Finite Group Databasebeta

← #7509278

Subgroups — #7509278

Order 512.

1580 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35630C211yesmaximal, characteristic, complemented
2#35620C211yesmaximal, characteristic
3#32451C214yesmaximal, complemented
4#32450C214yesmaximal, complemented
5#32435C214yesmaximal, complemented
6#32434C214yesmaximal, complemented
7C2×#2097C211yesmaximal, characteristic, complemented
8#23633C211yesmaximal, characteristic, complemented
9#23627C211yesmaximal, characteristic
10#23556C211yesmaximal, characteristic, complemented
11#23545C211yesmaximal, characteristic
12#23457C211yesmaximal, characteristic, complemented
13#21237C211yesmaximal, characteristic
14#21173C211yesmaximal, characteristic, complemented
15#21068C211yesmaximal, characteristic
16#21022C211yesmaximal, characteristic, complemented
17#20931C211yesmaximal, characteristic
18#20881C211yesmaximal, characteristic
19#20765C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#416C2211yescharacteristic
30C2×#416C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C2×C4×M16C2211yescharacteristic
34#2568C2212yes—
35#2566C2212yes—
36#2565C2212yes—
37#2563C2212yes—
38#2514—22no—
39#2514C2214yes—
40#2511—22no—
41#2511C2214yes—
42#2493—24no—
43#2450—24no—
44#2442—22no—
45#2439—22no—
46#2317C2211yescharacteristic
47#2317C2211yescharacteristic
48#2311C2211yescharacteristic
49#2310C2211yescharacteristic
50#2310C2211yescharacteristic
51#2297C2211yescharacteristic
52#2296C2211yescharacteristic
53C2×#391C2211yescharacteristic
54#2285C2211yescharacteristic
55C2×#382C2211yescharacteristic
56#2274C2211yescharacteristic
57#2097C2214yes—
58#2097C2214yes—
59#2074C2212yes—
60#2074C2214yes—
61#2073C2212yes—
62#2072C2212yes—
63#2072C2214yes—
64#2071C2212yes—
65#2059C2214yes—
66#2058C2214yes—
67#2056C2214yes—
68#2055C2214yes—
69#2026C2214yes—
70#2024C2214yes—
71#1973C2212yes—
72#1971C2212yes—
73#1969C2212yes—
74#1968C2212yes—
75#1949C2214yes—
76#1947C2214yes—
77#1927C2214yes—
78#1926C2214yes—
79#1917C2214yes—
80#1916C2214yes—
81#1878C2214yes—
82#1877C2214yes—
83#1863C2214yes—
84#1862C2214yes—
85#1836C2214yes—
86#1835C2214yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
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#544—42no—
109#544—42no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2312yes—
112C2×#125—21no—
113C2×#125C2312yes—
114C2×C22⋊Q8—22no—
115#520—42no—
116#520—42no—
117C2×C4×Q8C2312yes—
118C2×C4×D4—22no—
119C2×C4×D4C2312yes—
120C2×#117—21no—
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#428—42no—
132C2×#107—42no—
133C2×#107C2312yes—
134C2×#107C2314yes—
135C2×#106—42no—
136C2×#106C2312yes—
137C2×#106C2314yes—
138#424—24no—
139C2×C4⋊C8C2314yes—
140#419—42no—
141#417—42no—
142#416—24no—
143#416—24no—
144C2×#103—42no—
145C2×#103—42no—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C4×M16—24no—
157C2×#97C2312yes—
158C2×C4×C8C2312yes—
159#401—24no—
160#400—24no—
161#400—24no—
162#400—24no—
163#400—24no—
164#398—24no—
165#396—24no—
166#393—24no—
167#391—21no—
168#391—22no—
169#391—21no—
170#391—24no—
171#391—22no—
172#391C2311yescharacteristic
173#391C2312yes—
174#391C2314yes—
175#391C2314yes—
176#391C2311yescharacteristic
177#391C2312yes—
178#391C2312yes—
179#390C2311yescharacteristic
180#390C2312yes—
181#390C2311yescharacteristic
182#389—22no—
183#389D412yes—
184#389D412yes—
185#389C2314yes—
186#387—24no—
187#387—24no—
188#387—24no—
189#387—24no—
190#386—22no—
191#386—22no—
192#385—22no—
193#385—22no—
194#385—24no—
195#385D414yes—
196#384—21no—
197#384C2312yes—
198#384C2312yes—
199#383—24no—
200#382—21no—
201#382—22no—
202#382C2312yes—
203#382C2312yes—
204#382C2312yes—
205#382C2312yes—
206#381—22no—
207#381—22no—
208#380—22no—
209#380—24no—
210#380—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C2×#95—22no—
219C2×#95—22no—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#336C2314yes—
229#336C2312yes—
230#336C2314yes—
231#336C2312yes—
232#336C2314yes—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41#4114yes—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8C2412yes—
248C22×Q8C2411yescharacteristic
249C22×D4—21no—
250C22×D4C2411yescharacteristic
