C2211yescharacteristic
52C2×#382C2211yescharacteristic
53#2274C2211yescharacteristic
54C4×#117C2211yescharacteristic
55#2101C2212yescomplemented
56#2101C2214yescomplemented
57#2101C2212yescomplemented
58#2101C2214yescomplemented
59#2100C2212yescomplemented
60#2099C2212yescomplemented
61#2094C2212yes—
62#2093C2212yes—
63#2092C2212yes—
64#2092C2214yes—
65#2092C2212yes—
66#2092C2214yes—
67#2059C2214yescomplemented
68#2059C2214yescomplemented
69#2058C2214yescomplemented
70#2057C2214yescomplemented
71#2025C2214yes—
72#2025C2214yes—
73#2023C2214yes—
74#2023C2214yes—
75#1949C2214yescomplemented
76#1949C2214yescomplemented
77#1948C2214yescomplemented
78#1947C2214yescomplemented
79#1832C2214yescomplemented
80#1831C2214yescomplemented
81#1804C2214yes—
82#1804C2214yes—
83#1802C2214yes—
84#1802C2214yes—
85#1736C2211yescharacteristic
86#1736C2211yescharacteristic
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1735C2211yescharacteristic
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#544—42no—
103#544—42no—
104#544—42no—
105#544—42no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×#125—21no—
111C2×#125—21no—
112C2×#125C2312yes—
113C2×#125C2312yes—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116#518—24no—
117#518Q818yescomplemented
118C2×C4×D4—22no—
119C2×C4×D4C2314yes—
120C2×#117—24no—
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—22no—
131C22×C22⋊C4—22no—
132C22×C42C2311yescharacteristic
133#428—42no—
134#428—42no—
135C2×#107—42no—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×#107C2312yes—
139C2×#106—42no—
140C2×#106C2312yes—
141C2×#106C2314yes—
142C2×#106C2312yes—
143#424—24no—
144#424—24no—
145C2×C4⋊C8C2314yes—
146C2×C4⋊C8C2314yes—
147#418—42no—
148#418—42no—
149C2×#102—42no—
150C2×#102—42no—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2312yes—
156C2×#102C2312yes—
157#398—24no—
158#398—24no—
159#395—24no—
160#395—24no—
161#391—24no—
162#391—24no—
163#390—21no—
164#390—21no—
165#390—22no—
166#390C2314yes—
167#390C2312yes—
168#390C2312yes—
169#389—22no—
170#389—22no—
171#389—24no—
172#389—24no—
173#389C2314yes—
174#389C2312yes—
175#389C2312yes—
176#388—24no—
177#388—24no—
178#388—24no—
179#388—24no—
180#385—22no—
181#385—22no—
182#385—22no—
183#385—22no—
184#384—21no—
185#384C2312yes—
186#384C2312yes—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#382—21no—
190#382—22no—
191#382C2312yes—
192#382C2312yes—
193#382C2311yescharacteristic
194#382C2312yes—
195#382C2311yescharacteristic
196#382C2312yes—
197#381—24no—
198#380—22no—
199#380—24no—
200#379—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4C2314yes—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—24no—
209C4×C22⋊C4D414yes—
210#376C2312yes—
211C2×#95—22no—
212C43C2312yes—
213#340C2312yes—
214#340C2314yes—
215#340C2312yes—
216#340C2312yes—
217#340C2314yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#336C2314yes—
226#336C2314yes—
227#336C2312yes—
228#336C2312yes—
229#336C2312yes—
230#336C2312yes—
231C25—21no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×D4—21no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4C2×Q814yes—
241C22×D4C2412yes—
242C23×C4—24no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4C2×D412yes—
247C23×C4C2411yescharacteristic
248C2×M16—22no—
249C2×M16—22no—
250C2×M16—22no—
251C2×M16—22no—
252C2×M16—22no—
253C2×M16—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C4⋊Q8—42no—
265C4⋊Q8—42no—
266C4⋊Q8—42no—
267C4⋊Q8—42no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270#125—42no—
271#125—42no—
272#125—42no—
273#125—42no—
274#125—44no—
275#125—44no—
276C22⋊Q8—48no—
277C22⋊Q8—48no—
278C4×D4—216no—
279C4×D4—28no—
280C4×D4C2×Q8116yes—
281#117—22no—
282#117—22no—
283#117—22no—
284#117—22no—
285#117—22no—
286#117—22no—
287#117—22no—
288#117—22no—
289#117—22no—
290#117—88no—
291#117C2×Q814yes—
292#117C2×Q814yes—
293#117#4114yes—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—216no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—84no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—24no—
348C2×C42C2×D412yes—
349C2×C42C2×D414yes—
