Finite Group Databasebeta

← #6461979

Subgroups — #6461979

Order 512.

1604 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35339C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic
3#29588C214yesmaximal, complemented
4#29584C214yesmaximal, complemented
5#29582C214yesmaximal, complemented
6#29576C214yesmaximal, complemented
7#23744C211yesmaximal, characteristic, complemented
8#23731C211yesmaximal, characteristic
9#23659C211yesmaximal, characteristic, complemented
10C2×#2073C211yesmaximal, characteristic
11#21160C212yesmaximal, complemented
12#21089C212yesmaximal
13#21035C212yesmaximal, complemented
14#20876C212yesmaximal
15#20368C211yesmaximal, characteristic, complemented
16C2×#518C2211yescharacteristic
17C2×#428C2212yes—
18C2×#428C2212yes—
19C2×#418C2212yes—
20C22×#102C2212yes—
21C2×#405C2212yes—
22#2864C2214yescomplemented
23#2647C2212yescomplemented
24#2647C2212yescomplemented
25#2570—22no—
26#2570—22no—
27#2569—24no—
28#2357—22no—
29#2354—22no—
30#2342—24no—
31#2319C2214yes—
32#2317C2212yes—
33#2317C2211yescharacteristic
34#2316C2212yes—
35#2309C2211yescharacteristic
36#2296C2211yescharacteristic
37#2293C2214yes—
38C2×#390C2211yescharacteristic
39#2285C2211yescharacteristic
40C2×#382C2211yescharacteristic
41#2274C2211yescharacteristic
42C4×#117C2211yescharacteristic
43#2101C2212yescomplemented
44#2101C2212yescomplemented
45#2100C2212yescomplemented
46#2099C2212yescomplemented
47#2094C2212yes—
48#2093C2212yes—
49#2092C2214yes—
50#2085C2214yescomplemented
51#2085C2214yescomplemented
52#2073C2212yes—
53#2073C2212yes—
54#2073C2214yes—
55#1944C2214yescomplemented
56#1943C2214yescomplemented
57#1942C2214yescomplemented
58#1941C2214yescomplemented
59#1936C2214yes—
60#1936C2214yes—
61#1934C2214yes—
62#1934C2214yes—
63#1921C2214yescomplemented
64#1920C2214yescomplemented
65#1919C2214yescomplemented
66#1918C2214yescomplemented
67#1864C2214yes—
68#1864C2214yes—
69#1862C2214yes—
70#1862C2214yes—
71#1746C2214yescomplemented
72#1745C2212yescomplemented
73#1744C2212yescomplemented
74#1736C2212yes—
75#1735C2212yes—
76#1735C2212yes—
77C2×#340C2212yes—
78#1729C2212yes—
79#1728C2211yescharacteristic
80C2×#336C2211yescharacteristic
81C23×D4C2311yescharacteristic
82C22×M16C2312yes—
83C23×C8C2312yes—
84#518—24no—
85#518—24no—
86C2×C4×D4—22no—
87C2×C4×D4C2314yes—
88C2×#117—24no—
89C2×#117C2311yescharacteristic
90C2×#117C2312yes—
91C2×#117C2311yescharacteristic
92C2×#117C2311yescharacteristic
93C2×#117C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C22⋊C4—22no—
97C22×C22⋊C4—22no—
98C22×C42C2311yescharacteristic
99#428—28no—
100#428—28no—
101C2×#107C2314yes—
102C2×#107C2314yes—
103C2×#106C2314yes—
104C2×#106C2314yes—
105#418—28no—
106C2×#102—28no—
107C2×#102C2314yes—
108C2×#102C2314yes—
109C2×#102C2314yes—
110C2×#102C2314yes—
111#405—28no—
112C2×#97C2314yes—
113C2×C4×C8C2314yes—
114#400—24no—
115#400—24no—
116#398—24no—
117#398—24no—
118#390—21no—
119#390—21no—
120#390—22no—
121#390C2314yes—
122#390C2312yes—
