Finite Group Databasebeta

← #6448183

Subgroups — #6448183

Order 512.

1698 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35338C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#26331C214yesmaximal, complemented
4#26330C214yesmaximal, complemented
5#26329C214yesmaximal, complemented
6#26328C214yesmaximal, complemented
7#23736C211yesmaximal, characteristic, complemented
8#23735C211yesmaximal, characteristic
9#23633C211yesmaximal, characteristic, complemented
10#23632C211yesmaximal, characteristic
11#21301C211yesmaximal, characteristic, complemented
12#21261C211yesmaximal, characteristic
13C2×#1959C211yesmaximal, characteristic
14#21171C211yesmaximal, characteristic, complemented
15#21066C211yesmaximal, characteristic, complemented
16#21065C211yesmaximal, characteristic
17#21064C211yesmaximal, characteristic, complemented
18#21063C211yesmaximal, characteristic
19#20451C211yesmaximal, characteristic, complemented
20C2×#533C2211yescharacteristic
21C22×#124C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#97C2211yescharacteristic
36#2861—22no—
37#2640—22no—
38#2592C2214yes—
39#2592C2214yes—
40#2561—24no—
41#2360—24no—
42#2355C2214yescomplemented
43Q8×C4⋊C4C2214yescomplemented
44#2330—22no—
45#2317C2211yescharacteristic
46#2317C2211yescharacteristic
47#2310C2211yescharacteristic
48#2310C2211yescharacteristic
49#2309C2211yescharacteristic
50#2309C2211yescharacteristic
51C4×C4⋊Q8—22no—
52C2×#384C2211yescharacteristic
53#2274C2211yescharacteristic
54#2274C2211yescharacteristic
55C4×#117C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2212yescomplemented
58#2097C2212yescomplemented
59#2096C2212yescomplemented
60#2095C2212yescomplemented
61#2094C2212yes—
62#2093C2212yes—
63#2092C2212yes—
64#2092C2212yes—
65#2074C2214yescomplemented
66#2073C2214yes—
67#2072C2214yescomplemented
68#2071C2214yes—
69#1985C2214yescomplemented
70#1985C2214yescomplemented
71#1983C2214yes—
72#1983C2214yes—
73#1959C2214yes—
74#1959C2214yes—
75#1948C2214yescomplemented
76#1947C2214yescomplemented
77#1928C2214yescomplemented
78#1928C2214yescomplemented
79#1927C2214yescomplemented
80#1927C2214yescomplemented
81#1926C2214yes—
82#1926C2214yes—
83#1925C2214yes—
84#1925C2214yes—
85#1776C2212yescomplemented
86#1776C2212yescomplemented
87#1776C2212yescomplemented
88#1776C2212yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1729C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#533—24no—
104#533—24no—
105C2×C4⋊Q8D412yes—
106C2×#125D412yes—
107C2×#124—21no—
108C2×#124—21no—
109C2×#124—24no—
110C2×#124—24no—
111C2×#124C2314yes—
112C2×C22⋊Q8D414yes—
113C2×C4×Q8—24no—
114C2×C4×Q8C2314yes—
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—22no—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—22no—
125C22×C42C2311yescharacteristic
126#428—42no—
127#428—42no—
128#427—24no—
129#427—24no—
130C2×#107C2311yescharacteristic
131C2×#107C2311yescharacteristic
132C2×#107C2312yes—
133C2×#107C2312yes—
134C2×#107C2312yes—
135C2×#106C2311yescharacteristic
136C2×#106C2311yescharacteristic
137C2×#106C2312yes—
138C2×#106C2312yes—
139C2×#106C2312yes—
140#423—24no—
141C2×C4⋊C8C2314yes—
142#419—24no—
143#419—42no—
144#418—24no—
145#418—42no—
146C2×#103—24no—
147C2×#103—42no—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#102—24no—
155C2×#102—42no—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×#102C2314yes—
159C2×#102C2312yes—
160C2×#102C2314yes—
161C2×#102C2312yes—
162C2×#97—24no—
163C2×#97C2312yes—
164C2×#97C2312yes—
165#397—24no—
166#397—24no—
167#393—24no—
168#393—24no—
169#391—24no—
170#391—22no—
171#391—22no—
172#391C2314yes—
173#391C2312yes—
174#391C2312yes—
175#390—22no—
176#390—22no—
177#390C2314yes—
178#390C2312yes—
179#390C2312yes—
180#389—22no—
181#389—22no—
182#389—24no—
183#389—24no—
184#389—24no—
185#388—24no—
186#388—24no—
187#388—24no—
188#388—24no—
189#387—24no—
190#387—24no—
191#385—22no—
192#385—22no—
193#384—21no—
194#384—21no—
195#384C2312yes—
196#384C2312yes—
197#384C2314yes—
198#382C2312yes—
199#382C2312yes—
200#381—22no—
201#381—22no—
202#379—22no—
203#379—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—24no—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217#376—21no—
218#376—21no—
219C43—21no—
220C43—21no—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#336C2314yes—
228#336C2314yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
233C25—21no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8C2×D414yes—
242C22×Q8C2412yes—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4C2412yes—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4C2×D412yes—
255C23×C4C2411yescharacteristic
256C2×M16—22no—
257C2×M16—22no—
258C2×M16—22no—
