Finite Group Databasebeta

← #6470481

Subgroups — #6470481

Order 512.

2880 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
1#37480C211yesmaximal, characteristic, complemented
2#34171C211yesmaximal, characteristic
3#32962C214yesmaximal, complemented
4#29959C214yesmaximal, complemented
5#25974C218yesmaximal, complemented
6#25966C218yesmaximal, complemented
7#21943C212yesmaximal, complemented
8#21935C211yesmaximal, characteristic, complemented
9#20598C212yesmaximal
10C24×D4C2211yescharacteristic
11C2×#518C2212yes—
12C2×#476C2214yescomplemented
13C2×#474C2214yescomplemented
14C2×#418C2214yes—
15C2×#418C2214yes—
16C2×#407C2212yes—
17#2842—21no—
18#2840—21no—
19#2665—24no—
20#2628C2214yescomplemented
21#2607—22no—
22#2439C2218yescomplemented
23D4×C22⋊C4—24no—
24D4×C22⋊C4C2214yes—
25D4×C22⋊C4C2214yes—
26#2309C2212yes—
27#2308—22no—
28#2280C2212yes—
29#2280C2211yescharacteristic
30#2280C2211yescharacteristic
31C2×#379—21no—
32C2×#379—21no—
33#2092C2218yescomplemented
34#2067C2218yescomplemented
35#2065C2218yescomplemented
36#2023C2218yescomplemented
37#1864C2218yescomplemented
38#1862C2218yescomplemented
39#1854C2218yescomplemented
40#1854C2218yescomplemented
41#1851C2218yescomplemented
42#1809C2218yescomplemented
43#1804C2214yescomplemented
44#1804C2214yescomplemented
45#1798C2214yescomplemented
46#1786C2214yes—
47#1786C2214yes—
48#1786C2214yes—
49#1786C2214yes—
50C26—21no—
51C23×D4—21no—
52C23×D4—22no—
53C23×D4—24no—
54C23×D4—44no—
55C23×D4—44no—
56C23×D4Q818yescomplemented
57C23×D4C2311yescharacteristic
58C23×D4C2312yes—
59C23×D4C2312yes—
60C23×D4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C22×M16C2314yes—
63C2×C4⋊Q8C2314yescomplemented
64#518—44no—
65#518—44no—
66C2×C4×D4—24no—
67C2×C4×D4C2314yes—
68C2×C4×D4C2314yes—
69C2×#117C2312yes—
70C2×#117C2314yes—
71C2×#117C2312yes—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—24no—
75C22×C22⋊C4—24no—
76#476—48no—
77#474—48no—
78C2×#107C2318yescomplemented
79C2×#106C2318yescomplemented
80C2×C4⋊C8C2314yescomplemented
81C2×C4⋊C8C2314yescomplemented
82#418—48no—
83#418—48no—
84C2×#102C2318yes—
85C2×#102C2318yes—
86C2×#102C2318yes—
87C2×#102C2318yes—
88#407—28no—
89C2×C22⋊C8C2314yes—
90C2×C22⋊C8C2314yes—
91#400—24no—
92#399—24no—
93#396—28no—
94#395—24no—
95#393—24no—
96#393—28no—
97#391—28no—
98#390—24no—
99#390C2318yes—
100#389C2318yescomplemented
101#387—28no—
102#386—28no—
103#386—28no—
104#382C2314yescomplemented
105#381—21no—
106#381—22no—
107#381D412yes—
108#380D414yes—
109#380C2314yes—
110#380C2314yes—
111#380C2314yes—
112#379—22no—
113#379—21no—
114#379—22no—
115#379—22no—
116#379—24no—
117#379—24no—
118#379D411yescharacteristic
119#379D411yescharacteristic
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4C2314yes—
124#340C2318yescomplemented
125#340C2318yescomplemented
126#336C2318yescomplemented
127C25—21no—
128C25—22no—
129C25—22no—
130C25—28no—
131C25—21no—
132C25—48no—
133C25—88no—
134C25C2×D411yescharacteristic
135C22×Q8—24no—
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—28no—
