Finite Group Databasebeta

← #6434873

Subgroups — #6434873

Order 512.

1539 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35267C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#30160C214yesmaximal, complemented
4#30157C214yesmaximal, complemented
5#30152C212yesmaximal, complemented
6#30149C212yesmaximal, complemented
7#30147C212yesmaximal, complemented
8#30146C212yesmaximal, complemented
9#23671C212yesmaximal, complemented
10#23657C211yesmaximal, characteristic, complemented
11#23627C211yesmaximal, characteristic
12#23610C212yesmaximal
13C2×#2058C211yesmaximal, characteristic, complemented
14#23452C211yesmaximal, characteristic
15#21062C212yesmaximal
16#21038C212yesmaximal, complemented
17#20754C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#428C2212yes—
20C22×#106C2212yes—
21C2×#424C2212yes—
22C2×#417C2212yes—
23C2×#416C2212yes—
24#2863C2212yes—
25#2860C2212yes—
26#2645C2212yes—
27#2644C2212yes—
28#2510—22no—
29#2497—21no—
30#2496—21no—
31#2495—21no—
32#2494—21no—
33#2481—22no—
34#2369—21no—
35#2369—21no—
36#2365—21no—
37#2365—21no—
38#2345—22no—
39Q8×C22⋊C4—22no—
40#2331C2214yes—
41#2317C2211yescharacteristic
42#2316C2211yescharacteristic
43#2316C2212yes—
44#2311C2211yescharacteristic
45#2311C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48#2297C2211yescharacteristic
49C2×#389C2211yescharacteristic
50#2285C2211yescharacteristic
51#2274C2212yes—
52C4×#117C2211yescharacteristic
53#2088C2214yes—
54#2088C2214yes—
55#2087C2214yes—
56#2086C2214yes—
57#2085C2214yes—
58#2084C2212yes—
59#2083C2212yes—
60#2074C2212yes—
61#2073C2212yes—
62#2072C2211yescharacteristic
63#2072C2211yescharacteristic
64#2071C2211yescharacteristic
65#2071C2211yescharacteristic
66#2066C2214yes—
67#2066C2214yes—
68#2065C2214yes—
69#2065C2214yes—
70#2058C2212yes—
71#2058C2212yes—
72#2058C2214yes—
73#2024C2212yes—
74#2024C2212yes—
75#2023C2212yes—
76#2023C2212yes—
77#1928C2214yes—
78#1927C2214yes—
79#1926C2214yes—
80#1925C2214yes—
81#1920C2214yes—
82#1920C2214yes—
83#1919C2214yes—
84#1919C2214yes—
85#1832C2214yes—
86#1831C2212yes—
87#1831C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8—21no—
97C22×M16C2312yes—
98C23×C8C2312yes—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×#126—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×#117C2311yescharacteristic
108C2×#117C2312yes—
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C42C2311yescharacteristic
117#428—44no—
118C2×#107C2314yes—
119C2×#106—28no—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123#424—44no—
124C2×C4⋊C8C2314yes—
125C2×C4⋊C8C2314yes—
126#417—28no—
127#416—44no—
128C2×#103C2314yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#102C2314yes—
132C2×#102C2312yes—
133C2×#102C2312yes—
134#401—22no—
135#400—22no—
136#400—22no—
137#400—22no—
138#397—22no—
139#397—22no—
140#397—22no—
141#396—22no—
142#391—22no—
143#391—22no—
144#391—24no—
145#391—24no—
146#391C2312yes—
147#391C2312yes—
148#391C2311yescharacteristic
149#391C2311yescharacteristic
150#390C2312yes—
151#390C2312yes—
152#390C2311yescharacteristic
153#390C2311yescharacteristic
154#389—22no—
155#389—22no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389C2314yes—
163#389C2314yes—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#388—22no—
171#388—22no—
172#388—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—24no—
177#387—24no—
178#386—21no—
179#386—21no—
180#386—22no—
181#386—22no—
182#385—22no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#383—22no—
187#382—21no—
188#382C2312yes—
189#380—21no—
190#380—21no—
191#380—21no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380—24no—
196#380—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2314yes—
209C4×C22⋊C4—24no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4D412yes—
214#376C2312yes—
215C2×#95—21no—
216C43C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2314yes—
233#336C2314yes—
234C2×#41—24no—
235C2×#41—22no—
236C2×#41—24no—
237C2×#41—22no—
238C22×Q8—28no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—82no—
242C22×Q8—82no—
243C22×Q8C2411yescharacteristic
244C22×D4—21no—
245C22×D4C2411yescharacteristic
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×M16—24no—
