Finite Group Databasebeta

← #6564214

Subgroups — #6564214

Order 512.

1579 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36480C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic
3#30147C212yesmaximal, complemented
4#29938C211yesmaximal, characteristic
5#29512C212yesmaximal, complemented
6#29273C212yesmaximal, complemented
7#27456C214yesmaximal
8#27328C214yesmaximal
9#26444C212yesmaximal, complemented
10#25811C211yesmaximal, characteristic
11#25453C212yesmaximal, complemented
12#24301C211yesmaximal, characteristic
13#23837C212yesmaximal, complemented
14#23756C211yesmaximal, characteristic
15#23156C212yesmaximal, complemented
16#23062C212yesmaximal, complemented
17#22632C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#519—21no—
20C2×#498C2212yes—
21C2×#475C2212yes—
22#2863C2212yes—
23#2857C2211yescharacteristic
24#2605—22no—
25#2567C2211yescharacteristic
26#2551C2211yescharacteristic
27#2509—21no—
28#2497C2212yes—
29#2492C2212yes—
30#2465—22no—
31#2439—21no—
32#2369C2211yescharacteristic
33#2365—21no—
34#2365—21no—
35#2365—21no—
36#2365C2212yes—
37#2365C2212yes—
38#2355C2211yescharacteristic
39#2347—21no—
40#2344—22no—
41#2344C2214yes—
42#2327—22no—
43#2310—21no—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×C22⋊Q8C2211yescharacteristic
48#2285C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2099C2212yes—
51#2099C2211yescharacteristic
52#2096C2211yescharacteristic
53#2088C2214yes—
54#2084C2211yescharacteristic
55#2084C2214yes—
56#2084C2212yes—
57#2077C2212yes—
58#2077C2214yes—
59#2072C2211yescharacteristic
60#2066C2214yes—
61#2058C2212yes—
62#2058C2214yes—
63#2058C2212yes—
64#2058C2212yes—
65#2035C2212yes—
66#2024C2212yes—
67#1990C2212yes—
68#1988C2212yes—
69#1987C2214yes—
70#1987C2212yes—
71#1948C2212yes—
72#1948C2214yes—
73#1941C2214yes—
74#1939C2212yes—
75#1933C2214yes—
76#1927C2214yes—
77#1927C2214yes—
78#1923C2214yes—
79#1920C2214yes—
80#1920C2214yes—
81#1919C2214yes—
82#1919C2214yes—
83#1872C2212yes—
84#1853C2212yes—
85#1853C2214yes—
86#1831C2212yes—
87#1797C2211yescharacteristic
88C4×#106C2214yes—
89C4×#106C2214yes—
90C4×#106C2212yes—
91C8×C4⋊C4C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yes—
95C23×Q8—21no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2311yescharacteristic
106#519—48no—
107C2×C4×Q8—22no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8C2311yescharacteristic
113C2×#117—21no—
114C2×#117—24no—
115C2×#117D411yescharacteristic
116C2×#117D411yescharacteristic
117C22×C4⋊C4—21no—
118C22×C4⋊C4C2312yes—
119C22×C42C2311yescharacteristic
120#498—28no—
121#475—28no—
122C2×#106C2314yes—
123C2×#106C2312yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×#106C2312yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C4×C8C2312yes—
136C2×C4×C8C2314yes—
137#400—22no—
138#400—22no—
139#396—22no—
140#395—22no—
141#395—22no—
142#393—22no—
143#391—22no—
144#391—22no—
145#391—22no—
146#391—22no—
147#391—22no—
148#391—21no—
149#391—22no—
150#391—22no—
151#391C2312yes—
152#391C2311yescharacteristic
153#391C2312yes—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—21no—
162#389C2311yescharacteristic
163#389C2312yes—
164#389C2314yes—
165#389C2314yes—
166#389C2312yes—
167#389C2312yes—
168#388—22no—
169#388—22no—
170#387—21no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—24no—
175#387—24no—
176#387—24no—
177#387—24no—
178#387—22no—
179#387—22no—
180#385—21no—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2312yes—
184#384C2311yescharacteristic
185#383—22no—
186#382—21no—
187#382—21no—
188#381—24no—
189#380—21no—
190#380—21no—
191#380C2312yes—
192#380C2312yes—
193#380C2311yescharacteristic
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—21no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4C2312yes—
214#376D412yes—
215C43C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2314yes—
231#336C2314yes—
232C22×Q8—28no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—44no—
238C22×Q8—44no—
239C22×Q8C2411yescharacteristic
240C23×C4—22no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244C22×C8C2412yes—
245C22×C8C2414yes—
246C22×C8C2412yes—
247C4⋊Q8—28no—
248C4⋊Q8—28no—
249C4⋊Q8—28no—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—42no—
