Finite Group Databasebeta

← C3×D8×C22⋊C9 #13152223

Subgroups — C3×D8×C22⋊C9 #13152223

Order 1728.

514 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C3×D4×C22⋊C9C212yesmaximal, complemented
2C3×C8×C22⋊C9C211yesmaximal, characteristic, complemented
3C22×C32×D8C311yesmaximal, characteristic
4D8×C22⋊C9C313yesmaximal, direct product factor
5C22×C3×C22⋊C9—22no—
6C3×C4×C22⋊C9C2211yescharacteristic
7C3×C9×D8—41nomaximal
8C22×C32×D4C2×C312yes—
9C2×C32×D8—31no—
10C2×C32×D8—31no—
11C2×C32×D8—32no—
12C22×C32×C8C2×C311yescharacteristic
13D4×C22⋊C9C2×C316yescomplemented
14C8×C22⋊C9C2×C313yescomplemented
15C2×C3×C22⋊C9—42no—
16C2×C3×C22⋊C9D411yescharacteristic
17C3×D4×C9—42no—
18C3×C8×C9—41no—
19C22×C3×D8C913yescomplemented
20C22×C3×D8C3211yescharacteristic
21C24×C32—22no—
22C2×C32×D4—32no—
23C2×C32×D4—32no—
24C2×C32×D4—62no—
25C22×C32×C4C22×C311yescharacteristic
26C22×C22⋊C9—26no—
27C32×D8—32no—
28C32×D8—32no—
29C32×D8—31no—
30C32×D8A411yescharacteristic
31C2×C32×C8—31no—
32C2×C32×C8—31no—
33C4×C22⋊C9C22×C313yes—
34C9×D8—43no—
35C22×C3×C9—82no—
36C3×C22⋊C9D811yescharacteristic, direct product factor
37C3×C4×C9—41no—
38C22×C3×D4C2×C916yescomplemented
39C22×C3×D4C2×C3212yes—
40C2×C3×D8—33no—
41C2×C3×D8—31no—
42C2×C3×D8—33no—
43C2×C3×D8—36no—
44C2×C3×D8—31no—
45C2×C3×D8—32no—
46C22×C3×C8C2×C913yescomplemented
47C22×C3×C8C2×C3211yescharacteristic
48C23×C32—42no—
49C23×C32—62no—
50C23×C32—62no—
51C23×C32—122no—
52C23×C32C3×D411yescharacteristic
53C32×D4—32no—
54C32×D4—62no—
55C32×D4—62no—
56C32×D4C2×A412yes—
57C2×C32×C4—31no—
58C2×C32×C4—31no—
59C2×C22⋊C9—46no—
60C2×C22⋊C9C3×D413yes—
61C32×C8—31no—
62C32×C8C2×A411yescharacteristic
63D4×C9—46no—
64C8×C9—43no—
65C22×D8C3×C911yescharacteristic, complemented, Sylow
66C2×C3×C9—41no—
67C2×C3×C9—162no—
68C24×C3—26no—
69C24×C3—22no—
70C2×C3×D4—36no—
71C2×C3×D4—32no—
72C2×C3×D4—36no—
73C2×C3×D4—32no—
74C2×C3×D4—66no—
75C2×C3×D4—62no—
76C22×C3×C4C22×C913yes—
77C22×C3×C4C22×C3211yescharacteristic
78C3×D8—32no—
79C3×D8—36no—
80C3×D8—36no—
81C3×D8—32no—
82C3×D8—33no—
83C3×D8—31no—
84C3×D8C22⋊C913yesdirect product factor
85C3×D8C3×A411yescharacteristic
86C2×C3×C8—33no—
87C2×C3×C8—31no—
88C2×C3×C8—33no—
89C2×C3×C8—31no—
90C22×C32—22no—
91C22×C32—31no—
92C22×C32—31no—
93C22×C32—62no—
94C22×C32—122no—
95C22×C32—122no—
96C22×C32—122no—
97C22×C32—122no—
98C22×C32C3×D811yescharacteristic
99C32×C4—31no—
100C32×C4C22×A411yescharacteristic
101C22×C9—86no—
102C22⋊C9C3×D813yesdirect product factor
103C4×C9—43no—
104C22×D4C2×C3×C912yescomplemented
105C2×D8—31no—
106C2×D8—31no—
107C2×D8—32no—
108C22×C8C2×C3×C911yescharacteristic, complemented
109C3×C9—41noSylow
110C23×C3—46no—
111C23×C3—42no—
112C23×C3—62no—
113C23×C3—66no—
114C23×C3—62no—
115C23×C3—66no—
116C23×C3—126no—
117C23×C3—122no—
118C23×C3D4×C913yes—
119C23×C3C32×D411yescharacteristic
120C3×D4—32no—
121C3×D4—36no—
122C3×D4—62no—
123C3×D4—66no—
124C3×D4—62no—
125C3×D4—66no—
126C3×D4C2×C22⋊C916yescomplemented
127C3×D4C2×C3×A412yes—
128C2×C3×C4—31no—
129C2×C3×C4—33no—
130C2×C3×C4—33no—
131C2×C3×C4—31no—
132C3×C8—31no—
133C3×C8—33no—
134C3×C8C2×C22⋊C913yescomplemented
135C3×C8C2×C3×A411yescharacteristic
136C2×C32—31no—
137C2×C32—31no—
138C2×C32—42no—
139C2×C32—122no—
140C2×C32D4×A411yescharacteristic
141C2×C9—43no—
142C2×C9—166no—
143C24—22no—
144C2×D4—32no—
145C2×D4—32no—
146C2×D4—62no—
147C22×C4C22×C3×C911yescharacteristic
148D8—32no—
149D8—32no—
150D8—31no—
151D8C3×C22⋊C911yescharacteristic, direct product factor
152C2×C8—31no—
153C2×C8—31no—
154C22×C3—22no—
155C22×C3—26no—
156C22×C3—33no—
157C22×C3—31no—
158C22×C3—31no—
159C22×C3—33no—
160C22×C3—62no—
161C22×C3—66no—
162C22×C3—126no—
163C22×C3—122no—
164C22×C3—126no—
165C22×C3—122no—
166C22×C3—126no—
167C22×C3—122no—
168C22×C3—126no—
169C22×C3—122no—
170C22×C3C9×D813yescomplemented
171C22×C3C32×D811yescharacteristic
172C3×C4—31no—
173C3×C4—33no—
174C3×C4C22×C22⋊C913yes—
175C3×C4C22×C3×A411yescharacteristic
176C32A4×D811yescharacteristic
177C9—43no—
178C23—42no—
179C23—62no—
180C23—62no—
181C23—122no—
182C23C3×D4×C911yescharacteristic
183D4—32no—
184D4—62no—
185D4—62no—
186D4C2×C3×C22⋊C912yescomplemented
187C2×C4—31no—
188C2×C4—31no—
189C8—31no—
190C8C2×C3×C22⋊C911yescharacteristic, complemented
191C2×C3—33no—
192C2×C3—33no—
193C2×C3—31no—
194C2×C3—31no—
195C2×C3—46no—
196C2×C3—42no—
197C2×C3—126no—
198C2×C3—122no—
199C2×C3D4×C22⋊C913yes—
200C2×C3C3×D4×A411yescharacteristic
201C22—22no—
202C22—31no—
203C22—31no—
204C22—62no—
205C22—122no—
206C22—122no—
207C22—122no—
208C22—122no—
209C22C3×C9×D811yescharacteristic, complemented
210C4—31no—
211C4C22×C3×C22⋊C911yescharacteristic
212C3D8×C22⋊C913yesdirect product factor
213C3C3×A4×D811yescharacteristic
214C2—31no—
215C2—31no—
216C2—42no—
217C2—122no—
218C2C3×D4×C22⋊C911yescharacteristic
subgroupG/H
centerC2×C32D4×A4
commutatorC22×C4C22×C3×C9
FrattiniC3×C4C22×C3×A4
Fitting, prime radicalC22×C32×D8C3
socleC23×C32C3×D4
cosocleC22×C3×C4C22×C32
hypercenterC32×D8A4
hypocenterC22C3×C9×D8
LCSG ▷ C22×C4 ▷ C23 ▷ C22
UCSC1 ◁ C2×C32 ◁ C32×C4 ◁ C32×D8

