Finite Group Databasebeta

← #6528591

Subgroups — #6528591

Order 512.

1648 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35180C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#29336C214yesmaximal, complemented
4#27978C214yesmaximal, complemented
5#27758C214yesmaximal, complemented
6#27756C214yesmaximal, complemented
7#27737C214yesmaximal, complemented
8#27735C214yesmaximal, complemented
9#21860C211yesmaximal, characteristic, complemented
10#21859C211yesmaximal, characteristic
11#21386C211yesmaximal, characteristic
12#21385C211yesmaximal, characteristic, complemented
13#20536C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C2×#532C2211yescharacteristic
16C22×#125—21no—
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2850C2214yes—
20#2639—22no—
21#2592—22no—
22#2568C2212yes—
23#2563C2212yes—
24#2562—22no—
25#2552—22no—
26#2552—22no—
27#2552C2214yes—
28#2552C2214yes—
29#2548C2212yes—
30#2547C2212yes—
31C2×#398—22no—
32#2311—21no—
33#2311—21no—
34C2×#389—21no—
35C2×#389C2212yes—
36C2×#384C2212yes—
37C2×#384C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40#2101C2218yes—
41#2072C2218yes—
42#2071C2218yes—
43#2042C2218yes—
44#1990C2212yes—
45#1990C2212yes—
46#1988C2212yes—
47#1987C2212yes—
48#1982C2212yes—
49#1981C2212yes—
50#1959C2218yes—
51#1957C2212yes—
52#1956C2212yes—
53#1936C2218yes—
54#1935C2218yes—
55#1874C2212yes—
56#1874C2214yes—
57#1874C2212yes—
58#1872C2212yes—
59#1872C2214yes—
60#1872C2212yes—
61#1836C2218yes—
62#1803C2214yes—
63#1802C2214yes—
64#1770C2214yes—
65#1770C2214yes—
66C4×#103C2214yes—
67C4×#102C2214yes—
68#1749C2214yes—
69C22×#41—21no—
70#533—24no—
71#533—24no—
72#532—24no—
73#532—24no—
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×#127C2311yescharacteristic
78C2×#125—24no—
79C2×#125—24no—
80C2×#125—21no—
81C2×#125—48no—
82C2×#124—21no—
83C2×#124C2314yes—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8C2312yes—
86C2×#121—21no—
87C2×#121C2312yes—
88C2×C4×Q8—21no—
89C2×C4×Q8C2312yes—
90C2×C4×D4—21no—
91C2×C4×D4C2312yes—
92C2×#117—21no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4D411yescharacteristic
99C22×C4⋊C4D411yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C42—21no—
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106#431—24no—
107#431—24no—
108C2×C4⋊C8C2312yes—
109C2×C4⋊C8C2312yes—
110C2×#103C2318yes—
111C2×#103C2314yes—
112C2×#103C2314yes—
113C2×#102C2318yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×C22⋊C8C2312yes—
117C2×C22⋊C8C2312yes—
118C2×#97C2318yes—
119C2×C4×C8C2312yes—
120C2×C4×C8C2312yes—
121#400—24no—
122#398—28no—
123#398—28no—
124#397—24no—
125#394—24no—
126#393—24no—
127#391—22no—
128#391—22no—
129#391C2314yes—
130#390—22no—
131#390—22no—
132#390C2314yes—
133#389—24no—
134#389—24no—
135#389—24no—
136#389C2318yes—
137#388—24no—
138#388—24no—
139#386—24no—
140#386—24no—
141#384—24no—
142#384—21no—
143#384—21no—
144#384C2312yes—
145#384C2312yes—
146#384C2318yes—
147#382C2314yes—
148#380—22no—
149#380—22no—
150#380—22no—
151#380—22no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4D412yes—
155C4×C4⋊C4D412yes—
156C4×C4⋊C4C2312yes—
157C4×C4⋊C4C2312yes—
158C4×C4⋊C4C2312yes—
159C4×C4⋊C4C2312yes—
160C2×#95—21no—
161C2×#95—22no—
162C2×#95D412yes—
163#340C2318yes—
164#340C2318yes—
165#340C2314yes—
166#340C2314yes—
167#340C2314yes—
168#340C2314yes—
169#336C2318yes—
170C2×#41—44no—
171C2×#41—44no—
172C2×#41—44no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8C2412yes—
176C22×D4—22no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4C2412yes—
180C23×C4—22no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2411yescharacteristic
189C22×C8C2414yes—
190C22×C8C2414yes—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197#127—22no—
198#127—22no—
199#125—416no—
200#125—416no—
201#125—44no—
202#124—48no—
203#124—44no—
204C22⋊Q8—24no—
205C22⋊Q8—24no—
206C22⋊Q8—44no—
207#121—24no—
208#121—24no—
