Finite Group Databasebeta

← #6508966

Subgroups — #6508966

Order 512.

1866 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3#27616C214yesmaximal, complemented
4#27602C214yesmaximal, complemented
5#24945C218yesmaximal, complemented
6#24941C218yesmaximal, complemented
7#21940C211yesmaximal, characteristic, complemented
8#21938C211yesmaximal, characteristic, complemented
9#21738C211yesmaximal, characteristic
10C2×#1823C211yesmaximal, characteristic, complemented
11C2×#1823C211yesmaximal, characteristic, complemented
12C22×#123—21no—
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17#2486—22no—
18#2486—22no—
19#2486C2214yes—
20#2486C2214yes—
21C2×#398—21no—
22C2×#397—21no—
23#2353—22no—
24#2353—22no—
25#2352C2214yes—
26#2352C2214yes—
27#2313—21no—
28#2312—21no—
29#2308—21no—
30C2×#389C2212yes—
31#2283C2212yes—
32#2280C2212yes—
33C2×#381—21no—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2212yes—
38#2273—21no—
39#2101C2218yescomplemented
40#2085C2218yescomplemented
41#2077C2218yescomplemented
42#2076C2218yescomplemented
43#1921C2218yescomplemented
44#1920C2218yescomplemented
45#1869C2214yes—
46#1869C2214yes—
47#1854C2218yescomplemented
48#1853C2214yes—
49#1853C2214yescomplemented
50#1852C2214yes—
51#1852C2214yescomplemented
52#1825C2218yescomplemented
53#1823C2214yes—
54#1823C2214yes—
55#1823C2214yescomplemented
56#1823C2214yescomplemented
57#1817C2212yes—
58#1808C2214yescomplemented
59#1807C2214yescomplemented
60#1779C2212yes—
61C2×#340C2214yescomplemented
62C2×#340C2214yescomplemented
63C23×D4D411yescharacteristic
64C24×C4—21no—
65C24×C4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C23×C8C2312yes—
68C23×C8C2312yes—
69C2×#123—24no—
70C2×#123—24no—
71C2×#123—48no—
72C2×C4×D4—24no—
73C2×C4×D4—24no—
74C2×C4×D4D412yes—
75C2×C4×D4D412yes—
76C2×C4×D4C2312yes—
77C2×C4×D4C2312yes—
78C2×#117C2314yes—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—22no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4D411yescharacteristic
97C22×C22⋊C4D411yescharacteristic
98C22×C42C2311yescharacteristic
99C2×#107C2318yes—
100C2×#106C2318yes—
101C2×C22⋊C8—24no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8C2312yes—
104C2×C22⋊C8C2312yes—
105C2×C22⋊C8C2312yes—
106C2×C22⋊C8C2312yes—
107C2×C22⋊C8C2318yes—
108#398—24no—
109#398—24no—
110#397—24no—
111#397—24no—
112#389—24no—
113#389C2318yes—
114#388—24no—
115#388—24no—
116#387—24no—
117#387—24no—
118#386—24no—
119#386—24no—
120#383—22no—
121#382C2314yes—
122#381—22no—
123#381—22no—
124#381—24no—
125#381—24no—
126#380—21no—
127#380—21no—
128#380—21no—
129#380—21no—
130#380—24no—
131#380—21no—
132#380—21no—
133#380C2312yes—
134#380C2312yes—
135#380C2312yes—
136#380C2312yes—
137#380C2314yes—
138#380C2318yes—
139#380C2312yes—
140#380C2312yes—
141#379—22no—
142C4×C4⋊C4C2314yes—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—24no—
148#376—22no—
149C2×#95—21no—
150C2×#95—21no—
151C2×#95—21no—
152C2×#95—22no—
153C2×#95—22no—
154#340C2318yes—
155#340C2318yes—
156#340C2318yes—
157#340C2318yes—
158#340C2318yes—
159#340C2318yes—
160#336C2318yes—
161C25C2×D411yescharacteristic
162C25C2×D411yescharacteristic
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4C2×D412yes—
167C22×D4C2×D412yes—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4#4114yes—
189C23×C4#4114yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C22×C8—24no—
195C22×C8—24no—
196C22×C8—44no—
197C22×C8—44no—
198C22×C8C2414yes—
199C22×C8C2414yes—
200#123—416no—
201#123—416no—
202C4×D4—28no—
203C4×D4—28no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—24no—
207C4×D4—24no—
208#117—48no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—48no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2414yes—
233C2×C4⋊C4C2414yes—
234C2×C4⋊C4C2414yes—
235C2×C4⋊C4C2414yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4C2×D412yes—
252C2×C22⋊C4C2×D412yes—
253C2×C22⋊C4C2×D412yes—
