Finite Group Databasebeta

← #6434869

Subgroups — #6434869

Order 512.

1511 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35267C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#30164C214yesmaximal, complemented
4#30161C214yesmaximal, complemented
5#30152C212yesmaximal, complemented
6#30151C212yesmaximal, complemented
7#30146C212yesmaximal, complemented
8#30145C212yesmaximal, complemented
9#23667C212yesmaximal, complemented
10C2×#2083C211yesmaximal, characteristic, complemented
11#23630C211yesmaximal, characteristic
12#23616C212yesmaximal
13#23554C211yesmaximal, characteristic, complemented
14#23451C211yesmaximal, characteristic
15#21067C212yesmaximal
16#21036C212yesmaximal, complemented
17#20754C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#428C2212yes—
20C22×#107C2212yes—
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#2360—22no—
35#2360—22no—
36#2355—21no—
37#2355—21no—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4—21no—
40#2330C2214yes—
41#2317C2212yes—
42#2317C2211yescharacteristic
43#2316C2211yescharacteristic
44#2311C2211yescharacteristic
45#2311C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48#2297C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#384C2211yescharacteristic
51#2274C2212yes—
52C4×#117C2211yescharacteristic
53#2087C2214yes—
54#2087C2214yes—
55#2086C2214yes—
56#2086C2214yes—
57#2083C2212yes—
58#2083C2212yes—
59#2083C2214yes—
60#2072C2212yes—
61#2072C2212yes—
62#2071C2212yes—
63#2071C2212yes—
64#2068C2214yes—
65#2067C2214yes—
66#2066C2214yes—
67#2065C2214yes—
68#2059C2214yes—
69#2058C2212yes—
70#2057C2212yes—
71#2026C2212yes—
72#2025C2212yes—
73#2024C2211yescharacteristic
74#2024C2211yescharacteristic
75#2023C2211yescharacteristic
76#2023C2211yescharacteristic
77#1928C2214yes—
78#1928C2214yes—
79#1925C2214yes—
80#1925C2214yes—
81#1921C2214yes—
82#1920C2214yes—
83#1919C2214yes—
84#1918C2214yes—
85#1832C2214yes—
86#1831C2212yes—
87#1831C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98#533—28no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#126—22no—
102C2×#125C2312yes—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—21no—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—22no—
108C2×C4×Q8—22no—
109C2×#117C2311yescharacteristic
110C2×#117C2312yes—
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C42C2311yescharacteristic
120#428—28no—
121C2×#107—44no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106C2314yes—
126#424—44no—
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2314yes—
129#417—44no—
130#416—28no—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#103C2314yes—
134C2×#102C2312yes—
135C2×#102C2312yes—
136C2×#102C2314yes—
137#401—22no—
138#400—22no—
139#400—22no—
140#400—22no—
141#397—22no—
142#397—22no—
143#397—22no—
144#396—22no—
145#391—22no—
146#391—22no—
147#391C2312yes—
148#391C2312yes—
149#391C2311yescharacteristic
150#391C2311yescharacteristic
151#390C2312yes—
152#390C2312yes—
153#390C2311yescharacteristic
154#390C2311yescharacteristic
155#389—21no—
156#389—21no—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—24no—
162#389—24no—
163#389C2314yes—
164#389C2314yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#388—22no—
172#388—22no—
173#388—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—24no—
178#387—24no—
179#386—21no—
180#386—21no—
181#386—22no—
182#386—22no—
183#385—24no—
184#385—22no—
185#384—22no—
186#384—21no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2312yes—
191#383—22no—
192#382—22no—
193#382—21no—
194#380—21no—
195#380—21no—
196#380—21no—
197#380—21no—
198#380—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
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#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#336C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2314yes—
233#336C2314yes—
234C2×#41—24no—
235C2×#41—22no—
