Finite Group Databasebeta

← C22×C3×#3773 #13181047

Subgroups — C22×C3×#3773 #13181047

Order 1728.

1416 conjugacy classes of subgroups; only the 370 normal subgroups are shown.

normal subgroups only

This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 370 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×C3×#157C211yesmaximal, characteristic
2C22×C3×#624C212yesmaximal
3C2×C3×#3773C2112yesmaximal, direct product factor
4C22×#3773C311yesmaximal, characteristic, direct product factor
5C23×C3×C3⋊D3C2211yescharacteristic
6C22×C3×#157C418yescomplemented
7C22×C3×#157C2216yes—
8C22×C32×Dic3—22no—
9C2×C3×#624—26no—
10C2×C3×#624C22112yes—
11C3×#3773C22116yesdirect product factor
12C23×#157C2×C311yescharacteristic
13C22×C3×C4×D3—62no—
14C22×#624C2×C312yes—
15C2×#3773C2×C3112yesdirect product factor
16C23×C33D411yescharacteristic
17C22×C3×C3⋊D3C2×C414yes—
18C22×C3×C3⋊D3D413yes—
19C22×C3×C3⋊D3Q814yes—
20C22×C3×C3⋊D3C2313yes—
21C2×C3×#157C2×C4124yescomplemented
22C2×C3×#157C2314yes—
23C2×C32×Dic3—212no—
24C3×#624—212no—
25C3×#624C2318yes—
26C22×C3×C4⋊C4—91nomaximal
27C23×C3⋊D3C22×C311yescharacteristic
28C23×C3×D3—62no—
29C22×#157C3×C418yescomplemented
30C22×#157C22×C316yes—
31C22×C32×C4—62no—
32C22×C3×Dic3—22no—
33C22×C3×Dic3—22no—
34C22×C3×Dic3—42no—
35C2×C3×C4×D3—612no—
36C2×C3×C4×D3—612no—
37C2×#624—26no—
38C2×#624C22×C3112yes—
39#3773C22×C3116yesdirect product factor
40C22×C33C4⋊C414yescomplemented
41C22×C33C2×D413yes—
42C2×C3×C3⋊D3C4⋊C4112yescomplemented
43C2×C3×C3⋊D3C22×C416yes—
44C2×C3×C3⋊D3C2×D413yes—
45C2×C3×C3⋊D3C2×Q816yes—
46C2×C3×C3⋊D3C2411yescharacteristic
47C3×#157C22×C418yescomplemented
48C32×Dic3—28no—
49C23×C3×C4—92no—
50C23×C3×C4—91no—
51C22×C4×D3—62no—
52C2×C3×C4⋊C4—912no—
53C23×C32—22no—
54C23×C32—42no—
55C23×C32C3×D411yescharacteristic
56C22×C3⋊D3C2×C3×C414yes—
57C22×C3⋊D3C3×D413yes—
58C22×C3⋊D3C3×Q814yes—
59C22×C3⋊D3C23×C313yes—
60C22×C3×D3—68no—
61C22×C3×D3—66no—
62C22×C3×D3—68no—
63C22×C3×D3—66no—
64C2×#157C2×C3×C4124yescomplemented
65C2×#157C23×C314yes—
66C2×C32×C4—612no—
67C2×C3×Dic3—212no—
68C2×C3×Dic3—212no—
69C2×C3×Dic3—412no—
70C3×C4×D3—68no—
71C3×C4×D3—624no—
72#624—212no—
73#624C23×C318yes—
74C22×C4⋊C4—91noSylow
75C2×C33C2×C4⋊C416yescomplemented
76C2×C33C22×D411yescharacteristic
77C3×C3⋊D3C2×C4⋊C416yescomplemented
78C3×C3⋊D3C23×C411yescharacteristic
79C3×C3⋊D3C22×Q811yescharacteristic
80C24×C3—91no—
81C23×D3—62no—
82C22×C3×C4—62no—
83C22×C3×C4—62no—
84C22×C3×C4—912no—
85C22×C3×C4—98no—
86C22×C3×C4—96no—
87C22×C3×C4—122no—
88C22×C3×C4—186no—
89C22×Dic3—22no—
90C2×C4×D3—612no—
91C2×C4×D3—612no—
92C3×C4⋊C4—916no—
