Finite Group Databasebeta

← #6479526

Subgroups — #6479526

Order 512.

1938 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic
3#33099C214yesmaximal, complemented
4#29909C214yesmaximal, complemented
5#27858C218yesmaximal, complemented
6#27614C214yesmaximal, complemented
7#27586C214yesmaximal, complemented
8#21940C212yesmaximal
9#21759C212yesmaximal
10C2×#1817C211yesmaximal, characteristic
11C22×C22⋊Q8C2211yescharacteristic
12C22×#117—21no—
13C23×C42C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15#2855—21no—
16#2855—21no—
17#2855—22no—
18#2855C2212yes—
19#2855C2214yes—
20#2855C2212yes—
21C2×#391C2212yes—
22C2×#389C2212yes—
23C2×#387—22no—
24C2×#385—21no—
25#2285—22no—
26#2285—22no—
27#2285C2214yes—
28#2285C2214yes—
29C2×#384C2212yes—
30C2×#382—21no—
31C2×#381—21no—
32C2×#380—21no—
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×C4×C4⋊C4—21no—
36C2×C4×C4⋊C4C2212yes—
37#2100C2218yescomplemented
38#2095C2218yescomplemented
39#2083C2218yescomplemented
40#2072C2218yescomplemented
41#1928C22116yescomplemented
42#1921C22116yescomplemented
43#1892C2218yes—
44#1872C2218yes—
45#1852C2218yes—
46#1852C2218yes—
47#1824C2214yes—
48#1817C2212yes—
49#1817C2214yes—
50#1817C2212yes—
51#1808C2218yes—
52#1787C2214yes—
53C2×#336C2214yes—
54C23×Q8C2311yescharacteristic
55C24×C4D411yescharacteristic
56C24×C4D411yescharacteristic
57C24×C4C2311yescharacteristic
58C23×C8C2314yes—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—28no—
62C2×C22⋊Q8C2312yes—
63C2×C22⋊Q8C2312yes—
64C2×#117—24no—
65C2×#117—24no—
66C2×#117—48no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2314yes—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42—28no—
78C22×C42—22no—
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D412yes—
82C22×C42D412yes—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2312yes—
86C22×C42C2312yes—
87C2×#107C23116yes—
88C2×#103C23116yes—
89C2×C22⋊C8—28no—
90C2×C22⋊C8C2314yes—
91C2×C22⋊C8C2314yes—
92#391—22no—
93#391—22no—
94#391C2314yes—
95#391C2314yes—
96#389—22no—
97#389—22no—
98#389C2314yes—
99#389C2314yes—
100#387—24no—
101#387—24no—
102#387—24no—
103#387—24no—
104#385—22no—
105#385—24no—
106#385—22no—
107#384—22no—
108#384—22no—
109#384C2314yes—
110#384C2314yes—
111#382—24no—
112#382—24no—
113#381—22no—
114#381—24no—
115#381—22no—
116#380—21no—
117#380—21no—
118#380—24no—
119#380—24no—
120#380—22no—
121#380—22no—
122#380C2312yes—
123#380C2314yes—
124#380C2314yes—
125#380C2312yes—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—24no—
129C4×C4⋊C4C2318yes—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136#340C23116yes—
137#340C23116yes—
138#336C2318yes—
139#336C2318yes—
140C25C2×D411yescharacteristic
141C22×Q8—21no—
142C22×Q8—24no—
143C22×Q8—44no—
144C22×Q8C2412yes—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—21no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4Q1618yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4#4118yes—
167C23×C4C2411yescharacteristic
168C23×C4C2411yescharacteristic
169C23×C4C2411yescharacteristic
170C22×C8—28no—
171C22×C8—48no—
172C22×C8C2418yes—
173C22⋊Q8—28no—
174C22⋊Q8—28no—
175C22⋊Q8—44no—
176C22⋊Q8—44no—
177#117—416no—
178#117—416no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—48no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—48no—
194C2×C4⋊C4—48no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4C2412yes—
198C2×C4⋊C4C2418yes—
199C2×C4⋊C4C2414yes—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—48no—
209C2×C42—24no—
210C2×C42—28no—
211C2×C42—24no—
212C2×C42—28no—
213C2×C42—24no—
214C2×C42—28no—
215C2×C42—24no—
216C2×C42—28no—
217C2×C42—28no—
218C2×C42—48no—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2412yes—
