Finite Group Databasebeta

← #6460653

Subgroups — #6460653

Order 512.

1704 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35662C211yesmaximal, characteristic, complemented
2#35558C211yesmaximal, characteristic
3#29094C214yesmaximal, complemented
4#29093C214yesmaximal, complemented
5#29086C214yesmaximal, complemented
6#29085C214yesmaximal, complemented
7#23667C211yesmaximal, characteristic, complemented
8#23629C211yesmaximal, characteristic, complemented
9#23541C211yesmaximal, characteristic
10#23454C211yesmaximal, characteristic
11#21321C211yesmaximal, characteristic, complemented
12C2×#1973C211yesmaximal, characteristic
13#21091C211yesmaximal, characteristic
14#21090C211yesmaximal, characteristic, complemented
15#21089C211yesmaximal, characteristic
16#21037C211yesmaximal, characteristic, complemented
17#20936C211yesmaximal, characteristic
18#20436C211yesmaximal, characteristic, complemented
19#20417C211yesmaximal, characteristic, complemented
20C23×SD16C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#519C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C2×#424C2211yescharacteristic
25C2×#419C2211yescharacteristic
26C2×#419C2211yescharacteristic
27C2×#418C2211yescharacteristic
28C2×#418C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C2×#405C2211yescharacteristic
32C2×#405C2211yescharacteristic
33C22×#97C2211yescharacteristic
34#2582—22no—
35#2581—22no—
36#2513—22no—
37#2513C2214yes—
38#2512—22no—
39#2512C2214yes—
40#2480—24no—
41#2453C2214yescomplemented
42#2451C2214yescomplemented
43#2438—24no—
44#2317C2211yescharacteristic
45#2317C2211yescharacteristic
46#2316C2211yescharacteristic
47#2310C2211yescharacteristic
48#2309C2211yescharacteristic
49#2309C2211yescharacteristic
50#2296C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#390C2211yescharacteristic
53#2285C2211yescharacteristic
54#2274C2211yescharacteristic
55#2087C2214yescomplemented
56#2086C2214yescomplemented
57#2074C2214yescomplemented
58#2072C2214yescomplemented
59#2055C2214yes—
60#2055C2214yes—
61#2025C2214yes—
62#2023C2214yes—
63#1993C2212yescomplemented
64#1993C2212yescomplemented
65#1992C2212yescomplemented
66#1992C2212yescomplemented
67#1973C2212yes—
68#1973C2212yes—
69#1973C2212yes—
70#1973C2212yes—
71#1936C2214yes—
72#1936C2214yes—
73#1935C2214yescomplemented
74#1934C2214yes—
75#1934C2214yes—
76#1933C2214yescomplemented
77#1921C2214yescomplemented
78#1918C2214yescomplemented
79#1880C2214yes—
80#1880C2214yes—
81#1771C2212yescomplemented
82#1771C2212yescomplemented
83#1770C2212yescomplemented
84#1770C2212yescomplemented
85#1763C2214yescomplemented
86C4×#103C2214yescomplemented
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×SD16—24no—
100C22×SD16—24no—
101C22×SD16—42no—
102C22×SD16C2312yes—
103C22×SD16C2312yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#533—24no—
109#533—42no—
110C2×C4⋊Q8C2312yes—
111C2×#126—24no—
112C2×#125C2312yes—
113C2×C22⋊Q8—22no—
114#519—24no—
115#519—42no—
116C2×C4×Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8C2314yes—
119C2×#117—22no—
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—22no—
129C22×C42C2311yescharacteristic
130C2×#107—24no—
131C2×#107C2312yes—
132C2×#107C2312yes—
133#424—24no—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136#419—24no—
137#419—24no—
138#418—24no—
139#418—42no—
140C2×#103—24no—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#102—24no—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150#405—24no—
151#405—42no—
152C2×#97—42no—
153C2×#97C2312yes—
154C2×#97C2311yescharacteristic
155C2×#97C2311yescharacteristic
156C2×#97C2311yescharacteristic
157C2×#97C2311yescharacteristic
158C2×#97C2312yes—
159C2×C4×C8C2312yes—
160C2×C4×C8C2312yes—
161#400—24no—
162#398—24no—
163#398—24no—
164#397—24no—
165#397—24no—
166#397—24no—
167#396—24no—
168#395—24no—
169#391—21no—
170#391—21no—
171#391—22no—
172#391—24no—
173#391C2314yes—
174#391C2312yes—
175#391C2312yes—
176#390—21no—
177#390—22no—
178#390—21no—
179#390—22no—
180#390C2314yes—
181#390C2314yes—
182#390C2312yes—
183#390C2312yes—
184#389—22no—
185#389—24no—
186#389D412yes—
187#389D412yes—
188#389C2314yes—
189#389C2314yes—
190#389C2314yes—
191#388—24no—
192#388—24no—
193#388—24no—
194#387—24no—
195#387—24no—
196#385—22no—
197#385—22no—
198#385D414yes—
199#384—21no—
200#384—21no—
201#384C2312yes—
202#382—21no—
203#382—21no—
204#382C2312yes—
205#381—22no—
206#380—22no—
207#380—22no—
208#379—22no—
209#379—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—22no—
216C2×#95—22no—
217#340C2312yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
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×Q8#4114yes—
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×SD16—24no—
257C2×SD16—24no—
258C2×SD16—24no—
259C2×SD16—24no—
