Finite Group Databasebeta

← #6543865

Subgroups — #6543865

Order 512.

1519 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36288C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#30042C212yesmaximal, complemented
4#29664C211yesmaximal, characteristic
5#29394C212yesmaximal, complemented
6#29347C212yesmaximal, complemented
7#27972C211yesmaximal, characteristic
8#27648C211yesmaximal, characteristic
9#26726C214yesmaximal
10#26621C214yesmaximal
11#26026C212yesmaximal, complemented
12#24632C212yesmaximal, complemented
13#23679C212yesmaximal, complemented
14#23571C211yesmaximal, characteristic
15#22998C212yesmaximal, complemented
16#22771C212yesmaximal, complemented
17#22639C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#484C2212yes—
20C2×#462C2212yes—
21C2×C4×Q16C2212yes—
22C2×C4×Q16C2214yes—
23#2862C2212yes—
24#2857C2211yescharacteristic
25#2612—22no—
26#2574—22no—
27#2551C2211yescharacteristic
28#2546C2211yescharacteristic
29#2508—21no—
30#2492C2212yes—
31#2439—21no—
32#2429C2212yes—
33#2369—21no—
34#2365C2211yescharacteristic
35#2360—22no—
36#2360C2214yes—
37#2355—21no—
38#2355C2212yes—
39#2355C2211yescharacteristic
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4C2212yes—
42#2346—21no—
43#2330—22no—
44#2317—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×C22⋊Q8C2211yescharacteristic
48#2295C2212yes—
49C2×#384C2211yescharacteristic
50#2274—21no—
51C4×#117C2211yescharacteristic
52#2083C2212yes—
53#2072C2212yes—
54#2072C2212yes—
55#2072C2212yes—
56#2068C2214yes—
57#2057C2211yescharacteristic
58#2056C2214yes—
59#2043C2211yescharacteristic
60#2036C2212yes—
61#2024C2211yescharacteristic
62#2024C2214yes—
63#2024C2212yes—
64#2016C2211yescharacteristic
65#2016C2214yes—
66#2016C2212yes—
67#2001C2212yes—
68#2001C2214yes—
69#1987C2214yes—
70#1981C2214yes—
71#1956C2212yes—
72#1947C2212yes—
73#1939C2212yes—
74#1933C2214yes—
75#1933C2214yes—
76#1928C2214yes—
77#1923C2214yes—
78#1921C2214yes—
79#1921C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1872C2214yes—
83#1863C2214yes—
84#1839C2212yes—
85#1834C2212yes—
86#1824C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1797C2211yescharacteristic
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2212yes—
92C4×#103C2212yes—
93C4×#103C2214yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×Q16C2314yes—
96C22×Q16C2312yes—
97#533—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#125—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2311yescharacteristic
108C2×C22⋊Q8C2312yes—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2314yes—
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—
124C4×Q16—28no—
125C4×Q16—48no—
126C2×#107C2312yes—
127C2×#107C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2314yes—
143#400—22no—
144#398—22no—
145#398—22no—
146#396—22no—
147#395—22no—
148#393—22no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391—22no—
153#391C2314yes—
154#391C2312yes—
155#391C2312yes—
156#391C2311yescharacteristic
157#391C2312yes—
158#391C2312yes—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—21no—
167#389—21no—
168#389—24no—
169#389—24no—
170#389—22no—
171#389C2314yes—
172#389C2311yescharacteristic
173#389C2312yes—
174#388—22no—
175#388—22no—
176#387—22no—
177#387—21no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—24no—
182#387—24no—
183#385—21no—
184#385—24no—
185#384—22no—
186#384—21no—
187#384—21no—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2312yes—
191#382—22no—
192#382—21no—
193#382—22no—
194#382—22no—
195#380—21no—
196#380—21no—
197#380C2311yescharacteristic
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—21no—
215C4×C22⋊C4—24no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218#376—22no—
219#376D412yes—
220C43C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2314yes—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8C2411yescharacteristic
