Finite Group Databasebeta

← #6470353

Subgroups — #6470353

Order 512.

3462 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic, complemented
2C22×#407C211yesmaximal, characteristic
3C2×#2848C212yesmaximal
4#32937C214yesmaximal, complemented
5#29733C214yesmaximal, complemented
6#24556C218yesmaximal, complemented
7#24386C218yesmaximal, complemented
8#21938C211yesmaximal, characteristic, complemented
9#21759C212yesmaximal, complemented
10C25×C4C2211yescharacteristic
11C23×M16C2212yes—
12C22×C22⋊Q8C2212yes—
13C22×C22⋊Q8C2214yes—
14#3189—28no—
15C2×#484C2214yescomplemented
16C2×#470C2218yescomplemented
17C2×#467C2214yescomplemented
18C2×#451C2214yescomplemented
19C2×#407—28no—
20C2×#407C2218yes—
21C22×C22⋊C8C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24#2848—22no—
25#2848—22no—
26#2848—24no—
27#2848C2214yes—
28#2848C2218yes—
29#2848C2214yes—
30C2×#385—24no—
31C2×#385—22no—
32C2×#380C2212yes—
33C2×#380C2214yes—
34C2×#379—22no—
35#2075C2218yescomplemented
36#2035C22116yescomplemented
37#2001C2218yescomplemented
38#1894C22116yescomplemented
39#1892C2218yescomplemented
40#1872C2218yescomplemented
41#1853C2218yescomplemented
42#1824C2214yescomplemented
43#1809C2218yescomplemented
44#1807C2214yescomplemented
45#1787C2214yescomplemented
46C26D411yescharacteristic
47C23×Q8C2312yes—
48C24×C4—24no—
49C24×C4—24no—
50C24×C4—48no—
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4D414yes—
54C24×C4D411yescharacteristic
55C24×C4C2311yescharacteristic
56C24×C4C2314yes—
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C22×Q16C2314yescomplemented
60C22×M16—28no—
61C22×M16—48no—
62C22×M16C2318yes—
63C23×C8C2312yes—
64C23×C8C2312yes—
65C2×C22⋊Q8—24no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—28no—
70C2×C22⋊Q8—28no—
71C2×C22⋊Q8—416no—
72C2×C22⋊Q8C2314yes—
73C2×C22⋊Q8C2314yes—
74C2×C22⋊Q8C2318yes—
75C2×C22⋊Q8C2318yes—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C22⋊C4—24no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—24no—
83C22×C22⋊C4—22no—
84#484—216no—
85#470—416no—
86#467—216no—
87#451—216no—
88C2×#107C2314yescomplemented
89C2×#106C2318yescomplemented
90C2×#103C2318yescomplemented
91C2×#103C23116yescomplemented
92#407—232no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—28no—
95C2×C22⋊C8D418yes—
96C2×C22⋊C8C2318yes—
97C2×C22⋊C8C2314yes—
98C2×C22⋊C8C2314yes—
99#385—28no—
100#385—24no—
101#385—28no—
102#385—28no—
103#385—28no—
104#385—24no—
105#385—28no—
106#380—24no—
107#380—24no—
108#380—22no—
109#380—22no—
110#380C2314yes—
111#380C2318yes—
112#380C2318yes—
113#380C2314yes—
114#379—28no—
115#379—24no—
116#379—24no—
117C2×#95—24no—
118C2×#95—22no—
119#340C2318yescomplemented
120#340C23116yescomplemented
121C25—24no—
122C25—28no—
123C25—44no—
124C25—44no—
125C25C2×D414yes—
126C25C2×D411yescharacteristic
127C25C2×D411yescharacteristic
128C25C2×D411yescharacteristic
129C22×Q8—22no—
130C22×Q8—24no—
131C22×Q8—24no—
132C22×Q8—48no—
133C22×Q8C2414yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—28no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—216no—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—28no—