350C2×C42C2×D414yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2412yes—
355#107—44no—
356#107—42no—
357#107—42no—
358#106—44no—
359#106—42no—
360#106—42no—
361C4⋊C8—28no—
362C4⋊C8—28no—
363#102—44no—
364#102—44no—
365#102—42no—
366#102—42no—
367#102—42no—
368#102—42no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95C2×D414yes—
380#95C2×D414yes—
381C24—22no—
382C24—24no—
383C24—42no—
384C24—42no—
385C24—82no—
386C24—82no—
387C24C22×D411yescharacteristic
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—84no—
391C2×Q8—84no—
392C2×D4—24no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4C22×Q814yes—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—216no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D414yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×Q812yes—
433C22×C4C2×#4112yes—
434C22×C4H(5,𝔽2)12yes—
435C22×C4H(5,𝔽2)12yes—
436C22×C42−1+412yes—
437C22×C42−1+412yes—
438C22×C4C2511yescharacteristic
439M16—44no—
440M16—44no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—44no—
454C2×C8—44no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—84no—
490C4⋊C4C22⋊Q818yes—
491C4⋊C4C22×Q814yes—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—216no—
495C22⋊C4—48no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4C22⋊Q818yes—
503C42—24no—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—44no—
512C42—44no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—84no—
517C42C22×Q814yes—
518C42C2×#4114yes—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—48no—
527C23—88no—
528C23—162no—
529C23—162no—
530C23—164no—
531C23C2×C22⋊Q812yes—
532C23#53512yes—
533C23#53612yes—
534C23C23×D411yescharacteristic
535Q8—164no—
536Q8—164no—
537D4—24no—
538D4—24no—
539D4—48no—
540D4—42no—
541D4—42no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—816no—
575C2×C4—84no—
576C2×C4—162no—
577C2×C4—162no—
578C2×C4—162no—
579C2×C4C2×C22⋊Q818yes—
580C2×C4C2×C22⋊Q812yes—
581C2×C4#53512yes—
582C2×C4#53612yes—
583C2×C4#54214yes—
584C2×C4#54214yes—
585C2×C4#54414yes—
586C2×C4#54414yes—
587C2×C4C23×Q811yescharacteristic
588C2×C4C22×#4111yescharacteristic
589C2×C4C2×H(5,𝔽2)11yescharacteristic
590C2×C4C2×H(5,𝔽2)11yescharacteristic
591C2×C4C2×2−1+411yescharacteristic
592C2×C4C2×2−1+411yescharacteristic
593C8—42no—
594C8—42no—
595C8—42no—
596C8—42no—
597C22—24no—
598C22—22no—
599C22—22no—
600C22—24no—
601C22—44no—
602C22—48no—
603C22—82no—
604C22—82no—
605C22—84no—
606C22—168no—
607C22#262814yes—
608C22C22×C22⋊Q811yescharacteristic
609C22C2×#53511yescharacteristic
610C22C2×#53611yescharacteristic
611C22#358412yes—
612C22#358512yes—
613C4—28no—
614C4—44no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—81no—
620C4—82no—
621C4—81no—
622C4—81no—
623C4—82no—
624C4—82no—
625C4—81no—
626C4—82no—
627C4#262814yes—
628C4C2×#54211yescharacteristic
629C4C2×#54211yescharacteristic
630C4C2×#54411yescharacteristic
631C4C2×#54411yescharacteristic
632C2—24no—
633C2—84no—
634C2C2×#262811yescharacteristic
635C2#6049612yes—
636C2#6081911yescharacteristic
637C2#6082511yescharacteristic
638C2#6087512yes—
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
1. #35330C2semidirect
3. #29991C2semidirect
4. #29990C2semidirect
5. #29986C2semidirect
6. #29982C2semidirect
7. #23746C2semidirect
8. #23744C2semidirect
11. #23556C2semidirect
12. #23555C2semidirect
15. #21174C2semidirect
16. #21173C2semidirect
17. #20754C2semidirect
33. #2862C22semidirect
34. #2859C22semidirect
35. #2660C22semidirect
36. #2660C22semidirect
55. #2101C22semidirect
56. #2101C22semidirect
57. #2101C22semidirect
58. #2101C22semidirect
59. #2100C22semidirect
60. #2099C22semidirect
67. #2059C22semidirect
68. #2059C22semidirect
69. #2058C22semidirect
70. #2057C22semidirect
75. #1949C22semidirect
76. #1949C22semidirect
77. #1948C22semidirect
78. #1947C22semidirect
79. #1832C22semidirect
80. #1831C22semidirect
117. #518Q8semidirect

Maximal subgroups

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

subgroupindex
#353302
#342552
#299912
#299902
#299862
#299822
#237462
#237442
#237312
C2×#20922
#235562
#235552
#234562
#211742
#211732
#207542
#206502