Finite Group Databasebeta

← #6376391

Subgroups — #6376391

Order 512.

1854 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×M16C212yesmaximal
2C2×#2399C211yesmaximal, characteristic
3#22245C218yesmaximal, complemented
4#22243C212yesmaximal, complemented
5#22241C214yesmaximal, complemented
6#22239C212yesmaximal, complemented
7#21303C212yesmaximal
8#21023C218yesmaximal
9#20753C211yesmaximal, characteristic
10#20365C211yesmaximal, characteristic
11C22×C4×Q8C2212yes—
12Q8×M16—216no—
13C2×#446C2214yes—
14C2×#446C2218yes—
15C2×C8×Q8C2214yes—
16C2×#423C2212yes—
17C2×#423C2214yes—
18C2×C4×M16C2214yes—
19C2×C4×M16C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21#2399C414yes—
22#2399C414yes—
23#2399C2211yescharacteristic
24#2399C2212yes—
25#2399C2211yescharacteristic
26#2399C2214yes—
27C2×#398C414yes—
28C2×#395C414yes—
29C2×C4×C4⋊C4C2214yes—
30C2×#376C2211yescharacteristic
31#1985C2214yes—
32#1985C2214yes—
33#1984C2218yes—
34#1917C22116yes—
35#1917C22116yes—
36#1916C22116yes—
37#1916C22116yes—
38#1832C2212yes—
39#1832C2212yes—
40#1832C2214yes—
41#1746C2212yes—
42#1746C2212yes—
43#1746C2214yes—
44#1728C2214yes—
45#1728C2218yes—
46#1728C2214yes—
47C23×Q8C2×C412yes—
48C22×M16C2314yes—
49C22×M16C2312yes—
50C22×M16C2312yes—
51C2×C4×Q8—28no—
52C2×C4×Q8—28no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C414yes—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2312yes—
57C2×C4×Q8C2314yes—
58C2×C4×Q8C2312yes—
59C22×C4⋊C4C2×C414yes—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C418yes—
62C22×C42C2314yes—
63C22×C42C2312yes—
64C22×C42C2311yescharacteristic
65#446—232no—
66#446—216no—
67C8×Q8—216no—
68#423—216no—
69#423—28no—
70C2×C4⋊C8C2314yes—
71C2×C4⋊C8C2314yes—
72C2×C4⋊C8C2318yes—
73C2×C4⋊C8C2318yes—
74C4×M16—216no—
75C4×M16—24no—
76C4×M16D418yes—
77C2×#97C2318yes—
78C2×#97C2312yes—
79C2×#97C2312yes—
80C2×C4×C8C2318yes—
81C2×C4×C8C2312yes—
82C2×C4×C8C2312yes—
83#398C2×C414yes—
84#398C2×C414yes—
85#398C2×C418yes—
86#398C2×C414yes—
87#398C2×C414yes—
88#398C2×C418yes—
89#395C2×C414yes—
90#395C2×C414yes—
91#395C2×C418yes—
92#395C2×C418yes—
93#395C2×C414yes—
94#395C2×C414yes—
95C4×C4⋊C4C2×C414yes—
96C4×C4⋊C4C2×C418yes—
97C4×C4⋊C4C2×C414yes—
98C4×C4⋊C4C2314yes—
99C4×C4⋊C4C2318yes—
100C4×C4⋊C4C2314yes—
101#376C2×C414yes—
102#376C2311yescharacteristic
103#376C2311yescharacteristic
104#376C2312yes—
105C2×#95C2×C414yes—
106C2×#95C2×C414yes—
107C2×#95C2×C414yes—
108#336C2318yes—
109#336C2318yes—
110#336C23116yes—
111#336C2318yes—
112#336C2318yes—
113#336C23116yes—
114C22×Q8—28no—
115C22×Q8—48no—
116C22×Q8C22×C412yes—
117C22×Q8C22×C414yes—
118C22×Q8C22×C412yes—
119C23×C4C22×C414yes—
120C23×C4C22×C414yes—
121C23×C4C22×C414yes—
122C23×C4C22×C412yes—
123C23×C4C2411yescharacteristic
124C2×M16—28no—
125C2×M16—28no—
126C2×M16—24no—
127C2×M16—216no—
128C2×M16—28no—
129C2×M16C2×D418yes—
130C22×C8C2414yes—
131C22×C8C2418yes—
132C22×C8C2414yes—
133C4×Q8—28no—
134C4×Q8—216no—
135C4×Q8—28no—
136C4×Q8—24no—
137C4×Q8—28no—
138C4×Q8—24no—
139C4×Q8—28no—
140C4×Q8—24no—
141C4×Q8—24no—
142C2×C4⋊C4—216no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—416no—
145C2×C4⋊C4C22×C414yes—
146C2×C4⋊C4C22×C414yes—
147C2×C4⋊C4C22×C414yes—
148C2×C4⋊C4C22×C414yes—
