Finite Group Databasebeta

← #7231172

Subgroups — #7231172

Order 512.

1539 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46147C211yesmaximal, characteristic, complemented
2#36597C211yesmaximal, characteristic
3#33067C212yesmaximal, complemented
4#32818C212yesmaximal, complemented
5#32693C212yesmaximal, complemented
6#32038C212yesmaximal
7#31508C214yesmaximal, complemented
8#30659C214yesmaximal, complemented
9#27054C214yesmaximal, complemented
10#26359C214yesmaximal, complemented
11#25659C212yesmaximal, complemented
12#24107C212yesmaximal, complemented
13#23353C211yesmaximal, characteristic, complemented
14C2×#563—21no—
15C2×#559C2211yescharacteristic
16C2×#556C2211yescharacteristic
17C2×#489C2212yes—
18C2×#484C2212yes—
19C2×#476C2212yes—
20C2×#428C2212yes—
21#2859C2212yes—
22#2858—21no—
23#2858C2211yescharacteristic
24#2822—21no—
25#2734—21no—
26#2724—21no—
27#2698—24no—
28#2554—24no—
29#2546C2212yes—
30#2546C2211yescharacteristic
31#2545—21no—
32#2544—21no—
33#2541C2211yescharacteristic
34#2496C2214yes—
35#2486C2212yes—
36#2440C2214yes—
37#2426C2214yes—
38#2363C2214yes—
39C4×#124—21no—
40C4×#124C2211yescharacteristic
41C42×Q8C2211yescharacteristic
42#2093C2214yes—
43#2087C2214yes—
44#2071C2214yes—
45#2067C2214yes—
46#2067C2214yes—
47#2057C2214yes—
48#2057C2214yes—
49#2042C2214yes—
50#2041C2214yes—
51#2036C2214yes—
52#2034C2214yes—
53#2034C2214yes—
54#1985C2212yes—
55#1983C2212yes—
56#1948C2212yes—
57#1948C2212yes—
58#1944C2214yes—
59#1924C2214yes—
60#1919C2214yes—
61#1919C2214yes—
62#1919C2214yes—
63#1895C2214yes—
64#1894C2212yes—
65#1874C2214yes—
66#1870C2212yes—
67#1865C2214yes—
68#1865C2214yes—
69#1864C2214yes—
70#1854C2214yes—
71#1851C2214yes—
72#1831C2212yes—
73#1825C2212yes—
74#1803C2212yes—
75#1803C2212yes—
76#1762C2212yes—
77C22×M16C2312yes—
78#563—48no—
79#559—42no—
80#559—42no—
81#556—42no—
82#556—42no—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2312yes—
87C2×#127—21no—
88C2×#127C2311yescharacteristic
89C2×#126—24no—
90C2×#126—22no—
91C2×#125—22no—
92C2×#124—21no—
93C2×#124—21no—
94C2×#124—21no—
95C2×#124—21no—
96C2×#124D411yescharacteristic
97C2×#124C2312yes—
98C2×#123—22no—
99C2×#123—22no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8C2312yes—
102C2×#121C2314yes—
103C2×C4×Q8—21no—
104C2×C4×Q8C2311yescharacteristic
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2311yescharacteristic
107C2×C4×D4—21no—
108C2×C4×D4C2312yes—
109C2×C4×D4C2311yescharacteristic
110C2×#117C2312yes—
111C22×C4⋊C4C2312yes—
112#489—44no—
113#484—44no—
114#476—44no—
115#428—44no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2312yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2314yes—
131C2×#97C2312yes—
132#400—24no—
133#398—22no—
134#398—22no—
135#398—22no—
136#398—22no—
137#398—24no—
138#397—22no—
139#397—24no—
140#396—24no—
141#395—24no—
142#394—24no—
143#392—22no—
144#391—22no—
145#391—21no—
146#391D411yescharacteristic
147#390—21no—
148#390—24no—
149#390—24no—
150#390—22no—
151#390D411yescharacteristic
152#390C2314yes—
153#389—22no—
154#389—22no—
155#389C2314yes—
156#389C2314yes—
157#389C2314yes—
158#389C2314yes—
159#388—24no—
160#388—24no—
161#388D414yes—
162#387—24no—
163#386—24no—
164#385—24no—
165#384C2312yes—
166#384C2312yes—
167#382—22no—
168#382D411yescharacteristic
169#381—22no—
170#380C2312yes—
171#380C2314yes—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2311yescharacteristic
176C4×C4⋊C4C2312yes—
177C4×C22⋊C4—24no—
178C4×C22⋊C4C2314yes—
179C43—21no—
180C43C2311yescharacteristic
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2314yes—
188#336C2314yes—
189#336C2314yes—
190C22×Q8—21no—
191C22×Q8—22no—
192C22×Q8C2×D411yescharacteristic
193C22×Q8C2411yescharacteristic
194C22×D4—24no—
195C22×D4—21no—
196C22×D4—21no—
197C22×D4C2×D411yescharacteristic
198C22×D4C2412yes—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4C2412yes—
202C2×M16—44no—
203C2×M16—42no—
204C22×C8C2414yes—
205C22×C8C2414yes—
206C4⋊Q8—41no—
207C4⋊Q8—41no—
208C4⋊Q8—42no—
209C4⋊Q8—42no—
210C4⋊Q8—48no—
211C4⋊Q8—82no—
212#127—41no—
213#127—41no—
214#127—82no—
215#126—416no—
216#126—48no—
217#125—48no—
218#124—24no—
219#124—24no—
220#124—42no—
221#124—44no—
222#124—44no—
223#124—44no—
224#124—82no—
225#123—48no—
226#123—48no—
227C22⋊Q8—24no—
228C22⋊Q8—416no—
