Finite Group Databasebeta

← #7231156

Subgroups — #7231156

Order 512.

1495 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46152C211yesmaximal, characteristic, complemented
2#36406C211yesmaximal, characteristic
3#33072C212yesmaximal, complemented
4#32820C212yesmaximal, complemented
5#32694C212yesmaximal, complemented
6#32029C212yesmaximal, complemented
7#31498C214yesmaximal, complemented
8#30655C214yesmaximal, complemented
9#27053C214yesmaximal, complemented
10#26340C214yesmaximal, complemented
11#25662C212yesmaximal, complemented
12#24089C212yesmaximal
13#23352C211yesmaximal, characteristic, complemented
14C2×#556C2211yescharacteristic
15C2×#547C2211yescharacteristic
16C2×#489C2212yes—
17C2×#481C2212yes—
18C2×#475C2212yes—
19C22×#106C2212yes—
20#2862C2211yescharacteristic
21#2859C2212yes—
22#2857—21no—
23#2844—21no—
24#2809—21no—
25#2720—24no—
26#2683—21no—
27#2567C2212yes—
28#2553—24no—
29#2546C2211yescharacteristic
30#2542C2211yescharacteristic
31#2499—21no—
32#2497C2214yes—
33#2487C2212yes—
34#2440C2214yes—
35#2432—21no—
36#2427C2214yes—
37#2379—21no—
38#2360C2214yes—
39C4×#124C2211yescharacteristic
40#2297—21no—
41C42×D4C2211yescharacteristic
42#2096C2214yes—
43#2088C2214yes—
44#2072C2214yes—
45#2066C2214yes—
46#2066C2214yes—
47#2058C2214yes—
48#2058C2214yes—
49#2043C2214yes—
50#2042C2214yes—
51#2036C2214yes—
52#2036C2214yes—
53#2034C2214yes—
54#1985C2212yes—
55#1983C2212yes—
56#1948C2212yes—
57#1947C2212yes—
58#1943C2214yes—
59#1924C2214yes—
60#1921C2214yes—
61#1919C2214yes—
62#1918C2214yes—
63#1893C2212yes—
64#1892C2214yes—
65#1872C2214yes—
66#1869C2212yes—
67#1865C2214yes—
68#1864C2214yes—
69#1862C2214yes—
70#1853C2214yes—
71#1850C2214yes—
72#1831C2212yes—
73#1823C2212yes—
74#1802C2212yes—
75#1802C2212yes—
76#1763C2212yes—
77C22×#41—21no—
78C23×C8C2312yes—
79#556—42no—
80#556—42no—
81#547—42no—
82#547—42no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2311yescharacteristic
87C2×#126—22no—
88C2×#125—22no—
89C2×#124D411yescharacteristic
90C2×#124C2312yes—
91C2×#123—22no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8C2314yes—
96C2×#121C2312yes—
97C2×C4×Q8C2312yes—
98C2×C4×Q8C2311yescharacteristic
99C2×C4×D4C2311yescharacteristic
100C2×C4×D4C2312yes—
101C2×C4×D4C2311yescharacteristic
102C2×#117—21no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C42—21no—
106#489—44no—
107#481—44no—
108#475—44no—
109C2×#107C2314yes—
110C2×#106—44no—
111C2×#106C2314yes—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116C2×C4⋊C8C2312yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×C22⋊C8C2312yes—
124C2×C22⋊C8C2314yes—
125C2×#97C2312yes—
126#400—22no—
127#400—24no—
128#399—22no—
129#399—22no—
130#398—22no—
131#398—24no—
132#397—24no—
133#396—24no—
134#395—22no—
135#395—22no—
136#395—24no—
137#394—24no—
138#391—22no—
139#391—21no—
140#391—21no—
141#391—21no—
142#391D411yescharacteristic
143#391C2314yes—
144#390—21no—
145#390D411yescharacteristic
146#389—22no—
147#389—22no—
148#389—24no—
149#389C2314yes—
150#389C2314yes—
151#389C2314yes—
152#389C2314yes—
153#388—24no—
154#388D414yes—
155#387—24no—
156#387—24no—
157#386—22no—
158#386—24no—
159#385—24no—
160#385—24no—
161#385—22no—
162#384—22no—
163#384—21no—
164#384C2312yes—
165#384C2312yes—
166#382—22no—
167#382—22no—
168#382D411yescharacteristic
169#380—24no—
170#380C2312yes—
171#380C2314yes—
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2311yescharacteristic
175C4×C22⋊C4—24no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4—22no—
178C4×C22⋊C4C2314yes—
179C43C2311yescharacteristic
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#336C2314yes—
187#336C2314yes—
188#336C2314yes—
189C2×#41—44no—
190C2×#41—44no—
191C2×#41—82no—
192C22×Q8—21no—
193C22×Q8—22no—
194C22×Q8C2×D411yescharacteristic
195C22×Q8C2412yes—
196C22×D4—22no—
197C22×D4C2×D411yescharacteristic
198C22×D4C2411yescharacteristic
199C23×C4—21no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4C2412yes—
204C22×C8—44no—
205C22×C8—42no—
206C22×C8C2414yes—
207C22×C8C2414yes—
208C4⋊Q8—41no—
209C4⋊Q8—41no—
210C4⋊Q8—42no—
211C4⋊Q8—42no—
212C4⋊Q8—41no—
213C4⋊Q8—41no—
214C4⋊Q8—82no—
215#126—48no—
216#125—84no—
217#124—24no—
218#124—24no—
219#124—42no—
220#123—48no—
221C22⋊Q8—28no—
222C22⋊Q8—44no—
