Finite Group Databasebeta

← #7542955

Subgroups — #7542955

Order 512.

1657 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic
2#36534C211yesmaximal, characteristic
3#32270C212yesmaximal, complemented
4#32258C212yesmaximal, complemented
5#29211C212yesmaximal, complemented
6#26312C214yesmaximal, complemented
7#25352C214yesmaximal, complemented
8#25024C214yesmaximal, complemented
9#24107C212yesmaximal, complemented
10#24097C212yesmaximal, complemented
11#23175C211yesmaximal, characteristic
12#22659C214yesmaximal, complemented
13#21616C212yesmaximal, complemented
14C2×#559—21no—
15C2×#556C2211yescharacteristic
16C2×#554C2211yescharacteristic
17C2×#484C2212yes—
18C2×#483C2212yes—
19C2×#477C2212yes—
20C2×#435C2214yes—
21C2×#428C2212yes—
22C22×#106C2212yes—
23#2862C2211yescharacteristic
24#2861—21no—
25#2724—21no—
26#2724—21no—
27#2582—21no—
28#2565—21no—
29#2563C2211yescharacteristic
30#2560—24no—
31#2547C2211yescharacteristic
32#2547C2214yes—
33#2544—21no—
34#2536C2214yes—
35#2486C2212yes—
36#2486C2212yes—
37#2447—24no—
38#2352C2214yes—
39D4×C4⋊C4C2214yes—
40C4×C4⋊Q8C2212yes—
41C4×#127C2211yescharacteristic
42C4×#124—21no—
43C42×Q8C2211yescharacteristic
44#2077C2214yes—
45#2076C2214yes—
46#2075C2214yes—
47#2075C2214yes—
48#2058C2214yes—
49#2057C2214yes—
50#2036C2214yes—
51#2035C2214yes—
52#1993C2214yes—
53#1987C2212yes—
54#1955C2214yes—
55#1949C2214yes—
56#1949C2214yes—
57#1948C2212yes—
58#1918C2214yes—
59#1894C2212yes—
60#1892C2212yes—
61#1870C2212yes—
62#1869C2212yes—
63#1865C2214yes—
64#1854C2214yes—
65#1853C2214yes—
66#1853C2214yes—
67#1852C2214yes—
68#1847C2214yes—
69#1825C2212yes—
70#1824C2212yes—
71#1809C2214yes—
72#1808C2214yes—
73#1803C2212yes—
74#1776C2214yes—
75#1767C2212yes—
76#1763C2214yes—
77C22×M16C2312yes—
78C23×C8C2312yes—
79#559—48no—
80#556—42no—
81#556—42no—
82#554—42no—
83#554—42no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2311yescharacteristic
87C2×#127D411yescharacteristic
88C2×#126—22no—
89C2×#125—21no—
90C2×#124—21no—
91C2×#124—24no—
92C2×#124—21no—
93C2×#123—22no—
94C2×#123—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2312yes—
98C2×C4×Q8—21no—
99C2×C4×Q8—22no—
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2311yescharacteristic
102C2×C4×D4—24no—
103C2×C4×D4—21no—
104C2×C4×D4C2314yes—
105C2×C4×D4C2311yescharacteristic
106C2×C4×D4C2311yescharacteristic
107C2×#117C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111#484—44no—
112#483—44no—
113#477—28no—
114#435—216no—
115#428—44no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×#106—44no—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2312yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×#97C2314yes—
132#398—24no—
133#398—22no—
134#398—22no—
135#397—22no—
136#397—22no—
137#397—22no—
138#397—24no—
139#397—24no—
140#394—24no—
141#393—22no—
142#391—21no—
143#391—24no—
144#391—22no—
145#391—22no—
146#391—21no—
147#391—21no—
148#390—21no—
149#390D414yes—
150#390D411yescharacteristic
151#390D411yescharacteristic
152#389—24no—
153#389—22no—
154#389C2314yes—
155#388—24no—
156#388—24no—
157#388—24no—
158#387—24no—
159#386—24no—
160#385—24no—
161#385—24no—
162#384D411yescharacteristic
163#384C2312yes—
164#382—21no—
165#382C2314yes—
166#381—22no—
167#381—22no—
168#380C2314yes—
169#380C2314yes—
170#380C2314yes—
171#380C2314yes—
172#380C2312yes—
173#380C2312yes—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4C2314yes—
178C4×C4⋊C4C2311yescharacteristic
179C4×C4⋊C4C2312yes—
180C4×C22⋊C4—24no—
181C43—21no—
182C43C2311yescharacteristic
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#336C2314yes—
192C22×Q8—22no—
193C22×Q8—21no—
194C22×Q8—21no—
195C22×Q8C2411yescharacteristic
196C22×D4C2×D414yes—
197C22×D4C2×D411yescharacteristic
198C22×D4C2×D411yescharacteristic
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C2×M16—28no—
205C2×M16—42no—
206C22×C8—28no—
207C22×C8—42no—
208C22×C8C2414yes—
209C22×C8C2414yes—
210C4⋊Q8—24no—
211C4⋊Q8—24no—
212C4⋊Q8—41no—
213C4⋊Q8—41no—
214C4⋊Q8—41no—
215C4⋊Q8—41no—
216#127—24no—
217#126—48no—