229C22⋊Q8—42no—
230#121—28no—
231#121—44no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—24no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×Q8—42no—
238C4×Q8—44no—
239C4×D4—24no—
240C4×D4—42no—
241C4×D4—41no—
242C4×D4—41no—
243C4×D4—44no—
244#117—24no—
245#117—42no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D411yescharacteristic
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2411yescharacteristic
269C2×C4⋊C4C2412yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4C2×D414yes—
281C2×C22⋊C4C2×D412yes—
282C2×C22⋊C4C2×D414yes—
283C2×C42—21no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D412yes—
291C2×C42C2×D411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2412yes—
294C2×C42C2412yes—
295#107—44no—
296#107—44no—
297#106—28no—
298#106—28no—
299C4⋊C8—42no—
300C4⋊C8—42no—
301C4⋊C8—42no—
302#103—28no—
303#103—44no—
304#102—44no—
305#102—44no—
306#102—44no—
307#102—44no—
308C22⋊C8—44no—
309C22⋊C8—42no—
310#97—42no—
311#95—24no—
312#95—22no—
313#95—21no—
314#95—22no—
315#95—24no—
316#95—22no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95C2×D414yes—
321#95C2×D414yes—
322C24—22no—
323C24C22×D412yes—
324C2×Q8—22no—
325C2×Q8—21no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—44no—
333C2×Q8—42no—
334C2×Q8—82no—
335C2×Q8—82no—
336C2×Q8H(5,𝔽2)12yes—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—48no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—82no—
350C2×D4—84no—
351C2×D4—84no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—84no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4C22×D412yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D412yes—
370C22×C4C22×D412yes—
371C22×C4C22×D412yes—
372C22×C4C22×D412yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C2511yescharacteristic
375M16—48no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C4⋊C4—24no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—48no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—44no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C4⋊C4—84no—
407C4⋊C4H(5,𝔽2)12yes—
408C4⋊C42−1+414yes—
409C22⋊C4—28no—
410C22⋊C4—44no—
411C22⋊C4—42no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—48no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—48no—
419C22⋊C4—44no—
420C22⋊C4—48no—
421C22⋊C4—84no—
422C42—24no—
423C42—22no—
424C42—22no—
425C42—21no—
426C42—21no—
427C42—22no—
428C42—22no—
429C42—42no—
430C42—41no—
431C42—44no—
432C42—44no—
433C42—44no—
434C42—42no—
435C42—82no—
436C42AGL(1,ℤ/8)14yes—
437C23—22no—
438C23—24no—
439C23—22no—
440C23—24no—
441C23—22no—
442C23—42no—
443C23—44no—
444C23—42no—
445C23—44no—
446C23—42no—
447C23C23×D411yescharacteristic
448Q8—24no—
449Q8—41no—
450Q8—41no—
451Q8—82no—
452Q8—84no—
453Q8—164no—
454D4—42no—
455D4—42no—
456D4—44no—
457D4—82no—
458D4—84no—
459D4—168no—
460D4—162no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—164no—
497C2×C4—164no—
498C2×C4C2×AGL(1,ℤ/8)14yes—
499C2×C4C2×AGL(1,ℤ/8)12yes—
500C2×C4C2×H(5,𝔽2)11yescharacteristic
501C2×C4C2×H(5,𝔽2)11yescharacteristic
502C2×C4C2×2−1+412yes—
503C2×C4#58512yes—
504C8—44no—
505C8—44no—
506C22—44no—
507C22—42no—
508C22—42no—
509C22—44no—
510C22—42no—
511C22—84no—
512C22—82no—
513C22—82no—
514C22—84no—
515C22—82no—
516C22#348411yescharacteristic
517C22#348912yes—
518C22C22×AGL(1,ℤ/8)11yescharacteristic
519C22#358511yescharacteristic
520C22#358612yes—
521C4—22no—
522C4—24no—
523C4—42no—
524C4—44no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—41no—
530C4—42no—
531C4—84no—
532C4—82no—
533C4—84no—
534C4#352911yescharacteristic
535C4#355011yescharacteristic
536C4#355912yes—
537C2—82no—
538C2—162no—
539C2#4622811yescharacteristic
540C2#6158711yescharacteristic
541C2#6167411yescharacteristic
542C2#6211112yes—
543C2#6223612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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. #33067C2semidirect
4. #32818C2semidirect
5. #32693C2semidirect
7. #31508C2semidirect
8. #30659C2semidirect
9. #27054C2semidirect
10. #26359C2semidirect
11. #25659C2semidirect
12. #24107C2semidirect
13. #23353C2semidirect

Maximal subgroups

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

subgroupindex
#461472
#365972
#330672
#328182
#326932
#320382
#315082
#306592
#270542
#263592
#256592
#241072
#233532