Finite Group Databasebeta

← #6570187

Subgroups — #6570187

Order 512.

1436 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36477C211yesmaximal, characteristic
2#36194C211yesmaximal, characteristic
3#30326C212yesmaximal, complemented
4#30305C211yesmaximal, characteristic, complemented
5#29123C211yesmaximal, characteristic, complemented
6#28925C214yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#28581C212yesmaximal, complemented
9#28232C212yesmaximal
10#28151C212yesmaximal
11#28088C214yesmaximal, complemented
12#27864C212yesmaximal, complemented
13#23801C211yesmaximal, characteristic, complemented
14#23762C212yesmaximal, complemented
15#22814C212yesmaximal, complemented
16#22763C212yesmaximal, complemented
17#20568C211yesmaximal, characteristic
18C2×#487C2212yes—
19C2×#470C2214yes—
20C2×C4×Q16C2212yes—
21C2×C4×Q16C2212yes—
22C2×#436C2212yes—
23#2868C2211yescharacteristic
24#2859C2211yescharacteristic
25#2857C2212yes—
26#2855C2212yes—
27#2608—22no—
28#2587—22no—
29#2571—21no—
30#2570—21no—
31#2566—21no—
32#2563—21no—
33#2538C2211yescharacteristic
34#2538C2212yes—
35#2538C2211yescharacteristic
36#2497—22no—
37#2497C2212yes—
38#2492C2214yes—
39#2439—22no—
40#2429C2212yes—
41#2370—21no—
42#2369—21no—
43#2357—21no—
44#2355C2211yescharacteristic
45#2355C2212yes—
46#2354—21no—
47Q8×C4⋊C4C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×C22⋊Q8C2212yes—
51#2285C2212yes—
52C42×D4C2211yescharacteristic
53#2100C2212yes—
54#2095C2212yes—
55#2095C2211yescharacteristic
56#2095C2211yescharacteristic
57#2084C2212yes—
58#2084C2212yes—
59#2075C2214yes—
60#2072C2212yes—
61#2072C2212yes—
62#2058C2214yes—
63#2043C2212yes—
64#2035C2214yes—
65#2035C2214yes—
66#2016C2211yescharacteristic
67#2016C2211yescharacteristic
68#2016C2212yes—
69#2001C2214yes—
70#1990C2212yes—
71#1971C2214yes—
72#1935C2214yes—
73#1933C2214yes—
74#1927C2214yes—
75#1927C2214yes—
76#1918C2214yes—
77#1894C2214yes—
78#1894C2214yes—
79#1878C2214yes—
80#1844C2212yes—
81#1834C2212yes—
82#1809C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1776C2214yes—
88#1770C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90#1763C2214yes—
91#1753C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94#1728C2212yes—
95C22×Q16C2312yes—
96C22×Q16C2312yes—
97C22×M16C2314yes—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#125—21no—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2314yes—
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×D4D411yescharacteristic
112C2×C4×D4D411yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C2×#117—22no—
115C22×C42C2312yes—
116#487—44no—
117#470—48no—
118C4×Q16—28no—
119C4×Q16—28no—
120#436—28no—
121C2×#107C2312yes—
122C2×#107C2312yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2312yes—
126C2×#103C2314yes—
127C2×#103C2312yes—
128C2×#103C2312yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×C22⋊C8C2314yes—
135C2×#97C2314yes—
136C2×C4×C8C2312yes—
137C2×C4×C8C2312yes—
138#400—22no—
139#400—24no—
140#398—22no—
141#396—22no—
142#396—22no—
143#396—22no—
144#396—22no—
145#396—24no—
146#393—24no—
147#391—24no—
148#391C2311yescharacteristic
149#391C2311yescharacteristic
150#391C2312yes—
151#391C2312yes—
152#391C2314yes—
153#390—22no—
154#390—21no—
155#390—21no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389—21no—
165#389—22no—
166#389—21no—
167#389—22no—
168#389—22no—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#388—22no—
173#388—22no—
174#388—24no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—24no—
181#385—22no—
182#385—22no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2312yes—
192#382—22no—
193#382—21no—
194#382—21no—
195#382—22no—
196#381—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380C2312yes—
201#380C2314yes—
202#380C2312yes—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4C2312yes—
221C43C2311yescharacteristic
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#336C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233C22×Q8—22no—
234C22×Q8C2411yescharacteristic
235C22×Q8C2412yes—
236C22×Q8C2411yescharacteristic
