Finite Group Databasebeta

← #6564223

Subgroups — #6564223

Order 512.

1698 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36473C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic, complemented
3#30145C212yesmaximal, complemented
4#29936C211yesmaximal, characteristic, complemented
5#29514C212yesmaximal, complemented
6#29272C212yesmaximal, complemented
7#27454C214yesmaximal, complemented
8#27326C214yesmaximal
9#26440C212yesmaximal, complemented
10#25813C211yesmaximal, characteristic, complemented
11#25449C212yesmaximal, complemented
12#24298C211yesmaximal, characteristic
13#23835C212yesmaximal, complemented
14#23754C211yesmaximal, characteristic
15#23157C212yesmaximal, complemented
16#23060C212yesmaximal, complemented
17#22632C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C2×#519—21no—
21C2×#500C2212yes—
22C2×#474C2212yes—
23#2860C2212yes—
24#2857C2211yescharacteristic
25#2604—22no—
26#2551C2211yescharacteristic
27#2542C2211yescharacteristic
28#2521—22no—
29#2496C2212yes—
30#2492—21no—
31#2491C2212yes—
32#2415—21no—
33#2365—21no—
34#2365—21no—
35#2365C2212yes—
36#2365C2212yes—
37#2365C2211yescharacteristic
38#2362—21no—
39#2354C2211yescharacteristic
40#2344—22no—
41#2344C2214yes—
42#2327—22no—
43#2310—21no—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×#121C2211yescharacteristic
48#2285C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2099C2212yes—
51#2099C2211yescharacteristic
52#2093C2211yescharacteristic
53#2086C2214yes—
54#2083C2211yescharacteristic
55#2083C2214yes—
56#2083C2212yes—
57#2075C2212yes—
58#2075C2214yes—
59#2071C2211yescharacteristic
60#2067C2214yes—
61#2057C2214yes—
62#2057C2212yes—
63#2057C2212yes—
64#2057C2212yes—
65#2033C2212yes—
66#2023C2212yes—
67#1990C2212yes—
68#1988C2214yes—
69#1988C2212yes—
70#1988C2212yes—
71#1947C2212yes—
72#1947C2214yes—
73#1942C2214yes—
74#1939C2212yes—
75#1934C2214yes—
76#1925C2214yes—
77#1925C2214yes—
78#1923C2214yes—
79#1918C2214yes—
80#1918C2214yes—
81#1918C2214yes—
82#1918C2214yes—
83#1874C2212yes—
84#1852C2212yes—
85#1852C2214yes—
86#1831C2212yes—
87#1797C2211yescharacteristic
88C4×#107C2214yes—
89C4×#107C2214yes—
90C4×#107C2212yes—
91C8×C4⋊C4C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yes—
95C23×Q8—21no—
96#556—48no—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#126—22no—
106C2×#126—22no—
107C2×#124—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×#121C2311yescharacteristic
112#519—48no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×D4—21no—
120C2×C4×D4C2311yescharacteristic
121C2×#117—21no—
122C2×#117—21no—
123C2×#117—24no—
124C2×#117D411yescharacteristic
125C2×#117D411yescharacteristic
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2312yes—
128C22×C42C2311yescharacteristic
129#500—28no—
130#474—28no—
131C2×#107C2312yes—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#107C2312yes—
136C2×#107C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×#102C2314yes—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C4×C8C2312yes—
145C2×C4×C8C2314yes—
146#400—22no—
147#399—22no—
148#397—22no—
149#395—22no—
150#394—22no—
151#393—22no—
152#391—22no—
153#391—22no—
154#391—21no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391—21no—
159#390C2312yes—
160#390C2311yescharacteristic
161#390C2312yes—
162#389—21no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—21no—
167#389C2311yescharacteristic
168#389C2312yes—
169#389C2314yes—
170#389C2314yes—
171#389C2312yes—
172#389C2312yes—
173#388—21no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—24no—
180#387—24no—
181#387—24no—
182#387—24no—
183#387—22no—
184#387—22no—
185#386—21no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2311yescharacteristic
190#382—21no—
191#381—24no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380C2312yes—
196#380C2312yes—
197#380C2311yescharacteristic
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215#376D412yes—
216C43C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2314yes—
233C22×Q8—28no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—44no—
