Finite Group Databasebeta

← #6557826

Subgroups — #6557826

Order 512.

1552 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#36476C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic
3#30367C212yesmaximal, complemented
4#30205C211yesmaximal, characteristic, complemented
5#29206C211yesmaximal, characteristic, complemented
6#29166C212yesmaximal, complemented
7#29001C214yesmaximal, complemented
8#28373C211yesmaximal, characteristic, complemented
9#25504C212yesmaximal, complemented
10#25394C212yesmaximal
11#25349C212yesmaximal
12#25047C214yesmaximal, complemented
13#23816C211yesmaximal, characteristic, complemented
14#23577C212yesmaximal, complemented
15#23004C212yesmaximal, complemented
16#22971C212yesmaximal, complemented
17#20568C211yesmaximal, characteristic
18C2×#533C2212yes—
19C2×#477C2212yes—
20C2×#470C2214yes—
21C2×#462C2212yes—
22C2×#441C2214yes—
23C2×#436C2212yes—
24#2868C2211yescharacteristic
25#2862C2211yescharacteristic
26#2857C2212yes—
27#2855C2212yes—
28#2607—22no—
29#2589—22no—
30#2580—21no—
31#2570—21no—
32#2563C2211yescharacteristic
33#2552—21no—
34#2542—21no—
35#2542C2211yescharacteristic
36#2538C2212yes—
37#2492C2212yes—
38#2441—22no—
39#2429—22no—
40#2423C2214yes—
41#2423C2212yes—
42#2369C2211yescharacteristic
43#2365—21no—
44#2365C2211yescharacteristic
45#2357—21no—
46#2355—21no—
47#2354—21no—
48Q8×C4⋊C4C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×C22⋊Q8C2212yes—
52C42×D4C2211yescharacteristic
53#2100C2211yescharacteristic
54#2100C2211yescharacteristic
55#2095C2212yes—
56#2075C2214yes—
57#2072C2212yes—
58#2072C2214yes—
59#2058C2212yes—
60#2058C2212yes—
61#2043C2211yescharacteristic
62#2043C2211yescharacteristic
63#2035C2214yes—
64#2035C2214yes—
65#2024C2212yes—
66#2024C2212yes—
67#2024C2212yes—
68#2016C2212yes—
69#2001C2214yes—
70#1990C2212yes—
71#1990C2212yes—
72#1969C2214yes—
73#1956C2212yes—
74#1948C2212yes—
75#1948C2212yes—
76#1943C2214yes—
77#1935C2214yes—
78#1918C2214yes—
79#1894C2214yes—
80#1894C2214yes—
81#1878C2214yes—
82#1865C2214yes—
83#1865C2214yes—
84#1844C2212yes—
85#1809C2214yes—
86#1770C2212yes—
87C4×C4⋊C8C2211yescharacteristic
88#1763C2214yes—
89C4×#103C2212yes—
90C4×#103C2212yes—
91#1753C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94#1728C2212yes—
95C22×Q16C2314yes—
96C22×M16C2314yes—
97#533—48no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#125—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2314yes—
110C2×C22⋊Q8C2312yes—
111C2×C4×Q8—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2314yes—
115C2×C4×Q8C2312yes—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×#117—22no—
120C22×C42C2312yes—
121#477—28no—
122#470—48no—
123#462—28no—
124#441—48no—
125#436—28no—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2312yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×C22⋊C8C2314yes—
139C2×#97C2314yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2312yes—
142#400—22no—
143#399—22no—
144#399—22no—
145#399—24no—
146#398—22no—
147#398—24no—
148#396—24no—
149#393—22no—
150#393—22no—
151#391—21no—
152#391—21no—
153#391—22no—
154#391—21no—
155#391—22no—
156#391—22no—
157#391C2312yes—
158#391C2312yes—
159#391C2312yes—
160#391C2312yes—
161#390—21no—
162#390—21no—
163#390—22no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—24no—
172#389C2311yescharacteristic
173#389C2311yescharacteristic
174#389C2312yes—
175#389C2314yes—
176#388—22no—
177#388—22no—
178#388—24no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—24no—
185#385—22no—
186#385—22no—
187#384—21no—
188#384—22no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#382—21no—
196#382—21no—
197#382—21no—
198#381—22no—
199#380—22no—
200#380C2312yes—
201#380C2312yes—
202#380C2314yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—22no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220C43C2311yescharacteristic
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8C2412yes—
238C22×Q8C2412yes—