237C22×D4C2×D411yescharacteristic
238C23×C4C2×D412yes—
239C23×C4C2×D412yes—
240C23×C4C2412yes—
241C2×Q16—24no—
242C2×Q16—24no—
243C2×Q16—24no—
244C2×Q16—42no—
245C2×Q16—42no—
246C2×Q16—42no—
247C2×M16—28no—
248C2×M16—44no—
249C2×M16—44no—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C4⋊Q8—22no—
254C4⋊Q8—42no—
255C4⋊Q8—42no—
256C4⋊Q8—42no—
257C4⋊Q8—41no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260#125—44no—
261C22⋊Q8—24no—
262C22⋊Q8—42no—
263C22⋊Q8—44no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—44no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—24no—
274#117—84no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
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—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C42—22no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—44no—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2412yes—
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337#107—24no—
338#107—24no—
339#106—44no—
340#106—44no—
341C4⋊C8—24no—
342#103—28no—
343#103—24no—
344#103—24no—
345#103—24no—
346#103—24no—
347#103—44no—
348#103—44no—
349#103—44no—
350C22⋊C8—28no—
351#97—44no—
352C4×C8—42no—
353C4×C8#4118yes—
354#95—21no—
355#95—22no—
356#95—21no—
357#95—24no—
358#95—24no—
359#95—22no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—22no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24C22×D412yes—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—44no—
390C2×Q8—82no—
391C2×Q8—82no—
392C2×Q8H(5,𝔽2)12yes—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2511yescharacteristic
425Q16—48no—
426Q16—48no—
427M16—88no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8C2×#4114yes—
436C2×C8C2×#4114yes—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—84no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—44no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C22⋊C4—84no—
483C22⋊C4—82no—
484C22⋊C4—84no—
485C42—21no—
486C42—24no—
487C42—22no—
488C42—21no—
489C42—42no—
490C42—42no—
491C42—41no—
492C42—41no—
493C42—42no—
494C42—44no—
495C42—42no—
496C42—82no—
497C42—84no—
498C42—82no—
499C42#13714yes—
500C42#13714yes—
501C42C2×#4114yes—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23C23×D411yescharacteristic
510Q8—24no—
511Q8—42no—
512Q8—42no—
513Q8—42no—
514Q8—42no—
515Q8—42no—
516Q8—42no—
517Q8—42no—
518Q8—164no—
519D4—44no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—48no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—84no—
570C2×C4#53212yes—
571C2×C4#53212yes—
572C2×C4C2×#13712yes—
573C2×C4C2×#13712yes—
574C2×C4C2×#13712yes—
575C2×C4C2×#13712yes—
576C2×C4C2×#13712yes—
577C2×C4C2×#13712yes—
578C2×C4C22×#4112yes—
579C2×C4C2×H(5,𝔽2)11yescharacteristic
580C2×C4#58511yescharacteristic
581C2×C4#58512yes—
582C8—24no—
583C8—24no—
584C8—44no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22C2×#53211yescharacteristic
593C22#348511yescharacteristic
594C22#348811yescharacteristic
595C22C22×#13711yescharacteristic
596C22C22×#13711yescharacteristic
597C22#358511yescharacteristic
598C22#358611yescharacteristic
599C4—22no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—82no—
609C4—84no—
610C4—82no—
611C4—82no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—82no—
616C4#318514yes—
617C4#349312yes—
618C4#349412yes—
619C2—82no—
620C2#3606211yescharacteristic
621C2C2×#318511yescharacteristic
622C2#6072111yescharacteristic
623C2#6171911yescharacteristic
624C2#6172311yescharacteristic
625C2#6196411yescharacteristic
626C2#6196511yescharacteristic
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. #30326C2semidirect
4. #30305C2semidirect
5. #29123C2semidirect
6. #28925C2semidirect
7. #28717C2semidirect
8. #28581C2semidirect
11. #28088C2semidirect
12. #27864C2semidirect
13. #23801C2semidirect
14. #23762C2semidirect
15. #22814C2semidirect
16. #22763C2semidirect

Maximal subgroups

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

subgroupindex
#364772
#361942
#303262
#303052
#291232
#289252
#287172
#285812
#282322
#281512
#280882
#278642
#238012
#237622
#228142
#227632
#205682