Finite Group Databasebeta

← #7459824

Subgroups — #7459824

Order 512.

1681 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46147C211yesmaximal, characteristic, complemented
2#39360C211yesmaximal, characteristic
3#32682C212yesmaximal, complemented
4#32258C212yesmaximal, complemented
5#31686C214yesmaximal, complemented
6#31078C214yesmaximal, complemented
7#30304C212yesmaximal, complemented
8#29572C212yesmaximal, complemented
9#28841C214yesmaximal, complemented
10#26065C214yesmaximal, complemented
11#25838C212yesmaximal, complemented
12#24088C212yesmaximal
13#23269C211yesmaximal, characteristic, complemented
14C2×#563—21no—
15C2×D4×Q8C2211yescharacteristic
16C2×#547C2211yescharacteristic
17C2×#536—21no—
18C2×#500C2212yes—
19C2×#480C2212yes—
20C2×#474C2212yes—
21C22×#107C2212yes—
22#2868C2211yescharacteristic
23#2859C2212yes—
24#2822—21no—
25#2734—21no—
26#2724—21no—
27#2698—24no—
28#2680—24no—
29#2621—21no—
30#2547C2211yescharacteristic
31#2546C2212yes—
32#2546C2211yescharacteristic
33#2496C2214yes—
34#2486C2212yes—
35#2440C2214yes—
36#2440C2214yes—
37#2434—21no—
38#2432—21no—
39#2422C2214yes—
40C4×#124C2211yescharacteristic
41#2293C2211yescharacteristic
42#2099C2212yes—
43#2093C2212yes—
44#2086C2214yes—
45#2083C2214yes—
46#2083C2214yes—
47#2083C2212yes—
48#2075C2214yes—
49#2075C2214yes—
50#2071C2212yes—
51#2057C2214yes—
52#2057C2214yes—
53#2057C2214yes—
54#2057C2214yes—
55#2033C2214yes—
56#1988C2212yes—
57#1984C2212yes—
58#1942C2214yes—
59#1942C2214yes—
60#1934C2214yes—
61#1923C2214yes—
62#1918C2214yes—
63#1918C2214yes—
64#1893C2212yes—
65#1869C2212yes—
66#1862C2214yes—
67#1862C2214yes—
68#1852C2214yes—
69#1852C2214yes—
70#1845C2214yes—
71#1834C2214yes—
72#1823C2212yes—
73#1808C2214yes—
74C4×#102C2212yes—
75C8×C22⋊C4C2214yes—
76#1744C2212yes—
77C22×#41—21no—
78C23×Q8—21no—
79C23×C8C2312yes—
80#563—48no—
81D4×Q8—42no—
82D4×Q8—42no—
83#547—42no—
84#547—42no—
85#536—48no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2312yes—
94C2×#126—24no—
95C2×#125—22no—
96C2×#124—21no—
97C2×#124D411yescharacteristic
98C2×#123—22no—
99C2×#123—24no—
100C2×#123—22no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—24no—
106C2×#121C2312yes—
107C2×C4×Q8C2311yescharacteristic
108C2×C4×D4C2311yescharacteristic
109C2×C4×D4C2312yes—
110C2×C4×D4C2311yescharacteristic
111C2×#117—21no—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114#500—44no—
115#480—44no—
116#474—44no—
117C2×#107—44no—
118C2×#107C2314yes—
119C2×#107C2314yes—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×C4⋊C8C2312yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2312yes—
129C2×C4×C8C2314yes—
130C2×C4×C8C2312yes—
131#400—24no—
132#398—22no—
133#398—22no—
134#398—22no—
135#398—22no—
136#398—24no—
137#398—24no—
138#397—22no—
139#397—22no—
140#397—24no—
141#397—24no—
142#395—24no—
143#395—24no—
144#394—24no—
145#391—21no—
146#391—21no—
147#391—22no—
148#391D411yescharacteristic
149#390D411yescharacteristic
150#390C2312yes—
151#389—22no—
152#389—22no—
153#389C2314yes—
154#389C2314yes—
155#389C2314yes—
156#389C2314yes—
157#389C2312yes—
158#388—22no—
159#388D414yes—
160#387—24no—
161#385—24no—
162#385—24no—
163#385—24no—
164#384—22no—
165#384C2312yes—
166#384C2314yes—
167#384C2312yes—
168#382—22no—
169#382D411yescharacteristic
170#381—22no—
171#380C2314yes—
172#380C2314yes—
173#380C2312yes—
174C4×C4⋊C4C2311yescharacteristic
175C4×C4⋊C4C2312yes—
176C4×C22⋊C4—21no—
177C4×C22⋊C4C2314yes—
178C43C2311yescharacteristic
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2314yes—
188#336C2314yes—
189#336C2312yes—
190#336C2312yes—
191C2×#41—44no—
192C2×#41—44no—
193C2×#41—82no—
194C22×Q8—21no—
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—44no—
199C22×Q8—44no—
200C22×Q8—82no—
201C22×Q8C2×D411yescharacteristic
202C22×D4—22no—
203C22×D4C2×D411yescharacteristic
204C22×D4C2411yescharacteristic
205C23×C4—21no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4C2412yes—
210C22×C8—28no—
211C22×C8—42no—
212C22×C8C2414yes—
213C22×C8C2414yes—
214C4⋊Q8—24no—
215C4⋊Q8—41no—
216C4⋊Q8—41no—
217C4⋊Q8—42no—
