Finite Group Databasebeta

← #7487441

Subgroups — #7487441

Order 512.

1483 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46152C211yesmaximal, characteristic, complemented
2#39174C211yesmaximal, characteristic
3#32629C212yesmaximal, complemented
4#32037C212yesmaximal, complemented
5#31305C214yesmaximal, complemented
6#31132C214yesmaximal, complemented
7#30227C212yesmaximal, complemented
8#29581C212yesmaximal, complemented
9#29061C214yesmaximal, complemented
10#27274C214yesmaximal, complemented
11#25651C212yesmaximal, complemented
12#24046C212yesmaximal
13#23271C211yesmaximal, characteristic, complemented
14C2×#557C2211yescharacteristic
15C2×#550C2211yescharacteristic
16C2×#487C2212yes—
17C2×#480C2212yes—
18C2×#477C2212yes—
19C2×#417C2212yes—
20#2859C2212yes—
21#2858C2211yescharacteristic
22#2844—21no—
23#2809—21no—
24#2720—24no—
25#2711—24no—
26#2683—21no—
27#2641—21no—
28#2567C2212yes—
29#2566—21no—
30#2548—21no—
31#2548C2211yescharacteristic
32#2542C2211yescharacteristic
33#2497C2214yes—
34#2487C2212yes—
35#2440C2214yes—
36#2432C2214yes—
37#2426C2214yes—
38#2379—21no—
39#2326—21no—
40C4×#124C2211yescharacteristic
41#2295C2211yescharacteristic
42#2099C2212yes—
43#2096C2212yes—
44#2083C2212yes—
45#2072C2212yes—
46#2065C2214yes—
47#2043C2214yes—
48#2043C2214yes—
49#2042C2214yes—
50#2042C2214yes—
51#2041C2214yes—
52#2035C2214yes—
53#2035C2214yes—
54#2033C2214yes—
55#2023C2214yes—
56#1984C2212yes—
57#1981C2212yes—
58#1959C2214yes—
59#1959C2214yes—
60#1943C2214yes—
61#1935C2214yes—
62#1934C2214yes—
63#1926C2214yes—
64#1924C2214yes—
65#1920C2214yes—
66#1895C2212yes—
67#1894C2214yes—
68#1872C2214yes—
69#1870C2212yes—
70#1863C2214yes—
71#1863C2214yes—
72#1788C2212yes—
73#1788C2214yes—
74C4×#103C2212yes—
75#1753C2214yes—
76#1745C2212yes—
77C22×#41C2312yes—
78C22×M16C2312yes—
79#557—42no—
80#557—42no—
81#550—42no—
82#550—42no—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×#127—21no—
87C2×#127C2311yescharacteristic
88C2×#125—21no—
89C2×#125—22no—
90C2×#125—22no—
91C2×#124—21no—
92C2×#124D411yescharacteristic
93C2×#124C2314yes—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8C2314yes—
96C2×#121—22no—
97C2×#121C2312yes—
98C2×C4×Q8—21no—
99C2×C4×Q8C2311yescharacteristic
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2311yescharacteristic
102C2×C4×D4—21no—
103C2×C4×D4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105#487—44no—
106#480—44no—
107#477—44no—
108C2×#107C2314yes—
109C2×#106C2314yes—
110C2×C4⋊C8C2312yes—
111#417—44no—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×C22⋊C8C2312yes—
123C2×C22⋊C8C2314yes—
124C2×#97C2314yes—
125C2×C4×C8C2312yes—
126#400—22no—
127#400—24no—
128#399—22no—
129#399—22no—
130#398—24no—
131#396—24no—
132#396—24no—
133#395—22no—
134#395—22no—
135#395—24no—
136#394—22no—
137#394—24no—
138#394—24no—
139#394—24no—
140#391—22no—
141#391D411yescharacteristic
142#391C2312yes—
143#390—21no—
144#390—21no—
145#390—24no—
146#390—21no—
147#390—21no—
148#390—22no—
149#390D411yescharacteristic
150#390C2314yes—
151#389—22no—
152#389—22no—
153#389C2314yes—
154#389C2314yes—
155#389C2312yes—
156#389C2314yes—
157#388—24no—
158#388—24no—
159#388D414yes—
160#387—22no—
161#387—24no—
162#386—22no—
163#385—24no—
164#385—22no—
165#385—24no—
166#385—24no—
167#384—21no—
168#384—22no—
169#384C2312yes—
170#384C2312yes—
171#382—22no—
172#382D411yescharacteristic
173#380C2314yes—
174#380C2312yes—
175C4×C4⋊C4—21no—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2311yescharacteristic
178C4×C22⋊C4C2314yes—
179#376—21no—
180C43C2311yescharacteristic
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#336C2312yes—
186#336C2312yes—
187#336C2314yes—
188#336C2314yes—
189C2×#41—24no—
190C2×#41—42no—
191C2×#41—44no—
192C2×#41—44no—
193C22×Q8—22no—
194C22×Q8C2×D411yescharacteristic
195C22×Q8C2412yes—
196C22×Q8C2411yescharacteristic
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—21no—
200C22×D4—21no—
201C22×D4C2×D411yescharacteristic
202C22×D4C2412yes—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4C2412yes—
206C2×M16—44no—
207C2×M16—42no—
208C22×C8C2414yes—
209C22×C8C2414yes—
210C4⋊Q8—24no—
211C4⋊Q8—42no—
212C4⋊Q8—42no—
213C4⋊Q8—42no—
214C4⋊Q8—41no—
215C4⋊Q8—41no—
216#127—41no—
217#127—41no—
218#127—44no—
219#125—44no—
