Finite Group Databasebeta

← #6477608

Subgroups — #6477608

Order 512.

2040 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C212yesmaximal
2#35011C211yesmaximal, characteristic
3C2×#2345C211yesmaximal, characteristic
4#33109C214yesmaximal, complemented
5#29757C214yesmaximal, complemented
6#24582C218yesmaximal, complemented
7#24482C218yesmaximal, complemented
8#21930C211yesmaximal, characteristic
9#21759C212yesmaximal
10C22×C22⋊Q8C2212yes—
11C23×C4⋊C4C2211yescharacteristic
12C2×#484—216no—
13C2×#484C2218yes—
14C2×#484C2214yes—
15C2×#484C2214yes—
16C22×#107C2214yes—
17C22×#103C2214yes—
18C22×C22⋊C8C2212yes—
19#2857C2214yes—
20#2659—22no—
21#2583—24no—
22#2502C2214yes—
23#2434C2214yes—
24#2345—24no—
25#2345C2218yes—
26#2315—21no—
27#2315—21no—
28C2×#389C2212yes—
29C2×#387—22no—
30C2×#380—21no—
31C2×#380C2211yescharacteristic
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C22⋊C4—21no—
34C2×C4×C22⋊C4C2211yescharacteristic
35#2083C2218yescomplemented
36#2068C22116yescomplemented
37#2043C2218yescomplemented
38#1981C2218yescomplemented
39#1947C2218yescomplemented
40#1928C22116yescomplemented
41#1892C2218yes—
42#1872C2218yes—
43#1850C2218yes—
44#1824C2214yes—
45#1797C2214yes—
46#1787C2214yes—
47C23×Q8C2312yes—
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C23×C8C2314yes—
51C2×C4⋊Q8C2314yes—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—28no—
55C2×C22⋊Q8—28no—
56C2×C22⋊Q8C2314yes—
57C2×C22⋊Q8C2314yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D418yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4D411yescharacteristic
72C22×C22⋊C4D411yescharacteristic
73C22×C42—22no—
74C22×C42C2312yes—
75#484—232no—
76#484—216no—
77#484—216no—
78C2×#107—216no—
79C2×#107C2318yes—
80C2×#107C2318yes—
81C2×C4⋊C8C2318yes—
82C2×#103—216no—
83C2×#103C2318yes—
84C2×#103C2318yes—
85C2×#103C23116yes—
86C2×C22⋊C8—28no—
87C2×C22⋊C8C2314yes—
88C2×C22⋊C8C2314yes—
89#398—28no—
90#395—28no—
91#391C2318yes—
92#389—24no—
93#389—24no—
94#389—24no—
95#389C2318yes—
96#387—216no—
97#385—24no—
98#385—24no—
99#385—24no—
100#385—24no—
101#384C2314yes—
102#380—24no—
103#380—24no—
104#380—22no—
105#380—28no—
106#380C2314yes—
107C4×C4⋊C4—24no—
108C4×C4⋊C4C2318yes—
109C4×C22⋊C4—28no—
110C4×C22⋊C4D414yes—
111C4×C22⋊C4C2314yes—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95D412yes—
116#340C23116yes—
117#336C2318yes—
118C25C2×D411yescharacteristic
119C22×Q8—22no—
120C22×Q8—24no—
121C22×Q8—24no—
122C22×Q8—48no—
123C22×Q8C2414yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—28no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—44no—
132C23×C4—84no—
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D414yes—
137C23×C4C2×D412yes—
138C23×C4#4114yes—
139C23×C4C2411yescharacteristic
140C23×C4C2412yes—
141C22×C8—28no—
142C22×C8—48no—
143C22×C8C2418yes—
144C4⋊Q8—48no—
145C22⋊Q8—28no—
146C22⋊Q8—28no—
147C22⋊Q8—28no—
148C22⋊Q8—28no—
149C22⋊Q8—48no—
150C22⋊Q8—48no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—216no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4—84no—
173C2×C4⋊C4C2×D4116yes—
174C2×C4⋊C4C2×D414yes—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4C2414yes—
177C2×C4⋊C4C2414yes—
178C2×C4⋊C4C2412yes—
179C2×C4⋊C4C2412yes—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—48no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4Q1618yes—
187C2×C22⋊C4C2×D414yes—
188C2×C22⋊C4C2×D414yes—
189C2×C42—28no—
190C2×C42—44no—
