Finite Group Databasebeta

← #7505194

Subgroups — #7505194

Order 512.

1624 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35626C211yesmaximal, characteristic
2#35555C211yesmaximal, characteristic, complemented
3#31593C214yesmaximal, complemented
4#31590C214yesmaximal, complemented
5#31589C214yesmaximal, complemented
6#31586C214yesmaximal, complemented
7#23736C211yesmaximal, characteristic, complemented
8#23735C211yesmaximal, characteristic
9#23658C211yesmaximal, characteristic, complemented
10#23633C211yesmaximal, characteristic, complemented
11#23615C211yesmaximal, characteristic, complemented
12#23506C211yesmaximal, characteristic
13C2×#1959C211yesmaximal, characteristic
14#21154C211yesmaximal, characteristic, complemented
15#21066C211yesmaximal, characteristic, complemented
16#21063C211yesmaximal, characteristic
17#21021C211yesmaximal, characteristic, complemented
18#20879C211yesmaximal, characteristic
19#20650C211yesmaximal, characteristic
20C22×#124C2211yescharacteristic
21C2×#518C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#97C2211yescharacteristic
36#2570C2214yescomplemented
37#2552C2214yescomplemented
38#2507—22no—
39#2506—22no—
40#2483—24no—
41#2473—24no—
42#2447—22no—
43#2447—22no—
44#2447C2214yes—
45#2447C2214yes—
46#2317C2211yescharacteristic
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2310C2211yescharacteristic
50#2310C2211yescharacteristic
51#2309C2211yescharacteristic
52#2296C2211yescharacteristic
53#2296C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#384C2211yescharacteristic
56#2274C2211yescharacteristic
57#2097C2212yescomplemented
58#2097C2212yescomplemented
59#2096C2212yescomplemented
60#2095C2212yescomplemented
61#2094C2212yes—
62#2093C2212yes—
63#2092C2212yes—
64#2092C2212yes—
65#2084C2214yescomplemented
66#2083C2214yescomplemented
67#2074C2214yescomplemented
68#2072C2214yescomplemented
69#2068C2214yescomplemented
70#2066C2214yescomplemented
71#2042C2214yes—
72#2042C2214yes—
73#1959C2214yes—
74#1959C2214yes—
75#1944C2212yescomplemented
76#1944C2212yescomplemented
77#1943C2212yescomplemented
78#1943C2212yescomplemented
79#1928C2214yescomplemented
80#1927C2214yescomplemented
81#1926C2214yes—
82#1925C2214yes—
83#1917C2214yescomplemented
84#1917C2214yescomplemented
85#1864C2214yes—
86#1862C2214yes—
87#1804C2214yes—
88#1802C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103C2×#125—24no—
104C2×#124—24no—
105C2×#124—21no—
106C2×#124—21no—
107C2×#124—42no—
108C2×#124C2314yes—
109#518—24no—
110#518—42no—
111C2×C4×D4—22no—
112C2×C4×D4C2314yes—
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—22no—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—22no—
123C22×C22⋊C4—22no—
124C22×C42C2311yescharacteristic
125#428—42no—
126#428—42no—
127#427—24no—
128#427—42no—
129C2×#107C2311yescharacteristic
130C2×#107C2311yescharacteristic
131C2×#107C2312yes—
132C2×#107C2312yes—
133C2×#107C2312yes—
134C2×#106C2311yescharacteristic
135C2×#106C2311yescharacteristic
136C2×#106C2312yes—
137C2×#106C2312yes—
138C2×#106C2312yes—
139#424—42no—
140C2×C4⋊C8C2314yes—
141#419—42no—
142#419—42no—
143#418—24no—
144#418—42no—
145C2×#103—24no—
146C2×#103—42no—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#102—24no—
154C2×#102—42no—
155C2×#102C2312yes—
156C2×#102C2312yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#97—42no—
162C2×#97C2312yes—
163C2×#97C2312yes—
164#400—24no—
165#398—24no—
166#398—24no—
167#398—24no—
168#397—24no—
169#397—24no—
170#397—24no—
171#397—24no—
172#391—22no—
173#391—22no—
174#391C2312yes—
175#391C2312yes—
176#391C2314yes—
177#390—22no—
178#390C2312yes—
179#390C2312yes—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—24no—
184#389D412yes—
185#389D412yes—
186#389C2312yes—
187#389C2314yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#387—24no—
192#387—24no—
193#385—22no—
194#385—22no—
195#385—24no—
196#385—24no—
197#385—22no—
198#385—22no—
199#385D414yes—
200#384—21no—
201#384—21no—
202#384C2312yes—
203#384C2314yes—
204#384C2312yes—
205#382C2312yes—
206#382C2312yes—
207#381—22no—
208#381—22no—
209#381—24no—
210#381—24no—
211#380—22no—
212#379—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2314yes—
220C4×C22⋊C4—22no—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
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×Q8C2412yes—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4#4114yes—
249C22×D4C2412yes—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—24no—
