Finite Group Databasebeta

← #508370

Subgroups — #508370

Order 512.

734 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 170 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
1C2×#2208C212yesmaximal
2#33302C211yesmaximal, characteristic, complemented
3#21275C211yesmaximal, characteristic
4#13409C214yesmaximal, complemented
5#13327C214yesmaximal
6#13253C212yesmaximal, complemented
7#12879C211yesmaximal, characteristic, complemented
8C22×SD32C2214yes—
9C2×#494—22no—
10C2×#494—21no—
11C2×#494C2212yes—
12C2×#494C2214yes—
13C2×C4×D8C2211yescharacteristic
14#2868—21no—
15#2293—21no—
16#2208—28no—
17C2×C4⋊C16C2212yes—
18C2×#358C2214yes—
19C2×#358C2218yes—
20#1970C2212yes—
21#1942C2212yes—
22#1934—22no—
23C4×#107C2211yescharacteristic
24C4×#102—21no—
25C42×C8C2211yescharacteristic
26#1374C2214yes—
27C22×Q16—22no—
28C22×Q16—24no—
29C22×Q16C2314yes—
30C22×D8C2311yescharacteristic
31C2×C4⋊Q8—24no—
32C2×C4⋊Q8—22no—
33C2×C4×D4—21no—
34C2×#117—41no—
35C2×SD32—28no—
36C2×SD32—48no—
37C22×C16C2314yes—
38#494—24no—
39#494—24no—
40#494—48no—
41#494—48no—
42#494—44no—
43C4×D8—22no—
44C4×D8D414yes—
45C2×#107C2312yes—
46C2×#107C2311yescharacteristic
47C2×#102—21no—
48C2×#102—22no—
49C2×C4×C8D412yes—
50C2×C4×C8D411yescharacteristic
51C2×C4×C8C2312yes—
52C2×C4×C8C2311yescharacteristic
53#390—22no—
54#389—22no—
55#388—42no—
56#384—41no—
57#384—42no—
58#384—41no—
59C4×C4⋊C4—21no—
60C4×C22⋊C4—41no—
61C43D411yescharacteristic
62C4⋊C16—24no—
63#358—28no—
64#358—48no—
65#340—21no—
66#340—22no—
67C22×Q8—24no—
68C22×D4—21no—
69C23×C4—41no—
70C2×Q16—24no—
71C2×Q16—24no—
72C2×Q16—28no—
73C2×Q16—28no—
74C2×Q16—48no—
75C2×Q16—48no—
76C2×Q16—48no—
77C2×Q16—44no—
78C2×Q16—48no—
79C2×D8—21no—
80C2×D8—24no—
81C2×D8C2×D412yes—
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D412yes—
84C22×C8C2411yescharacteristic
85C4⋊Q8—48no—
86C4⋊Q8—48no—
87C4⋊Q8—44no—
88C4⋊Q8—44no—
89C4×D4—24no—
90C4×D4—42no—
91#117—84no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—44no—
96C2×C4⋊C4—44no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C42—41no—
103C2×C42D811yescharacteristic
104C2×C42D812yes—
105C2×C42D811yescharacteristic
106C2×C42C2×D412yes—
107C2×C42C2×D411yescharacteristic
108SD32—416no—
109C2×C16—24no—
110C2×C16—28no—
111#107—21no—
112#107—24no—
113#107C2×D412yes—
114#102—24no—
115#102—44no—
116C4×C8—24no—
117C4×C8—24no—
118C4×C8—22no—
119C4×C8C2×D412yes—
120C4×C8#4112yes—
121#95—42no—
122#95—41no—
123C24—41no—
124C2×Q8—44no—
125C2×Q8—48no—
126C2×Q8—48no—
127C2×Q8—44no—
128C2×D4—22no—
129C2×D4—42no—
130C2×D4—42no—
131C2×D4—41no—
132C22×C4—41no—
133C22×C4—44no—
134C22×C4—42no—
135C22×C4—44no—
136C22×C4—82no—
137C22×C4—82no—
138C22×C4C2×D811yescharacteristic
139C22×C4C2×D812yes—
140C22×C4C2×D812yes—
141C22×C4C2×D811yescharacteristic
142C22×C4C22×D411yescharacteristic
143Q16—44no—
144Q16—44no—
145Q16—48no—
146Q16—816no—
147Q16—88no—
148D8—24no—
149D8—42no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8#12112yes—
154C2×C8AGL(1,ℤ/8)14yes—
155C2×C8C22×D411yescharacteristic
156C2×C8C2×#4111yescharacteristic
157C2×C8H(5,𝔽2)14yes—
158C4⋊C4—22no—
159C4⋊C4—41no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—88no—
163C4⋊C4—88no—
164C4⋊C4—84no—
165C4⋊C4—84no—
166C4⋊C4—84no—
167C22⋊C4—44no—
168C22⋊C4—84no—
169C22⋊C4—84no—
170C42—22no—
171C42—22no—
172C42—24no—
173C42—22no—
174C42—82no—
175C42SD3214yescomplemented
176C42#12112yes—
177C42AGL(1,ℤ/8)12yes—
178C16—44no—
179C23—42no—
180C23—81no—
181C23—82no—
182C23—81no—
183C23—82no—
184C23C22×D811yescharacteristic
185Q8—84no—
186Q8—88no—
187Q8—84no—
188D4—42no—
189D4—42no—
190D4—82no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—88no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—88no—
204C2×C4—84no—
205C2×C4#45912yes—
206C2×C4#45912yes—
207C2×C4C2×SD3214yes—
208C2×C4C2×SD3212yes—
209C2×C4#51014yes—
210C2×C4#51012yes—
211C2×C4C2×#12111yescharacteristic
212C2×C4#53512yes—
213C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C4C2×AGL(1,ℤ/8)12yes—
215C8—24no—
216C8—22no—
217C8#53512yes—
218C8#54012yes—
219C22—82no—
220C22—82no—
221C22—82no—
222C22—162no—
223C22—162no—
224C22C2×#45911yescharacteristic
225C22#320512yes—
226C22C22×SD3211yescharacteristic
227C22C2×#51011yescharacteristic
228C22C2×#51012yes—
229C4—22no—
230C4—24no—
231C4—24no—
232C4—82no—
233C4—164no—
234C4—162no—
235C4#220814yes—
236C4#261911yescharacteristic
237C4#303811yescharacteristic
238C4#321712yes—
239C2—162no—
240C2#2979411yescharacteristic
241C2C2×#220811yescharacteristic
242C2#3358311yescharacteristic
243C2#3377212yes—
244C2#3378412yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33302C2semidirect
4. #13409C2semidirect
6. #13253C2semidirect
7. #12879C2semidirect
175. C42SD32semidirect

Maximal subgroups

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

subgroupindex
C2×#22082
#333022
#212752
#134092
#133272
#132532
#128792