Finite Group Databasebeta

← C2×D4×Q8⋊C9 #11919717

Subgroups — C2×D4×Q8⋊C9 #11919717

Order 1152.

861 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 861 conjugacy classes of subgroups — too many to draw. Only its 213 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
1C23×Q8⋊C9C212yesmaximal, complemented
2D4×Q8⋊C9C214yesmaximal, direct product factor
3C2×C4×Q8⋊C9C211yesmaximal, characteristic, complemented
4C2×C3×D4×Q8C311yesmaximal, characteristic
5C22×D4×C9—41nomaximal
6C22×Q8⋊C9—24no—
7C22×Q8⋊C9C2211yescharacteristic
8C22×Q8⋊C9C2214yescomplemented
9C4×Q8⋊C9C2212yescomplemented
10C23×C3×Q8C2×C312yes—
11C3×D4×Q8—34no—
12C3×D4×Q8C2×C314yes—
13C2×C3×C4⋊Q8—31no—
14C2×C3×C22⋊Q8—32no—
15C2×C3×C4×Q8C2×C311yescharacteristic
16C2×C3×C4×D4—31no—
17C24×C9—42no—
18C2×D4×C9—48no—
19C2×D4×C9—44no—
20C22×C4×C9—41no—
21C2×Q8⋊C9—24no—
22C2×Q8⋊C9D412yesdirect product factor
23C2×Q8⋊C9C2311yescharacteristic
24C2×D4×Q8C911yescharacteristic, complemented, Sylow
25C22×C3×Q8—24no—
26C22×C3×Q8—32no—
27C22×C3×Q8—34no—
28C22×C3×Q8—64no—
29C22×C3×Q8C22×C311yescharacteristic
30C22×C3×Q8C22×C314yes—
31C22×C3×D4A411yescharacteristic
32C23×C3×C4—32no—
33C3×C4⋊Q8—34no—
34C3×C4⋊Q8—34no—
35C3×C22⋊Q8—38no—
36C3×C22⋊Q8—38no—
37C3×C4×Q8—32no—
38C3×C4×Q8C22×C312yes—
39C3×C4×D4—34no—
40C3×C4×D4—34no—
41C2×C3×C4⋊C4—31no—
42C2×C3×C4⋊C4—32no—
43C2×C3×C4⋊C4—31no—
44C2×C3×C22⋊C4—32no—
45C2×C3×C42—31no—
46C23×C9—48no—
47C23×C9—44no—
48C23×C9—41no—
49C23×C9—84no—
50C23×C9—84no—
51D4×C9—416no—
52C2×C4×C9—44no—
53C2×C4×C9—42no—
54Q8⋊C9C2×D411yescharacteristic, direct product factor
55C23×Q8C2×C912yescomplemented
56D4×Q8—34no—
57D4×Q8C2×C914yescomplemented
58C2×C4⋊Q8—31no—
59C2×C22⋊Q8—32no—
60C2×C4×Q8C2×C911yescharacteristic, complemented
61C2×C4×D4—31no—
62C24×C3C2×A412yes—
63C2×C3×Q8—24no—
64C2×C3×Q8—34no—
65C2×C3×Q8—31no—
66C2×C3×Q8—32no—
67C2×C3×Q8—34no—
68C2×C3×Q8—64no—
69C2×C3×Q8—64no—
70C2×C3×Q8—68no—
71C2×C3×Q8—64no—
72C2×C3×Q8—68no—
73C2×C3×Q8C3×D412yes—
74C2×C3×Q8C23×C311yescharacteristic
75C2×C3×D4SL(2,𝔽3)18yes—
76C2×C3×D4C2×A414yes—
77C22×C3×C4—31no—
78C22×C3×C4—34no—
79C22×C3×C4—34no—
80C22×C3×C4—31no—
81C22×C3×C4—32no—
82C22×C3×C4—64no—
83C22×C3×C4—64no—
84C22×C3×C4C2×A411yescharacteristic
85C3×C4⋊C4—32no—
86C3×C4⋊C4—32no—
87C3×C4⋊C4—38no—
88C3×C4⋊C4—34no—
89C3×C22⋊C4—38no—
90C3×C42—32no—
91C3×C42—32no—
92C22×C9—42no—
93C22×C9—42no—
94C22×C9—48no—
95C22×C9—42no—
96C22×C9—41no—
97C22×C9—84no—
98C22×C9—84no—
99C22×C9—816no—
100C4×C9—44no—
101C22×Q8—24no—
102C22×Q8—32no—
103C22×Q8—34no—
104C22×Q8—64no—
105C22×Q8C22×C911yescharacteristic
106C22×Q8C22×C914yescomplemented
107C22×D4C22⋊C911yescharacteristic
108C23×C4—32no—
109C4⋊Q8—34no—
110C4⋊Q8—34no—
111C22⋊Q8—38no—
112C22⋊Q8—38no—
113C4×Q8—32no—
114C4×Q8C22×C912yescomplemented
115C4×D4—34no—
116C4×D4—34no—
117C2×C4⋊C4—31no—
118C2×C4⋊C4—32no—
119C2×C4⋊C4—31no—
120C2×C22⋊C4—32no—
121C2×C42—31no—
122C23×C3—24no—
123C23×C3—24no—
124C23×C3C2×SL(2,𝔽3)18yes—
125C23×C3C22×A414yes—
126C23×C3C22×A411yescharacteristic
127C3×Q8—32no—
128C3×Q8—31no—
129C3×Q8—32no—
130C3×Q8—64no—
131C3×Q8—64no—
132C3×Q8—68no—
133C3×Q8C2×C3×D411yescharacteristic
134C3×D4C2×SL(2,𝔽3)116yes—
135C2×C3×C4—32no—
136C2×C3×C4—32no—
137C2×C3×C4—31no—
138C2×C3×C4—31no—
139C2×C3×C4—32no—
140C2×C3×C4—34no—
141C2×C3×C4—64no—
142C2×C3×C4—68no—
143C2×C3×C4—64no—
144C2×C3×C4—64no—
145C2×C3×C4—62no—
146C2×C3×C4C2×SL(2,𝔽3)14yes—
147C2×C3×C4C22×A412yes—
148C2×C9—41no—
149C2×C9—41no—
150C2×C9—41no—
151C2×C9—44no—
152C2×C9—88no—
153C24C2×C22⋊C912yes—
154C2×Q8—24no—
155C2×Q8—34no—
156C2×Q8—31no—
157C2×Q8—32no—
158C2×Q8—34no—
159C2×Q8—64no—
160C2×Q8—64no—
161C2×Q8—68no—
162C2×Q8—64no—
163C2×Q8—68no—
164C2×Q8D4×C912yescomplemented
165C2×Q8C23×C911yescharacteristic
166C2×D4Q8⋊C918yesdirect product factor
167C2×D4C2×C22⋊C914yes—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—32no—
171C22×C4—34no—
172C22×C4—34no—
173C22×C4—64no—
174C22×C4—64no—
175C22×C4C2×C22⋊C911yescharacteristic
176C4⋊C4—32no—
177C4⋊C4—32no—
178C4⋊C4—38no—
179C4⋊C4—34no—
180C22⋊C4—38no—
181C42—32no—
182C42—32no—
183C22×C3—24no—
184C22×C3—24no—
185C22×C3—216no—
186C22×C3D4×A412yes—
187C22×C3C22×SL(2,𝔽3)12yes—
188C22×C3C22×SL(2,𝔽3)18yes—
189C22×C32−1+4⋊C312yes—
190C22×C3C23×A411yescharacteristic
191C3×C4—31no—
192C3×C4—32no—
193C3×C4—31no—
194C3×C4—64no—
195C3×C4—62no—
196C3×C4C22×SL(2,𝔽3)14yes—
197C9—41noSylow
198C23—24no—
199C23—24no—
200C23C2×Q8⋊C918yescomplemented
201C23C22×C22⋊C914yes—
202C23C22×C22⋊C911yescharacteristic
203Q8—32no—
204Q8—31no—
205Q8—32no—
206Q8—64no—
207Q8—64no—
208Q8—68no—
209Q8C2×D4×C911yescharacteristic, complemented
210D4C2×Q8⋊C9116yesdirect product factor
211C2×C4—31no—
212C2×C4—31no—
213C2×C4—32no—
214C2×C4—32no—
215C2×C4—32no—
216C2×C4—34no—
217C2×C4—64no—
218C2×C4—64no—
219C2×C4—68no—
220C2×C4—64no—
221C2×C4—62no—
222C2×C4C2×Q8⋊C914yescomplemented
223C2×C4C22×C22⋊C912yes—
224C2×C3—28no—
225C2×C3D4×SL(2,𝔽3)14yes—
226C2×C3C2×D4×A411yescharacteristic
227C2×C3C23×SL(2,𝔽3)11yescharacteristic
228C2×C3C2×2−1+4⋊C311yescharacteristic
229C22—24no—
230C22—24no—
231C22—216no—
232C22D4×C22⋊C912yes—
233C22C22×Q8⋊C912yes—
234C22C22×Q8⋊C918yescomplemented
235C222−1+4⋊C912yes—
236C22C23×C22⋊C911yescharacteristic
237C4—31no—
238C4—31no—
239C4—32no—
240C4—64no—
241C4—62no—
242C4C22×Q8⋊C914yescomplemented
243C3C2×D4×SL(2,𝔽3)11yescharacteristic
244C2—28no—
245C2D4×Q8⋊C914yesdirect product factor
246C2C2×D4×C22⋊C911yescharacteristic
247C2C23×Q8⋊C911yescharacteristic
248C2C2×2−1+4⋊C911yescharacteristic
subgroupG/H
center, socleC23×C3C22×A4
commutatorC2×Q8C23×C9
FrattiniC22×C3C23×A4
FittingC2×C3×D4×Q8C3
prime radical, hypercenterC22×C3×D4A4
cosocleC2×C3×Q8C23×C3
hypocenterQ8C2×D4×C9
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C23×C3 ◁ C22×C3×D4

