Finite Group Databasebeta

← #7231118

Subgroups — #7231118

Order 512.

1703 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46152C211yesmaximal, characteristic, complemented
2#36125C211yesmaximal, characteristic
3#33068C212yesmaximal, complemented
4#32819C212yesmaximal, complemented
5#32683C212yesmaximal, complemented
6#32028C212yesmaximal, complemented
7#31490C214yesmaximal, complemented
8#30631C214yesmaximal, complemented
9#27032C214yesmaximal, complemented
10#26338C214yesmaximal, complemented
11#25649C212yesmaximal, complemented
12#24089C212yesmaximal, complemented
13#23346C211yesmaximal, characteristic, complemented
14C2×#550C2211yescharacteristic
15C2×#547C2211yescharacteristic
16C2×#532—21no—
17C2×#486C2212yes—
18C2×#481C2212yes—
19C2×#476C2212yes—
20C22×#106C2212yes—
21#2859C2212yes—
22#2858C2211yescharacteristic
23#2850—21no—
24#2844—21no—
25#2809—21no—
26#2720—24no—
27#2683—21no—
28#2567C2212yes—
29#2546C2211yescharacteristic
30#2541C2211yescharacteristic
31#2497C2214yes—
32#2487C2212yes—
33#2464—24no—
34#2440C2214yes—
35#2432—21no—
36#2431—21no—
37#2428C2214yes—
38#2379—21no—
39#2361C2214yes—
40C4×#124C2211yescharacteristic
41C42×D4C2211yescharacteristic
42#2094C2214yes—
43#2088C2214yes—
44#2071C2214yes—
45#2065C2214yes—
46#2065C2214yes—
47#2058C2214yes—
48#2058C2214yes—
49#2041C2214yes—
50#2041C2214yes—
51#2034C2214yes—
52#2034C2214yes—
53#2034C2214yes—
54#1984C2212yes—
55#1982C2212yes—
56#1948C2212yes—
57#1948C2212yes—
58#1941C2214yes—
59#1923C2214yes—
60#1920C2214yes—
61#1920C2214yes—
62#1919C2214yes—
63#1893C2212yes—
64#1893C2214yes—
65#1874C2214yes—
66#1869C2212yes—
67#1864C2214yes—
68#1864C2214yes—
69#1864C2214yes—
70#1853C2214yes—
71#1850C2214yes—
72#1831C2212yes—
73#1823C2212yes—
74#1802C2212yes—
75#1802C2212yes—
76C4×#102C2212yes—
77C22×#41—21no—
78C22×#41—21no—
79C23×C8C2312yes—
80#550—42no—
81#550—42no—
82#547—42no—
83#547—42no—
84#532—48no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×#127—21no—
89C2×#127C2312yes—
90C2×#127C2311yescharacteristic
91C2×#125—22no—
92C2×#124—21no—
93C2×#124—21no—
94C2×#124D411yescharacteristic
95C2×#123—22no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—24no—
98C2×#121—21no—
99C2×#121—22no—
100C2×#121—24no—
101C2×#121C2312yes—
102C2×#121C2314yes—
103C2×C4×Q8C2311yescharacteristic
104C2×C4×D4—24no—
105C2×C4×D4C2311yescharacteristic
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×C4×D4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C42—21no—
112#486—44no—
113#481—44no—
114#476—44no—
115C2×#106—44no—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2312yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2314yes—
131C2×C4×C8C2312yes—
132#400—22no—
133#400—24no—
134#399—22no—
135#398—22no—
136#398—24no—
137#396—24no—
138#395—22no—
139#395—22no—
140#395—24no—
141#394—24no—
142#392—24no—
143#392—22no—
144#391—22no—
145#391—21no—
146#391D411yescharacteristic
147#390—21no—
148#390—22no—
149#390D411yescharacteristic
150#390C2314yes—
151#389—22no—
152#389—22no—
153#389C2314yes—
154#389C2314yes—
155#389C2314yes—
156#389C2314yes—
157#388—24no—
158#388—24no—
159#388D414yes—
160#387—24no—
161#386—22no—
162#386—24no—
163#386—24no—
164#385—24no—
165#385—22no—
166#385—24no—
167#384—22no—
168#384C2312yes—
169#384C2312yes—
170#382—22no—
171#382D411yescharacteristic
172#380C2312yes—
173#380C2314yes—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2311yescharacteristic
177C4×C22⋊C4—21no—
178C4×C22⋊C4—22no—
179C4×C22⋊C4C2314yes—
180C43C2311yescharacteristic
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2314yes—
188#336C2314yes—
189#336C2314yes—
190C2×#41—44no—
191C2×#41—44no—
192C2×#41—44no—
193C2×#41—44no—
194C2×#41—82no—
195C2×#41—82no—
196C22×Q8—21no—
197C22×Q8—22no—
198C22×Q8C2×D411yescharacteristic
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—21no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—21no—
205C22×D4C2×D411yescharacteristic
206C22×D4C2412yes—
207C22×D4C2411yescharacteristic
208C23×C4—21no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4C2412yes—
214C22×C8—44no—
215C22×C8—42no—
216C22×C8C2414yes—
217C22×C8C2414yes—
218C4⋊Q8—41no—
219C4⋊Q8—41no—
220C4⋊Q8—42no—
221C4⋊Q8—42no—
222C4⋊Q8—44no—
223#127—24no—
224#127—42no—
225#127—41no—
226#127—41no—
227#127—44no—
228#125—84no—
229#124—44no—
230#124—82no—
231#124SD1618yes—
232#123—48no—
233C22⋊Q8—44no—
234C22⋊Q8—88no—
235#121—24no—
236#121—28no—
237#121—48no—
238#121—44no—
239#121—42no—
240#121—44no—
241#121—416no—
242C4×Q8—41no—
243C4×Q8—41no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—42no—
