Finite Group Databasebeta

← #6451260

Subgroups — #6451260

Order 512.

1627 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35629C211yesmaximal, characteristic
2#34254C211yesmaximal, characteristic, complemented
3#27517C214yesmaximal, complemented
4#27506C214yesmaximal, complemented
5#27494C214yesmaximal, complemented
6#27489C214yesmaximal, complemented
7#23737C211yesmaximal, characteristic
8C2×#2097C211yesmaximal, characteristic, complemented
9#23661C211yesmaximal, characteristic, complemented
10#23547C211yesmaximal, characteristic
11#21234C211yesmaximal, characteristic
12#21172C211yesmaximal, characteristic, complemented
13#21068C211yesmaximal, characteristic
14#21064C211yesmaximal, characteristic, complemented
15#21058C211yesmaximal, characteristic
16#21027C211yesmaximal, characteristic, complemented
17#20930C211yesmaximal, characteristic
18#20880C211yesmaximal, characteristic, complemented
19#20455C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C2×#519C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#416C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C2×#405C2211yescharacteristic
35#2514C2214yes—
36#2513C2214yes—
37#2510—24no—
38#2495—22no—
39#2494—22no—
40#2356C2214yes—
41#2355C2214yes—
42#2343—24no—
43#2320—22no—
44#2317C2211yescharacteristic
45#2316C2211yescharacteristic
46#2311C2211yescharacteristic
47#2311C2211yescharacteristic
48#2310C2211yescharacteristic
49#2297C2211yescharacteristic
50#2296C2211yescharacteristic
51#2295—22no—
52C2×#391C2211yescharacteristic
53C2×#382C2211yescharacteristic
54#2274C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2214yes—
58#2097C2212yes—
59#2097C2214yes—
60#2095C2212yes—
61#2093C2212yes—
62#2092C2212yes—
63#2085C2214yes—
64#2084C2214yes—
65#2056C2214yes—
66#2055C2214yes—
67#1973C2212yes—
68#1971C2212yes—
69#1969C2212yes—
70#1968C2212yes—
71#1949C2214yes—
72#1949C2214yes—
73#1928C2212yes—
74#1928C2214yes—
75#1927C2212yes—
76#1927C2214yes—
77#1927C2214yes—
78#1926C2212yes—
79#1926C2214yes—
80#1925C2212yes—
81#1917C2214yes—
82#1916C2214yes—
83#1880C2214yes—
84#1878C2214yes—
85#1865C2214yes—
86#1863C2214yes—
87#1776C2214yes—
88C4×#107C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C2×#137—24no—
100C2×#137—24no—
101C2×#137—42no—
102C22×Q16C2312yes—
103C22×SD16C2312yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#519—24no—
109#519—24no—
110C2×C4×Q8—22no—
111C2×C4×Q8C2314yes—
112C2×#117—22no—
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#428—42no—
123#428—42no—
124#427—24no—
125C2×#107—24no—
126C2×#107C2312yes—
127C2×#107C2312yes—
128C2×#107C2312yes—
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#106—42no—
132C2×#106C2312yes—
133C2×#106C2312yes—
134C2×#106C2314yes—
135C2×#106C2312yes—
136#423—24no—
137C2×C4⋊C8C2314yes—
138#419—24no—
139#417—42no—
140#416—24no—
141#416—24no—
142C2×#103—24no—
143C2×#103—42no—
144C2×#103C2314yes—
145C2×#103C2311yescharacteristic
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2311yescharacteristic
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#102C2311yescharacteristic
153C2×#102C2311yescharacteristic
154C2×#102C2312yes—
155C2×#102C2312yes—
156#405—24no—
157C2×#97C2312yes—
158C2×C4×C8C2312yes—
159#401—24no—
160#400—24no—
161#397—24no—
162#397—24no—
163#391—22no—
164#391—21no—
165#391—21no—
166#391C2312yes—
167#391C2312yes—
168#391C2314yes—
169#391C2312yes—
170#391C2312yes—
171#391C2312yes—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#389—22no—
176#389—22no—
177#389D412yes—
178#389D412yes—
179#389C2314yes—
180#388—24no—
181#388—24no—
182#387—24no—
183#387—24no—
184#387—24no—
185#387—24no—
186#387—24no—
187#386—22no—
188#386—22no—
189#386—22no—
190#385—22no—
191#385D414yes—
192#384C2312yes—
193#383—24no—
194#383—24no—
195#382—24no—
196#382—21no—
197#382—21no—
198#382C2312yes—
199#382C2314yes—
200#381—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—24no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2314yes—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—21no—
217#376—21no—
218#376—24no—
219C2×#95—22no—
220C2×#95—22no—
221C43—21no—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#336C2314yes—
228#336C2312yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2314yes—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8#4114yes—
246C22×Q8C2411yescharacteristic
247C22×Q8C2412yes—