239C22×Q8—44no—
240C22×D4—21no—
241C22×D4C2411yescharacteristic
242C23×C4—22no—
243C23×C4—21no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C22×C8C2412yes—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C4⋊Q8—28no—
251C4⋊Q8—28no—
252C4⋊Q8—28no—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—42no—
257C4⋊Q8—44no—
258C4⋊Q8#4118yes—
259#126—48no—
260#126—48no—
261#124—48no—
262C22⋊Q8—44no—
263C22⋊Q8—44no—
264C22⋊Q8—44no—
265#121—22no—
266#121—41no—
267C4×Q8—48no—
268C4×Q8—48no—
269C4×Q8—44no—
270C4×Q8—48no—
271C4×Q8—48no—
272C4×Q8—48no—
273C4×D4—24no—
274C4×D4—44no—
275#117—42no—
276#117—44no—
277#117—44no—
278#117—416no—
279#117D818yes—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343#107—24no—
344#107—28no—
345#107—28no—
346#107—28no—
347#107—24no—
348#107—44no—
349C4⋊C8—24no—
350C4⋊C8—44no—
351#102—24no—
352#102—28no—
353#102—42no—
354C22⋊C8—24no—
355C22⋊C8—42no—
356C4×C8—28no—
357C4×C8—24no—
358#95—21no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24—21no—
378C24C22×D411yescharacteristic
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—432no—
384C2×Q8—42no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—82no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—81no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4C2×D814yes—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C2511yescharacteristic
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—22no—
441C2×C82−1+414yes—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—28no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—81no—
474C4⋊C4C2×D814yes—
475C4⋊C4C2×D814yes—
476C4⋊C4C2×#4118yes—
477C4⋊C42−1+412yes—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—42no—
485C22⋊C4—44no—
486C22⋊C4—42no—
487C22⋊C4—44no—
488C22⋊C4—42no—
489C22⋊C4—82no—
490C22⋊C4C2×D818yes—
491C42—22no—
492C42—22no—
493C42—24no—
494C42—22no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—42no—
499C42—42no—
500C42—48no—
501C42—48no—
502C42—44no—
503C42—44no—
504C42—42no—
505C42—44no—
506C42C2×D818yes—
507C42C2×#4114yes—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C22×D812yes—
522C23C23×D411yescharacteristic
523Q8—416no—
524Q8—416no—
525Q8—88no—
526Q8—84no—
527D4—42no—
528D4—42no—
529D4—84no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—44no—
553C2×C4—416no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—84no—
579C2×C4#52512yes—
580C2×C4#52512yes—
581C2×C4C22×D814yes—
582C2×C4C22×D814yes—
583C2×C4C22×D812yes—
584C2×C4C22×D812yes—
585C2×C4#57512yes—
586C2×C4#57812yes—
587C2×C4C22×#4112yes—
588C2×C4C2×2−1+412yes—
589C2×C4C2×2−1+411yescharacteristic
590C2×C4#58511yescharacteristic
591C8—24no—
592C8—42no—
593C8—44no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22C2×#52511yescharacteristic
609C22#348911yescharacteristic
610C22#349911yescharacteristic
611C22C23×D811yescharacteristic
612C22C2×#57511yescharacteristic
613C22C2×#57811yescharacteristic
614C22#358611yescharacteristic
615C4—24no—
616C4—22no—
617C4—24no—
618C4—22no—
619C4—22no—
620C4—42no—
621C4—48no—
622C4—42no—
623C4—41no—
624C4—42no—
625C4—82no—
626C4—81no—
627C4—81no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4#310014yes—
632C4#351812yes—
633C4#352312yes—
634C2—42no—
635C2—161no—
636C2#3600711yescharacteristic
637C2C2×#310011yescharacteristic
638C2#6058311yescharacteristic
639C2#6209511yescharacteristic
640C2#6214611yescharacteristic
641C2#6247111yescharacteristic
642C2#6250711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. #34209C2semidirect
3. #30145C2semidirect
4. #29936C2semidirect
5. #29514C2semidirect
6. #29272C2semidirect
7. #27454C2semidirect
9. #26440C2semidirect
10. #25813C2semidirect
11. #25449C2semidirect
13. #23835C2semidirect
15. #23157C2semidirect
16. #23060C2semidirect
17. #22632C2semidirect

Maximal subgroups

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

subgroupindex
#364732
#342092
#301452
#299362
#295142
#292722
#274542
#273262
#264402
#258132
#254492
#242982
#238352
#237542
#231572
#230602
#226322