Finite Group Databasebeta

← #6434240

Subgroups — #6434240

Order 512.

1952 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35266C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic, complemented
3#30141C218yesmaximal, complemented
4#30129C214yesmaximal, complemented
5#30126C214yesmaximal, complemented
6#23745C211yesmaximal, characteristic, complemented
7C2×#2094C211yesmaximal, characteristic, complemented
8#23610C212yesmaximal, complemented
9#23544C212yesmaximal, complemented
10#23504C211yesmaximal, characteristic, complemented
11#23428C211yesmaximal, characteristic
12#20932C212yesmaximal, complemented
13#20882C212yesmaximal, complemented
14#20754C211yesmaximal, characteristic, complemented
15C22×AGL(1,ℤ/8)C2212yes—
16C2×#532C2211yescharacteristic
17C2×#520C2212yes—
18C22×#106C2212yes—
19C2×#424C2212yes—
20C2×#417C2212yes—
21C2×#416C2212yes—
22C2×#416C2212yes—
23C22×#102C2212yes—
24#2860C2214yes—
25#2644C2214yes—
26#2513—22no—
27#2512—22no—
28#2480—24no—
29#2361—24no—
30#2354—22no—
31D4×C4⋊C4—22no—
32#2326C2214yescomplemented
33#2317C2211yescharacteristic
34#2316C2211yescharacteristic
35#2311C2212yes—
36#2309C2211yescharacteristic
37#2309C2211yescharacteristic
38C4×C4⋊Q8C2212yescomplemented
39C4×#127C2212yescomplemented
40#2296C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C4×#117C2211yescharacteristic
44#2100C2214yescomplemented
45#2099C2214yescomplemented
46#2094C2214yes—
47#2094C2214yes—
48#2066C2218yescomplemented
49#2065C2218yescomplemented
50#2055C2218yes—
51#2054C2218yes—
52#2043C2212yescomplemented
53#2042C2214yescomplemented
54#2041C2212yescomplemented
55#2013C2212yes—
56#2012C2214yes—
57#2011C2212yes—
58#1880C2218yes—
59#1879C2218yes—
60#1865C2218yescomplemented
61#1864C2218yescomplemented
62#1832C2214yescomplemented
63#1831C2214yescomplemented
64#1736C2212yes—
65#1736C2212yes—
66#1729C2212yes—
67#1728C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C22×#41C2312yes—
70C23×D4C2311yescharacteristic
71C2×AGL(1,ℤ/8)—28no—
72C2×AGL(1,ℤ/8)—44no—
73C2×AGL(1,ℤ/8)—44no—
74C22×SD16C2314yes—
75C22×D8C2314yes—
76C22×M16C2312yes—
77C23×C8C2312yes—
78#532—28no—
79C2×C4⋊Q8C2311yescharacteristic
80C2×#127C2311yescharacteristic
81C2×#126—24no—
82C2×#124C2312yes—
83C2×#121—22no—
84#520—44no—
85#520—44no—
86C2×C4×Q8C2314yes—
87C2×C4×D4—24no—
88C2×C4×D4—22no—
89C2×C4×D4—24no—
90C2×C4×D4C2314yes—
91C2×#117—22no—
92C2×#117C2311yescharacteristic
93C2×#117C2311yescharacteristic
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C22×C4⋊C4—22no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4—22no—
101C22×C42C2311yescharacteristic
102C2×#106—44no—
103C2×#106C2318yes—
104#424—44no—
105C2×C4⋊C8C2318yes—
106#417—44no—
107#416—28no—
108#416—44no—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×#103C2314yes—
112C2×#102—44no—
113C2×#102C2314yes—
114C2×#102C2318yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117#397—24no—
118#397—24no—
119#397—24no—
120#396—24no—
121#391C2314yes—
122#390—21no—
123#390—22no—
124#390—21no—
125#390—24no—
126#390—22no—
127#390C2314yes—
128#390C2314yes—
129#390C2314yes—
130#390C2312yes—
131#390C2312yes—
132#389—22no—
133#389—24no—
134#389C2314yes—
135#389C2314yes—
136#389C2312yes—
137#389C2312yes—
138#388—24no—
139#388—24no—
140#388—24no—
141#388—28no—
142#386—24no—
143#385—22no—
144#385—22no—
145#385—28no—
146#384—21no—
147#384—22no—
148#384C2312yes—
149#384C2312yes—
150#382—21no—
151#380—24no—
152#380—22no—
153#379—22no—
154#379—22no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—22no—
157C4×C4⋊C4D412yes—
158C4×C4⋊C4D412yes—
159C4×C22⋊C4D414yes—
160#376C2312yes—
161C43C2312yes—
162#340C2318yes—
163#340C2318yes—
164#336C2312yes—
165#336C2312yes—
166#336C2318yes—
167#336C2312yes—
168#336C2312yes—
169C25—21no—
170C2×#41—24no—
171C2×#41—24no—
172C2×#41—24no—
173C2×#41—24no—
174C2×#41—24no—
175C2×#41#4118yes—
176C22×Q8C2412yes—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—21no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—24no—
183C22×D4—22no—
184C22×D4C2412yes—
185C22×D4C2412yes—
186C23×C4—22no—
187C23×C4—21no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4C2×D412yes—
191C23×C4C2411yescharacteristic
192AGL(1,ℤ/8)—432no—
193C2×SD16—28no—
194C2×SD16—44no—
195C2×SD16—44no—
196C2×D8—28no—
197C2×D8—44no—
198C2×D8—44no—
199C2×M16—24no—
200C2×M16—24no—
201C2×M16—24no—
202C22×C8—24no—
203C22×C8—24no—
204C22×C8—24no—
205C22×C8C2414yes—
206C22×C8C2414yes—
207C4⋊Q8—24no—
208#127—24no—
209#126—88no—
210#124—24no—
211#124—24no—
212#121—216no—
213C4×Q8—44no—
214C4×Q8—44no—
215C4×D4—216no—
216C4×D4—44no—
217C4×D4—44no—
218C4×D4—416no—
219C4×D4—88no—
220#117—24no—
221#117—22no—
222#117—22no—
223#117—24no—
224#117—22no—
225#117—22no—
226#117—48no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4—84no—