218C4⋊Q8—41no—
219C4⋊Q8—41no—
220C4⋊Q8—42no—
221C4⋊Q8—42no—
222C4⋊Q8—44no—
223C4⋊Q8—48no—
224C4⋊Q8—44no—
225C4⋊Q8—82no—
226#126—416no—
227#125—48no—
228#124—44no—
229#124D818yes—
230#123—48no—
231#123—48no—
232#123—416no—
233C22⋊Q8—48no—
234C22⋊Q8—44no—
235C22⋊Q8—44no—
236C22⋊Q8—416no—
237C22⋊Q8—416no—
238#121—24no—
239#121—42no—
240C4×Q8—41no—
241C4×Q8—41no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—24no—
245C4×D4—42no—
246C4×D4—41no—
247C4×D4—41no—
248#117—44no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4C2×D411yescharacteristic
271C2×C4⋊C4C2×D411yescharacteristic
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2414yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2411yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4C2×D414yes—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D414yes—
290C2×C42—21no—
291C2×C42—22no—
292C2×C42C2×D412yes—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2411yescharacteristic
296C2×C42C2412yes—
297C2×C42C2412yes—
298#107—28no—
299#107—28no—
300#107—44no—
301#107—44no—
302#107—44no—
303#107—44no—
304C4⋊C8—42no—
305#102—28no—
306#102—28no—
307C22⋊C8—28no—
308C22⋊C8—42no—
309C4×C8—28no—
310C4×C8—42no—
311#95—22no—
312#95—21no—
313#95—22no—
314#95—24no—
315#95—24no—
316#95—22no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95C2×D414yes—
322#95C2×D414yes—
323C24—21no—
324C24C22×D412yes—
325#41—816no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—44no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—44no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—44no—
337C2×Q8—82no—
338C2×Q8—82no—
339C2×Q8—82no—
340C2×Q8—816no—
341C2×Q8C2×D814yes—
342C2×D4—21no—
343C2×D4—24no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—82no—
349C2×D4C2×D814yes—
350C2×D4H(5,𝔽2)12yes—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D412yes—
380C22×C4C2511yescharacteristic
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—48no—
387C2×C82−1+418yes—
388C4⋊C4—21no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—84no—
418C4⋊C4H(5,𝔽2)12yes—
419C22⋊C4—24no—
420C22⋊C4—28no—
421C22⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4C2×D8116yes—
430C42—24no—
431C42—22no—
432C42—22no—
433C42—21no—
434C42—21no—
435C42—24no—
436C42—41no—
437C42—42no—
438C42—44no—
439C42C2×D814yes—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—24no—
444C23—22no—
445C23—41no—
446C23—42no—
447C23—41no—
448C23—42no—
449C23C22×D814yes—
450C23C23×D411yescharacteristic
451Q8—44no—
452Q8—88no—
453Q8—84no—
454Q8—84no—
455Q8—164no—
456D4—24no—
457D4—44no—
458D4—42no—
459D4—88no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—48no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—88no—
497C2×C4—88no—
498C2×C4—164no—
499C2×C4C22×D812yes—
500C2×C4C22×D814yes—
501C2×C4C22×D814yes—
502C2×C4#57814yes—
503C2×C4C2×H(5,𝔽2)11yescharacteristic
504C2×C4C2×H(5,𝔽2)11yescharacteristic
505C2×C4C2×2−1+414yes—
506C8—28no—
507C8—44no—
508C22—28no—
509C22—24no—
510C22—44no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—82no—
517C22—82no—
518C22#348411yescharacteristic
519C22#349912yes—
520C22C23×D811yescharacteristic
521C22C2×#57812yes—
522C22#358511yescharacteristic
523C4—24no—
524C4—24no—
525C4—22no—
526C4—24no—
527C4—42no—
528C4—42no—
529C4—41no—
530C4—42no—
531C4—82no—
532C4—84no—
533C4—82no—
534C4—82no—
535C4—84no—
536C4#353812yes—
537C4#354412yes—
538C2—44no—
539C2—82no—
540C2#5154211yescharacteristic
541C2#6155911yescharacteristic
542C2#6166511yescharacteristic
543C2#6247112yes—
544C2#6251612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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. #46147C2semidirect
3. #32682C2semidirect
4. #32258C2semidirect
5. #31686C2semidirect
6. #31078C2semidirect
7. #30304C2semidirect
8. #29572C2semidirect
9. #28841C2semidirect
10. #26065C2semidirect
11. #25838C2semidirect
13. #23269C2semidirect

Maximal subgroups

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

subgroupindex
#461472
#393602
#326822
#322582
#316862
#310782
#303042
#295722
#288412
#260652
#258382
#240882
#232692