236C2×#41—24no—
237C2×#41—22no—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8C2411yescharacteristic
241C22×D4—21no—
242C22×D4C2411yescharacteristic
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×M16—24no—
251C2×M16—24no—
252C2×M16—24no—
253C22×C8—24no—
254C22×C8—24no—
255C22×C8—24no—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261#126—84no—
262#125—24no—
263#125—24no—
264C22⋊Q8—28no—
265C22⋊Q8—28no—
266C4×Q8—28no—
267C4×Q8—28no—
268C4×Q8—84no—
269C4×Q8—84no—
270#117—24no—
271#117—22no—
272#117—24no—
273#117—22no—
274#117—24no—
275#117#4118yes—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—21no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—28no—
342#107—44no—
343#107—44no—
344#106—28no—
345C4⋊C8—44no—
346C4⋊C8—44no—
347#103—24no—
348#103—24no—
349#103—44no—
350#102—24no—
351#102—24no—
352#102—44no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—24no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24—21no—
376C24C22×D411yescharacteristic
377#41—48no—
378#41—48no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—22no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—42no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4—82no—
419C22×C4—82no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C2×#4114yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C42−1+414yes—
434C22×C42−1+412yes—
435C22×C4C2511yescharacteristic
436M16—48no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—48no—
447C4⋊C4—28no—
448C4⋊C4—28no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4C2×#4114yes—
480C4⋊C4C2×#4114yes—
481C22⋊C4—28no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—82no—
488C22⋊C4—81no—
489C22⋊C4—82no—
490C22⋊C4—81no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C42—22no—
494C42—22no—
495C42—28no—
496C42—24no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—24no—
502C42—22no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42—84no—
507C42—84no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23#52512yes—
520C23#53612yes—
521C23#53712yes—
522C23C23×D411yescharacteristic
523Q8—48no—
524Q8—48no—
525Q8—42no—
526Q8—42no—
527D4—42no—
528D4—42no—
529D4—84no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—162no—
581C2×C4—164no—
582C2×C4#52512yes—
583C2×C4#53612yes—
584C2×C4#53712yes—
585C2×C4#54118yes—
586C2×C4#54218yes—
587C2×C4#57612yes—
588C2×C4#57812yes—
589C2×C4C22×#4112yes—
590C2×C4C2×H(5,𝔽2)11yescharacteristic
591C2×C4C2×2−1+411yescharacteristic
592C2×C4C2×2−1+412yes—
593C8—42no—
594C8—42no—
595C8—44no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22#263714yes—
610C22C2×#52511yescharacteristic
611C22C2×#53611yescharacteristic
612C22C2×#53711yescharacteristic
613C22C2×#57611yescharacteristic
614C22C2×#57811yescharacteristic
615C22#358511yescharacteristic
616C22#358611yescharacteristic
617C4—22no—
618C4—22no—
619C4—41no—
620C4—44no—
621C4—41no—
622C4—44no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—81no—
628C4—82no—
629C4—81no—
630C4—81no—
631C4—82no—
632C4—84no—
633C4#263714yes—
634C4C2×#54112yes—
635C4C2×#54212yes—
636C2—24no—
637C2—161no—
638C2C2×#263711yescharacteristic
639C2#6058411yescharacteristic
640C2#6058911yescharacteristic
641C2#6088011yescharacteristic
642C2#6088711yescharacteristic
643C2#6092211yescharacteristic
644C2#6092511yescharacteristic
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. #34214C2semidirect
3. #30164C2semidirect
4. #30161C2semidirect
5. #30152C2semidirect
6. #30151C2semidirect
7. #30146C2semidirect
8. #30145C2semidirect
9. #23667C2semidirect
10. C2×#2083C2semidirect
13. #23554C2semidirect
16. #21036C2semidirect
17. #20754C2semidirect

Maximal subgroups

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

subgroupindex
#352672
#342142
#301642
#301612
#301522
#301512
#301462
#301452
#236672
C2×#20832
#236302
#236162
#235542
#234512
#210672
#210362
#207542