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257#137—416no—
258C2×Q16—24no—
259C2×Q16—24no—
260C2×Q16—42no—
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—42no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16#4114yes—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8#4114yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2411yescharacteristic
279C22×C8C2411yescharacteristic
280C22×C8C2412yes—
281C4×Q8—216no—
282C4×Q8—28no—
283C4×Q8—28no—
284#117—216no—
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—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C42—24no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2412yes—
341C2×C42C2412yes—
342#107—24no—
343#107—24no—
344#107—24no—
345#107—42no—
346#107—42no—
347#106—24no—
348#106—44no—
349#106—42no—
350#106—42no—
351C4⋊C8—28no—
352#103—28no—
353#103—24no—
354#103—24no—
355#103—22no—
356#103—22no—
357#103—24no—
358#103—44no—
359#103—42no—
360#102—22no—
361#102—22no—
362#102—24no—
363#102—42no—
364#97—24no—
365C4×C8—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95C2×D414yes—
383#95C2×D414yes—
384C24—21no—
385C24C22×D411yescharacteristic
386#41—44no—
387#41—44no—
388#41—44no—
389#41—44no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—28no—
393C2×Q8—28no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8C2×#4114yes—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
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×C4—82no—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
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
454Q16—48no—
455SD16—48no—
456M16—28no—
457M16—44no—
458C2×C8—28no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—44no—
472C2×C8C2×#4114yes—
473C2×C8C2×#4112yes—
474C2×C8C2×#4112yes—
475C4⋊C4—216no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
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—21no—
490C4⋊C4—21no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
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—82no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—82no—
508C42—216no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—44no—
519C42—42no—
520C42—42no—
521C42—84no—
522C42—84no—
523C42—82no—
524C23—22no—
525C23—22no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23#52512yes—
532C23#54712yes—
533C23#55012yes—
534C23C23×D411yescharacteristic
535Q8—24no—
536Q8—24no—
537Q8—42no—
538Q8—42no—
539Q8—48no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—84no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—48no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—162no—
587C2×C4—162no—
588C2×C4—162no—
589C2×C4#52512yes—
590C2×C4#52914yes—
591C2×C4#54214yes—
592C2×C4#54712yes—
593C2×C4#55012yes—
594C2×C4#55114yes—
595C2×C4#55214yes—
596C2×C4C22×#4111yescharacteristic
597C2×C4C22×#4111yescharacteristic
598C2×C4C22×#4111yescharacteristic
599C2×C4C22×#4111yescharacteristic
600C2×C4C2×H(5,𝔽2)11yescharacteristic
601C2×C4C2×2−1+411yescharacteristic
602C8—22no—
603C8—22no—
604C8—24no—
605C8—42no—
606C8—42no—
607C22—24no—
608C22—24no—
609C22—22no—
610C22—22no—
611C22—82no—
612C22—81no—
613C22—82no—
614C22—81no—
615C22—81no—
616C22#247914yes—
617C22C2×#52511yescharacteristic
618C22C2×#54711yescharacteristic
619C22C2×#55011yescharacteristic
620C22#358412yes—
621C22#358612yes—
622C4—44no—
623C4—42no—
624C4—42no—
625C4—44no—
626C4—44no—
627C4—44no—
628C4—81no—
629C4—82no—
630C4—81no—
631C4—82no—
632C4—82no—
633C4—81no—
634C4—82no—
635C4#247914yes—
636C4C2×#52911yescharacteristic
637C4C2×#54211yescharacteristic
638C4C2×#55111yescharacteristic
639C4C2×#55211yescharacteristic
640C2—24no—
641C2—161no—
642C2C2×#247911yescharacteristic
643C2#6059812yes—
644C2#6127512yes—
645C2#6141311yescharacteristic
646C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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. #34254C2semidirect
3. #27517C2semidirect
4. #27506C2semidirect
5. #27494C2semidirect
6. #27489C2semidirect
8. C2×#2097C2semidirect
9. #23661C2semidirect
12. #21172C2semidirect
14. #21064C2semidirect
16. #21027C2semidirect
18. #20880C2semidirect
19. #20455C2semidirect

Maximal subgroups

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

subgroupindex
#356292
#342542
#275172
#275062
#274942
#274892
#237372
C2×#20972
#236612
#235472
#212342
#211722
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