218#125—44no—
219#124—44no—
220#124—416no—
221#124—44no—
222#123—48no—
223#123—48no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C22⋊Q8—42no—
227C22⋊Q8—42no—
228C22⋊Q8—48no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—44no—
234C4×Q8—48no—
235C4×D4—28no—
236C4×D4—28no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—41no—
240C4×D4—41no—
241C4×D4—44no—
242C4×D4—416no—
243#117—24no—
244#117—42no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2414yes—
272C2×C4⋊C4C2411yescharacteristic
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4C2×D414yes—
283C2×C22⋊C4C2×D414yes—
284C2×C22⋊C4C2×D412yes—
285C2×C22⋊C4C2×D412yes—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—21no—
291C2×C42—22no—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D412yes—
295C2×C42C2411yescharacteristic
296C2×C42C2412yes—
297#107—28no—
298#107—44no—
299#107—44no—
300#106—28no—
301#106—44no—
302#106—44no—
303C4⋊C8—24no—
304C4⋊C8#4118yes—
305#103—28no—
306#103—28no—
307C22⋊C8—28no—
308C22⋊C8—28no—
309C22⋊C8—42no—
310C22⋊C8—42no—
311#97—28no—
312#95—24no—
313#95—24no—
314#95—22no—
315#95—21no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—22no—
322#95—22no—
323#95—24no—
324C24C22×D412yes—
325C24C22×D412yes—
326C2×Q8—21no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—44no—
330C2×Q8—42no—
331C2×Q8—44no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—44no—
335C2×Q8—42no—
336C2×Q8—82no—
337C2×Q8H(5,𝔽2)12yes—
338C2×D4—28no—
339C2×D4—28no—
340C2×D4—22no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—44no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4AGL(1,ℤ/8)14yes—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—84no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C2511yescharacteristic
377M16—48no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—48no—
384C2×C8C2×#4118yes—
385C4⋊C4—24no—
386C4⋊C4—28no—
387C4⋊C4—21no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—48no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4H(5,𝔽2)12yes—
413C22⋊C4—28no—
414C22⋊C4—28no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—48no—
421C22⋊C4—44no—
422C22⋊C4—48no—
423C22⋊C4—44no—
424C22⋊C4—48no—
425C22⋊C4—84no—
426C42—24no—
427C42—22no—
428C42—21no—
429C42—21no—
430C42—42no—
431C42—41no—
432C42—44no—
433C42—48no—
434C42—44no—
435C42—42no—
436C42—82no—
437C42C2×#4114yes—
438C422−1+414yes—
439C23—24no—
440C23—22no—
441C23—24no—
442C23—22no—
443C23—24no—
444C23—22no—
445C23—24no—
446C23—22no—
447C23—22no—
448C23C2×AGL(1,ℤ/8)14yes—
449C23C23×D411yescharacteristic
450Q8—24no—
451Q8—41no—
452Q8—41no—
453Q8—88no—
454Q8—84no—
455Q8—162no—
456D4—44no—
457D4—416no—
458D4—82no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—164no—
496C2×C4—164no—
497C2×C4—164no—
498C2×C4#54714yes—
499C2×C4#54714yes—
500C2×C4C2×AGL(1,ℤ/8)12yes—
501C2×C4C22×#4112yes—
502C2×C4C2×H(5,𝔽2)11yescharacteristic
503C2×C4C2×H(5,𝔽2)11yescharacteristic
504C2×C4C2×2−1+412yes—
505C8—28no—
506C8—44no—
507C22—28no—
508C22—24no—
509C22—44no—
510C22—42no—
511C22—42no—
512C22—44no—
513C22—44no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22C2×#54712yes—
518C22#348411yescharacteristic
519C22C22×AGL(1,ℤ/8)11yescharacteristic
520C22#358612yes—
521C22#358611yescharacteristic
522C4—24no—
523C4—24no—
524C4—22no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—41no—
529C4—42no—
530C4—82no—
531C4—84no—
532C4—84no—
533C4—82no—
534C4—82no—
535C4#348111yescharacteristic
536C4#352212yes—
537C4#355011yescharacteristic
538C2—44no—
539C2—82no—
540C2#5907011yescharacteristic
541C2#6125212yes—
542C2#6129212yes—
543C2#6157211yescharacteristic
544C2#6167411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #32270C2semidirect
4. #32258C2semidirect
5. #29211C2semidirect
6. #26312C2semidirect
7. #25352C2semidirect
8. #25024C2semidirect
9. #24107C2semidirect
10. #24097C2semidirect
12. #22659C2semidirect
13. #21616C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#365342
#322702
#322582
#292112
#263122
#253522
#250242
#241072
#240972
#231752
#226592
#216162