239C22×D4C2×D411yescharacteristic
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2412yes—
243C2×Q16—28no—
244C2×Q16—28no—
245C2×Q16—44no—
246C2×M16—28no—
247C2×M16—44no—
248C2×M16—44no—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2414yes—
252C4⋊Q8—24no—
253C4⋊Q8—22no—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—41no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260#125—48no—
261C22⋊Q8—28no—
262C22⋊Q8—24no—
263C22⋊Q8—42no—
264C22⋊Q8—44no—
265C22⋊Q8—48no—
266C22⋊Q8—44no—
267C22⋊Q8—84no—
268C4×Q8—28no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—48no—
274C4×Q8—44no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—24no—
278#117—48no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C42—24no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—44no—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340#107—28no—
341#106—44no—
342#106—44no—
343C4⋊C8#4118yes—
344#103—24no—
345#103—24no—
346#103—28no—
347#103—28no—
348#103—24no—
349#103—42no—
350#103—44no—
351#103—44no—
352C22⋊C8—28no—
353#97—44no—
354C4×C8—24no—
355C4×C8—42no—
356#95—21no—
357#95—22no—
358#95—24no—
359#95—22no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24C22×D412yes—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—81no—
394C2×Q8—81no—
395C2×Q8—81no—
396C2×Q8H(5,𝔽2)14yes—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2511yescharacteristic
429Q16—416no—
430M16—88no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8C2×#4118yes—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—81no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—84no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C42−1+412yes—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—48no—
483C22⋊C4—44no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C22⋊C4—84no—
488C42—24no—
489C42—22no—
490C42—21no—
491C42—21no—
492C42—42no—
493C42—42no—
494C42—41no—
495C42—41no—
496C42—48no—
497C42—44no—
498C42—42no—
499C42—42no—
500C42—82no—
501C42—82no—
502C42#13714yes—
503C42#13714yes—
504C42C2×#4114yes—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23C23×D411yescharacteristic
513Q8—24no—
514Q8—24no—
515Q8—42no—
516Q8—42no—
517Q8—44no—
518Q8—44no—
519Q8—84no—
520Q8—88no—
521Q8—162no—
522D4—44no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—48no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—84no—
573C2×C4#53212yes—
574C2×C4#53212yes—
575C2×C4C2×#13712yes—
576C2×C4C2×#13712yes—
577C2×C4C2×#13712yes—
578C2×C4C2×#13712yes—
579C2×C4C2×#13712yes—
580C2×C4C2×#13712yes—
581C2×C4C22×#4112yes—
582C2×C4C2×H(5,𝔽2)12yes—
583C2×C4C2×2−1+411yescharacteristic
584C2×C4#58511yescharacteristic
585C8—28no—
586C8—42no—
587C8—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22C2×#53211yescharacteristic
596C22#348411yescharacteristic
597C22#348911yescharacteristic
598C22C22×#13711yescharacteristic
599C22C22×#13711yescharacteristic
600C22#358511yescharacteristic
601C22#358611yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—42no—
608C4—44no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4—81no—
616C4—81no—
617C4—84no—
618C4—81no—
619C4#318514yes—
620C4#351312yes—
621C4#352212yes—
622C2—82no—
623C2#3591211yescharacteristic
624C2C2×#318511yescharacteristic
625C2#6072711yescharacteristic
626C2#6158411yescharacteristic
627C2#6158811yescharacteristic
628C2#6210811yescharacteristic
629C2#6211311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30367C2semidirect
4. #30205C2semidirect
5. #29206C2semidirect
6. #29166C2semidirect
7. #29001C2semidirect
8. #28373C2semidirect
9. #25504C2semidirect
12. #25047C2semidirect
13. #23816C2semidirect
14. #23577C2semidirect
15. #23004C2semidirect
16. #22971C2semidirect

Maximal subgroups

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

subgroupindex
#364762
#361342
#303672
#302052
#292062
#291662
#290012
#283732
#255042
#253942
#253492
#250472
#238162
#235772
#230042
#229712
#205682