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8#4114yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2411yescharacteristic
272C22×C8C2411yescharacteristic
273C4⋊Q8—28no—
274#125—28no—
275#124—28no—
276#124—28no—
277#124—44no—
278#124—44no—
279C22⋊Q8—216no—
280C4×Q8—28no—
281C4×Q8—28no—
282C4×Q8—88no—
283#117—22no—
284#117—22no—
285#117—22no—
286#117—22no—
287#117—22no—
288#117—22no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117#4114yes—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—84no—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—48no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42#4114yes—
345C2×C42#4114yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2412yes—
349#107—22no—
350#107—22no—
351#107—24no—
352#107—42no—
353#107—42no—
354#106—22no—
355#106—22no—
356#106—24no—
357#106—42no—
358#106—42no—
359C4⋊C8—28no—
360#103—24no—
361#103—24no—
362#103—28no—
363#103—44no—
364#103—42no—
365#103—42no—
366#102—24no—
367#102—24no—
368#102—28no—
369#102—44no—
370#102—42no—
371#102—42no—
372#97—24no—
373#97—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386C24—22no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—82no—
391C24—82no—
392C24C22×D411yescharacteristic
393C2×Q8—28no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—44no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4#12318yes—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C42−1+412yes—
452C22×C4C2511yescharacteristic
453M16—44no—
454M16—44no—
455C2×C8—28no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—44no—
470C2×C8C2×#4112yes—
471C2×C8C2×#4112yes—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—28no—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—48no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4C2×#4114yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—44no—
517C42—44no—
518C42—42no—
519C42—42no—
520C42—42no—
521C42—82no—
522C42—82no—
523C42—84no—
524C42—84no—
525C42#12314yes—
526C42#12314yes—
527C42C2×#4114yes—
528C42C2×#4114yes—
529C23—22no—
530C23—22no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—88no—
537C23—84no—
538C23—164no—
539C23—162no—
540C23—162no—
541C23C2×#12312yes—
542C23#54812yes—
543C23#55012yes—
544C23C23×D411yescharacteristic
545Q8—416no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550Q8—84no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—84no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—48no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—84no—
589C2×C4—162no—
590C2×C4—162no—
591C2×C4C2×#12314yes—
592C2×C4C2×#12314yes—
593C2×C4C2×#12312yes—
594C2×C4#52914yes—
595C2×C4#54214yes—
596C2×C4#54812yes—
597C2×C4#55012yes—
598C2×C4#55614yes—
599C2×C4#55914yes—
600C2×C4C22×#4111yescharacteristic
601C2×C4C22×#4111yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C2×H(5,𝔽2)11yescharacteristic
605C2×C4C2×2−1+411yescharacteristic
606C8—22no—
607C8—22no—
608C8—42no—
609C8—42no—
610C8—42no—
611C22—24no—
612C22—24no—
613C22—22no—
614C22—22no—
615C22—84no—
616C22—82no—
617C22—84no—
618C22—82no—
619C22—82no—
620C22—168no—
621C22#244814yes—
622C22C22×#12311yescharacteristic
623C22C2×#54811yescharacteristic
624C22C2×#55011yescharacteristic
625C22#358512yes—
626C22#358612yes—
627C4—24no—
628C4—24no—
629C4—44no—
630C4—44no—
631C4—44no—
632C4—82no—
633C4—82no—
634C4—81no—
635C4—81no—
636C4—82no—
637C4—82no—
638C4—81no—
639C4—81no—
640C4#244814yes—
641C4C2×#52911yescharacteristic
642C4C2×#54211yescharacteristic
643C4C2×#55611yescharacteristic
644C4C2×#55911yescharacteristic
645C2—24no—
646C2—162no—
647C2C2×#244811yescharacteristic
648C2#6051712yes—
649C2#6133612yes—
650C2#6141711yescharacteristic
651C2#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. #26331C2semidirect
4. #26330C2semidirect
5. #26329C2semidirect
6. #26328C2semidirect
7. #23736C2semidirect
9. #23633C2semidirect
11. #21301C2semidirect
14. #21171C2semidirect
15. #21066C2semidirect
17. #21064C2semidirect
19. #20451C2semidirect
42. #2355C22semidirect
43. Q8×C4⋊C4C22semidirect
57. #2097C22semidirect
58. #2097C22semidirect
59. #2096C22semidirect
60. #2095C22semidirect
65. #2074C22semidirect
67. #2072C22semidirect
69. #1985C22semidirect
70. #1985C22semidirect
75. #1948C22semidirect
76. #1947C22semidirect
77. #1928C22semidirect
78. #1928C22semidirect
79. #1927C22semidirect
80. #1927C22semidirect
85. #1776C22semidirect
86. #1776C22semidirect
87. #1776C22semidirect
88. #1776C22semidirect

Maximal subgroups

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

subgroupindex
#353382
#342142
#263312
#263302
#263292
#263282
#237362
#237352
#236332
#236322
#213012
#212612
C2×#19592
#211712
#210662
#210652
#210642
#210632
#204512