Complements

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

NH ≅ G/Ntype
1. C3×D4×C22⋊C9C2semidirect
2. C3×C8×C22⋊C9C2semidirect
4. D8×C22⋊C9C3direct (N × H)
13. D4×C22⋊C9C2×C3semidirect
14. C8×C22⋊C9C2×C3semidirect
19. C22×C3×D8C9semidirect
36. C3×C22⋊C9D8direct (N × H)
38. C22×C3×D4C2×C9semidirect
46. C22×C3×C8C2×C9semidirect
65. C22×D8C3×C9semidirect
84. C3×D8C22⋊C9direct (N × H)
102. C22⋊C9C3×D8direct (N × H)
104. C22×D4C2×C3×C9semidirect
108. C22×C8C2×C3×C9semidirect
126. C3×D4C2×C22⋊C9semidirect
134. C3×C8C2×C22⋊C9semidirect
151. D8C3×C22⋊C9direct (N × H)
170. C22×C3C9×D8semidirect
186. D4C2×C3×C22⋊C9semidirect
190. C8C2×C3×C22⋊C9semidirect
209. C22C3×C9×D8semidirect
212. C3D8×C22⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D4×C22⋊C92
C3×C8×C22⋊C92
C22×C32×D83
D8×C22⋊C93
C3×C9×D84