Finite Group Databasebeta

← #6572339

Subgroups — #6572339

Order 512.

1509 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36481C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#30151C212yesmaximal, complemented
4#29664C211yesmaximal, characteristic
5#29495C212yesmaximal, complemented
6#28161C212yesmaximal, complemented
7#27972C211yesmaximal, characteristic
8#27648C211yesmaximal, characteristic
9#26932C214yesmaximal
10#26662C214yesmaximal
11#26112C212yesmaximal, complemented
12#24732C212yesmaximal, complemented
13#23816C212yesmaximal, complemented
14#23571C211yesmaximal, characteristic
15#23096C212yesmaximal, complemented
16#22771C212yesmaximal, complemented
17#22632C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#484C2212yes—
20C2×#462C2212yes—
21C2×#462C2214yes—
22C2×C4×Q16C2212yes—
23#2863C2212yes—
24#2857C2211yescharacteristic
25#2606—22no—
26#2552C2211yescharacteristic
27#2546C2211yescharacteristic
28#2522—22no—
29#2507—21no—
30#2441—21no—
31#2429C2212yes—
32#2423C2212yes—
33#2370—21no—
34#2365C2211yescharacteristic
35#2360—22no—
36#2360C2214yes—
37#2355—21no—
38#2355C2212yes—
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4C2212yes—
41Q8×C4⋊C4C2211yescharacteristic
42#2347—21no—
43#2330—22no—
44#2317—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×C22⋊Q8C2211yescharacteristic
49C2×#384C2211yescharacteristic
50#2274—21no—
51C4×#117C2211yescharacteristic
52#2100C2214yes—
53#2087C2214yes—
54#2083C2212yes—
55#2072C2212yes—
56#2072C2212yes—
57#2072C2212yes—
58#2068C2214yes—
59#2057C2211yescharacteristic
60#2043C2214yes—
61#2043C2211yescharacteristic
62#2036C2212yes—
63#2024C2211yescharacteristic
64#2024C2212yes—
65#2016C2211yescharacteristic
66#2016C2212yes—
67#2001C2212yes—
68#2001C2214yes—
69#1969C2214yes—
70#1956C2212yes—
71#1947C2212yes—
72#1939C2212yes—
73#1933C2214yes—
74#1928C2214yes—
75#1923C2214yes—
76#1921C2214yes—
77#1878C2214yes—
78#1878C2214yes—
79#1878C2214yes—
80#1872C2214yes—
81#1863C2214yes—
82#1834C2212yes—
83#1831C2212yes—
84#1824C2212yes—
85#1803C2214yes—
86#1803C2212yes—
87#1803C2212yes—
88#1797C2211yescharacteristic
89C8×C4⋊C4C2212yes—
90C4×#103C2212yes—
91C4×#103C2214yes—
92C4×#103C2214yes—
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yes—
95C22×Q16C2312yes—
96C22×Q16C2314yes—
97#533—28no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125—21no—
103C2×#125—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2311yescharacteristic
107C2×C22⋊Q8C2312yes—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2314yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×#117—21no—
116C2×#117D411yescharacteristic
117C2×#117D411yescharacteristic
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2312yes—
121C22×C42C2311yescharacteristic
122#484—44no—
123#462—28no—
124#462—48no—
125C4×Q16—28no—
126C2×#107C2312yes—
127C2×#107C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2314yes—
143#399—22no—
144#398—22no—
145#398—22no—
146#398—22no—
147#398—22no—
148#396—22no—
149#391—22no—
150#391C2311yescharacteristic
151#391C2312yes—
152#391C2314yes—
153#391C2312yes—
154#391C2312yes—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—24no—
167#389—24no—
168#389—22no—
169#389—22no—
170#389C2314yes—
171#389C2311yescharacteristic
172#389C2312yes—
173#389C2312yes—
174#388—22no—
175#388—22no—
176#387—21no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—24no—
181#387—24no—
182#387—22no—
183#387—22no—
184#385—21no—
185#385—24no—
186#384—22no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384C2312yes—
191#384C2312yes—
192#384C2311yescharacteristic
193#384C2312yes—
194#383—22no—
195#382—22no—
196#380—21no—
197#380—21no—
198#380C2311yescharacteristic
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—21no—
216C4×C22⋊C4—24no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219#376D412yes—
220C43C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2314yes—
