Finite Group Databasebeta

← #513199

Subgroups — #513199

Order 512.

370 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 122 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#33901C211yesmaximal, characteristic
2D4×Q32C211yesmaximal, characteristic, complemented
3#13720C211yesmaximal, characteristic, complemented
4#13714C212yesmaximal, complemented
5#13704C211yesmaximal, characteristic, complemented
6#13696C212yesmaximal, complemented
7#13687C211yesmaximal, characteristic, complemented
8#13662C212yesmaximal, complemented
9C4×SD64C211yesmaximal, characteristic, complemented
10D4×C32C211yesmaximal, characteristic, complemented
11C2×#1416C212yesmaximal, complemented
12C22×Q32C2212yes—
13#3334—21no—
14D4×Q16—21no—
15C2×SD64C2211yescharacteristic
16C2×SD64C2212yes—
17C22×C32C2212yes—
18#2252C2211yescharacteristic
19#2247C2211yescharacteristic
20#2232—22no—
21#2224—21no—
22#2216C2212yes—
23#2215C2212yes—
24#2209—21no—
25#2188—22no—
26C4×Q32C2211yescharacteristic
27C4×D16C2211yescharacteristic
28D4×C16C2211yescharacteristic
29C2×#366C2212yes—
30#1424C2211yescharacteristic
31#1424C2212yes—
32C4⋊C32C2211yescharacteristic
33#1416C2211yescharacteristic
34#1416C2212yes—
35#1416C2212yes—
36#1416C2212yes—
37#1416C2212yes—
38#1415C2211yescharacteristic
39C22⋊C32C2212yes—
40C4×C32C2211yescharacteristic
41C22×Q16—22no—
42D4×Q8—41no—
43#547—41no—
44C2×Q32—24no—
45C2×Q32—22no—
46C2×Q32—21no—
47C2×Q32—42no—
48C2×Q32C2312yes—
49C2×Q32C2311yescharacteristic
50C2×D16C2311yescharacteristic
51C22×C16C2312yes—
52#500—21no—
53#494—21no—
54#480—42no—
55#474—41no—
56#467—22no—
57#466—22no—
58#462—41no—
59#451—42no—
60C4×Q16—21no—
61C4×D8—21no—
62C8×D4D411yescharacteristic
63C2×#107—22no—
64SD64—22no—
65SD64—42no—
66C2×C32—42no—
67C2×C32C2311yescharacteristic
68C2×C32C2311yescharacteristic
69C2×C32C2312yes—
70#366—22no—
71#366—21no—
72#366C2311yescharacteristic
73#366C2312yes—
74#366C2311yescharacteristic
75C4⋊C16D411yescharacteristic
76#358—21no—
77#358—22no—
78#357—21no—
79#357—22no—
80C22⋊C16D412yes—
81C4×C16C2311yescharacteristic
82C2×#41—81no—
83C22×Q8—42no—
84C2×Q16—21no—
85C2×Q16—22no—
86C2×Q16—44no—
87C2×Q16—42no—
88C2×Q16—41no—
89C2×Q16—82no—
90C2×D8—21no—
91C22×C8C2×D412yes—
92C4⋊Q8—41no—
93C4⋊Q8—41no—
94C4⋊Q8—81no—
95#123—82no—
96C22⋊Q8—42no—
97C22⋊Q8—81no—
98C22⋊Q8—82no—
99#121—42no—
100C4×Q8—41no—
101C4×D4—41no—
102C4×D4—81no—
103C4×D4D811yescharacteristic
104C2×C4⋊C4—42no—
105Q32—22no—
106Q32—21no—
107Q32—22no—
108Q32—44no—
109Q32—81no—
110Q32#4112yescomplemented
111D16—21no—
112D16—21no—
113C2×C16—22no—
114C2×C16—22no—
115C2×C16C2×D411yescharacteristic
116C2×C16C2×D412yes—
117C2×C16C2411yescharacteristic
118#107—21no—
119#107—22no—
120#107—21no—
121#107—42no—
122#107—41no—
123C4⋊C8D811yescharacteristic
124#103—41no—
125#103—42no—
126#102—41no—
127#102—42no—
128C22⋊C8D812yes—
129C4×C8C2×D411yescharacteristic
130C32—21no—
131C32—21no—
132C32—22no—
133#41—164no—
134C2×Q8—41no—
135C2×Q8—42no—
136C2×Q8—84no—
137C2×Q8—81no—
138C2×Q8—82no—
139C2×Q8—162no—
140C2×D4—41no—
141C2×D4—82no—
142C2×D4D1611yescharacteristic
143C22×C4—82no—
144C22×C4—82no—
145C22×C4—81no—
146C22×C4C2×D812yes—
147Q16—22no—
148Q16—42no—
149Q16—41no—
150Q16—42no—
151Q16—84no—
152Q16—161no—
153D8—41no—
154D8—41no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8C2×D811yescharacteristic
158C2×C8C2×D812yes—
159C2×C8C22×D411yescharacteristic
160C4⋊C4—41no—
161C4⋊C4—42no—
162C4⋊C4—41no—
163C4⋊C4—82no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C4⋊C4—82no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4D1611yescharacteristic
170C22⋊C4—82no—
171C22⋊C4—81no—
172C22⋊C4—82no—
173C22⋊C4D1612yes—
174C42—81no—
175C42C2×D811yescharacteristic
176C16—22no—
177C16—21no—
178C16C2×#4111yescharacteristic
179C16H(5,𝔽2)11yescharacteristic
180C23—81no—
181C23C2×D1612yes—
182Q8—42no—
183Q8—81no—
184Q8—82no—
185Q8—82no—
186Q8—164no—
187Q8—162no—
188D4—81no—
189D4—81no—
190D4—164no—
191D4SD6414yescomplemented
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—164no—
200C2×C4—164no—
201C2×C4—162no—
202C2×C4C2×D1611yescharacteristic
203C2×C4C2×D1612yes—
204C2×C4C2×D1611yescharacteristic
205C2×C4C22×D811yescharacteristic
206C8—21no—
207C8—22no—
208C8#54611yescharacteristic
209C8#57711yescharacteristic
210C22—22no—
211C22—22no—
212C22—161no—
213C22—161no—
214C22—161no—
215C22C2×SD6414yes—
216C22C22×D1611yescharacteristic
217C4—22no—
218C4—21no—
219C4—82no—
220C4—162no—
221C4—161no—
222C4—162no—
223C4—161no—
224C4C2×SD6412yes—
225C4#329411yescharacteristic
226C4#341611yescharacteristic
227C2—24no—
228C2—321no—
229C2#3386111yescharacteristic
230C2C22×SD6411yescharacteristic
231C2#3398411yescharacteristic
subgroupG/H
center, socleC22C22×D16
commutator, Frattini, cosocleC2×C16C24
prime radical#33901C2
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. D4×Q32C2semidirect
3. #13720C2semidirect
4. #13714C2semidirect
5. #13704C2semidirect
6. #13696C2semidirect
7. #13687C2semidirect
8. #13662C2semidirect
9. C4×SD64C2semidirect
10. D4×C32C2semidirect
11. C2×#1416C2semidirect
110. Q32#41semidirect
191. D4SD64semidirect

Maximal subgroups

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

subgroupindex
#339012
D4×Q322
#137202
#137142
#137042
#136962
#136872
#136622
C4×SD642
D4×C322
C2×#14162