143C23×C4—28no—
144C23×C4—22no—
145C23×C4—216no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—44no—
150C23×C4—48no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—48no—
154C23×C4—44no—
155C23×C4—48no—
156C23×C4—44no—
157C23×C4—432no—
158C23×C4—432no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D414yes—
162C23×C4C2×D414yes—
163C23×C4C2×D414yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D414yes—
169C23×C4#4114yes—
170C23×C4C2412yes—
171C23×C4C2412yes—
172C23×C4C2411yescharacteristic
173C23×C4C2412yes—
174C2×Q16—28no—
175C2×Q16—48no—
176C2×M16—216no—
177C2×M16—216no—
178C2×M16—48no—
179C2×M16—432no—
180C22×C8—24no—
181C22×C8—44no—
182C22×C8—44no—
183C22×C8C2×D418yes—
184C22×C8C2414yes—
185C22×C8C2414yes—
186C22⋊Q8—28no—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C22⋊Q8—28no—
190C22⋊Q8—232no—
191C22⋊Q8—416no—
192C22⋊Q8—416no—
193C22⋊Q8—416no—
194C22⋊Q8—48no—
195C22⋊Q8—48no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—48no—
205C2×C4⋊C4—416no—
206C2×C4⋊C4—48no—
207C2×C4⋊C4C2418yes—
208C2×C4⋊C4C2414yes—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—416no—
218C2×C22⋊C4—416no—
219C2×C22⋊C4—48no—
220C2×C22⋊C4—48no—
221#107—28no—
222#106—48no—
223#103—216no—
224#103—416no—
225C22⋊C8—28no—
226C22⋊C8—216no—
227C22⋊C8C2×D4116yes—
228#95—28no—
229#95—24no—
230#95—24no—
231#95—28no—
232C24—22no—
233C24—22no—
234C24—28no—
235C24—24no—
236C24—28no—
237C24—28no—
238C24—24no—
239C24—28no—
240C24—24no—
241C24—24no—
242C24—28no—
243C24—42no—
244C24—44no—
245C24—42no—
246C24—42no—
247C24—44no—
248C24—42no—
249C24—416no—
250C24—416no—
251C24—464no—
252C24—416no—
253C24#12114yes—
254C24#13718yes—
255C24C22×D412yes—
256C24C22×D412yes—
257C24C22×D411yescharacteristic
258C24C22×D412yes—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—216no—
262C2×Q8—48no—
263C2×Q8—44no—
264C2×Q8—88no—
265C2×Q8—88no—
266C22×C4—216no—
267C22×C4—216no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—28no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—28no—
279C22×C4—22no—
280C22×C4—28no—
281C22×C4—28no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—28no—
285C22×C4—28no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—28no—
289C22×C4—24no—
290C22×C4—28no—
291C22×C4—24no—
292C22×C4—48no—
293C22×C4—48no—
294C22×C4—42no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—432no—
302C22×C4—432no—
303C22×C4—416no—
304C22×C4—464no—
305C22×C4—416no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—416no—
309C22×C4—48no—
310C22×C4—48no—
311C22×C4—48no—
312C22×C4—48no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—48no—
316C22×C4#12118yes—
317C22×C4#12114yes—
318C22×C4C22×D412yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D412yes—
321C22×C4C22×D414yes—
322C22×C4C22×D412yes—
323C22×C4C22×D412yes—
324C22×C4C2×#4114yes—
325C22×C4C2×#4112yes—
326C22×C4H(5,𝔽2)18yes—
327C22×C4C2511yescharacteristic
328Q16—88no—
329M16—432no—
330C2×C8—28no—
331C2×C8—24no—
332C2×C8—28no—
333C2×C8—48no—
334C2×C8—48no—