233C22×Q8—21no—
234C22×Q8C2412yes—
235C22×Q8C2412yes—
236C22×Q8C2411yescharacteristic
237C23×C4—22no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C2×Q16—24no—
242C2×Q16—24no—
243C2×Q16—28no—
244C2×Q16—42no—
245C2×Q16—44no—
246C2×Q16—44no—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—42no—
255#125—28no—
256#125—44no—
257C22⋊Q8—28no—
258C22⋊Q8—24no—
259C22⋊Q8—24no—
260C22⋊Q8—22no—
261C22⋊Q8—41no—
262C22⋊Q8—44no—
263C4×Q8—28no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—44no—
267C4×Q8—44no—
268C4×Q8—44no—
269C4×Q8—44no—
270C4×Q8—84no—
271C4×Q8#4118yes—
272#117—28no—
273#117—42no—
274#117Q1618yescomplemented
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—24no—
342#107—44no—
343C4⋊C8—24no—
344C4⋊C8—44no—
345#103—24no—
346#103—28no—
347#103—24no—
348#103—28no—
349#103—42no—
350#103—44no—
351#103—42no—
352#103—44no—
353#103—44no—
354C22⋊C8—42no—
355C22⋊C8—42no—
356C4×C8—24no—
357C4×C8—44no—
358#95—24no—
359#95—21no—
360#95—24no—
361#95—21no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—21no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24C22×D411yescharacteristic
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8C2×#4114yes—
396C2×Q8H(5,𝔽2)12yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—41no—
415C22×C4—44no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4C2×Q1614yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2511yescharacteristic
429Q16—48no—
430Q16—88no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—28no—
443C4⋊C4—24no—
444C4⋊C4—28no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—82no—
469C4⋊C4—81no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—84no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—84no—
477C4⋊C4—82no—
478C4⋊C4C2×Q1614yes—
479C4⋊C4C2×Q1614yes—
480C4⋊C4C2×#4114yes—
481C22⋊C4—28no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—81no—
485C22⋊C4—82no—
486C22⋊C4C2×Q1618yes—
487C42—22no—
488C42—22no—
489C42—28no—
490C42—24no—
491C42—22no—
492C42—22no—
493C42—24no—
494C42—22no—
495C42—42no—
496C42—42no—
497C42—42no—
498C42—42no—
499C42—84no—
500C42—82no—
501C42—82no—
502C42—84no—
503C42C2×Q1618yes—
504C42C2×#4114yes—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23C22×Q1612yes—
512C23C23×D411yescharacteristic
513Q8—24no—
514Q8—24no—
515Q8—24no—
516Q8—48no—
517Q8—42no—
518Q8—42no—
519Q8—42no—
520Q8—44no—
521Q8—44no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—164no—
573C2×C4#52512yes—
574C2×C4#52512yes—
575C2×C4C22×Q1612yes—
576C2×C4C22×Q1614yes—
577C2×C4C22×Q1614yes—
578C2×C4C22×Q1612yes—
579C2×C4#57512yes—
580C2×C4#57712yes—
581C2×C4C22×#4112yes—
582C2×C4C2×H(5,𝔽2)11yescharacteristic
583C2×C4#58512yes—
584C2×C4#58511yescharacteristic
585C8—42no—
586C8—44no—
587C8—42no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C2×#52511yescharacteristic
596C22#348811yescharacteristic
597C22#350011yescharacteristic
598C22C23×Q1611yescharacteristic
599C22C2×#57511yescharacteristic
600C22C2×#57711yescharacteristic
601C22#358611yescharacteristic
602C4—24no—
603C4—24no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—42no—
608C4—42no—
609C4—44no—
610C4—41no—
611C4—44no—
612C4—42no—
613C4—81no—
614C4—84no—
615C4—81no—
616C4—82no—
617C4—81no—
618C4—82no—
619C4#310414yes—
620C4#349412yes—
621C4#351212yes—
622C2—42no—
623C2#3608311yescharacteristic
624C2C2×#310411yescharacteristic
625C2#6058511yescharacteristic
626C2#6195211yescharacteristic
627C2#6199711yescharacteristic
628C2#6253011yescharacteristic
629C2#6258711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30151C2semidirect
5. #29495C2semidirect
6. #28161C2semidirect
11. #26112C2semidirect
12. #24732C2semidirect
13. #23816C2semidirect
15. #23096C2semidirect
16. #22771C2semidirect
274. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#364812
#342142
#301512
#296642
#294952
#281612
#279722
#276482
#269322
#266622
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