123#390C2312yes—
124#389—22no—
125#389—22no—
126#389C2314yes—
127#389C2314yes—
128#389C2314yes—
129#389C2314yes—
130#389C2314yes—
131#389C2312yes—
132#389C2312yes—
133#388—24no—
134#388—24no—
135#388—24no—
136#388—24no—
137#387—24no—
138#387—24no—
139#387—24no—
140#387—24no—
141#385—22no—
142#385—24no—
143#385—22no—
144#384—22no—
145#384—21no—
146#384C2311yescharacteristic
147#384C2311yescharacteristic
148#382—22no—
149#382—21no—
150#382—22no—
151#382C2312yes—
152#382C2312yes—
153#382C2312yes—
154#381—24no—
155#380—22no—
156#380—24no—
157#380—24no—
158#379—22no—
159C4×C4⋊C4—22no—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4D412yes—
163C4×C4⋊C4D412yes—
164C4×C4⋊C4C2314yes—
165C4×C22⋊C4—22no—
166C4×C22⋊C4—24no—
167C4×C22⋊C4D414yes—
168#376C2312yes—
169C2×#95—22no—
170C43C2312yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2312yes—
176#340C2312yes—
177#340C2312yes—
178#340C2312yes—
179#340C2312yes—
180#340C2312yes—
181#340C2312yes—
182#340C2312yes—
183#336C2312yes—
184#336C2311yescharacteristic
185#336C2311yescharacteristic
186#336C2312yes—
187#336C2312yes—
188#336C2314yes—
189#336C2314yes—
190C25—21no—
191C22×D4—21no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—24no—
195C22×D4#4114yes—
196C22×D4C2412yes—
197C23×C4—24no—
198C23×C4—21no—
199C23×C4—22no—
200C23×C4C2×D412yes—
201C23×C4C2411yescharacteristic
202C2×M16—24no—
203C2×M16—24no—
204C2×M16#4118yes—
205C22×C8—24no—
206C22×C8—24no—
207C22×C8—24no—
208C22×C8C2412yes—
209C22×C8C2412yes—
210C22×C8C2414yes—
211C4×D4—216no—
212C4×D4—28no—
213C4×D4—28no—
214#117—22no—
215#117—22no—
216#117—22no—
217#117—24no—
218#117—24no—
219#117—24no—
220#117—24no—
221#117—88no—
222#117#4114yes—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2412yes—
241C2×C4⋊C4C2414yes—
242C2×C4⋊C4C2412yes—
243C2×C4⋊C4C2412yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2412yes—
246C2×C22⋊C4—216no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—84no—
256C2×C22⋊C4C2×D412yes—
257C2×C22⋊C4C2×D412yes—
258C2×C42—21no—
259C2×C42—24no—
260C2×C42—21no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42C2×D412yes—
264C2×C42C2×D414yes—
265C2×C42C2×D414yes—
266C2×C42C2412yes—
267C2×C42C2412yes—
268C2×C42C2411yescharacteristic
269C2×C42C2411yescharacteristic
270#107—28no—
271#107—28no—
272#106—28no—
273#106—28no—
274#102—28no—
275#102—28no—
276#102—28no—
277#102—28no—
278#97—28no—
279C4×C8—28no—
280#95—24no—
281#95—24no—
282#95—22no—
283#95—24no—
284#95—24no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95C2×D414yes—
293#95C2×D414yes—
294C24—22no—
295C24—24no—
296C24—42no—
297C24—42no—
298C24—82no—
299C24—82no—
300C24C22×D411yescharacteristic
301C2×D4—24no—
302C2×D4—216no—
303C2×D4—24no—
304C2×D4—22no—
305C2×D4—24no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—48no—
309C2×D4C2×#4114yes—
310C22×C4—22no—
311C22×C4—216no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—84no—
323C22×C4—84no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4C22×D412yes—
