Finite Group Databasebeta

← #7459858

Subgroups — #7459858

Order 512.

1505 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46147C211yesmaximal, characteristic, complemented
2#39489C211yesmaximal, characteristic
3#32694C212yesmaximal, complemented
4#32270C212yesmaximal, complemented
5#31691C214yesmaximal, complemented
6#31086C214yesmaximal, complemented
7#30305C212yesmaximal, complemented
8#29572C212yesmaximal, complemented
9#28869C214yesmaximal, complemented
10#26078C214yesmaximal, complemented
11#25851C212yesmaximal, complemented
12#24104C212yesmaximal
13#23272C211yesmaximal, characteristic, complemented
14C2×#563—21no—
15C2×#555C2211yescharacteristic
16C2×#554C2211yescharacteristic
17C2×#501C2212yes—
18C2×#481C2212yes—
19C2×#476C2212yes—
20C2×#428C2212yes—
21#2868C2211yescharacteristic
22#2859C2212yes—
23#2822—21no—
24#2734—21no—
25#2724—21no—
26#2698—24no—
27#2674—24no—
28#2620—21no—
29#2567C2211yescharacteristic
30#2563C2211yescharacteristic
31#2546C2212yes—
32#2501—21no—
33#2499—21no—
34#2496C2214yes—
35#2486C2212yes—
36#2441C2214yes—
37#2440C2214yes—
38#2422C2214yes—
39#2317—21no—
40C4×#124C2211yescharacteristic
41#2293C2211yescharacteristic
42#2099C2212yes—
43#2093C2212yes—
44#2088C2214yes—
45#2084C2214yes—
46#2084C2214yes—
47#2083C2212yes—
48#2077C2214yes—
49#2076C2214yes—
50#2071C2212yes—
51#2059C2214yes—
52#2058C2214yes—
53#2058C2214yes—
54#2058C2214yes—
55#2034C2214yes—
56#1989C2212yes—
57#1985C2212yes—
58#1943C2214yes—
59#1942C2214yes—
60#1934C2214yes—
61#1923C2214yes—
62#1921C2214yes—
63#1918C2214yes—
64#1895C2212yes—
65#1870C2212yes—
66#1864C2214yes—
67#1862C2214yes—
68#1854C2214yes—
69#1853C2214yes—
70#1844C2214yes—
71#1834C2214yes—
72#1825C2212yes—
73#1809C2214yes—
74#1762C2212yes—
75#1753C2214yes—
76#1744C2212yes—
77C22×M16C2312yes—
78#563—48no—
79#555—42no—
80#555—42no—
81#554—42no—
82#554—42no—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×#126—24no—
89C2×#125—22no—
90C2×#125C2312yes—
91C2×#124—22no—
92C2×#124D411yescharacteristic
93C2×#123—22no—
94C2×#123—24no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—24no—
98C2×#121—21no—
99C2×#121C2312yes—
100C2×C4×Q8C2311yescharacteristic
101C2×C4×D4C2311yescharacteristic
102C2×C4×D4C2312yes—
103C2×C4×D4C2311yescharacteristic
104C2×#117—21no—
105C2×#117—21no—
106C2×#117C2312yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4C2312yes—
109#501—44no—
110#481—44no—
111#476—44no—
112#428—44no—
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×C4⋊C8C2312yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×C22⋊C8C2314yes—
123C2×C22⋊C8C2312yes—
124C2×#97C2312yes—
125C2×C4×C8C2314yes—
126#400—24no—
127#399—24no—
128#398—22no—
129#398—22no—
130#398—22no—
131#398—24no—
132#397—22no—
133#397—24no—
134#397—24no—
135#396—24no—
136#395—22no—
137#395—24no—
138#394—24no—
139#393—22no—
140#391—21no—
141#391—22no—
142#391D411yescharacteristic
143#390—21no—
144#390D411yescharacteristic
145#390C2312yes—
146#389—22no—
147#389—22no—
148#389—24no—
149#389C2314yes—
150#389C2314yes—
151#389C2314yes—
152#389C2312yes—
153#389C2314yes—
154#388—22no—
155#388D414yes—
156#387—24no—
157#387—24no—
158#385—24no—
159#385—24no—
160#384—21no—
161#384—21no—
162#384—21no—
163#384C2312yes—
164#384C2314yes—
165#384C2312yes—
166#382—21no—
167#382—22no—
168#382D411yescharacteristic
169#381—22no—
170#380C2314yes—
171#380C2314yes—
172#380C2312yes—
173C4×C4⋊C4C2311yescharacteristic
174C4×C4⋊C4C2312yes—
175C4×C22⋊C4—21no—
176C4×C22⋊C4C2314yes—
177C43C2311yescharacteristic
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#336C2314yes—
187#336C2314yes—
188#336C2312yes—
189#336C2312yes—
190C22×Q8—21no—
191C22×Q8—22no—
192C22×Q8—21no—
193C22×Q8C2×D411yescharacteristic
194C22×D4—22no—
195C22×D4C2×D411yescharacteristic
196C22×D4C2411yescharacteristic
197C23×C4—21no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4C2412yes—
201C2×M16—28no—
202C2×M16—42no—
203C22×C8C2414yes—
204C22×C8C2414yes—
205C4⋊Q8—41no—
206C4⋊Q8—41no—
207C4⋊Q8—41no—
208C4⋊Q8—41no—
209C4⋊Q8—42no—
210C4⋊Q8—42no—
211C4⋊Q8—48no—
212C4⋊Q8—84no—
213#126—416no—
214#125—24no—