249C4×D4—42no—
250C4×D4—41no—
251C4×D4—41no—
252C4×D4—416no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4C2×D411yescharacteristic
270C2×C4⋊C4C2×D411yescharacteristic
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2414yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2411yescharacteristic
276C2×C4⋊C4C2412yes—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D414yes—
288C2×C22⋊C4C2×D414yes—
289C2×C42—22no—
290C2×C42—21no—
291C2×C42—22no—
292C2×C42—44no—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D412yes—
295C2×C42C2×D411yescharacteristic
296C2×C42C2411yescharacteristic
297C2×C42C2412yes—
298C2×C42C2412yes—
299#106—28no—
300#106—28no—
301#106—44no—
302#106—44no—
303C4⋊C8—42no—
304C4⋊C8—42no—
305C4⋊C8—42no—
306#102—28no—
307#102—44no—
308#102—44no—
309#102—44no—
310#102—44no—
311#102—44no—
312C22⋊C8—44no—
313C22⋊C8—42no—
314C4×C8—42no—
315#95—22no—
316#95—21no—
317#95—22no—
318#95—24no—
319#95—22no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95C2×D414yes—
325#95C2×D414yes—
326C24—21no—
327C24—22no—
328C24C22×D412yes—
329#41—816no—
330#41—816no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—84no—
337C2×Q8—82no—
338C2×Q8C2×SD1614yes—
339C2×D4—21no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—24no—
344C2×D4—22no—
345C2×D4—24no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—48no—
349C2×D4—44no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—44no—
353C2×D4—42no—
354C2×D4—44no—
355C2×D4—48no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—42no—
359C2×D4—82no—
360C2×D4—82no—
361C2×D4—84no—
362C2×D4—84no—
363C2×D4—82no—
364C2×D4C2×SD1614yes—
365C2×D4H(5,𝔽2)12yes—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—84no—
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C2511yescharacteristic
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—48no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4—42no—
419C4⋊C4—48no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—84no—
423C4⋊C4—82no—
424C4⋊C4—82no—
425C4⋊C4—84no—
426C4⋊C4—82no—
427C4⋊C4H(5,𝔽2)12yes—
428C4⋊C42−1+414yes—
429C22⋊C4—24no—
430C22⋊C4—42no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—48no—
435C22⋊C4—44no—
436C22⋊C4—48no—
437C22⋊C4—84no—
438C22⋊C4—82no—
439C22⋊C4—84no—
440C22⋊C4C2×SD16116yes—
441C42—22no—
442C42—22no—
443C42—21no—
444C42—21no—
445C42—22no—
446C42—22no—
447C42—41no—
448C42—42no—
449C42—44no—
450C42—42no—
451C42—82no—
452C42C2×SD1614yes—
453C23—24no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—22no—
458C23—41no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23—42no—
463C23—44no—
464C23—42no—
465C23—44no—
466C23—42no—
467C23C22×SD1614yes—
468C23C23×D411yescharacteristic
469Q8—44no—
470Q8—84no—
471Q8—164no—
472D4—24no—
473D4—44no—
474D4—41no—
475D4—42no—
476D4—41no—
477D4—42no—
478D4—44no—
479D4—88no—
480D4—84no—
481D4—816no—
482D4—88no—
483D4—164no—
484D4—162no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—48no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—88no—
520C2×C4—88no—
521C2×C4—88no—
522C2×C4—164no—
523C2×C4C22×SD1614yes—
524C2×C4C22×SD1612yes—
525C2×C4C22×SD1614yes—
526C2×C4C2×H(5,𝔽2)11yescharacteristic
527C2×C4C2×H(5,𝔽2)11yescharacteristic
528C2×C4C2×2−1+412yes—
529C2×C4#58512yes—
530C8—44no—
531C8—44no—
532C22—24no—
533C22—28no—
534C22—44no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—82no—
541C22—82no—
542C22—84no—
543C22—82no—
544C22—84no—
545C22—82no—
546C22—82no—
547C22#348411yescharacteristic
548C22#348912yes—
549C22C23×SD1611yescharacteristic
550C22#358511yescharacteristic
551C22#358512yes—
552C4—22no—
553C4—24no—
554C4—24no—
555C4—42no—
556C4—42no—
557C4—44no—
558C4—42no—
559C4—41no—
560C4—42no—
561C4—84no—
562C4—82no—
563C4—84no—
564C4#352911yescharacteristic
565C4#355011yescharacteristic
566C4#355912yes—
567C2—44no—
568C2—82no—
569C2—162no—
570C2#4622811yescharacteristic
571C2#6156011yescharacteristic
572C2#6167311yescharacteristic
573C2#6209612yes—
574C2#6224212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36125C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #46152C2semidirect
3. #33068C2semidirect
4. #32819C2semidirect
5. #32683C2semidirect
6. #32028C2semidirect
7. #31490C2semidirect
8. #30631C2semidirect
9. #27032C2semidirect
10. #26338C2semidirect
11. #25649C2semidirect
12. #24089C2semidirect
13. #23346C2semidirect

Maximal subgroups

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

subgroupindex
#461522
#361252
#330682
#328192
#326832
#320282
#314902
#306312
#270322
#263382
#256492
#240892
#233462