Finite Group Databasebeta

← #513118

Subgroups — #513118

Order 512.

391 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 118 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#33735C211yesmaximal, characteristic
2C2×#2247C211yesmaximal, characteristic, complemented
3#13730C212yesmaximal, complemented
4#13724C212yesmaximal, complemented
5#13688C212yesmaximal, complemented
6#13687C212yesmaximal, complemented
7#13665C212yesmaximal, complemented
8#13662C212yesmaximal
9#13640C211yesmaximal, characteristic, complemented
10C22×Q32—21no—
11C22×Q32C2212yes—
12#3168—21no—
13C2×#494—21no—
14C2×Q64C2214yes—
15C2×SD64C2214yes—
16#2253C2211yescharacteristic
17#2249C2211yescharacteristic
18#2247—21no—
19#2247—21no—
20#2247C2212yes—
21#2247C2212yes—
22#2242—21no—
23#2217—21no—
24#2216C2211yescharacteristic
25#2215C2211yescharacteristic
26C4×Q32C2211yescharacteristic
27C4×SD32—21no—
28C4×D16C2211yescharacteristic
29C2×C4×C16C2211yescharacteristic
30C4⋊C32C2212yes—
31#1416C2214yes—
32#1416C2214yes—
33C22⋊C32C2212yes—
34#1397C2212yes—
35C22×Q16—22no—
36C22×Q16—41no—
37C2×C4⋊Q8—41no—
38#529—41no—
39C2×Q32—22no—
40C2×Q32—22no—
41C2×Q32—22no—
42C2×Q32—44no—
43C2×Q32—44no—
44C2×Q32C2314yes—
45C2×Q32C2311yescharacteristic
46C2×SD32—21no—
47C2×D16C2311yescharacteristic
48C22×C16D411yescharacteristic
49C22×C16C2311yescharacteristic
50#499—21no—
51#495—21no—
52#494—22no—
53#494—22no—
54#494—41no—
55#494—41no—
56#488—41no—
57#467—21no—
58#466—21no—
59#465—41no—
60C4×Q16—21no—
61C4×SD16—41no—
62C4×D8—21no—
63C2×C4×C8D411yescharacteristic
64Q64—44no—
65SD64—44no—
66C2×C32C2314yes—
67#367—21no—
68#366C2311yescharacteristic
69#366C2311yescharacteristic
70#358—21no—
71#358—21no—
72#357—21no—
73#357—21no—
74C4×C16D411yescharacteristic
75C4×C16D411yescharacteristic
76C4×C16C2311yescharacteristic
77C4×C16C2311yescharacteristic
78C22×Q8—42no—
79C2×Q16—24no—
80C2×Q16—21no—
81C2×Q16—44no—
82C2×Q16—44no—
83C2×Q16—42no—
84C2×Q16—42no—
85C2×Q16—42no—
86C2×Q16—84no—
87C2×SD16—41no—
88C2×D8—21no—
89C22×C8D811yescharacteristic
90C22×C8C2×D411yescharacteristic
91C4⋊Q8—42no—
92C4⋊Q8—42no—
93C4⋊Q8—81no—
94C4⋊Q8—81no—
95#126—81no—
96#125—41no—
97#124—41no—
98#123—81no—
99C22⋊Q8—41no—
100#121—41no—
101C4×Q8—41no—
102C4×D4—41no—
103C4×D4—81no—
104#117—81no—
105C2×C4⋊C4—81no—
106C2×C4⋊C4—81no—
107C2×C42D811yescharacteristic
108Q32—24no—
109Q32—22no—
110Q32—24no—
111Q32—44no—
112Q32—44no—
113Q32—44no—
114SD32—42no—
115D16—22no—
116C2×C16—22no—
117C2×C16—22no—
118C2×C16C2×D411yescharacteristic
119C2×C16C2×D411yescharacteristic
120C2×C16C2×D411yescharacteristic
121C2×C16C2411yescharacteristic
122#107—21no—
123#107—21no—
124#106—41no—
125#103—41no—
126#103—41no—
127#102—41no—
128#102—41no—
129C4×C8D811yescharacteristic
130C4×C8D811yescharacteristic
131C4×C8C2×D411yescharacteristic
132C4×C8C2×D411yescharacteristic
133C32—24no—
134C2×Q8—41no—
135C2×Q8—44no—
136C2×Q8—84no—
137C2×Q8—84no—
138C2×Q8—82no—
139C2×D4—41no—
140C2×D4—81no—
141C22×C4—82no—
142C22×C4—81no—
143C22×C4D1611yescharacteristic
144C22×C4D1611yescharacteristic
145C22×C4C2×D811yescharacteristic
146Q16—44no—
147Q16—41no—
148Q16—41no—
149Q16—44no—
150Q16—84no—
151Q16—84no—
152Q16—84no—
153SD16—82no—
154D8—41no—
155D8—41no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—22no—
159C2×C8C2×D811yescharacteristic
160C2×C8C2×D811yescharacteristic
161C2×C8C2×D811yescharacteristic
162C2×C8C22×D411yescharacteristic
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—82no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C22⋊C4—81no—
174C22⋊C4—81no—
175C22⋊C4—81no—
176C22⋊C4—81no—
177C22⋊C4—81no—
178C42—81no—
179C42D1611yescharacteristic
180C42D1611yescharacteristic
181C42C2×D811yescharacteristic
182C42C2×D811yescharacteristic
183C16—21no—
184C16—21no—
185C16AGL(1,ℤ/8)12yes—
186C16H(5,𝔽2)12yes—
187C23—81no—
188C23C2×D1611yescharacteristic
189Q8—81no—
190Q8—84no—
191Q8—84no—
192Q8—81no—
193Q8—164no—
194D4—81no—
195D4—81no—
196D4—162no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—84no—
205C2×C4—164no—
206C2×C4—162no—
207C2×C4C2×D1611yescharacteristic
208C2×C4C2×D1611yescharacteristic
209C2×C4C2×D1611yescharacteristic
210C2×C4C2×D1611yescharacteristic
211C2×C4C2×D1611yescharacteristic
212C2×C4C22×D811yescharacteristic
213C8—21no—
214C8—21no—
215C8#50912yes—
216C8#53511yescharacteristic
217C8C2×AGL(1,ℤ/8)11yescharacteristic
218C22—21no—
219C22—21no—
220C22—161no—
221C22—161no—
222C22—161no—
223C22#226412yes—
224C22C22×D1611yescharacteristic
225C4—21no—
226C4—21no—
227C4—21no—
228C4—21no—
229C4—161no—
230C4—161no—
231C4—164no—
232C4—161no—
233C4#226412yes—
234C4#320511yescharacteristic
235C4C2×#50911yescharacteristic
236C2—22no—
237C2—321no—
238C2#3377111yescharacteristic
239C2C2×#226411yescharacteristic
240C2C2×#226411yescharacteristic
subgroupG/H
center, socleC22C22×D16
commutator, Frattini, cosocleC2×C16C24
prime radical#2215C22
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2247C2semidirect
3. #13730C2semidirect
4. #13724C2semidirect
5. #13688C2semidirect
6. #13687C2semidirect
7. #13665C2semidirect
9. #13640C2semidirect

Maximal subgroups

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

subgroupindex
#337352
C2×#22472
#137302
#137242
#136882
#136872
#136652
#136622
#136402