Finite Group Databasebeta

← #6510796

Subgroups — #6510796

Order 512.

2314 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3193C212yesmaximal
2C2×#3095C212yesmaximal, complemented
3#35011C211yesmaximal, characteristic, complemented
4C2×#2500C211yesmaximal, characteristic
5#28367C218yesmaximal, complemented
6#28311C212yesmaximal, complemented
7#28302C212yesmaximal, complemented
8#27945C218yesmaximal, complemented
9#27919C212yesmaximal, complemented
10#27910C212yesmaximal, complemented
11#20524C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2212yes—
13C22×#121C2211yescharacteristic
14C22×#117C2211yescharacteristic
15#3193—216no—
16#3095—216no—
17C2×#484C2214yes—
18C2×#484C2214yes—
19C2×#485C2218yescomplemented
20C2×#483C2214yes—
21C2×#480C2214yes—
22C2×#468C2218yes—
23C2×#448C2214yes—
24C2×#450C2214yes—
25C2×#428C2214yescomplemented
26C2×#419C2214yescomplemented
27C2×#407C2212yes—
28#2857C2212yes—
29#2857C2212yes—
30#2659—22no—
31#2583—24no—
32#2502C2214yescomplemented
33#2500—24no—
34#2500C2218yes—
35#2434C2214yescomplemented
36C2×#394—22no—
37#2315—22no—
38C2×#390C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#388—22no—
41C2×#384C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44#2097C2218yescomplemented
45#2095C2218yescomplemented
46#2094C2214yescomplemented
47#2093C2214yescomplemented
48#2013C2214yescomplemented
49#2013C2214yescomplemented
50#1973C2218yescomplemented
51#1968C2218yescomplemented
52#1960C2214yescomplemented
53#1956C2214yescomplemented
54#1847C2214yescomplemented
55#1845C2214yescomplemented
56#1753C2218yescomplemented
57#1749C2212yes—
58C4×C22⋊C8C2212yes—
59C23×Q8C2312yes—
60C23×D4—21no—
61C23×D4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C22×SD16C2318yes—
64C22×M16C2314yes—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—28no—
70C2×C22⋊Q8—28no—
71C2×C22⋊Q8C2314yes—
72C2×C22⋊Q8C2314yes—
73C2×#121—22no—
74C2×#121—28no—
75C2×#121C2314yes—
76C2×#117—22no—
77C2×#117—28no—
78C2×#117C2314yes—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C22⋊C4—22no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4D412yes—
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#485—232no—
95#484—216no—
96#484—216no—
97#483—216no—
98#480—216no—
99#468—232no—
100#450—216no—
101#448—216no—
102#428—216no—
103C2×#107C2318yes—
104C2×#107C2318yes—
105C2×#106C2318yes—
106#419—216no—
107C2×#103C2318yes—
108C2×#103C2318yes—
109C2×#103C2318yes—
110C2×#102C2318yes—
111#407D4116yes—
112C2×C22⋊C8C2314yes—
113C2×C22⋊C8C2314yes—
114C2×#97C2314yes—
115C2×C4×C8C2314yes—
116#398—28no—
117#395—28no—
118#394—216no—
119#391C2314yes—
120#391C2314yes—
121#390—22no—
122#390—22no—
123#390C2314yes—
124#390C2314yes—
125#389—24no—
126#389—24no—
127#388—216no—
128#385—28no—
129#385—28no—
130#384—22no—
131#384C2312yes—
132#384C2312yes—
133#384C2314yes—
134#382—22no—
135#382C2314yes—
136#380—24no—
137#380—24no—
138C4×C22⋊C4D414yes—
139C4×C22⋊C4C2314yes—
140C2×#95—22no—
141C2×#95D412yes—
142#336C2314yes—
143#336C2314yes—
144C25—21no—
145C25C2×D411yescharacteristic
146C22×Q8—22no—
147C22×Q8—24no—
148C22×Q8—24no—
149C22×Q8—44no—
150C22×Q8—44no—
151C22×Q8C2414yes—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—21no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4—44no—
158C22×D4—84no—
159C22×D4C2412yes—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D418yes—
171C23×C4C2×D412yes—
172C23×C4C2411yescharacteristic
173C23×C4C2412yes—
174C2×SD16—216no—
175C2×SD16—48no—
176C2×SD16—48no—
177C2×M16—44no—
178C2×M16—44no—
179C2×M16C2×D4116yes—
180C22×C8C2418yes—
181C4⋊Q8—44no—
182C4⋊Q8—44no—
183C22⋊Q8—28no—
184C22⋊Q8—28no—
185C22⋊Q8—28no—
186C22⋊Q8—28no—
187C22⋊Q8—48no—
188C22⋊Q8—48no—
189#121—28no—
190#121—28no—
191#121—48no—
192#117—28no—
193#117—28no—
194#117—48no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—48no—
212C2×C4⋊C4—84no—
213C2×C4⋊C4C2412yes—
214C2×C4⋊C4C2412yes—
215C2×C4⋊C4C2412yes—
216C2×C4⋊C4C2414yes—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—48no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—48no—
228C2×C22⋊C4—84no—
229C2×C22⋊C4C2×D418yes—