236C22×Q8C2412yes—
237C22×Q8C2412yes—
238C23×C4—22no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2411yescharacteristic
242C2×Q16—24no—
243C2×Q16—28no—
244C2×Q16—42no—
245C2×Q16—42no—
246C2×Q16—44no—
247C2×Q16—44no—
248C22×C8C2412yes—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—44no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257#125—28no—
258C22⋊Q8—28no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—22no—
262C22⋊Q8—41no—
263C22⋊Q8—44no—
264C4×Q8—28no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—42no—
269C4×Q8—44no—
270C4×Q8—44no—
271C4×Q8—84no—
272C4×Q8#4118yes—
273#117—28no—
274#117—42no—
275#117Q1618yescomplemented
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2412yes—
336C2×C42C2411yescharacteristic
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339#107—24no—
340#107—28no—
341C4⋊C8—24no—
342C4⋊C8—44no—
343#103—24no—
344#103—24no—
345#103—28no—
346#103—24no—
347#103—44no—
348#103—44no—
349#103—42no—
350#103—44no—
351#103—44no—
352C22⋊C8—42no—
353C22⋊C8—42no—
354C4×C8—24no—
355C4×C8—44no—
356#95—21no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95—21no—
363#95—22no—
364#95—21no—
365#95—21no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—22no—
372#95—21no—
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—22no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8C2×#4114yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—41no—
418C22×C4—44no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4C2×Q1614yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C2511yescharacteristic
432Q16—48no—
433Q16—416no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—28no—
446C4⋊C4—24no—
447C4⋊C4—28no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—44no—
458C4⋊C4—42no—
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—81no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4C2×Q1614yes—
478C4⋊C4C2×Q1614yes—
479C4⋊C4C2×#4114yes—
480C4⋊C4H(5,𝔽2)12yes—
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—44no—
498C42—42no—
499C42—42no—
500C42—82no—
501C42—84no—
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—48no—
516Q8—42no—
517Q8—42no—
518Q8—42no—
519Q8—44no—
520Q8—44no—
521Q8—42no—
522Q8—164no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—164no—
574C2×C4#52512yes—
575C2×C4#52512yes—
576C2×C4C22×Q1612yes—
577C2×C4C22×Q1614yes—
578C2×C4C22×Q1614yes—
579C2×C4C22×Q1612yes—
580C2×C4#57512yes—
581C2×C4#57712yes—
582C2×C4C22×#4112yes—
583C2×C4C2×H(5,𝔽2)11yescharacteristic
584C2×C4#58511yescharacteristic
585C2×C4#58512yes—
586C8—42no—
587C8—42no—
588C8—44no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22C2×#52511yescharacteristic
597C22#348811yescharacteristic
598C22#350011yescharacteristic
599C22C23×Q1611yescharacteristic
600C22C2×#57511yescharacteristic
601C22C2×#57711yescharacteristic
602C22#358611yescharacteristic
603C4—24no—
604C4—24no—
605C4—22no—
606C4—22no—
607C4—22no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—44no—
612C4—41no—
613C4—44no—
614C4—84no—
615C4—82no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—82no—
620C4#310414yes—
621C4#349312yes—
622C4#350612yes—
623C2—42no—
624C2#3575611yescharacteristic
625C2C2×#310411yescharacteristic
626C2#6058511yescharacteristic
627C2#6195211yescharacteristic
628C2#6198611yescharacteristic
629C2#6253011yescharacteristic
630C2#6259311yescharacteristic
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. #30042C2semidirect
5. #29394C2semidirect
6. #29347C2semidirect
11. #26026C2semidirect
12. #24632C2semidirect
13. #23679C2semidirect
15. #22998C2semidirect
16. #22771C2semidirect
275. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#362882
#342142
#300422
#296642
#293942
#293472
#279722
#276482
#267262
#266212
#260262
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