139C22×D4—28no—
140C22×D4—216no—
141C22×D4—28no—
142C22×D4—21no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—24no—
149C22×D4—24no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—44no—
153C22×D4—44no—
154C22×D4—416no—
155C22×D4—416no—
156C22×D4—432no—
157C22×D4—416no—
158C22×D4—48no—
159C22×D4C2×Q814yes—
160C22×D4C2×Q818yes—
161C22×D4C2×Q818yes—
162C22×D4C2×Q814yes—
163C22×D4C2412yes—
164C22×D4C2411yescharacteristic
165C22×D4C2411yescharacteristic
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×Q814yes—
173C23×C4#4114yes—
174C23×C4C2411yescharacteristic
175C23×C4C2412yes—
176C2×M16—28no—
177C2×M16—44no—
178C2×M16—44no—
179C22×C8C2418yes—
180C4⋊Q8—44no—
181C4⋊Q8—44no—
182C4×D4—28no—
183C4×D4—28no—
184C4×D4—44no—
185C4×D4—44no—
186C4×D4—416no—
187#117—42no—
188#117—42no—
189#117—42no—
190#117—42no—
191#117—44no—
192#117—44no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4C2×D414yes—
196C2×C4⋊C4C2414yes—
197C2×C4⋊C4C2418yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—48no—
208C2×C22⋊C4—48no—
209C2×C22⋊C4—88no—
210C2×C22⋊C4—88no—
211C2×C22⋊C4C2×D412yes—
212C2×C22⋊C4C2×D412yes—
213C2×C22⋊C4C2×D414yes—
214C2×C22⋊C4C2×D414yes—
215C2×C42—22no—
216C2×C42—24no—
217C2×C42C2414yes—
218#107—48no—
219#106—48no—
220C4⋊C8—44no—
221C4⋊C8—44no—
222#102—216no—
223#102—216no—
224#102—48no—
225#102—48no—
226C22⋊C8—28no—
227C22⋊C8—28no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—28no—
234#95C2×D418yes—
235C24—28no—
236C24—21no—
237C24—21no—
238C24—24no—
239C24—22no—
240C24—28no—
241C24—28no—
242C24—24no—
243C24—42no—
244C24—44no—
245C24—48no—
246C24—48no—
247C24—48no—
248C24—48no—
249C24—432no—
250C24—44no—
251C24—84no—
252C24—84no—
253C24—84no—
254C24—84no—
255C24—832no—
256C24—864no—
257C24C22⋊Q814yes—
258C24C22×D412yes—
259C24C22×D411yescharacteristic
260C2×Q8—84no—
261C2×Q8—88no—
262C2×D4—28no—
263C2×D4—216no—
264C2×D4—28no—
265C2×D4—28no—
266C2×D4—216no—
267C2×D4—216no—
268C2×D4—24no—
269C2×D4—28no—
270C2×D4—28no—
271C2×D4—24no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—48no—
280C2×D4—48no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—416no—
284C2×D4—416no—
285C2×D4—48no—
286C2×D4—48no—
287C2×D4—44no—
288C2×D4—48no—
289C2×D4—464no—
290C2×D4—432no—
291C2×D4C22×Q814yes—
292C2×D4C22×Q812yes—
293C2×D4C22×Q812yes—
294C2×D4H(5,𝔽2)14yes—
295C2×D4H(5,𝔽2)14yes—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—28no—
301C22×C4—24no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—48no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—88no—
310C22×C4—88no—
311C22×C4C22⋊Q818yes—
312C22×C4C22⋊Q814yes—
313C22×C4C22×D414yes—
314C22×C4C22×D414yes—
315C22×C4C22×D412yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×Q812yes—
318C22×C4C22×Q814yes—
319C22×C4C2×#4112yes—
320C22×C4C2×#4114yes—
321C22×C4C2511yescharacteristic
322M16—88no—
323C2×C8—28no—
324C2×C8—28no—
325C2×C8—28no—
326C4⋊C4—28no—
327C4⋊C4—28no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—44no—
331C4⋊C4—88no—
332C4⋊C4—84no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—48no—