230C2×C42—24no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—44no—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2412yes—
237C2×C42C2412yes—
238#107—216no—
239#107—216no—
240#106—216no—
241#103—216no—
242#103—216no—
243#103—216no—
244#102—216no—
245C22⋊C8C2×D4116yes—
246C22⋊C8C2×D4116yes—
247#97—44no—
248C4×C8—44no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95C2×D418yes—
253C24—24no—
254C24—22no—
255C24—41no—
256C24—41no—
257C24—42no—
258C24—44no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—44no—
263C24#12714yes—
264C24C22×D412yes—
265C24C22×D411yescharacteristic
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—28no—
269C2×Q8—28no—
270C2×Q8—416no—
271C2×Q8—44no—
272C2×Q8—82no—
273C2×Q8—82no—
274C2×Q8—88no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—48no—
281C2×D4—42no—
282C2×D4—88no—
283C2×D4—88no—
284C2×D4—816no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—416no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—84no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4#12714yes—
316C22×C4#12718yes—
317C22×C4#13718yes—
318C22×C4C22×D414yes—
319C22×C4C22×D412yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D414yes—
322C22×C4C22×D414yes—
323C22×C4C22×D414yes—
324C22×C4C22×D414yes—
325C22×C4C2511yescharacteristic
326SD16—816no—
327M16—416no—
328C2×C8—28no—
329C2×C8—28no—
330C2×C8C22×D4116yes—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4—48no—
341C4⋊C4—48no—
342C4⋊C4—84no—
343C4⋊C4—84no—
344C4⋊C4—88no—
345C4⋊C4—84no—
346C4⋊C4—84no—
347C22⋊C4—216no—
348C22⋊C4—48no—
349C22⋊C4—48no—
350C22⋊C4—48no—
351C22⋊C4—48no—
352C22⋊C4—416no—
353C22⋊C4—88no—
354C42—24no—
355C42—24no—
356C42—42no—
357C42—42no—
358C42—48no—
359C23—22no—
360C23—21no—
361C23—21no—
362C23—24no—
363C23—22no—
364C23—22no—
365C23—24no—
366C23—28no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—44no—
374C23—84no—
375C23—84no—
376C23—84no—
377C23—84no—
378C23—88no—
379C23—88no—
380C23—84no—
381C23C2×#12714yes—
382C23C2×#12712yes—
383C23C2×#13714yes—
384C23C23×D411yescharacteristic
385Q8—48no—
386Q8—48no—
387Q8—88no—
388D4—48no—
389D4—84no—
390D4—816no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—28no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—48no—
404C2×C4—44no—
405C2×C4—48no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—48no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—84no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4—88no—
421C2×C4—88no—
422C2×C4—84no—
423C2×C4C2×#12714yes—
424C2×C4C2×#12712yes—
425C2×C4C2×#13718yes—
426C2×C4C23×D412yes—
427C2×C4#58512yes—
428C2×C4#58512yes—
429C8—216no—
430C22—24no—
431C22—24no—
432C22—22no—
433C22—22no—
434C22—24no—
435C22—24no—
436C22—24no—
437C22—44no—
438C22—44no—
439C22—88no—
440C22—84no—
441C22—84no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—168no—
446C22#315214yes—
447C22C22×#12711yescharacteristic
448C22#348511yescharacteristic
449C22#348511yescharacteristic
450C22C22×#13712yes—
451C22#358512yes—
452C4—28no—
453C4—22no—
454C4—24no—
455C4—44no—
456C4—84no—
457C4—82no—
458C4—82no—
459C4—84no—
460C4—82no—
461C4#348514yes—
462C2—24no—
463C2—44no—
464C2—162no—
465C2#3546311yescharacteristic
466C2C2×#315211yescharacteristic
467C2#6064411yescharacteristic
468C2#6170812yes—
469C2#6172112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3095C2semidirect
3. #35011C2semidirect
5. #28367C2semidirect
6. #28311C2semidirect
7. #28302C2semidirect
8. #27945C2semidirect
9. #27919C2semidirect
10. #27910C2semidirect
11. #20524C2semidirect
19. C2×#485C22semidirect
25. C2×#428C22semidirect
26. C2×#419C22semidirect
32. #2502C22semidirect
35. #2434C22semidirect
44. #2097C22semidirect
45. #2095C22semidirect
46. #2094C22semidirect
47. #2093C22semidirect
48. #2013C22semidirect
49. #2013C22semidirect
50. #1973C22semidirect
51. #1968C22semidirect
52. #1960C22semidirect
53. #1956C22semidirect
54. #1847C22semidirect
55. #1845C22semidirect
56. #1753C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31932
C2×#30952
#350112
C2×#25002
#283672
#283112
#283022
#279452
#279192
#279102
#205242