215#125—42no—
216#125—48no—
217#124—24no—
218#124—48no—
219#123—48no—
220#123—88no—
221C22⋊Q8—48no—
222C22⋊Q8—416no—
223C22⋊Q8—44no—
224#121—24no—
225#121—42no—
226#121—44no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—24no—
232C4×D4—42no—
233C4×D4—41no—
234C4×D4—41no—
235#117—24no—
236#117—42no—
237#117—44no—
238#117—44no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2412yes—
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—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4C2×D414yes—
278C2×C22⋊C4C2×D414yes—
279C2×C22⋊C4C2×D412yes—
280C2×C42—22no—
281C2×C42—21no—
282C2×C42—22no—
283C2×C42C2×D412yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2411yescharacteristic
287C2×C42C2412yes—
288C2×C42C2412yes—
289#107—28no—
290#107—28no—
291#106—44no—
292#106—44no—
293#106—44no—
294#106—44no—
295C4⋊C8—42no—
296#102—28no—
297#102—28no—
298C22⋊C8—28no—
299C22⋊C8—42no—
300#97—42no—
301C4×C8—28no—
302#95—22no—
303#95—21no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95C2×D414yes—
313#95C2×D414yes—
314C24—21no—
315C24C22×D412yes—
316C2×Q8—22no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—42no—
320C2×Q8—44no—
321C2×Q8—42no—
322C2×Q8—82no—
323C2×Q8—82no—
324C2×Q8—82no—
325C2×Q8—82no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—24no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—44no—
332C2×D4—84no—
333C2×D4H(5,𝔽2)12yes—
334C22×C4—24no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—84no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4C22×D412yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D412yes—
355C22×C4C22×D412yes—
356C22×C4C22×D412yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D412yes—
359C22×C4C2511yescharacteristic
360M16—48no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C82−1+418yes—
367C4⋊C4—21no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—48no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—48no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—84no—
395C4⋊C4—82no—
396C4⋊C4—84no—
397C4⋊C4H(5,𝔽2)12yes—
398C22⋊C4—24no—
399C22⋊C4—28no—
400C22⋊C4—28no—
401C22⋊C4—42no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—48no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—84no—
410C42—24no—
411C42—22no—
412C42—22no—
413C42—21no—
414C42—21no—
415C42—24no—
416C42—41no—
417C42—44no—
418C42—44no—
419C42—42no—
420C42AGL(1,ℤ/8)14yes—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—22no—
426C23—24no—
427C23—41no—
428C23—42no—
429C23—41no—
430C23—42no—
431C23C23×D411yescharacteristic
432Q8—82no—
433Q8—164no—
434Q8—162no—
435Q8—162no—
436D4—24no—
437D4—42no—
438D4—82no—
439D4—164no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—48no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—164no—
476C2×C4—164no—
477C2×C4C2×AGL(1,ℤ/8)12yes—
478C2×C4C2×AGL(1,ℤ/8)14yes—
479C2×C4#57714yes—
480C2×C4C2×H(5,𝔽2)11yescharacteristic
481C2×C4C2×H(5,𝔽2)11yescharacteristic
482C2×C4C2×2−1+414yes—
483C8—28no—
484C8—44no—
485C22—44no—
486C22—44no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—82no—
494C22—82no—
495C22#348411yescharacteristic
496C22#349912yes—
497C22C22×AGL(1,ℤ/8)11yescharacteristic
498C22C2×#57712yes—
499C22#358611yescharacteristic
500C4—24no—
501C4—24no—
502C4—22no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—41no—
508C4—84no—
509C4—82no—
510C4—84no—
511C4—82no—
512C4—82no—
513C4#353812yes—
514C4#354412yes—
515C2—82no—
516C2—82no—
517C2#5154211yescharacteristic
518C2#6158511yescharacteristic
519C2#6166211yescharacteristic
520C2#6247712yes—
521C2#6251312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
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. #32694C2semidirect
4. #32270C2semidirect
5. #31691C2semidirect
6. #31086C2semidirect
7. #30305C2semidirect
8. #29572C2semidirect
9. #28869C2semidirect
10. #26078C2semidirect
11. #25851C2semidirect
13. #23272C2semidirect

Maximal subgroups

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

subgroupindex
#461472
#394892
#326942
#322702
#316912
#310862
#303052
#295722
#288692
#260782
#258512
#241042
#232722