226C2×C42C2412yes—
227#107—416no—
228#103—416no—
229C22⋊C8—28no—
230C22⋊C8—28no—
231#95—24no—
232#95—24no—
233#95—28no—
234#95—22no—
235#95—22no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242C24—42no—
243C24—42no—
244C24#12314yes—
245C24C2×Q1614yes—
246C24AGL(1,ℤ/8)14yes—
247C24C22×D411yescharacteristic
248C24C22×D411yescharacteristic
249C24C22×D411yescharacteristic
250C2×Q8—24no—
251C2×Q8—28no—
252C2×Q8—44no—
253C2×Q8—42no—
254C2×Q8—88no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—232no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—28no—
277C22×C4—22no—
278C22×C4—416no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—416no—
286C22×C4—48no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—48no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—48no—
295C22×C4—44no—
296C22×C4#12314yes—
297C22×C4#12314yes—
298C22×C4#12314yes—
299C22×C4C2×Q1614yes—
300C22×C4C2×Q1618yes—
301C22×C4C2×Q1618yes—
302C22×C4C2×Q1614yes—
303C22×C4C22×D412yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D412yes—
308C22×C4C22×D412yes—
309C22×C4C22×D412yes—
310C22×C4C2×#4114yes—
311C22×C4C2×#4114yes—
312C22×C4C2×#4114yes—
313C22×C4C2511yescharacteristic
314C2×C8—28no—
315C2×C8—216no—
316C2×C8—88no—
317C4⋊C4—24no—
318C4⋊C4—216no—
319C4⋊C4—28no—
320C4⋊C4—48no—
321C4⋊C4—44no—
322C4⋊C4—48no—
323C4⋊C4—48no—
324C4⋊C4—48no—
325C4⋊C4—48no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—48no—
330C4⋊C4—48no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—48no—
336C22⋊C4—48no—
337C42—24no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—24no—
344C42—24no—
345C42—416no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—44no—
353C23—44no—
354C23—48no—
355C23—48no—
356C23#484116yes—
357C23C2×#12312yes—
358C23C2×#12312yes—
359C23C2×#12312yes—
360C23C22×Q1612yes—
361C23C22×Q1612yes—
362C23C22×Q1612yes—
363C23C2×AGL(1,ℤ/8)12yes—
364C23C2×AGL(1,ℤ/8)12yes—
365C23C2×AGL(1,ℤ/8)12yes—
366C23C23×D411yescharacteristic
367Q8—48no—
368Q8—84no—
369C2×C4—28no—
370C2×C4—28no—
371C2×C4—28no—
372C2×C4—216no—
373C2×C4—216no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—28no—
384C2×C4—28no—
385C2×C4—28no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—48no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—48no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—48no—
404C2×C4—44no—
405C2×C4—816no—
406C2×C4—88no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4C2×#12312yes—
410C2×C4C2×#12312yes—
411C2×C4C2×#12312yes—
412C2×C4C22×Q1614yes—
413C2×C4C22×Q1614yes—
414C2×C4#57514yes—
415C2×C4C22×#4112yes—
416C2×C4#58514yes—
417C8—48no—
418C22—24no—
419C22—24no—
420C22—24no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—24no—
425C22—24no—
426C22—24no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—416no—
431C22C2×#48414yes—
432C22C2×#48414yes—
433C22C2×#48414yes—
434C22C22×#12311yescharacteristic
435C22#344912yes—
436C22C23×Q1611yescharacteristic
437C22C22×AGL(1,ℤ/8)11yescharacteristic
438C22C2×#57512yes—
439C4—22no—
440C4—22no—
441C4—22no—
442C4—24no—
443C4—24no—
444C4—24no—
445C4—42no—
446C4—82no—
447C4—88no—
448C4—84no—
449C4—82no—
450C4#349114yes—
451C2—24no—
452C2—24no—
453C2—24no—
454C2#3509711yescharacteristic
455C2C22×#48411yescharacteristic
456C2#6051111yescharacteristic
457C2#6094312yes—
458C2#6095812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33099C2semidirect
4. #29909C2semidirect
5. #27858C2semidirect
6. #27614C2semidirect
7. #27586C2semidirect
37. #2100C22semidirect
38. #2095C22semidirect
39. #2083C22semidirect
40. #2072C22semidirect
41. #1928C22semidirect
42. #1921C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#22852
#330992
#299092
#278582
#276142
#275862
#219402
#217592
C2×#18172