209#121—44no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—44no—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—44no—
216#117—44no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—48no—
239C2×C4⋊C4—48no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—82no—
242C2×C4⋊C4—84no—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2412yes—
249C2×C4⋊C4C2414yes—
250C2×C4⋊C4C2412yes—
251C2×C4⋊C4C2412yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42C2×D414yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42#4114yes—
268C2×C42#4114yes—
269C2×C42C2414yes—
270C2×C42C2411yescharacteristic
271C2×C42C2411yescharacteristic
272C4⋊C8—24no—
273C4⋊C8—24no—
274#103—48no—
275#103—44no—
276#103—44no—
277#102—44no—
278#102—44no—
279#102—48no—
280C22⋊C8—24no—
281C22⋊C8—24no—
282#97—48no—
283C4×C8—24no—
284C4×C8—24no—
285#95—22no—
286#95—22no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95C2×D414yes—
293#95C2×D414yes—
294C24—22no—
295C24C22×D411yescharacteristic
296#41—816no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—81no—
306C2×Q8—81no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—81no—
319C2×D4—81no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4#12414yes—
349C22×C4#12414yes—
350C22×C4AGL(1,ℤ/8)14yes—
351C22×C4#13714yes—
352C22×C4C22×D414yes—
353C22×C4C22×D414yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4112yes—
360C22×C4C2×#4112yes—
361C22×C4C2511yescharacteristic
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—28no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—416no—
385C4⋊C4—416no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4—88no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—84no—
401C22⋊C4—84no—
402C22⋊C4—84no—
403C22⋊C4—84no—
404C42—28no—
405C42—44no—
406C42—44no—
407C42—44no—
408C42—44no—
409C42#12414yes—
410C42#12414yes—
411C42C2×#4112yes—
412C42C2×#4112yes—
413C42C2×#4112yes—
414C42C2×#4112yes—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23C2×#12412yes—
427C23C2×AGL(1,ℤ/8)12yes—
428C23C2×#13712yes—
429C23C23×D411yescharacteristic
430Q8—44no—
431Q8—44no—
432Q8—84no—
433Q8—84no—
434D4—44no—
435D4—44no—
436D4—84no—
437D4—88no—
438D4—84no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—44no—
454C2×C4—48no—
455C2×C4—48no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—84no—
475C2×C4—84no—
476C2×C4—88no—
477C2×C4—88no—
478C2×C4#489116yes—
479C2×C4C2×#12412yes—
480C2×C4C2×#12412yes—
481C2×C4C2×#12412yes—
482C2×C4C2×#12412yes—
483C2×C4C2×#12412yes—
484C2×C4C2×AGL(1,ℤ/8)12yes—
485C2×C4C2×AGL(1,ℤ/8)12yes—
486C2×C4C2×#13712yes—
487C2×C4C2×#13712yes—
488C2×C4C22×#4111yescharacteristic
489C2×C4C22×#4111yescharacteristic
490C2×C4#58512yes—
491C2×C4#58512yes—
492C8—44no—
493C8—44no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22C2×#48914yes—
508C22C22×#12411yescharacteristic
509C22#348511yescharacteristic
510C22#348711yescharacteristic
511C22C22×AGL(1,ℤ/8)11yescharacteristic
512C22C22×#13711yescharacteristic
513C22#358511yescharacteristic
514C22#358611yescharacteristic
515C4—22no—
516C4—22no—
517C4—22no—
518C4—44no—
519C4—44no—
520C4—82no—
521C4—82no—
522C4—82no—
523C4—82no—
524C4—84no—
525C4—82no—
526C4C2×#48914yes—
527C4C2×#48914yes—
528C4#351212yes—
529C4#351212yes—
530C2—24no—
531C2—162no—
532C2#3564911yescharacteristic
533C2C22×#48911yescharacteristic
534C2#6062111yescharacteristic
535C2#6171711yescharacteristic
536C2#6171911yescharacteristic
537C2#6189011yescharacteristic
538C2#6189311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2552C2semidirect
3. #29336C2semidirect
4. #27978C2semidirect
5. #27758C2semidirect
6. #27756C2semidirect
7. #27737C2semidirect
8. #27735C2semidirect
9. #21860C2semidirect
12. #21385C2semidirect
13. #20536C2semidirect

Maximal subgroups

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

subgroupindex
#351802
C2×#25522
#293362
#279782
#277582
#277562
#277372
#277352
#218602
#218592
#213862
#213852
#205362