Finite Group Databasebeta

← #7459828

Subgroups — #7459828

Order 512.

1501 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46147C211yesmaximal, characteristic
2#39617C211yesmaximal, characteristic
3#32683C212yesmaximal, complemented
4#32259C212yesmaximal
5#31688C214yesmaximal, complemented
6#31089C214yesmaximal, complemented
7#30305C212yesmaximal, complemented
8#29575C212yesmaximal, complemented
9#28847C214yesmaximal, complemented
10#26060C214yesmaximal, complemented
11#25837C212yesmaximal, complemented
12#24097C212yesmaximal
13#23269C211yesmaximal, characteristic
14C2×#563—21no—
15C2×Q82C2211yescharacteristic
16C2×#554C2211yescharacteristic
17C2×#498C2212yes—
18C2×#483C2212yes—
19C2×#475C2212yes—
20C22×#106C2212yes—
21#2868C2211yescharacteristic
22#2859C2212yes—
23#2822—21no—
24#2734—21no—
25#2724—21no—
26#2698—24no—
27#2677—24no—
28#2660—21no—
29#2567C2211yescharacteristic
30#2565—21no—
31#2563C2211yescharacteristic
32#2546C2212yes—
33#2542—21no—
34#2496C2214yes—
35#2486C2212yes—
36#2440C2214yes—
37#2440C2214yes—
38#2423C2214yes—
39#2326—21no—
40C4×#124C2211yescharacteristic
41#2295C2211yescharacteristic
42#2099C2212yes—
43#2096C2212yes—
44#2088C2214yes—
45#2084C2214yes—
46#2084C2214yes—
47#2084C2212yes—
48#2077C2214yes—
49#2077C2214yes—
50#2072C2212yes—
51#2058C2214yes—
52#2058C2214yes—
53#2058C2214yes—
54#2058C2214yes—
55#2035C2214yes—
56#1987C2212yes—
57#1984C2212yes—
58#1941C2214yes—
59#1941C2214yes—
60#1933C2214yes—
61#1923C2214yes—
62#1920C2214yes—
63#1919C2214yes—
64#1892C2212yes—
65#1869C2212yes—
66#1865C2214yes—
67#1865C2214yes—
68#1853C2214yes—
69#1853C2214yes—
70#1844C2214yes—
71#1834C2214yes—
72#1824C2212yes—
73#1807C2214yes—
74C4×#103C2212yes—
75C8×C22⋊C4C2214yes—
76#1744C2212yes—
77C23×C8C2312yes—
78#563—48no—
79Q82—42no—
80Q82—42no—
81#554—42no—
82#554—42no—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#126—24no—
92C2×#125—21no—
93C2×#125—22no—
94C2×#124D411yescharacteristic
95C2×#123—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8C2312yes—
100C2×C4×Q8—21no—
101C2×C4×Q8C2311yescharacteristic
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2311yescharacteristic
104C2×C4×D4—21no—
105C2×C4×D4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108#498—44no—
109#483—44no—
110#475—44no—
111C2×#106—44no—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×#106C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×C4⋊C8C2312yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×C22⋊C8C2314yes—
122C2×C22⋊C8C2312yes—
123C2×C4×C8C2314yes—
124C2×C4×C8C2312yes—
125#400—24no—
126#399—24no—
127#398—22no—
128#398—22no—
129#398—22no—
130#398—24no—
131#398—24no—
132#398—24no—
133#397—22no—
134#397—24no—
135#395—22no—
136#395—24no—
137#395—24no—
138#393—22no—
139#391—21no—
140#391—22no—
141#391D411yescharacteristic
142#391C2312yes—
143#390—21no—
144#390D411yescharacteristic
145#389—22no—
146#389—22no—
147#389C2314yes—
148#389C2314yes—
149#389C2314yes—
150#389C2314yes—
151#389C2312yes—
152#388—24no—
153#388D414yes—
154#387—22no—
155#387—24no—
156#387—24no—
157#385—24no—
158#385—24no—
159#384—21no—
160#384—21no—
161#384—21no—
162#384C2312yes—
163#384C2314yes—
164#384C2312yes—
165#382—21no—
166#382—22no—
167#382D411yescharacteristic
168#381—22no—
169#380C2314yes—
170#380C2314yes—
171#380C2312yes—
172C4×C4⋊C4—21no—
173C4×C4⋊C4C2311yescharacteristic
174C4×C4⋊C4C2312yes—
175C4×C22⋊C4C2314yes—
176#376—21no—
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—22no—
191C22×Q8—21no—
192C22×Q8—22no—
193C22×Q8—21no—
194C22×Q8C2×D411yescharacteristic
195C22×Q8C2411yescharacteristic
196C22×D4C2×D411yescharacteristic
197C23×C4—22no—
198C23×C4—22no—
199C23×C4C2412yes—
200C22×C8—28no—
201C22×C8—42no—
202C22×C8C2414yes—
203C22×C8C2414yes—
204C4⋊Q8—24no—
205C4⋊Q8—41no—
206C4⋊Q8—41no—
207C4⋊Q8—42no—
208C4⋊Q8—41no—
209C4⋊Q8—41no—
210C4⋊Q8—42no—
211C4⋊Q8—42no—
212C4⋊Q8—44no—
213C4⋊Q8—48no—
214C4⋊Q8—48no—
215C4⋊Q8—84no—