254C2×C22⋊C4C2×D412yes—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42C2×D412yes—
260C2×C42C2412yes—
261#107—48no—
262#106—48no—
263C22⋊C8—24no—
264C22⋊C8—24no—
265C22⋊C8—24no—
266C22⋊C8—24no—
267C22⋊C8—48no—
268#95—22no—
269#95—22no—
270#95—28no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—28no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24#12314yes—
288C24#13514yes—
289C24AGL(1,ℤ/8)14yes—
290C24C22×D412yes—
291C24C22×D411yescharacteristic
292C24C22×D411yescharacteristic
293C24C22×D411yescharacteristic
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—416no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—28no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—48no—
318C22×C4—48no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—84no—
340C22×C4—84no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—816no—
346C22×C4#12314yes—
347C22×C4#12314yes—
348C22×C4#12314yes—
349C22×C4C22×D412yes—
350C22×C4C22×D412yes—
351C22×C4C22×D412yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2×#4112yes—
357C22×C4C2×#4112yes—
358C22×C4C2×#4112yes—
359C22×C4C2×#4112yes—
360C22×C4C2×#4112yes—
361C22×C4C2511yescharacteristic
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—28no—
365C2×C8—28no—
366C2×C8—84no—
367C2×C8—84no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—28no—
373C4⋊C4—28no—
374C4⋊C4—28no—
375C4⋊C4—44no—
376C4⋊C4—416no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—416no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—84no—
394C22⋊C4—84no—
395C42—24no—
396C42—22no—
397C42—22no—
398C42—84no—
399C42—84no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—24no—
407C23—22no—
408C23—24no—
409C23—22no—
410C23—22no—
411C23—44no—
412C23—44no—
413C23—48no—
414C23—48no—
415C23—48no—
416C23—44no—
417C23—82no—
418C23—84no—
419C23—82no—
420C23#482116yes—
421C23C2×#12312yes—
422C23C2×#12312yes—
423C23C2×#12312yes—
424C23C2×#13512yes—
425C23C2×#13512yes—
426C23C2×#13512yes—
427C23C2×AGL(1,ℤ/8)12yes—
428C23C2×AGL(1,ℤ/8)12yes—
429C23C2×AGL(1,ℤ/8)12yes—
430C23C23×D411yescharacteristic
431D4—48no—
432D4—48no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—84no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—88no—
471C2×C4—88no—
472C2×C4—88no—
473C2×C4—88no—
474C2×C4—164no—
475C2×C4—164no—
476C2×C4C2×#12312yes—
477C2×C4C2×#12312yes—
478C2×C4C2×#12312yes—
479C2×C4C22×#4111yescharacteristic
480C2×C4C22×#4111yescharacteristic
481C2×C4#58514yes—
482C8—44no—
483C8—44no—
484C22—24no—
485C22—24no—
486C22—24no—
487C22—24no—
488C22—24no—
489C22—24no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—416no—
497C22—44no—
498C22—44no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—88no—
503C22C2×#48214yes—
504C22C2×#48214yes—
505C22C2×#48214yes—
506C22C22×#12311yescharacteristic
507C22#348512yes—
508C22C22×#13511yescharacteristic
509C22C22×AGL(1,ℤ/8)11yescharacteristic
510C22#358612yes—
511C4—22no—
512C4—22no—
513C4—22no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—84no—
518C4—84no—
519C4—84no—
520C4—84no—
521C4#349112yes—
522C4#349112yes—
523C2—24no—
524C2—24no—
525C2—24no—
526C2—82no—
527C2#3544811yescharacteristic
528C2C22×#48211yescharacteristic
529C2#6051911yescharacteristic
530C2#6169612yes—
531C2#6170612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27616C2semidirect
4. #27602C2semidirect
5. #24945C2semidirect
6. #24941C2semidirect
7. #21940C2semidirect
8. #21938C2semidirect
10. C2×#1823C2semidirect
11. C2×#1823C2semidirect
39. #2101C22semidirect
40. #2085C22semidirect
41. #2077C22semidirect
42. #2076C22semidirect
43. #1921C22semidirect
44. #1920C22semidirect
47. #1854C22semidirect
49. #1853C22semidirect
51. #1852C22semidirect
52. #1825C22semidirect
55. #1823C22semidirect
56. #1823C22semidirect
58. #1808C22semidirect
59. #1807C22semidirect
61. C2×#340C22semidirect
62. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
#341852
#276162
#276022
#249452
#249412
#219402
#219382
#217382
C2×#18232