254C2×M16—24no—
255C2×M16—24no—
256C22×C8—24no—
257C22×C8—24no—
258C22×C8—24no—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C4⋊Q8—28no—
263C4⋊Q8—28no—
264C4⋊Q8—28no—
265C4⋊Q8—28no—
266#126—84no—
267C4×Q8—84no—
268C4×Q8—84no—
269#117—22no—
270#117—24no—
271#117—22no—
272#117—24no—
273#117—24no—
274#117—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4#4118yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2414yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C42—24no—
323C2×C42—21no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2412yes—
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337#107—44no—
338#106—28no—
339#106—28no—
340#106—44no—
341C4⋊C8—44no—
342C4⋊C8—44no—
343#103—28no—
344#103—42no—
345#103—42no—
346#102—28no—
347#102—42no—
348#102—42no—
349#95—22no—
350#95—22no—
351#95—24no—
352#95—24no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C24—21no—
374C24C22×D411yescharacteristic
375#41—48no—
376#41—48no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—22no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—21no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—82no—
386C2×Q8—82no—
387C2×Q8—816no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C2×#4114yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C42−1+414yes—
433C22×C42−1+412yes—
434C22×C4C2511yescharacteristic
435M16—48no—
436C2×C8—24no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—48no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—28no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—84no—
474C4⋊C4—84no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4C2×#4118yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—42no—
481C22⋊C4—44no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—82no—
485C22⋊C4—81no—
486C22⋊C4—82no—
487C22⋊C4—81no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C42—22no—
491C42—22no—
492C42—24no—
493C42—24no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—24no—
498C42—22no—
499C42—42no—
500C42—42no—
501C42—42no—
502C42—44no—
503C42—84no—
504C42—84no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23#52512yes—
517C23#53612yes—
518C23#53712yes—
519C23C23×D411yescharacteristic
520Q8—416no—
521Q8—42no—
522Q8—42no—
523Q8—88no—
524D4—42no—
525D4—42no—
526D4—84no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—28no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—88no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—162no—
579C2×C4#52512yes—
580C2×C4#53612yes—
581C2×C4#53712yes—
582C2×C4#54118yes—
583C2×C4#54218yes—
584C2×C4#57712yes—
585C2×C4#57712yes—
586C2×C4C22×#4112yes—
587C2×C4C2×H(5,𝔽2)11yescharacteristic
588C2×C4C2×2−1+411yescharacteristic
589C2×C4C2×2−1+412yes—
590C8—42no—
591C8—42no—
592C8—44no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22#263714yes—
607C22C2×#52511yescharacteristic
608C22C2×#53611yescharacteristic
609C22C2×#53711yescharacteristic
610C22C2×#57711yescharacteristic
611C22C2×#57711yescharacteristic
612C22#358511yescharacteristic
613C22#358611yescharacteristic
614C4—22no—
615C4—22no—
616C4—42no—
617C4—42no—
618C4—41no—
619C4—48no—
620C4—41no—
621C4—42no—
622C4—42no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—81no—
627C4—82no—
628C4—81no—
629C4—81no—
630C4#263714yes—
631C4C2×#54112yes—
632C4C2×#54212yes—
633C2—24no—
634C2—161no—
635C2C2×#263711yescharacteristic
636C2#6057911yescharacteristic
637C2#6058211yescharacteristic
638C2#6088111yescharacteristic
639C2#6088711yescharacteristic
640C2#6092911yescharacteristic
641C2#6093111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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. #34216C2semidirect
3. #30160C2semidirect
4. #30157C2semidirect
5. #30152C2semidirect
6. #30149C2semidirect
7. #30147C2semidirect
8. #30146C2semidirect
9. #23671C2semidirect
10. #23657C2semidirect
13. C2×#2058C2semidirect
16. #21038C2semidirect
17. #20754C2semidirect

Maximal subgroups

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

subgroupindex
#352672
#342162
#301602
#301572
#301522
#301492
#301472
#301462
#236712
#236572
#236272
#236102
C2×#20582
#234522
#210622
#210382
#207542