Finite Group Databasebeta

← #6470951

Subgroups — #6470951

Order 512.

1962 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#2360C211yesmaximal, characteristic
3#33147C212yesmaximal, complemented
4#32954C212yesmaximal, complemented
5#29967C212yesmaximal, complemented
6#29951C212yesmaximal, complemented
7#26412C218yesmaximal, complemented
8#26340C218yesmaximal, complemented
9#21930C211yesmaximal, characteristic
10#21759C212yesmaximal
11C2×#1853C212yesmaximal
12C2×#543—21no—
13C22×C22⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#475C2214yescomplemented
17C22×#106C2214yes—
18C22×C22⋊C8C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2659—21no—
22#2659—21no—
23#2549—24no—
24#2440C2214yes—
25#2440C2214yes—
26#2360—24no—
27#2360C2218yes—
28#2315—21no—
29C2×#391C2211yescharacteristic
30C2×#389—21no—
31C2×#389C2211yescharacteristic
32C2×#387—21no—
33C2×#385—21no—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4—21no—
37C2×C4×C22⋊C4C2211yescharacteristic
38#2099C2214yescomplemented
39#2096C2214yescomplemented
40#2088C2218yescomplemented
41#2066C2218yescomplemented
42#2058C2214yescomplemented
43#2024C2214yescomplemented
44#1988C2214yescomplemented
45#1948C2214yescomplemented
46#1919C2218yescomplemented
47#1892C2218yes—
48#1872C2214yes—
49#1872C2214yes—
50#1865C2218yescomplemented
51#1853C2214yes—
52#1853C2218yes—
53#1853C2214yes—
54#1850C2218yes—
55#1824C2214yes—
56#1803C2214yescomplemented
57#1797C2212yes—
58#1797C2212yes—
59#1787C2214yes—
60C2×#340C2214yes—
61C23×Q8C2311yescharacteristic
62C24×C4—21no—
63C24×C4C2311yescharacteristic
64C23×C8C2314yes—
65#543—48no—
66C2×C4⋊Q8C2312yes—
67C2×C4⋊Q8C2312yes—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—24no—
73C2×C22⋊Q8—24no—
74C2×C22⋊Q8Q818yescomplemented
75C2×C22⋊Q8C2314yes—
76C2×C4×Q8—22no—
77C2×C4×Q8—44no—
78C2×C4×Q8C2314yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—44no—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—22no—
93C22×C22⋊C4D411yescharacteristic
94C22×C22⋊C4D411yescharacteristic
95C22×C42—21no—
96C22×C42—21no—
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99#475—48no—
100C2×#106—48no—
101C2×#106C2318yes—
102C2×#106C2318yes—
103C2×C4⋊C8C2314yes—
104C2×C4⋊C8C2314yes—
105C2×#103C2318yes—
106C2×#103C2318yes—
107C2×C22⋊C8—28no—
108C2×C22⋊C8C2314yes—
109C2×C22⋊C8C2314yes—
110#398—24no—
111#398—24no—
112#395—24no—
113#395—24no—
114#391—22no—
115#391—22no—
116#391—22no—
117#391C2314yes—
118#389—22no—
119#389—22no—
120#389—22no—
121#389—28no—
122#389C2314yes—
123#389C2314yes—
124#389C2314yes—
125#387—28no—
126#385—24no—
127#385—24no—
128#385—28no—
129#384C2312yes—
130#384C2312yes—
131#380—22no—
132#380—22no—
133#380—22no—
134#380—22no—
135#380C2314yes—
136#380C2314yes—
137C4×C4⋊C4—22no—
138C4×C4⋊C4C2314yes—
139C4×C22⋊C4—28no—
140C4×C22⋊C4D414yes—
141C4×C22⋊C4C2314yes—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—22no—
146C2×#95—21no—
147C2×#95D411yescharacteristic
148C2×#95D411yescharacteristic
149#340C2318yes—
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#336C2314yes—
154#336C2314yes—
155C25C2×D411yescharacteristic
156C22×Q8—21no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—42no—
161C22×Q8—42no—
162C22×Q8C2×Q814yes—
163C22×Q8C2412yes—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—82no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×Q814yes—
183C23×C4#4114yes—
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C22×C8—28no—
187C22×C8—44no—
188C22×C8—44no—
189C22×C8C2418yes—
190C4⋊Q8—44no—
191C4⋊Q8—44no—
192C22⋊Q8—216no—
193C22⋊Q8—28no—
194C22⋊Q8—48no—
195C22⋊Q8—48no—
196C22⋊Q8—416no—
197C22⋊Q8—416no—
198C22⋊Q8C2×Q8116yes—
199C4×Q8—28no—
200C4×Q8—44no—
201C4×Q8—48no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—48no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4C2×Q814yes—
232C2×C4⋊C4C2412yes—
233C2×C4⋊C4C2414yes—
234C2×C4⋊C4C2414yes—
235C2×C4⋊C4C2412yes—
236C2×C4⋊C4C2412yes—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—48no—
243C2×C22⋊C4SD1618yes—
244C2×C22⋊C4C2×D414yes—
245C2×C22⋊C4C2×D414yes—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—44no—
251C2×C42—82no—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2412yes—
255C2×C42C2412yes—
256#106—216no—
257#106—48no—
258C4⋊C8—44no—