328C22×C4C22×D412yes—
329C22×C4C22×D412yes—
330C22×C4C22×D414yes—
331C22×C4C22×D412yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C2×#4112yes—
334C22×C4C2×#4112yes—
335C22×C4C2×#4114yes—
336C22×C42−1+414yes—
337C22×C4C2511yescharacteristic
338M16—216no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—48no—
348C2×C8C2×#4118yes—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—42no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—84no—
369C4⋊C4—84no—
370C4⋊C4—82no—
371C4⋊C4—84no—
372C4⋊C4C2×#4114yes—
373C22⋊C4—216no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—48no—
378C22⋊C4—44no—
379C22⋊C4—42no—
380C22⋊C4—84no—
381C22⋊C4—84no—
382C22⋊C4—84no—
383C42—24no—
384C42—24no—
385C42—24no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—44no—
391C42—44no—
392C42—44no—
393C42—42no—
394C42—84no—
395C23—22no—
396C23—22no—
397C23—24no—
398C23—44no—
399C23—44no—
400C23—42no—
401C23—42no—
402C23—48no—
403C23—88no—
404C23—162no—
405C23—162no—
406C23—164no—
407C23#52512yes—
408C23#53212yes—
409C23#53612yes—
410C23C23×D411yescharacteristic
411D4—28no—
412D4—48no—
413D4—44no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—84no—
445C2×C4—816no—
446C2×C4—162no—
447C2×C4—164no—
448C2×C4#52512yes—
449C2×C4#53212yes—
450C2×C4#53612yes—
451C2×C4#55218yes—
452C2×C4#55218yes—
453C2×C4C22×#4111yescharacteristic
454C2×C4C22×#4111yescharacteristic
455C2×C4C22×#4112yes—
456C2×C4C2×2−1+412yes—
457C8—28no—
458C8—42no—
459C8—42no—
460C22—22no—
461C22—22no—
462C22—24no—
463C22—24no—
464C22—44no—
465C22—48no—
466C22—82no—
467C22—82no—
468C22—84no—
469C22—168no—
470C22#261114yes—
471C22C2×#52511yescharacteristic
472C22C2×#53211yescharacteristic
473C22C2×#53611yescharacteristic
474C22#358412yes—
475C22#358512yes—
476C4—42no—
477C4—42no—
478C4—44no—
479C4—44no—
480C4—42no—
481C4—42no—
482C4—82no—
483C4—82no—
484C4—82no—
485C4—84no—
486C4—82no—
487C4#261114yes—
488C4C2×#55212yes—
489C4C2×#55212yes—
490C2—24no—
491C2—84no—
492C2C2×#261111yescharacteristic
493C2#6059212yes—
494C2#6072411yescharacteristic
495C2#6072511yescharacteristic
496C2#6087512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #35339C2semidirect
3. #29588C2semidirect
4. #29584C2semidirect
5. #29582C2semidirect
6. #29576C2semidirect
7. #23744C2semidirect
9. #23659C2semidirect
11. #21160C2semidirect
13. #21035C2semidirect
15. #20368C2semidirect
22. #2864C22semidirect
23. #2647C22semidirect
24. #2647C22semidirect
43. #2101C22semidirect
44. #2101C22semidirect
45. #2100C22semidirect
46. #2099C22semidirect
50. #2085C22semidirect
51. #2085C22semidirect
55. #1944C22semidirect
56. #1943C22semidirect
57. #1942C22semidirect
58. #1941C22semidirect
63. #1921C22semidirect
64. #1920C22semidirect
65. #1919C22semidirect
66. #1918C22semidirect
71. #1746C22semidirect
72. #1745C22semidirect
73. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#353392
#342552
#295882
#295842
#295822
#295762
#237442
#237312
#236592
C2×#20732
#211602
#210892
#210352
#208762
#203682