335C2×C8C22×D418yes—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—216no—
339C4⋊C4—48no—
340C4⋊C4—416no—
341C4⋊C4—44no—
342C4⋊C4—48no—
343C4⋊C4—48no—
344C4⋊C4—416no—
345C22⋊C4—416no—
346C22⋊C4—416no—
347C22⋊C4—48no—
348C22⋊C4—48no—
349C22⋊C4—416no—
350C22⋊C4—48no—
351C22⋊C4—48no—
352C22⋊C4—416no—
353C23—24no—
354C23—22no—
355C23—24no—
356C23—22no—
357C23—28no—
358C23—28no—
359C23—232no—
360C23—22no—
361C23—22no—
362C23—24no—
363C23—22no—
364C23—24no—
365C23—24no—
366C23—24no—
367C23—22no—
368C23—44no—
369C23—44no—
370C23—416no—
371C23—416no—
372C23—416no—
373C23—416no—
374C23—416no—
375C23—416no—
376C23—416no—
377C23—44no—
378C23—44no—
379C23—416no—
380C23—4128no—
381C23—44no—
382C23—44no—
383C23—416no—
384C23#470116yescomplemented
385C23C2×#12112yes—
386C23C2×#12114yes—
387C23#53414yes—
388C23C2×#13714yes—
389C23C2×#13714yes—
390C23C2×#13714yes—
391C23#57514yes—
392C23C23×D411yescharacteristic
393Q8—48no—
394Q8—48no—
395Q8—88no—
396C2×C4—216no—
397C2×C4—28no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—24no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—48no—
411C2×C4—44no—
412C2×C4—416no—
413C2×C4—416no—
414C2×C4—416no—
415C2×C4—48no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—48no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—88no—
427C2×C4—816no—
428C2×C4—88no—
429C2×C4C2×#12112yes—
430C2×C4C2×#12114yes—
431C2×C4#53414yes—
432C2×C4C23×D411yescharacteristic
433C2×C4C22×#4111yescharacteristic
434C2×C4C2×H(5,𝔽2)14yes—
435C8—28no—
436C8—44no—
437C22—24no—
438C22—22no—
439C22—28no—
440C22—28no—
441C22—28no—
442C22—22no—
443C22—24no—
444C22—24no—
445C22—28no—
446C22—28no—
447C22—24no—
448C22—416no—
449C22—416no—
450C22—464no—
451C22—416no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—44no—
456C22—44no—
457C22—42no—
458C22C2×#47014yes—
459C22C2×#47018yes—
460C22C22×#12111yescharacteristic
461C22C2×#53412yes—
462C22C22×#13712yes—
463C22C2×#57512yes—
464C4—22no—
465C4—24no—
466C4—24no—
467C4—42no—
468C4—84no—
469C4—88no—
470C4—84no—
471C4#348412yes—
472C4#349012yes—
473C2—28no—
474C2—24no—
475C2—44no—
476C2—44no—
477C2#3498111yescharacteristic
478C2C22×#47011yescharacteristic
479C2#6044411yescharacteristic
480C2#6075114yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3189C2semidirect
4. #32937C2semidirect
5. #29733C2semidirect
6. #24556C2semidirect
7. #24386C2semidirect
8. #21938C2semidirect
9. #21759C2semidirect
15. C2×#484C22semidirect
16. C2×#470C22semidirect
17. C2×#467C22semidirect
18. C2×#451C22semidirect
35. #2075C22semidirect
36. #2035C22semidirect
37. #2001C22semidirect
38. #1894C22semidirect
39. #1892C22semidirect
40. #1872C22semidirect
41. #1853C22semidirect
42. #1824C22semidirect
43. #1809C22semidirect
44. #1807C22semidirect
45. #1787C22semidirect
59. C22×Q16C23semidirect
88. C2×#107C23semidirect
89. C2×#106C23semidirect
90. C2×#103C23semidirect
91. C2×#103C23semidirect
119. #340C23semidirect
120. #340C23semidirect
384. C23#470semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
C22×#4072
C2×#28482
#329372
#297332
#245562
#243862
#219382
#217592