260C2×SD16—42no—
261C2×SD16—42no—
262C2×SD16—416no—
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8#4114yes—
274C22×C8#4114yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2411yescharacteristic
278C22×C8C2411yescharacteristic
279C22×C8C2411yescharacteristic
280C22×C8C2411yescharacteristic
281C4⋊Q8—24no—
282C4⋊Q8—42no—
283#126—88no—
284#125—24no—
285#125—42no—
286C22⋊Q8—48no—
287C4×Q8—28no—
288C4×Q8—48no—
289C4×Q8—44no—
290C4×Q8—48no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302#117—48no—
303#117#4114yes—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—21no—
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—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—84no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—24no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—24no—
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—24no—
357#107—24no—
358C4⋊C8—24no—
359C4⋊C8—24no—
360#103—28no—
361#103—28no—
362#103—24no—
363#103—24no—
364#102—24no—
365#102—24no—
366#102—28no—
367#102—44no—
368#97—24no—
369#97—41no—
370#97—41no—
371#97—41no—
372#97—41no—
373#97—42no—
374C4×C8—24no—
375C4×C8—42no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95C2×D414yes—
387#95C2×D414yes—
388C24—22no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—82no—
393C24—82no—
394C24C22×D411yescharacteristic
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—28no—
398C2×Q8—28no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—82no—
410C2×Q8C2×#4114yes—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—44no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C42−1+412yes—
453C22×C4C2511yescharacteristic
454SD16—48no—
455SD16—48no—
456M16—44no—
457M16—44no—
458C2×C8—28no—
459C2×C8—28no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8C2×#4112yes—
475C2×C8C2×#4112yes—
476C2×C8C2×#4112yes—
477C2×C8C2×#4112yes—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4—82no—
504C4⋊C4—84no—
505C4⋊C4C2×#4114yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—48no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C42—24no—
519C42—24no—
520C42—22no—
521C42—22no—
522C42—21no—
523C42—21no—
524C42—22no—
525C42—22no—
526C42—21no—
527C42—21no—
528C42—48no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42—84no—
533C23—22no—
534C23—22no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—88no—
541C23—84no—
542C23—162no—
543C23—162no—
544C23—164no—
545C23#53212yes—
546C23#54612yes—
547C23#54812yes—
548C23C23×D411yescharacteristic
549Q8—24no—
550Q8—24no—
551Q8—48no—
552Q8—42no—
553Q8—42no—
554Q8—88no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—84no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—84no—
594C2×C4—162no—
595C2×C4—162no—
596C2×C4#52914yes—
597C2×C4#53212yes—
598C2×C4#53914yes—
599C2×C4#54612yes—
600C2×C4#54812yes—
601C2×C4#55114yes—
602C2×C4#55614yes—
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C2×H(5,𝔽2)11yescharacteristic
608C2×C4C2×2−1+411yescharacteristic
609C8—22no—
610C8—22no—
611C8—22no—
612C8—22no—
613C8—42no—
614C8—42no—
615C22—22no—
616C22—22no—
617C22—24no—
618C22—24no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—84no—
623C22—84no—
624C22—168no—
625C22#256014yes—
626C22C2×#53211yescharacteristic
627C22C2×#54611yescharacteristic
628C22C2×#54811yescharacteristic
629C22#358512yes—
630C22#358612yes—
631C4—44no—
632C4—42no—
633C4—42no—
634C4—44no—
635C4—42no—
636C4—42no—
637C4—44no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4#256014yes—
644C4C2×#52911yescharacteristic
645C4C2×#53911yescharacteristic
646C4C2×#55111yescharacteristic
647C4C2×#55611yescharacteristic
648C2—24no—
649C2—162no—
650C2C2×#256011yescharacteristic
651C2#6071911yescharacteristic
652C2#6072011yescharacteristic
653C2#6118212yes—
654C2#6133412yes—
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
1. #35662C2semidirect
3. #29094C2semidirect
4. #29093C2semidirect
5. #29086C2semidirect
6. #29085C2semidirect
7. #23667C2semidirect
8. #23629C2semidirect
11. #21321C2semidirect
14. #21090C2semidirect
16. #21037C2semidirect
18. #20436C2semidirect
19. #20417C2semidirect
41. #2453C22semidirect
42. #2451C22semidirect
55. #2087C22semidirect
56. #2086C22semidirect
57. #2074C22semidirect
58. #2072C22semidirect
63. #1993C22semidirect
64. #1993C22semidirect
65. #1992C22semidirect
66. #1992C22semidirect
73. #1935C22semidirect
76. #1933C22semidirect
77. #1921C22semidirect
78. #1918C22semidirect
81. #1771C22semidirect
82. #1771C22semidirect
83. #1770C22semidirect
84. #1770C22semidirect
85. #1763C22semidirect
86. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#356622
#355582
#290942
#290932
#290862
#290852
#236672
#236292
#235412
#234542
#213212
C2×#19732
#210912
#210902
#210892
#210372
#209362
#204362
#204172