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2414yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2412yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—84no—
261C2×C22⋊C4C2×D412yes—
262C2×C22⋊C4C2×D412yes—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42C2×D412yes—
268C2×C42C2×D414yes—
269C2×C42C2×D414yes—
270C2×C42C2412yes—
271C2×C42C2412yes—
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274#106—48no—
275C4⋊C8—48no—
276#103—28no—
277#103—44no—
278#103—44no—
279#102—28no—
280#102—48no—
281#102—44no—
282#102—44no—
283#95—28no—
284#95—22no—
285#95—22no—
286#95—24no—
287#95—24no—
288#95C2×D414yes—
289#95C2×D414yes—
290C24—22no—
291C24—22no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—82no—
296C24—82no—
297C24C22×D411yescharacteristic
298#41—216no—
299#41—216no—
300#41—48no—
301#41—48no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8C2×#4114yes—
308C2×D4—28no—
309C2×D4—28no—
310C2×D4—22no—
311C2×D4—24no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—24no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—44no—
328C2×D4C2×#4114yes—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—28no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—42no—
338C22×C4—48no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—88no—
344C22×C4—82no—
345C22×C4—84no—
346C22×C4C22×D414yes—
347C22×C4C22×D412yes—
348C22×C4C22×D412yes—
349C22×C4C22×D412yes—
350C22×C4C22×D412yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C2×#4114yes—
353C22×C4H(5,𝔽2)14yes—
354C22×C4H(5,𝔽2)12yes—
355C22×C42−1+412yes—
356C22×C4C2511yescharacteristic
357SD16—416no—
358D8—416no—
359M16—48no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—48no—
367C4⋊C4—28no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—48no—
378C4⋊C4—48no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—82no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C22⋊C4—216no—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—48no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—48no—
393C22⋊C4—84no—
394C22⋊C4—84no—
395C22⋊C4—84no—
396C22⋊C4—88no—
397C22⋊C4—84no—
398C22⋊C4—84no—
399C42—24no—
400C42—28no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—24no—
405C42—22no—
406C42—44no—
407C42—48no—
408C42—42no—
409C42—42no—
410C23—22no—
411C23—22no—
412C23—24no—
413C23—44no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—44no—
422C23—88no—
423C23—84no—
424C23—162no—
425C23—162no—
426C23—164no—
427C23#52512yes—
428C23#53512yes—
429C23#53712yes—
430C23C23×D411yescharacteristic
431Q8—24no—
432Q8—24no—
433Q8—42no—
434Q8—42no—
435D4—24no—
436D4—24no—
437D4—416no—
438D4—416no—
439D4—42no—
440D4—44no—
441D4—42no—
442D4—44no—
443D4—84no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—48no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—416no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—48no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—168no—
472C2×C4—162no—
473C2×C4—164no—
474C2×C4#52512yes—
475C2×C4#53512yes—
476C2×C4#53712yes—
477C2×C4#53818yes—
478C2×C4#53918yes—
479C2×C4#57714yes—
480C2×C4C22×#4112yes—
481C2×C4C2×H(5,𝔽2)11yescharacteristic
482C2×C4C2×H(5,𝔽2)12yes—
483C2×C4C2×2−1+411yescharacteristic
484C8—44no—
485C8—44no—
486C22—24no—
487C22—22no—
488C22—22no—
489C22—24no—
490C22—44no—
491C22—48no—
492C22—84no—
493C22—82no—
494C22—82no—
495C22—84no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—84no—
500C22—168no—
501C22#263614yes—
502C22C2×#52511yescharacteristic
503C22C2×#53511yescharacteristic
504C22C2×#53711yescharacteristic
505C22C2×#57712yes—
506C22#358512yes—
507C4—24no—
508C4—44no—
509C4—44no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—82no—
514C4—82no—
515C4—84no—
516C4—82no—
517C4#263614yes—
518C4C2×#53812yes—
519C4C2×#53912yes—
520C2—24no—
521C2—84no—
522C2—162no—
523C2C2×#263611yescharacteristic
524C2#6058812yes—
525C2#6082011yescharacteristic
526C2#6082711yescharacteristic
527C2#6092912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23428C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35266C2semidirect
2. #34207C2semidirect
3. #30141C2semidirect
4. #30129C2semidirect
5. #30126C2semidirect
6. #23745C2semidirect
7. C2×#2094C2semidirect
8. #23610C2semidirect
9. #23544C2semidirect
10. #23504C2semidirect
12. #20932C2semidirect
13. #20882C2semidirect
14. #20754C2semidirect
32. #2326C22semidirect
38. C4×C4⋊Q8C22semidirect
39. C4×#127C22semidirect
44. #2100C22semidirect
45. #2099C22semidirect
48. #2066C22semidirect
49. #2065C22semidirect
52. #2043C22semidirect
53. #2042C22semidirect
54. #2041C22semidirect
60. #1865C22semidirect
61. #1864C22semidirect
62. #1832C22semidirect
63. #1831C22semidirect

Maximal subgroups

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

subgroupindex
#352662
#342072
#301412
#301292
#301262
#237452
C2×#20942
#236102
#235442
#235042
#234282
#209322
#208822
#207542