Finite Group Databasebeta

← #6458240

Subgroups — #6458240

Order 512.

2596 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2C22×#384C211yesmaximal, characteristic, complemented
3#27681C214yesmaximal, complemented
4#27680C214yesmaximal, complemented
5#27678C214yesmaximal, complemented
6#27677C214yesmaximal, complemented
7#23745C212yesmaximal, complemented
8#23734C212yesmaximal, complemented
9#21171C212yesmaximal, complemented
10#20909C212yesmaximal
11#20844C212yesmaximal, complemented
12#20648C212yesmaximal, complemented
13#20626C211yesmaximal, characteristic, complemented
14C2×#525C2211yescharacteristic
15C2×C4×#41C2211yescharacteristic
16C23×C4⋊C4—22no—
17C23×C4⋊C4C2212yes—
18C23×C42C2211yescharacteristic
19C2×#428C2214yes—
20C2×#423C2212yes—
21C2×#417C2214yes—
22C2×#407C2212yes—
23C4×C22⋊Q8C2212yes—
24C4×C22⋊Q8C2212yes—
25C4×#123—22no—
26C4×#121C2212yes—
27C4×#121C2212yes—
28C4×#120—22no—
29#2297C2211yescharacteristic
30#2296C2211yescharacteristic
31#2290—22no—
32C2×#384—22no—
33C2×#384—22no—
34C2×#384—22no—
35C2×#384C2214yes—
36C2×#384C2214yes—
37C2×#384C2212yes—
38C2×#384C2212yes—
39#2280C2211yescharacteristic
40C42×D4—22no—
41C4×#117C2211yescharacteristic
42#2100C2218yescomplemented
43#2099C2218yescomplemented
44#2096C2214yescomplemented
45#2095C2214yescomplemented
46#2094C2214yescomplemented
47#2093C2214yescomplemented
48#1948C2218yescomplemented
49#1947C2218yescomplemented
50#1874C2218yes—
51#1872C2218yes—
52#1853C2218yes—
53#1852C2218yes—
54#1803C2218yescomplemented
55#1802C2218yescomplemented
56#1797C2214yes—
57#1797C2214yes—
58#1736C2214yes—
59#1736C2214yes—
60#1728C2212yes—
61C22×#41C2311yescharacteristic
62C24×C4—22no—
63C24×C4D412yes—
64C24×C4C2311yescharacteristic
65C22×M16C2314yes—
66#525—24no—
67#525Q818yes—
68C2×#123—21no—
69C2×C22⋊Q8C2312yes—
70C2×#121C2312yes—
71C2×#120—21no—
72C4×#41—28no—
73C2×C4×Q8C2312yes—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4C2312yes—
79C2×#117—21no—
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C22×C4⋊C4—24no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4—48no—
92C22×C4⋊C4—416no—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2318yes—
95C22×C42—28no—
96C22×C42—21no—
97C22×C42D412yes—
98C22×C42D412yes—
99C22×C42D412yes—
100C22×C42C2312yes—
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104#428—48no—
105C2×#107C2318yes—
106C2×#106C2318yes—
107#423—28no—
108C2×C4⋊C8C2318yes—
109#417—48no—
110C2×#103C2318yes—
111C2×#102C2318yes—
112#407—28no—
113C2×C22⋊C8C2318yes—
114#391C2311yescharacteristic
115#391C2311yescharacteristic
116#390C2311yescharacteristic
117#390C2311yescharacteristic
118#389C2312yes—
119#389C2312yes—
120#388—22no—
121#388—22no—
122#387—22no—
123#387—22no—
124#386—22no—
125#386—22no—
126#386—22no—
127#385—22no—
128#385—22no—
129#385—22no—
130#384—24no—
131#384—24no—
132#384—24no—
133#384—24no—
134#384—24no—
135#384—22no—
136#384—22no—
137#384C2318yes—
138#384C2314yes—
139#384C2314yes—
140#381—21no—
141#380C2312yes—
142#380C2312yes—
143#379—21no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—22no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4C2312yes—
148C4×C4⋊C4C2312yes—
149C4×C22⋊C4—22no—
150C4×C22⋊C4—22no—
