Finite Group Databasebeta

← #7228783

Subgroups — #7228783

Order 512.

1551 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46147C211yesmaximal, characteristic, complemented
2#36164C211yesmaximal, characteristic
3#33066C212yesmaximal, complemented
4#32820C212yesmaximal, complemented
5#32644C212yesmaximal, complemented
6#32270C212yesmaximal, complemented
7#32217C214yesmaximal, complemented
8#30803C214yesmaximal, complemented
9#27542C214yesmaximal, complemented
10#25978C214yesmaximal, complemented
11#25851C212yesmaximal, complemented
12#24054C212yesmaximal
13#23353C211yesmaximal, characteristic, complemented
14C2×#563—21no—
15C2×#554C2211yescharacteristic
16C2×#547C2211yescharacteristic
17C2×#501C2212yes—
18C2×#484C2212yes—
19C2×#476C2212yes—
20C2×#417C2212yes—
21#2862C2211yescharacteristic
22#2859C2212yes—
23#2850—21no—
24#2822—21no—
25#2734—21no—
26#2724—21no—
27#2698—24no—
28#2568C2211yescharacteristic
29#2563C2211yescharacteristic
30#2546C2212yes—
31#2504—21no—
32#2503—21no—
33#2496C2214yes—
34#2489C2214yes—
35#2488—24no—
36#2486C2212yes—
37#2440C2214yes—
38#2339C2214yes—
39C4×#124C2211yescharacteristic
40C2×#382—21no—
41C42×D4C2211yescharacteristic
42#2096C2214yes—
43#2077C2214yes—
44#2076C2214yes—
45#2072C2214yes—
46#2068C2214yes—
47#2066C2214yes—
48#2066C2214yes—
49#2059C2214yes—
50#2058C2214yes—
51#2043C2214yes—
52#2036C2214yes—
53#2024C2214yes—
54#1989C2212yes—
55#1985C2212yes—
56#1947C2212yes—
57#1947C2212yes—
58#1935C2214yes—
59#1928C2214yes—
60#1924C2214yes—
61#1918C2214yes—
62#1918C2214yes—
63#1894C2212yes—
64#1870C2212yes—
65#1864C2214yes—
66#1862C2214yes—
67#1862C2214yes—
68#1853C2214yes—
69#1851C2214yes—
70#1831C2212yes—
71#1809C2214yes—
72#1802C2212yes—
73#1802C2212yes—
74#1788C2214yes—
75#1788C2212yes—
76#1762C2212yes—
77C22×#41C2312yes—
78C22×M16C2312yes—
79#563—48no—
80#554—42no—
81#554—42no—
82#547—42no—
83#547—42no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×#126—24no—
89C2×#125—22no—
90C2×#125C2312yes—
91C2×#124D411yescharacteristic
92C2×#123—22no—
93C2×#123—22no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8C2312yes—
98C2×#121—21no—
99C2×C4×Q8C2311yescharacteristic
100C2×C4×D4C2311yescharacteristic
101C2×C4×D4C2312yes—
102C2×C4×D4C2312yes—
103C2×C4×D4C2311yescharacteristic
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2312yes—
107C22×C42—21no—
108#501—44no—
109#484—44no—
110#476—44no—
111C2×#107C2314yes—
112C2×#107C2314yes—
113C2×#106C2314yes—
114C2×#106C2314yes—
115C2×C4⋊C8C2312yes—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2312yes—
118#417—44no—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×C22⋊C8C2312yes—
126C2×C22⋊C8C2314yes—
127C2×#97C2312yes—
128#400—22no—
129#400—24no—
130#400—24no—
131#398—22no—
132#398—22no—
133#398—22no—
134#398—24no—
135#397—22no—
136#397—24no—
137#397—24no—
138#395—24no—
139#393—22no—
140#391—21no—
141#391—22no—
142#391—22no—
143#391D411yescharacteristic
144#391C2314yes—
145#391C2314yes—
146#390—21no—
147#390D411yescharacteristic
148#389—22no—
149#389—22no—
150#389C2314yes—
151#389C2314yes—
152#389C2314yes—
153#388—24no—
154#388D414yes—
155#387—24no—
156#386—24no—
157#386—24no—
158#385—24no—
159#384C2312yes—
160#384C2312yes—
161#382—21no—
162#382—21no—
163#382—21no—
164#382—22no—
165#382—21no—
166#382—24no—
167#382D411yescharacteristic
168#381—22no—
169#381—24no—
170#380C2314yes—
171#380C2314yes—
172#380C2312yes—
173C4×C4⋊C4C2311yescharacteristic
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2312yes—
176C4×C22⋊C4—22no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4C2314yes—
179C2×#95—24no—
180C43C2311yescharacteristic
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2314yes—
188#336C2314yes—
189#336C2314yes—
190C2×#41—24no—
191C2×#41—28no—
192C2×#41—42no—
193C2×#41—44no—
194C22×Q8—22no—
195C22×Q8—21no—
196C22×Q8C2×D411yescharacteristic
197C22×Q8C2412yes—
198C22×D4—22no—
199C22×D4C2×D411yescharacteristic
200C22×D4C2411yescharacteristic
201C23×C4—21no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4C2412yes—
207C2×M16—44no—
208C2×M16—42no—
209C22×C8C2414yes—
210C22×C8C2414yes—
211C4⋊Q8—41no—
212C4⋊Q8—41no—
213C4⋊Q8—42no—
214C4⋊Q8—42no—
215C4⋊Q8—41no—
216C4⋊Q8—41no—
217C4⋊Q8—48no—