216#126—416no—
217#125—44no—
218#125—48no—
219#124SD1618yes—
220#123—48no—
221C22⋊Q8—24no—
222C22⋊Q8—42no—
223C22⋊Q8—48no—
224C22⋊Q8—416no—
225C22⋊Q8—88no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—24no—
229C4×Q8—41no—
230C4×Q8—41no—
231C4×Q8—42no—
232C4×Q8—44no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—44no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2414yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2411yescharacteristic
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4C2×D414yes—
274C2×C22⋊C4C2×D412yes—
275C2×C22⋊C4C2×D414yes—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—21no—
280C2×C42C2×D412yes—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2411yescharacteristic
284C2×C42C2412yes—
285C2×C42C2412yes—
286#106—28no—
287#106—28no—
288#106—44no—
289#106—44no—
290#106—44no—
291#106—44no—
292C4⋊C8—42no—
293#103—28no—
294#103—28no—
295C22⋊C8—28no—
296C22⋊C8—42no—
297C4×C8—28no—
298C4×C8—42no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—21no—
303#95—22no—
304#95—24no—
305#95—24no—
306#95—22no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95C2×D414yes—
312#95C2×D414yes—
313C24C22×D412yes—
314C2×Q8—21no—
315C2×Q8—24no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—44no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—44no—
322C2×Q8—84no—
323C2×Q8—82no—
324C2×Q8—82no—
325C2×Q8—82no—
326C2×Q8—82no—
327C2×Q8C2×SD1614yes—
328C2×Q8H(5,𝔽2)12yes—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4C2×SD1614yes—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—82no—
346C22×C4—84no—
347C22×C4—82no—
348C22×C4C22×D412yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D412yes—
351C22×C4C22×D412yes—
352C22×C4C22×D412yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D412yes—
355C22×C4C2511yescharacteristic
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—48no—
362C2×C82−1+418yes—
363C4⋊C4—24no—
364C4⋊C4—21no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—48no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—48no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—84no—
394C4⋊C4—82no—
395C4⋊C4—84no—
396C4⋊C4H(5,𝔽2)12yes—
397C22⋊C4—28no—
398C22⋊C4—42no—
399C22⋊C4—44no—
400C22⋊C4—48no—
401C22⋊C4—44no—
402C22⋊C4—84no—
403C22⋊C4C2×SD16116yes—
404C42—24no—
405C42—24no—
406C42—22no—
407C42—22no—
408C42—21no—
409C42—21no—
410C42—24no—
411C42—41no—
412C42—44no—
413C42—42no—
414C42—44no—
415C42C2×SD1614yes—
416C23—22no—
417C23—22no—
418C23—24no—
419C23—24no—
420C23C22×SD1614yes—
421C23C23×D411yescharacteristic
422Q8—24no—
423Q8—44no—
424Q8—42no—
425Q8—164no—
426Q8—164no—
427Q8—162no—
428Q8—162no—
429D4—44no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—164no—
465C2×C4—164no—
466C2×C4C22×SD1614yes—
467C2×C4C22×SD1614yes—
468C2×C4C22×SD1612yes—
469C2×C4#57714yes—
470C2×C4C2×H(5,𝔽2)11yescharacteristic
471C2×C4C2×H(5,𝔽2)11yescharacteristic
472C2×C4C2×2−1+414yes—
473C8—28no—
474C8—44no—
475C22—28no—
476C22—24no—
477C22—44no—
478C22—42no—
479C22—42no—
480C22#348411yescharacteristic
481C22#349912yes—
482C22C23×SD1611yescharacteristic
483C22C2×#57712yes—
484C22#358611yescharacteristic
485C4—24no—
486C4—24no—
487C4—22no—
488C4—24no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—41no—
493C4—42no—
494C4—84no—
495C4—82no—
496C4—84no—
497C4—82no—
498C4—82no—
499C4#353812yes—
500C4#354412yes—
501C2—44no—
502C2#5154211yescharacteristic
503C2#6156211yescharacteristic
504C2#6166211yescharacteristic
505C2#6246912yes—
506C2#6251312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #32683C2semidirect
5. #31688C2semidirect
6. #31089C2semidirect
7. #30305C2semidirect
8. #29575C2semidirect
9. #28847C2semidirect
10. #26060C2semidirect
11. #25837C2semidirect

Maximal subgroups

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

subgroupindex
#461472
#396172
#326832
#322592
#316882
#310892
#303052
#295752
#288472
#260602
#258372
#240972
#232692