151C4×C22⋊C4—22no—
152C4×C22⋊C4—22no—
153C4×C22⋊C4—22no—
154C4×C22⋊C4—22no—
155C4×C22⋊C4—22no—
156#376—21no—
157C43—21no—
158#340C23116yes—
159#340C23116yes—
160#336C2314yes—
161#336C2314yes—
162C25C2×D411yescharacteristic
163C2×#41—22no—
164C2×#41—24no—
165C2×#41—24no—
166C2×#41C2×Q814yes—
167C22×Q8C2411yescharacteristic
168C22×D4—22no—
169C22×D4—21no—
170C22×D4C2411yescharacteristic
171C23×C4—28no—
172C23×C4—28no—
173C23×C4—28no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—21no—
177C23×C4—24no—
178C23×C4—28no—
179C23×C4—22no—
180C23×C4—48no—
181C23×C4—416no—
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4C2×D412yes—
190C23×C4C2×Q814yes—
191C23×C4#4114yes—
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194C2×M16—28no—
195C2×M16—28no—
196C2×M16—28no—
197C22×C8C2418yes—
198#123—28no—
199C22⋊Q8—24no—
200C22⋊Q8C2×Q818yes—
201#121—24no—
202#121C2×Q818yes—
203#120—28no—
204C4×Q8—28no—
205C4×D4—216no—
206C4×D4—28no—
207C4×D4—48no—
208C4×D4—48no—
209C4×D4—48no—
210#117—22no—
211#117—24no—
212#117—24no—
213#117—44no—
214#117C2×Q814yes—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—416no—
228C2×C4⋊C4—48no—
229C2×C4⋊C4—416no—
230C2×C4⋊C4—416no—
231C2×C4⋊C4—416no—
232C2×C4⋊C4—416no—
233C2×C4⋊C4—416no—
234C2×C4⋊C4—416no—
235C2×C4⋊C4—416no—
236C2×C4⋊C4C2418yes—
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2411yescharacteristic
239C2×C4⋊C4C2411yescharacteristic
240C2×C4⋊C4C2412yes—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C42—28no—
252C2×C42—21no—
253C2×C42—24no—
254C2×C42—28no—
255C2×C42—28no—
256C2×C42—28no—
257C2×C42—28no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—44no—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42#4118yes—
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274C2×C42C2412yes—
275#107—48no—
276#106—48no—
277C4⋊C8—216no—
278#103—48no—
279#102—48no—
280C22⋊C8—216no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289C24—22no—
290C24—22no—
291C24—24no—
292C24—24no—
293C24—21no—
294C24C22⋊Q814yes—
295C24C22×D411yescharacteristic
296C24C22×D412yes—
297#41—216no—
298#41—48no—
299C2×Q8—21no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8C22×Q812yes—
303C2×D4—28no—
304C2×D4—24no—
305C2×D4—22no—
306C2×D4—21no—
307C2×D4—22no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4C22×Q812yes—
313C22×C4—264no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—216no—
326C22×C4—216no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—28no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—416no—
347C22×C4—416no—
348C22×C4—416no—
349C22×C4—48no—
350C22×C4—416no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—832no—
355C22×C4C22⋊Q818yes—
356C22×C4C22⋊Q814yes—
357C22×C4#12318yes—
358C22×C4AGL(1,ℤ/8)18yes—
359C22×C4#13718yes—
360C22×C4C22×D412yes—
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D412yes—
366C22×C4C22×Q814yes—
367C22×C4C22×Q812yes—
368C22×C4C2×#4114yes—
369C22×C4C2×#4112yes—
370C22×C4C2×#4114yes—
371C22×C4H(5,𝔽2)14yes—
372C22×C4C2511yescharacteristic
373M16—416no—
374C2×C8—28no—
375C2×C8—28no—
376C2×C8—28no—
377C4⋊C4—28no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—28no—
381C4⋊C4—22no—
382C4⋊C4—28no—
383C4⋊C4—22no—
384C4⋊C4—416no—