338C22⋊C4—48no—
339C22⋊C4—44no—
340C22⋊C4—48no—
341C22⋊C4—88no—
342C22⋊C4—88no—
343C22⋊C4—84no—
344C22⋊C4—84no—
345C22⋊C4—84no—
346C42—24no—
347C42—24no—
348C42—44no—
349C42—84no—
350C23—216no—
351C23—24no—
352C23—24no—
353C23—24no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—24no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—44no—
362C23—48no—
363C23—416no—
364C23—416no—
365C23—48no—
366C23—416no—
367C23—416no—
368C23—416no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23—44no—
374C23—88no—
375C23—88no—
376C23—88no—
377C23—88no—
378C23—816no—
379C23—832no—
380C23—832no—
381C23—8128no—
382C23—816no—
383C23C2×C22⋊Q814yes—
384C23C2×C22⋊Q812yes—
385C23C23×D411yescharacteristic
386Q8—168no—
387D4—216no—
388D4—22no—
389D4—22no—
390D4—24no—
391D4—24no—
392D4—24no—
393D4—24no—
394D4—416no—
395D4—416no—
396D4—48no—
397D4—44no—
398D4#54314yes—
399D4#54314yes—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—28no—
407C2×C4—24no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—48no—
413C2×C4—44no—
414C2×C4—48no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—44no—
418C2×C4—48no—
419C2×C4—168no—
420C2×C4—168no—
421C2×C4C2×C22⋊Q812yes—
422C2×C4C2×C22⋊Q814yes—
423C2×C4C23×Q811yescharacteristic
424C2×C4C22×#4111yescharacteristic
425C2×C4C2×H(5,𝔽2)12yes—
426C2×C4C2×H(5,𝔽2)12yes—
427C8—48no—
428C22—24no—
429C22—24no—
430C22—24no—
431C22—22no—
432C22—24no—
433C22—24no—
434C22—24no—
435C22—416no—
436C22—44no—
437C22—44no—
438C22—48no—
439C22—48no—
440C22—48no—
441C22—44no—
442C22—44no—
443C22—44no—
444C22—44no—
445C22—816no—
446C22—832no—
447C22—832no—
448C22—816no—
449C22—88no—
450C22C22×C22⋊Q811yescharacteristic
451C22#348412yes—
452C22#358514yes—
453C4—22no—
454C4—24no—
455C4—44no—
456C4—44no—
457C4—84no—
458C4—88no—
459C4C2×#54311yescharacteristic
460C4C2×#54311yescharacteristic
461C4#349012yes—
462C2—44no—
463C2—42no—
464C2—44no—
465C2—44no—
466C2—84no—
467C2#3498211yescharacteristic
468C2#6049012yes—
469C2#6164414yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37480C2semidirect
3. #32962C2semidirect
4. #29959C2semidirect
5. #25974C2semidirect
6. #25966C2semidirect
7. #21943C2semidirect
8. #21935C2semidirect
12. C2×#476C22semidirect
13. C2×#474C22semidirect
20. #2628C22semidirect
22. #2439C22semidirect
33. #2092C22semidirect
34. #2067C22semidirect
35. #2065C22semidirect
36. #2023C22semidirect
37. #1864C22semidirect
38. #1862C22semidirect
39. #1854C22semidirect
40. #1854C22semidirect
41. #1851C22semidirect
42. #1809C22semidirect
43. #1804C22semidirect
44. #1804C22semidirect
45. #1798C22semidirect
56. C23×D4Q8semidirect
63. C2×C4⋊Q8C23semidirect
78. C2×#107C23semidirect
79. C2×#106C23semidirect
80. C2×C4⋊C8C23semidirect
81. C2×C4⋊C8C23semidirect
100. #389C23semidirect
104. #382C23semidirect
124. #340C23semidirect
125. #340C23semidirect
126. #336C23semidirect

Maximal subgroups

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

subgroupindex
#374802
#341712
#329622
#299592
#259742
#259662
#219432
#219352
#205982