220#125—48no—
221#125—84no—
222#124—24no—
223#124—44no—
224#124—44no—
225#124—44no—
226C22⋊Q8—44no—
227C22⋊Q8—44no—
228C22⋊Q8—88no—
229#121—24no—
230#121—42no—
231#121—48no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—24no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×Q8—42no—
238C4×Q8—44no—
239C4×D4—41no—
240C4×D4—41no—
241C4×D4—44no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2414yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2411yescharacteristic
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4C2×D414yes—
277C2×C22⋊C4C2×D412yes—
278C2×C22⋊C4C2×D414yes—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—21no—
282C2×C42—22no—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D412yes—
285C2×C42C2×D411yescharacteristic
286C2×C42C2411yescharacteristic
287C2×C42C2412yes—
288C2×C42C2412yes—
289#107—28no—
290#106—44no—
291C4⋊C8—42no—
292#103—28no—
293#103—44no—
294#103—44no—
295#103—44no—
296#103—44no—
297#103—44no—
298#102—44no—
299#102—44no—
300#102—44no—
301#102—44no—
302C22⋊C8—44no—
303C22⋊C8—42no—
304#97—44no—
305C4×C8—42no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—21no—
310#95—22no—
311#95—24no—
312#95—22no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95C2×D414yes—
317#95C2×D414yes—
318C24—22no—
319C24C22×D412yes—
320#41—48no—
321#41—84no—
322#41—88no—
323C2×Q8—21no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—84no—
332C2×Q8—82no—
333C2×Q8#13714yes—
334C2×Q8H(5,𝔽2)12yes—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—44no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—44no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—82no—
349C2×D4—84no—
350C2×D4—84no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—44no—
362C22×C4—84no—
363C22×C4—82no—
364C22×C4—84no—
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C22×D412yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D412yes—
372C22×C4C2511yescharacteristic
373M16—84no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C4⋊C4—24no—
381C4⋊C4—21no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—48no—
396C4⋊C4—42no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4—84no—
405C4⋊C4—82no—
406C4⋊C4H(5,𝔽2)12yes—
407C22⋊C4—28no—
408C22⋊C4—42no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—48no—
412C22⋊C4—84no—
413C22⋊C4—84no—
414C22⋊C4—84no—
415C22⋊C4—82no—
416C22⋊C4—84no—
417C22⋊C4—84no—
418C42—24no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—21no—
423C42—21no—
424C42—24no—
425C42—41no—
426C42—42no—
427C42—44no—
428C42—42no—
429C42—82no—
430C23—24no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—22no—
435C23—44no—
436C23—42no—
437C23—42no—
438C23—44no—
439C23—42no—
440C23C23×D411yescharacteristic
441Q8—24no—
442Q8—44no—
443Q8—42no—
444Q8—44no—
445Q8—42no—
446Q8—42no—
447Q8—164no—
448D4—42no—
449D4—42no—
450D4—44no—
451D4—82no—
452D4—84no—
453D4—88no—
454D4—164no—
455D4—162no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—88no—
491C2×C4—164no—
492C2×C4C2×#13712yes—
493C2×C4C2×#13714yes—
494C2×C4C2×H(5,𝔽2)11yescharacteristic
495C2×C4C2×H(5,𝔽2)11yescharacteristic
496C2×C4#58514yes—
497C8—44no—
498C8—44no—
499C22—42no—
500C22—44no—
501C22—44no—
502C22—42no—
503C22—42no—
504C22—82no—
505C22—84no—
506C22—84no—
507C22—82no—
508C22—82no—
509C22#348411yescharacteristic
510C22#350012yes—
511C22C22×#13711yescharacteristic
512C22#358512yes—
513C22#358611yescharacteristic
514C4—22no—
515C4—24no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—41no—
521C4—42no—
522C4—42no—
523C4—82no—
524C4—84no—
525C4—82no—
526C4—84no—
527C4#353312yes—
528C4#355112yes—
529C2—82no—
530C2—162no—
531C2#5197811yescharacteristic
532C2#6157311yescharacteristic
533C2#6167311yescharacteristic
534C2#6257412yes—
535C2#6259412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
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. #32629C2semidirect
4. #32037C2semidirect
5. #31305C2semidirect
6. #31132C2semidirect
7. #30227C2semidirect
8. #29581C2semidirect
9. #29061C2semidirect
10. #27274C2semidirect
11. #25651C2semidirect
13. #23271C2semidirect

Maximal subgroups

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

subgroupindex
#461522
#391742
#326292
#320372
#313052
#311322
#302272
#295812
#290612
#272742
#256512
#240462
#232712