385C4⋊C4—416no—
386C4⋊C4—416no—
387C4⋊C4—44no—
388C4⋊C4—48no—
389C4⋊C4—416no—
390C4⋊C4—416no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4C22×Q814yes—
394C22⋊C4—24no—
395C22⋊C4—28no—
396C22⋊C4—28no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C42—24no—
405C42—24no—
406C42—24no—
407C42—24no—
408C42—24no—
409C42—24no—
410C42—22no—
411C42—22no—
412C42—42no—
413C42—44no—
414C42—42no—
415C42—44no—
416C42—44no—
417C42#12314yes—
418C42#12314yes—
419C42C2×#4118yes—
420C23—216no—
421C23—28no—
422C23—28no—
423C23—216no—
424C23—22no—
425C23—22no—
426C23—28no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23C2×C22⋊Q814yes—
437C23C2×C22⋊Q812yes—
438C23C2×#12312yes—
439C23#53512yes—
440C23C2×AGL(1,ℤ/8)14yes—
441C23C2×#13714yes—
442C23C23×D411yescharacteristic
443Q8—24no—
444Q8—42no—
445D4—24no—
446D4—416no—
447D4—48no—
448D4—42no—
449C2×C4—216no—
450C2×C4—216no—
451C2×C4—216no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—28no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—216no—
462C2×C4—216no—
463C2×C4—216no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—216no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—48no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—48no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—416no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—816no—
484C2×C4—832no—
485C2×C4—88no—
486C2×C4C2×C22⋊Q814yes—
487C2×C4C2×C22⋊Q812yes—
488C2×C4C2×#12314yes—
489C2×C4C2×#12314yes—
490C2×C4C2×#12312yes—
491C2×C4#53512yes—
492C2×C4#55318yes—
493C2×C4#55418yes—
494C2×C4C2×AGL(1,ℤ/8)14yes—
495C2×C4C2×AGL(1,ℤ/8)14yes—
496C2×C4C2×#13714yes—
497C2×C4C2×#13714yes—
498C2×C4C23×Q811yescharacteristic
499C2×C4C22×#4111yescharacteristic
500C2×C4C22×#4112yes—
501C2×C4C2×H(5,𝔽2)12yes—
502C8—48no—
503C22—232no—
504C22—24no—
505C22—24no—
506C22—24no—
507C22—24no—
508C22—24no—
509C22—22no—
510C22—24no—
511C22—22no—
512C22—24no—
513C22—24no—
514C22—44no—
515C22—42no—
516C22—82no—
517C22—81no—
518C22—81no—
519C22#248714yes—
520C22#307918yes—
521C22C22×C22⋊Q811yescharacteristic
522C22C22×#12311yescharacteristic
523C22C2×#53511yescharacteristic
524C22C22×AGL(1,ℤ/8)12yes—
525C22C22×#13712yes—
526C4—24no—
527C4—24no—
528C4—24no—
529C4—24no—
530C4—44no—
531C4—42no—
532C4—82no—
533C4—81no—
534C4—88no—
535C4—81no—
536C4#248714yes—
537C4#309218yes—
538C4C2×#55312yes—
539C4C2×#55412yes—
540C2—28no—
541C2—24no—
542C2—82no—
543C2C2×#248711yescharacteristic
544C2C2×#307912yes—
545C2C2×#309212yes—
546C2#6079811yescharacteristic
547C2#6080011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#384C2semidirect
3. #27681C2semidirect
4. #27680C2semidirect
5. #27678C2semidirect
6. #27677C2semidirect
7. #23745C2semidirect
8. #23734C2semidirect
9. #21171C2semidirect
11. #20844C2semidirect
12. #20648C2semidirect
13. #20626C2semidirect
42. #2100C22semidirect
43. #2099C22semidirect
44. #2096C22semidirect
45. #2095C22semidirect
46. #2094C22semidirect
47. #2093C22semidirect
48. #1948C22semidirect
49. #1947C22semidirect
54. #1803C22semidirect
55. #1802C22semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
C22×#3842
#276812
#276802
#276782
#276772
#237452
#237342
#211712
#209092
#208442
#206482
#206262