254C4⋊Q8—44no—
255C4⋊Q8#4118yes—
256C22⋊Q8—22no—
257C22⋊Q8—41no—
258C22⋊Q8—82no—
259C4×Q8—24no—
260C4×Q8—48no—
261C4×Q8—48no—
262C4×Q8—48no—
263C4×Q8—48no—
264C4×Q8—84no—
265#117—42no—
266#117—44no—
267#117—416no—
268#117SD1618yes—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
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—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C22⋊C4—22no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D412yes—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42C2×D412yes—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2411yescharacteristic
325C2×C42C2412yes—
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328#106—24no—
329#106—28no—
330#106—28no—
331#106—28no—
332#106—24no—
333#106—44no—
334C4⋊C8—24no—
335C4⋊C8—44no—
336#103—24no—
337#103—28no—
338#103—42no—
339C22⋊C8—24no—
340C22⋊C8—42no—
341C4×C8—28no—
342C4×C8—24no—
343#95—21no—
344#95—21no—
345#95—22no—
346#95—22no—
347#95—21no—
348#95—24no—
349#95—24no—
350#95—22no—
351#95—22no—
352#95—21no—
353#95—22no—
354#95—24no—
355#95—24no—
356#95—22no—
357#95—21no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95C2×D412yes—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368C24C22×D411yescharacteristic
369C2×Q8—28no—
370C2×Q8—28no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—432no—
378C2×Q8—44no—
379C2×Q8—42no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—82no—
384C2×Q8—81no—
385C2×Q8—81no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—28no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—41no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4C2×SD1614yes—
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C2511yescharacteristic
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—22no—
426C2×C82−1+414yes—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—81no—
461C4⋊C4C2×SD1614yes—
462C4⋊C4C2×SD1614yes—
463C4⋊C4C2×#4118yes—
464C4⋊C42−1+412yes—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—44no—
468C22⋊C4—81no—
469C22⋊C4—82no—
470C22⋊C4C2×SD1618yes—
471C42—22no—
472C42—22no—
473C42—24no—
474C42—22no—
475C42—24no—
476C42—22no—
477C42—22no—
478C42—42no—
479C42—42no—
480C42—42no—
481C42—48no—
482C42—48no—
483C42—44no—
484C42—42no—
485C42—82no—
486C42—82no—
487C42C2×SD1618yes—
488C42C2×#4114yes—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23C22×SD1612yes—
496C23C23×D411yescharacteristic
497Q8—416no—
498Q8—416no—
499Q8—42no—
500Q8—42no—
501Q8—88no—
502Q8—162no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—28no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—44no—
526C2×C4—416no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4#52512yes—
555C2×C4#52512yes—
556C2×C4C22×SD1614yes—
557C2×C4C22×SD1614yes—
558C2×C4C22×SD1612yes—
559C2×C4C22×SD1612yes—
560C2×C4#57512yes—
561C2×C4#57712yes—
562C2×C4C22×#4112yes—
563C2×C4C2×2−1+412yes—
564C2×C4C2×2−1+411yescharacteristic
565C2×C4#58511yescharacteristic
566C8—24no—
567C8—42no—
568C8—44no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22C2×#52511yescharacteristic
577C22#348911yescharacteristic
578C22#349911yescharacteristic
579C22C23×SD1611yescharacteristic
580C22C2×#57511yescharacteristic
581C22C2×#57711yescharacteristic
582C22#358611yescharacteristic
583C4—24no—
584C4—22no—
585C4—24no—
586C4—22no—
587C4—22no—
588C4—42no—
589C4—48no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—82no—
594C4—81no—
595C4—82no—
596C4—81no—
597C4—82no—
598C4—81no—
599C4—82no—
600C4#310114yes—
601C4#351812yes—
602C4#352312yes—
603C2—42no—
604C2#3600711yescharacteristic
605C2C2×#310111yescharacteristic
606C2#6058211yescharacteristic
607C2#6209911yescharacteristic
608C2#6214411yescharacteristic
609C2#6246911yescharacteristic
610C2#6250711yescharacteristic
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. #30147C2semidirect
5. #29512C2semidirect
6. #29273C2semidirect
9. #26444C2semidirect
11. #25453C2semidirect
13. #23837C2semidirect
15. #23156C2semidirect
16. #23062C2semidirect

Maximal subgroups

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

subgroupindex
#364802
#342092
#301472
#299382
#295122
#292732
#274562
#273282
#264442
#258112
#254532
#243012
#238372
#237562
#231562
#230622
#226322