149C2×C4⋊C4C22×C412yes—
150C2×C4⋊C4C22×C414yes—
151C2×C4⋊C4C22×C412yes—
152C2×C4⋊C4C22×C418yes—
153C2×C4⋊C4C22×C418yes—
154C2×C4⋊C4C22×C418yes—
155C2×C4⋊C4C22×C418yes—
156C2×C42—28no—
157C2×C42—28no—
158C2×C42—28no—
159C2×C42C22×C414yes—
160C2×C42C22×C414yes—
161C2×C42C22×C414yes—
162C2×C42C22×C412yes—
163C2×C42C2×D414yes—
164C2×C42#4114yes—
165C2×C42C2414yes—
166C2×C42C2414yes—
167C2×C42C2412yes—
168C2×C42C2412yes—
169C2×C42C2411yescharacteristic
170C2×C42C2411yescharacteristic
171C4⋊C8—216no—
172C4⋊C8—216no—
173C4⋊C8—28no—
174C4⋊C8—28no—
175#97—216no—
176#97—24no—
177#97C2×D418yes—
178C4×C8—216no—
179C4×C8—24no—
180C4×C8C2×D418yes—
181#95C22×C414yes—
182#95C22×C414yes—
183#95C22×C418yes—
184#95C22×C414yes—
185#95C22×C418yes—
186#95C22×C414yes—
187#95C22×C414yes—
188#95C22×C418yes—
189#95C22×C414yes—
190C24C23×C411yescharacteristic
191C2×Q8—24no—
192C2×Q8—28no—
193C2×Q8—24no—
194C2×Q8—28no—
195C2×Q8—28no—
196C2×Q8—44no—
197C2×Q8—44no—
198C2×Q8—432no—
199C22×C4—28no—
200C22×C4—28no—
201C22×C4—28no—
202C22×C4—28no—
203C22×C4—48no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—48no—
207C22×C4C4×D418yes—
208C22×C4C23×C414yes—
209C22×C4C23×C414yes—
210C22×C4C23×C414yes—
211C22×C4C23×C414yes—
212C22×C4C23×C414yes—
213C22×C4C23×C412yes—
214C22×C4C23×C412yes—
215C22×C4C23×C414yes—
216C22×C4C23×C411yescharacteristic
217C22×C4C22×D412yes—
218C22×C4C2×#4112yes—
219C22×C42−1+418yes—
220C22×C4C2511yescharacteristic
221M16—216no—
222M16—416no—
223M16—48no—
224C2×C8—28no—
225C2×C8—28no—
226C2×C8—24no—
227C2×C8—216no—
228C2×C8—28no—
229C2×C8C22×D418yes—
230C4⋊C4—28no—
231C4⋊C4—28no—
232C4⋊C4—24no—
233C4⋊C4—28no—
234C4⋊C4—24no—
235C4⋊C4—28no—
236C4⋊C4—28no—
237C4⋊C4—216no—
238C4⋊C4—216no—
239C4⋊C4—48no—
240C4⋊C4—48no—
241C42—28no—
242C42—28no—
243C42—28no—
244C42—24no—
245C42—24no—
246C42—24no—
247C42—24no—
248C42—24no—
249C42—24no—
250C42C4×D418yes—
251C42C22×D412yes—
252C42C22×D412yes—
253C42C2×#4112yes—
254C42C2×#4112yes—
255C23—24no—
256C23C2×C4×D412yes—
257C23#51914yes—
258C23C24×C411yescharacteristic
259Q8—216no—
260Q8—416no—
261Q8—48no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—28no—
272C2×C4—28no—
273C2×C4—28no—
274C2×C4—28no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—48no—
278C2×C4—416no—
279C2×C4—416no—
280C2×C4—88no—
281C2×C4C2×C4×D414yes—
282C2×C4C2×C4×D414yes—
283C2×C4C2×C4×D412yes—
284C2×C4#51914yes—
285C2×C4#53618yes—
286C2×C4#54118yes—
287C2×C4#56818yes—
288C2×C4C23×D411yescharacteristic
289C2×C4C22×#4111yescharacteristic
290C2×C4C2×2−1+414yes—
291C8—28no—
292C8—48no—
293C8—44no—
294C22—28no—
295C22—24no—
296C22#232714yes—
297C22C22×C4×D411yescharacteristic
298C22C2×#51912yes—
299C22C2×#56814yes—
300C4—24no—
301C4—24no—
302C4—28no—
303C4—44no—
304C4—44no—
305C4—44no—
306C4—44no—
307C4—44no—
308C4#232714yes—
309C4C2×#53612yes—
310C4C2×#54112yes—
311C2—24no—
312C2C2×#232711yescharacteristic
313C2#6014112yes—
314C2#6024914yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC24C23×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22245C2semidirect
4. #22243C2semidirect
5. #22241C2semidirect
6. #22239C2semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×M162
C2×#23992
#222452
#222432
#222412
#222392
#213032
#210232
#207532
#203652