Complements

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

NH ≅ G/Ntype
1. C23×Q8⋊C9C2semidirect
2. D4×Q8⋊C9C2direct (N × H)
3. C2×C4×Q8⋊C9C2semidirect
8. C22×Q8⋊C9C22semidirect
9. C4×Q8⋊C9C22semidirect
22. C2×Q8⋊C9D4direct (N × H)
24. C2×D4×Q8C9semidirect
54. Q8⋊C9C2×D4direct (N × H)
55. C23×Q8C2×C9semidirect
57. D4×Q8C2×C9semidirect
60. C2×C4×Q8C2×C9semidirect
106. C22×Q8C22×C9semidirect
114. C4×Q8C22×C9semidirect
164. C2×Q8D4×C9semidirect
166. C2×D4Q8⋊C9direct (N × H)
200. C23C2×Q8⋊C9semidirect
209. Q8C2×D4×C9semidirect
210. D4C2×Q8⋊C9direct (N × H)
222. C2×C4C2×Q8⋊C9semidirect
234. C22C22×Q8⋊C9semidirect
242. C4C22×Q8⋊C9semidirect
245. C2D4×Q8⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C23×Q8⋊C92
D4×Q8⋊C92
C2×C4×Q8⋊C92
C2×C3×D4×Q83
C22×D4×C94