Finite Group Databasebeta

← #298433

Subgroups — #298433

Order 512.

1039 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 1039 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59832C211yesmaximal, characteristic
2#28511C212yesmaximal
3#28440C211yesmaximal, characteristic
4C2×#1633C214yesmaximal
5#10237C214yesmaximal
6#9673C212yesmaximal
7C4×#333C211yesmaximal, characteristic
8C2×Q82C2212yes—
9C2×#494C2212yes—
10C2×#477C2218yes—
11C2×#462—22no—
12#2868C2211yescharacteristic
13#2734—22no—
14#2582C2212yes—
15#2551—22no—
16#2546—22no—
17#2451—24no—
18#2365—22no—
19C4×C4⋊Q8C2211yescharacteristic
20C4×C4⋊Q8C2212yes—
21#2095—24no—
22#2016—22no—
23#1956—22no—
24#1956C2212yes—
25#1845—22no—
26C4×C4⋊C8C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28#1633—216no—
29C2×#333C2212yes—
30C2×Q8⋊C8C2218yes—
31#1296C2212yes—
32C22×Q16—24no—
33Q82—28no—
34Q82—44no—
35C2×C4⋊Q8—24no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—24no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8D412yes—
41C2×C4⋊Q8C2312yes—
42C2×C4⋊Q8C2311yescharacteristic
43C2×C4×Q8—24no—
44C2×C4×Q8D412yes—
45C2×C4×Q8C2314yes—
46#494—28no—
47#477—216no—
48#477—48no—
49#462—48no—
50C2×#107—28no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8C2314yes—
53C2×#103—28no—
54C2×#103—24no—
55C2×C4×C8—22no—
56C2×C4×C8C2312yes—
57#398—24no—
58#398—24no—
59#395—28no—
60#391—24no—
61#391—24no—
62#391D414yes—
63#391D412yes—
64#389—24no—
65#389—24no—
66#389—22no—
67#389—22no—
68#389—22no—
69#387—28no—
70#384—22no—
71#384—22no—
72#384D412yes—
73#384C2311yescharacteristic
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C43C2311yescharacteristic
79#336—22no—
80#336—21no—
81#333D418yes—
82Q8⋊C8—216no—
83C22×Q8—24no—
84C22×Q8—22no—
85C22×Q8C2×D414yes—
86C22×Q8C2×D412yes—
87C2×Q16—48no—
88C2×Q16—48no—
89C2×Q16—84no—
90C22×C8—22no—
91C22×C8—24no—
92C4⋊Q8—24no—
93C4⋊Q8—24no—
94C4⋊Q8—22no—
95C4⋊Q8—24no—
96C4⋊Q8—44no—
97C4⋊Q8—416no—
98C4⋊Q8—48no—
99C4⋊Q8—48no—
100C4⋊Q8—84no—
101C4⋊Q8—82no—
102C4⋊Q8Q1618yescomplemented
103C4⋊Q8C2×D414yes—
104C4×Q8—28no—
105C4×Q8—28no—
106C4×Q8—44no—
107C4×Q8—416no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4C2×D414yes—
123C2×C4⋊C4C2×D412yes—
124C2×C42—24no—
125C2×C42—22no—
126C2×C42—24no—
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D412yes—
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D412yes—
131C2×C42C2411yescharacteristic
132#107—416no—
133C4⋊C8—24no—
134C4⋊C8—24no—
135C4⋊C8C2×D418yes—
136#103—416no—
137#103—48no—
138C4×C8—24no—
139C4×C8—44no—
140#95—24no—
141#95—28no—
142#95—24no—
143#95—24no—
144#95—24no—
145C2×Q8—22no—
146C2×Q8—24no—
147C2×Q8—28no—
148C2×Q8—28no—
149C2×Q8—48no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—44no—
153C2×Q8—44no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—84no—
157C2×Q8—84no—
158C2×Q8#12714yes—
159C2×Q8C2×Q1618yes—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—42no—
165C22×C4#12011yescharacteristic
166C22×C4#12012yes—
167C22×C4#12011yescharacteristic
168C22×C4C22×D411yescharacteristic
169C22×C4C22×D412yes—
170Q16—816no—
171C2×C8—24no—
172C2×C8—28no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—44no—
176C2×C8—44no—
177C4⋊C4—28no—
178C4⋊C4—28no—
179C4⋊C4—28no—
180C4⋊C4—28no—
181C4⋊C4—42no—
182C4⋊C4—416no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—48no—
188C4⋊C4—82no—
189C4⋊C4—82no—
190C4⋊C4—82no—
191C4⋊C4—84no—
192C4⋊C4—84no—
193C4⋊C4—82no—
194C42—24no—
195C42—22no—
196C42—41no—
197C42—48no—
198C42—48no—
199C42—44no—
200C42C2×Q1614yes—
201C42#13714yes—
202C42C22×D412yes—
203C42H(5,𝔽2)12yes—
204C23C2×#12011yescharacteristic
205Q8—216no—
206Q8—44no—
207Q8—48no—
208Q8—82no—
209Q8—84no—
210Q8—84no—
211Q8—168no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—24no—
217C2×C4—28no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4#45114yes—
233C2×C4#45114yes—
234C2×C4#45312yes—
235C2×C4C2×#12711yescharacteristic
236C2×C4#53511yescharacteristic
237C2×C4D4214yes—
238C2×C4C22×Q1612yes—
239C2×C4C2×#13712yes—
240C2×C4#57814yes—
241C8—44no—
242C8—48no—
243C22#243311yescharacteristic
244C22C2×#45112yes—
245C22#301112yes—
246C22C2×#45311yescharacteristic
247C22#302111yescharacteristic
248C4—21no—
249C4—21no—
250C4—24no—
251C4—41no—
252C4—44no—
253C4—48no—
254C4—82no—
255C4—84no—
256C4—162no—
257C4#163314yescomplemented
258C4#327614yes—
259C4D4×Q1614yes—
260C2C2×#163311yescharacteristic
261C2#1871911yescharacteristic
262C2#1891512yes—
263C2#2573911yescharacteristic
264C2#2576012yes—
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
102. C4⋊Q8Q16semidirect
257. C4#1633semidirect

Maximal subgroups

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

subgroupindex
#598322
#285112
#284402
C2×#16332
#102372
#96732
C4×#3332