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
273#125—88no—
274#124—28no—
275#124—44no—
276#124—44no—
277#124—44no—
278C4×D4—28no—
279C4×D4—44no—
280C4×D4—48no—
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—22no—
291#117—22no—
292#117—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—84no—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—48no—
331C2×C22⋊C4—48no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—24no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2412yes—
346#107—22no—
347#107—22no—
348#107—42no—
349#107—42no—
350#107—42no—
351#106—22no—
352#106—22no—
353#106—42no—
354#106—42no—
355#106—42no—
356C4⋊C8—44no—
357#103—24no—
358#103—24no—
359#103—44no—
360#103—42no—
361#103—42no—
362#103—44no—
363#102—28no—
364#102—24no—
365#102—24no—
366#102—44no—
367#102—42no—
368#102—42no—
369#97—42no—
370#97—42no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95C2×D414yes—
384#95C2×D414yes—
385C24—22no—
386C24—24no—
387C24—42no—
388C24—42no—
389C24—82no—
390C24—82no—
391C24C22×D411yescharacteristic
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×D4—28no—
406C2×D4—24no—
407C2×D4—28no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4C2×#4114yes—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—48no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4C22×D414yes—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C42−1+412yes—
450C22×C42−1+412yes—
451C22×C4C2511yescharacteristic
452M16—44no—
453M16—44no—
454C2×C8—28no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—22no—
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—44no—
469C2×C8C2×#4112yes—
470C2×C8C2×#4112yes—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—84no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—44no—
530C23—44no—
531C23—42no—
532C23—42no—
533C23—48no—
534C23—88no—
535C23—164no—
536C23—162no—
537C23—162no—
538C23#53612yes—
539C23#54612yes—
540C23#55012yes—
541C23C23×D411yescharacteristic
542Q8—42no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—84no—
547D4—24no—
548D4—24no—
549D4—42no—
550D4—48no—
551D4—42no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—48no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—84no—
587C2×C4—168no—
588C2×C4—162no—
589C2×C4—162no—
590C2×C4#53612yes—
591C2×C4#53914yes—
592C2×C4#53914yes—
593C2×C4#54612yes—
594C2×C4#55012yes—
595C2×C4#55214yes—
596C2×C4#56214yes—
597C2×C4C22×#4111yescharacteristic
598C2×C4C22×#4111yescharacteristic
599C2×C4C2×H(5,𝔽2)11yescharacteristic
600C2×C4C2×H(5,𝔽2)11yescharacteristic
601C2×C4C2×2−1+411yescharacteristic
602C2×C4C2×2−1+411yescharacteristic
603C8—22no—
604C8—22no—
605C8—42no—
606C8—42no—
607C8—42no—
608C22—24no—
609C22—22no—
610C22—22no—
611C22—24no—
612C22—44no—
613C22—48no—
614C22—84no—
615C22—82no—
616C22—82no—
617C22—168no—
618C22#269414yes—
619C22C2×#53611yescharacteristic
620C22C2×#54611yescharacteristic
621C22C2×#55011yescharacteristic
622C22#358512yes—
623C22#358612yes—
624C4—44no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—82no—
629C4—81no—
630C4—81no—
631C4—82no—
632C4—82no—
633C4—81no—
634C4—81no—
635C4—82no—
636C4—82no—
637C4#269414yes—
638C4C2×#53911yescharacteristic
639C4C2×#53911yescharacteristic
640C4C2×#55211yescharacteristic
641C4C2×#56211yescharacteristic
642C2—24no—
643C2—84no—
644C2C2×#269411yescharacteristic
645C2#6088512yes—
646C2#6117712yes—
647C2#6141711yescharacteristic
648C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35555C2semidirect
3. #31593C2semidirect
4. #31590C2semidirect
5. #31589C2semidirect
6. #31586C2semidirect
7. #23736C2semidirect
9. #23658C2semidirect
10. #23633C2semidirect
11. #23615C2semidirect
14. #21154C2semidirect
15. #21066C2semidirect
17. #21021C2semidirect
36. #2570C22semidirect
37. #2552C22semidirect
57. #2097C22semidirect
58. #2097C22semidirect
59. #2096C22semidirect
60. #2095C22semidirect
65. #2084C22semidirect
66. #2083C22semidirect
67. #2074C22semidirect
68. #2072C22semidirect
69. #2068C22semidirect
70. #2066C22semidirect
75. #1944C22semidirect
76. #1944C22semidirect
77. #1943C22semidirect
78. #1943C22semidirect
79. #1928C22semidirect
80. #1927C22semidirect
83. #1917C22semidirect
84. #1917C22semidirect

Maximal subgroups

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

subgroupindex
#356262
#355552
#315932
#315902
#315892
#315862
#237362
#237352
#236582
#236332
#236152
#235062
C2×#19592
#211542
#210662
#210632
#210212
#208792
#206502