223C22⋊Q8—44no—
224C22⋊Q8—44no—
225C22⋊Q8—88no—
226#121—24no—
227#121—42no—
228C4×Q8—24no—
229C4×Q8—41no—
230C4×Q8—41no—
231C4×Q8—42no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—24no—
235C4×D4—42no—
236C4×D4—41no—
237C4×D4—41no—
238#117—44no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
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—42no—
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2414yes—
265C2×C4⋊C4C2411yescharacteristic
266C2×C4⋊C4C2412yes—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4C2×D414yes—
276C2×C22⋊C4C2×D414yes—
277C2×C22⋊C4C2×D412yes—
278C2×C42—22no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—82no—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D412yes—
285C2×C42C2×D411yescharacteristic
286C2×C42C2411yescharacteristic
287C2×C42C2412yes—
288C2×C42C2412yes—
289#107—28no—
290#106—28no—
291#106—44no—
292#106—44no—
293C4⋊C8—42no—
294C4⋊C8—42no—
295C4⋊C8—42no—
296#103—44no—
297#103—44no—
298#103—44no—
299#103—44no—
300#102—28no—
301#102—44no—
302C22⋊C8—44no—
303C22⋊C8—42no—
304#97—42no—
305#95—22no—
306#95—21no—
307#95—24no—
308#95—24no—
309#95—22no—
310#95—24no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95C2×D414yes—
317#95C2×D414yes—
318C24—21no—
319C24C22×D412yes—
320#41—816no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—84no—
332C2×Q8—82no—
333C2×D4—21no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—82no—
341C2×D4#13714yes—
342C2×D4H(5,𝔽2)12yes—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—44no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—82no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4—84no—
364C22×C4C22×D412yes—
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4C2511yescharacteristic
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—48no—
379C4⋊C4—21no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—48no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—48no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—84no—
401C4⋊C4—82no—
402C4⋊C4—84no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C4⋊C4—84no—
406C4⋊C4H(5,𝔽2)12yes—
407C4⋊C42−1+414yes—
408C22⋊C4—24no—
409C22⋊C4—28no—
410C22⋊C4—28no—
411C22⋊C4—42no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—48no—
416C22⋊C4—44no—
417C22⋊C4—82no—
418C22⋊C4—84no—
419C42—22no—
420C42—24no—
421C42—22no—
422C42—22no—
423C42—21no—
424C42—21no—
425C42—22no—
426C42—22no—
427C42—41no—
428C42—44no—
429C42—42no—
430C42—82no—
431C42—82no—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—22no—
437C23—41no—
438C23—42no—
439C23—41no—
440C23—42no—
441C23C2×#13714yes—
442C23C23×D411yescharacteristic
443Q8—42no—
444Q8—42no—
445Q8—44no—
446Q8—82no—
447Q8—84no—
448Q8—164no—
449D4—24no—
450D4—44no—
451D4—41no—
452D4—41no—
453D4—88no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—48no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—88no—
492C2×C4—88no—
493C2×C4—164no—
494C2×C4C2×#13712yes—
495C2×C4#57714yes—
496C2×C4C2×H(5,𝔽2)11yescharacteristic
497C2×C4C2×H(5,𝔽2)11yescharacteristic
498C2×C4C2×2−1+412yes—
499C2×C4#58512yes—
500C8—44no—
501C8—44no—
502C22—24no—
503C22—28no—
504C22—42no—
505C22—44no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—82no—
511C22—82no—
512C22#348411yescharacteristic
513C22#348912yes—
514C22C22×#13711yescharacteristic
515C22C2×#57712yes—
516C22#358511yescharacteristic
517C4—22no—
518C4—24no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—44no—
524C4—41no—
525C4—42no—
526C4—82no—
527C4—82no—
528C4—84no—
529C4—84no—
530C4#352911yescharacteristic
531C4#355011yescharacteristic
532C4#355912yes—
533C2—44no—
534C2—82no—
535C2#4622811yescharacteristic
536C2#6157011yescharacteristic
537C2#6166411yescharacteristic
538C2#6213112yes—
539C2#6216612yes—
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. #46152C2semidirect
3. #33072C2semidirect
4. #32820C2semidirect
5. #32694C2semidirect
6. #32029C2semidirect
7. #31498C2semidirect
8. #30655C2semidirect
9. #27053C2semidirect
10. #26340C2semidirect
11. #25662C2semidirect
13. #23352C2semidirect

Maximal subgroups

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

subgroupindex
#461522
#364062
#330722
#328202
#326942
#320292
#314982
#306552
#270532
#263402
#256622
#240892
#233522