259C4⋊C8—44no—
260#103—48no—
261#103—48no—
262C22⋊C8—28no—
263C22⋊C8—28no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95C2×D414yes—
273#95C2×D414yes—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24C22⋊Q814yes—
279C24C2×SD1614yes—
280C24#13714yes—
281C24C22×D412yes—
282C24C22×D411yescharacteristic
283C2×Q8—24no—
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—48no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—48no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—82no—
294C2×Q8—82no—
295C2×Q8—82no—
296C2×Q8—82no—
297C2×Q8C22×Q814yes—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—48no—
314C22×C4—48no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—48no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—48no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—84no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—816no—
346C22×C4C22⋊Q814yes—
347C22×C4C22⋊Q818yes—
348C22×C4C2×SD1614yes—
349C22×C4C2×SD1614yes—
350C22×C4C22×D412yes—
351C22×C4C22×D412yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D412yes—
354C22×C4C22×D412yes—
355C22×C4C22×D412yes—
356C22×C4C22×Q812yes—
357C22×C4C22×Q814yes—
358C22×C4C2×#4112yes—
359C22×C4C2×#4114yes—
360C22×C4C2511yescharacteristic
361C2×C8—28no—
362C2×C8—28no—
363C2×C8—28no—
364C2×C8—416no—
365C4⋊C4—24no—
366C4⋊C4—216no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—28no—
371C4⋊C4—24no—
372C4⋊C4—28no—
373C4⋊C4—48no—
374C4⋊C4—48no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4—84no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4C22×Q814yes—
388C4⋊C4H(5,𝔽2)14yes—
389C22⋊C4—216no—
390C22⋊C4—28no—
391C22⋊C4—48no—
392C22⋊C4—48no—
393C22⋊C4C2×SD16116yes—
394C42—24no—
395C42—24no—
396C42—42no—
397C42—42no—
398C42—48no—
399C42—84no—
400C23—21no—
401C23—22no—
402C23—21no—
403C23—22no—
404C23—21no—
405C23—22no—
406C23—21no—
407C23—22no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23#475116yescomplemented
415C23C2×C22⋊Q814yes—
416C23C2×C22⋊Q812yes—
417C23C22×SD1612yes—
418C23C22×SD1614yes—
419C23C2×#13714yes—
420C23C2×#13712yes—
421C23C23×D411yescharacteristic
422Q8—28no—
423Q8—48no—
424Q8—44no—
425Q8—164no—
426Q8—164no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—28no—
436C2×C4—24no—
437C2×C4—28no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—44no—
449C2×C4—48no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—48no—
453C2×C4—48no—
454C2×C4—48no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—816no—
473C2×C4—84no—
474C2×C4—164no—
475C2×C4—164no—
476C2×C4C2×C22⋊Q814yes—
477C2×C4C2×C22⋊Q812yes—
478C2×C4C22×SD1614yes—
479C2×C4C22×SD1614yes—
480C2×C4C23×Q811yescharacteristic
481C2×C4C22×#4111yescharacteristic
482C2×C4C2×H(5,𝔽2)12yes—
483C2×C4#58512yes—
484C8—48no—
485C22—22no—
486C22—24no—
487C22—22no—
488C22—24no—
489C22—22no—
490C22—24no—
491C22—22no—
492C22—24no—
493C22—22no—
494C22—24no—
495C22—22no—
496C22—24no—
497C22—416no—
498C22C2×#47518yes—
499C22C2×#47514yes—
500C22C22×C22⋊Q811yescharacteristic
501C22#348411yescharacteristic
502C22#348511yescharacteristic
503C22C23×SD1611yescharacteristic
504C22C22×#13711yescharacteristic
505C22#358511yescharacteristic
506C22#358611yescharacteristic
507C4—24no—
508C4—24no—
509C4—24no—
510C4—22no—
511C4—42no—
512C4—44no—
513C4—42no—
514C4—44no—
515C4—44no—
516C4—84no—
517C4—84no—
518C4—82no—
519C4#345912yes—
520C4#350112yes—
521C2—24no—
522C2—28no—
523C2#3498511yescharacteristic
524C2C22×#47511yescharacteristic
525C2#6049211yescharacteristic
526C2#6156211yescharacteristic
527C2#6157011yescharacteristic
528C2#6169711yescharacteristic
529C2#6170511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
3. #33147C2semidirect
4. #32954C2semidirect
5. #29967C2semidirect
6. #29951C2semidirect
7. #26412C2semidirect
8. #26340C2semidirect
16. C2×#475C22semidirect
38. #2099C22semidirect
39. #2096C22semidirect
40. #2088C22semidirect
41. #2066C22semidirect
42. #2058C22semidirect
43. #2024C22semidirect
44. #1988C22semidirect
45. #1948C22semidirect
46. #1919C22semidirect
50. #1865C22semidirect
56. #1803C22semidirect
74. C2×C22⋊Q8Q8semidirect
414. C23#475semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#23602
#331472
#329542
#299672
#299512
#264122
#263402
#219302
#217592
C2×#18532