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4C2×D412yes—
258C23×C4C2411yescharacteristic
259#137—416no—
260C2×Q16—24no—
261C2×Q16—24no—
262C2×Q16—42no—
263C2×SD16—24no—
264C2×SD16—24no—
265C2×SD16—42no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—22no—
271C2×M16#4114yes—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C4⋊Q8—42no—
283C4⋊Q8—42no—
284C4⋊Q8—44no—
285#125—42no—
286#125—42no—
287#125—44no—
288C22⋊Q8—48no—
289C4×Q8—42no—
290C4×Q8—42no—
291C4×D4—42no—
292C4×D4—42no—
293C4×D4—48no—
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—22no—
303#117—22no—
304#117—44no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
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—44no—
357#107—42no—
358#106—44no—
359#106—42no—
360C4⋊C8—28no—
361#103—24no—
362#103—24no—
363#103—44no—
364#103—44no—
365#103—42no—
366#103—42no—
367#103—42no—
368#102—24no—
369#102—24no—
370#102—42no—
371#97—24no—
372C4×C8—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95C2×D414yes—
390#95C2×D414yes—
391C24—21no—
392C24C22×D411yescharacteristic
393#41—28no—
394#41—28no—
395#41—44no—
396#41—44no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—84no—
417C2×Q8C2×#4112yes—
418C2×D4—24no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4C2×#4112yes—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—84no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4C22×D412yes—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C42−1+412yes—
464C22×C42−1+412yes—
465C22×C4C2511yescharacteristic
466Q16—48no—
467SD16—48no—
468M16—28no—
469M16—44no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—44no—
482C2×C8—44no—
483C2×C8C2×#4114yes—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—84no—
512C4⋊C4—82no—
513C4⋊C4—84no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—82no—
521C22⋊C4—84no—
522C22⋊C4—82no—
523C42—24no—
524C42—21no—
525C42—21no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—21no—
530C42—21no—
531C42—22no—
532C42—44no—
533C42—42no—
534C42—42no—
535C42—42no—
536C42—84no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23#53512yes—
545C23#54712yes—
546C23#54812yes—
547C23C23×D411yescharacteristic
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—84no—
557Q8—164no—
558D4—22no—
559D4—22no—
560D4—48no—
561D4—41no—
562D4—41no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—164no—
602C2×C4—162no—
603C2×C4—162no—
604C2×C4—162no—
605C2×C4#53512yes—
606C2×C4#53914yes—
607C2×C4#54214yes—
608C2×C4#54712yes—
609C2×C4#54812yes—
610C2×C4#55114yes—
611C2×C4#56314yes—
612C2×C4C22×#4111yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4C2×H(5,𝔽2)11yescharacteristic
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×2−1+411yescharacteristic
617C2×C4C2×2−1+411yescharacteristic
618C8—24no—
619C8—42no—
620C8—42no—
621C8—42no—
622C22—24no—
623C22—22no—
624C22—22no—
625C22—24no—
626C22—44no—
627C22—42no—
628C22—82no—
629C22—81no—
630C22—81no—
631C22#272314yes—
632C22C2×#53511yescharacteristic
633C22C2×#54711yescharacteristic
634C22C2×#54811yescharacteristic
635C22#358412yes—
636C22#358612yes—
637C4—44no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—81no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4—81no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4#272314yes—
652C4C2×#53911yescharacteristic
653C4C2×#54211yescharacteristic
654C4C2×#55111yescharacteristic
655C4C2×#56311yescharacteristic
656C2—24no—
657C2—82no—
658C2C2×#272311yescharacteristic
659C2#6081911yescharacteristic
660C2#6082611yescharacteristic
661C2#6127512yes—
662C2#6135012yes—
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
1. #35630C2semidirect
3. #32451C2semidirect
4. #32450C2semidirect
5. #32435C2semidirect
6. #32434C2semidirect
7. C2×#2097C2semidirect
8. #23633C2semidirect
10. #23556C2semidirect
12. #23457C2semidirect
14. #21173C2semidirect
16. #21022C2semidirect
19. #20765C2semidirect

Maximal subgroups

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

subgroupindex
#356302
#356202
#324512
#324502
#324352
#324342
C2×#20972
#236332
#236272
#235562
#235452
#234572
#212372
#211732
#210682
#210222
#209312
#208812
#207652