218#126—416no—
219#125—24no—
220#125—42no—
221#125—48no—
222#124—24no—
223#123—48no—
224#123—48no—
225C22⋊Q8—24no—
226C22⋊Q8—48no—
227C22⋊Q8—416no—
228C22⋊Q8—42no—
229C22⋊Q8—44no—
230#121—44no—
231C4×Q8—41no—
232C4×Q8—41no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—24no—
236C4×D4—24no—
237C4×D4—42no—
238C4×D4—42no—
239C4×D4—41no—
240C4×D4—41no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2411yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4C2×D414yes—
276C2×C22⋊C4C2×D412yes—
277C2×C22⋊C4C2×D414yes—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—21no—
281C2×C42—44no—
282C2×C42C2×D412yes—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2411yescharacteristic
286C2×C42C2412yes—
287C2×C42C2412yes—
288#107—28no—
289#107—44no—
290#106—44no—
291#106—44no—
292C4⋊C8—42no—
293C4⋊C8—42no—
294C4⋊C8—42no—
295#103—44no—
296#103—44no—
297#103—44no—
298#103—44no—
299#102—28no—
300#102—28no—
301C22⋊C8—44no—
302C22⋊C8—42no—
303#97—42no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—21no—
308#95—22no—
309#95—22no—
310#95—24no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95C2×D414yes—
318#95C2×D414yes—
319C24—21no—
320C24C22×D412yes—
321#41—48no—
322#41—84no—
323#41—88no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—42no—
332C2×Q8—44no—
333C2×Q8—82no—
334C2×Q8—82no—
335C2×Q8—82no—
336C2×D4—22no—
337C2×D4—21no—
338C2×D4—22no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—44no—
342C2×D4—48no—
343C2×D4H(5,𝔽2)12yes—
344C2×D42−1+414yes—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—48no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—82no—
365C22×C4—84no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4C22×D412yes—
369C22×C4C22×D412yes—
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C2511yescharacteristic
376M16—84no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C4⋊C4—21no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—48no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—48no—
405C4⋊C4—48no—
406C4⋊C4—44no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C4⋊C4H(5,𝔽2)12yes—
411C22⋊C4—24no—
412C22⋊C4—28no—
413C22⋊C4—44no—
414C22⋊C4—42no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—48no—
419C22⋊C4—44no—
420C22⋊C4—84no—
421C42—22no—
422C42—22no—
423C42—21no—
424C42—21no—
425C42—22no—
426C42—22no—
427C42—41no—
428C42—42no—
429C42—44no—
430C42—44no—
431C42—42no—
432C42AGL(1,ℤ/8)14yes—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—22no—
440C23—41no—
441C23—42no—
442C23—41no—
443C23C23×D411yescharacteristic
444Q8—42no—
445Q8—42no—
446Q8—44no—
447Q8—82no—
448Q8—164no—
449Q8—162no—
450D4—22no—
451D4—24no—
452D4—82no—
453D4—88no—
454D4#56314yescomplemented
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—48no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—88no—
492C2×C4—88no—
493C2×C4—164no—
494C2×C4—164no—
495C2×C4C2×AGL(1,ℤ/8)14yes—
496C2×C4C2×AGL(1,ℤ/8)12yes—
497C2×C4C2×H(5,𝔽2)11yescharacteristic
498C2×C4C2×H(5,𝔽2)11yescharacteristic
499C2×C4C2×2−1+412yes—
500C2×C4#58512yes—
501C8—44no—
502C8—44no—
503C22—24no—
504C22—44no—
505C22—42no—
506C22—44no—
507C22—42no—
508C22—44no—
509C22—42no—
510C22—42no—
511C22—44no—
512C22—42no—
513C22#348411yescharacteristic
514C22#348912yes—
515C22C22×AGL(1,ℤ/8)11yescharacteristic
516C22#358611yescharacteristic
517C22#358612yes—
518C4—22no—
519C4—24no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—41no—
526C4—82no—
527C4—84no—
528C4—84no—
529C4—82no—
530C4—82no—
531C4C2×#56311yescharacteristic
532C4#354511yescharacteristic
533C4#356112yes—
534C2—44no—
535C2—82no—
536C2#4613611yescharacteristic
537C2#6158511yescharacteristic
538C2#6167411yescharacteristic
539C2#6210712yes—
540C2#6223712yes—
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. #33066C2semidirect
4. #32820C2semidirect
5. #32644C2semidirect
6. #32270C2semidirect
7. #32217C2semidirect
8. #30803C2semidirect
9. #27542C2semidirect
10. #25978C2semidirect
11. #25851C2semidirect
13. #23353C2semidirect
454. D4#563semidirect

Maximal subgroups

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

subgroupindex
#461472
#361642
#330662
#328202
#326442
#322702
#322172
#308032
#275422
#259782
#258512
#240542
#233532