191C2×C42—84no—
192C2×C42C2×D414yes—
193C2×C42C2414yes—
194#107—216no—
195#107—216no—
196C4⋊C8—48no—
197#103—216no—
198#103—216no—
199#103—416no—
200C22⋊C8—28no—
201C22⋊C8—28no—
202#95—28no—
203#95—28no—
204#95—24no—
205#95—28no—
206#95—28no—
207#95C2×D418yes—
208C24—42no—
209C24—42no—
210C24#12114yes—
211C24C2×Q1614yes—
212C24#13714yes—
213C24C22×D412yes—
214C24C22×D411yescharacteristic
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—216no—
218C2×Q8—416no—
219C2×Q8—44no—
220C2×Q8—84no—
221C2×Q8—88no—
222C22×C4—24no—
223C22×C4—28no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—216no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—48no—
234C22×C4—48no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—84no—
249C22×C4—88no—
250C22×C4—84no—
251C22×C4—816no—
252C22×C4#12118yes—
253C22×C4#12114yes—
254C22×C4C2×Q1618yes—
255C22×C4C22×D414yes—
256C22×C4C22×D412yes—
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D412yes—
259C22×C4C22×D414yes—
260C22×C4C22×D414yes—
261C22×C4C2×#4114yes—
262C22×C4C2×#4112yes—
263C22×C4C2511yescharacteristic
264C2×C8—28no—
265C2×C8—216no—
266C2×C8—416no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—216no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—28no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—48no—
278C4⋊C4—48no—
279C4⋊C4—84no—
280C4⋊C4—84no—
281C4⋊C4—88no—
282C4⋊C4—84no—
283C4⋊C4—84no—
284C4⋊C4C22×D414yes—
285C4⋊C4C22×D414yes—
286C22⋊C4—28no—
287C22⋊C4—48no—
288C22⋊C4—48no—
289C22⋊C4—48no—
290C22⋊C4C2×Q16116yes—
291C42—28no—
292C42—44no—
293C42—88no—
294C23—22no—
295C23—24no—
296C23—22no—
297C23—24no—
298C23—44no—
299C23—44no—
300C23—48no—
301C23—48no—
302C23#462116yes—
303C23C2×#12112yes—
304C23C2×#12114yes—
305C23C22×Q1614yes—
306C23C22×Q1612yes—
307C23C2×#13714yes—
308C23C2×#13712yes—
309C23C23×D411yescharacteristic
310Q8—48no—
311Q8—48no—
312Q8—88no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—28no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—28no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—48no—
327C2×C4—44no—
328C2×C4—48no—
329C2×C4—48no—
330C2×C4—48no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—48no—
339C2×C4—44no—
340C2×C4—84no—
341C2×C4—816no—
342C2×C4—88no—
343C2×C4—88no—
344C2×C4—88no—
345C2×C4C2×#12112yes—
346C2×C4C2×#12114yes—
347C2×C4C22×Q1618yes—
348C2×C4C23×D411yescharacteristic
349C2×C4C22×#4111yescharacteristic
350C2×C4#58514yes—
351C8—48no—
352C22—24no—
353C22—28no—
354C22—22no—
355C22—24no—
356C22—24no—
357C22—28no—
358C22—24no—
359C22—22no—
360C22—416no—
361C22C2×#46214yes—
362C22C2×#46218yes—
363C22C22×#12111yescharacteristic
364C22#348512yes—
365C22C23×Q1611yescharacteristic
366C22C22×#13711yescharacteristic
367C22#358512yes—
368C4—28no—
369C4—22no—
370C4—24no—
371C4—44no—
372C4—44no—
373C4—44no—
374C4—84no—
375C4—84no—
376C4—84no—
377C4#344912yes—
378C4#349112yes—
379C2—24no—
380C2—28no—
381C2#3508911yescharacteristic
382C2C22×#46211yescharacteristic
383C2#6042311yescharacteristic
384C2#6169812yes—
385C2#6170812yes—
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
4. #33109C2semidirect
5. #29757C2semidirect
6. #24582C2semidirect
7. #24482C2semidirect
35. #2083C22semidirect
36. #2068C22semidirect
37. #2043C22semidirect
38. #1981C22semidirect
39. #1947C22semidirect
40. #1928C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
#350112
C2×#23452
#331092
#297572
#245822
#244822
#219302
#217592