93C22×C32—26no—
94C22×C32—28no—
95C22×C32—48no—
96C22×C32—46no—
97C22×C32C3×C4⋊C414yescomplemented
98C22×C32C2×C3×D413yes—
99C2×C3⋊D3C3×C4⋊C4112yescomplemented
100C2×C3⋊D3C22×C3×C416yes—
101C2×C3⋊D3C2×C3×D413yes—
102C2×C3⋊D3C2×C3×Q816yes—
103C2×C3⋊D3C24×C311yescharacteristic
104C2×C3×D3—624no—
105C2×C3×D3—612no—
106C2×C3×D3—66no—
107C2×C3×D3—612no—
108C2×C3×D3—62no—
109#157C22×C3×C418yescomplemented
110C32×C4—68no—
111C3×Dic3—28no—
112C3×Dic3—28no—
113C3×Dic3—48no—
114C23×C4—92no—
115C23×C4—91no—
116C2×C4⋊C4—912no—
117C33C22×C4⋊C411yescharacteristic, complemented, Sylow
118C23×C3—22no—
119C23×C3—42no—
120C23×C3—93no—
121C23×C3—93no—
122C23×C3—94no—
123C23×C3—94no—
124C23×C3GO+(4,𝔽2)11yescharacteristic
125C22×D3—68no—
126C22×D3—66no—
127C22×D3—68no—
128C22×D3—66no—
129C2×C3×C4—612no—
130C2×C3×C4—612no—
131C2×C3×C4—98no—
132C2×C3×C4—924no—
133C2×C3×C4—94no—
134C2×C3×C4—1212no—
135C2×C3×C4—1812no—
136C2×Dic3—212no—
137C4×D3—68no—
138C4×D3—624no—
139C2×C32—22no—
140C2×C32—212no—
141C2×C32—412no—
142C2×C32—42no—
143C2×C32C2×C3×C4⋊C416yescomplemented
144C2×C32C22×C3×D411yescharacteristic
145C3⋊D3C2×C3×C4⋊C416yescomplemented
146C3⋊D3C23×C3×C411yescharacteristic
147C3⋊D3C22×C3×Q811yescharacteristic
148C3×D3—612no—
149C3×D3—62no—
150C3×D3—62no—
151C24—91no—
152C22×C4—62no—
153C22×C4—912no—
154C22×C4—98no—
155C22×C4—96no—
156C22×C4—186no—
157C4⋊C4—916no—
158C22×C3—26no—
159C22×C3—28no—
160C22×C3—46no—
161C22×C3—48no—
162C22×C3—91no—
163C22×C3—93no—
164C22×C3—96no—
165C22×C3—96no—
166C22×C3—912no—
167C22×C3#377314yesdirect product factor
168C22×C3C2×GO+(4,𝔽2)13yes—
169C2×D3—66no—
170C2×D3—62no—
171C2×D3—624no—
172C2×D3—612no—
173C2×D3—612no—
174C3×C4—68no—
175C3×C4—68no—
176C3×C4—98no—
177C3×C4—128no—
178Dic3—28no—
179C32—22no—
180C32—42no—
181C32C22×C3×C4⋊C411yescharacteristic, complemented
182C23—93no—
183C23—93no—
184C23—94no—
185C23—94no—
186C23C3×GO+(4,𝔽2)11yescharacteristic
187C2×C4—612no—
188C2×C4—98no—
189C2×C4—94no—
190C2×C4—924no—
191C2×C4—1812no—
192C2×C3—212no—
193C2×C3—22no—
194C2×C3—412no—
195C2×C3—42no—
196C2×C3—91no—
197C2×C3—91no—
198C2×C3—96no—
199C2×C3C2×#377316yesdirect product factor
200C2×C3C22×GO+(4,𝔽2)11yescharacteristic
201D3—612no—
202D3—62no—
203D3—62no—
204C22—96no—
205C22—96no—
206C22—912no—
207C22—91no—
208C22—93no—
209C22C3×#377314yesdirect product factor
210C22C2×C3×GO+(4,𝔽2)13yes—
211C4—68no—
212C4—98no—
213C3—22no—
214C3—42no—
215C3C22×#377311yescharacteristic, direct product factor
216C2—91no—
217C2—96no—
218C2—91no—
219C2C2×C3×#377316yesdirect product factor
220C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC23×C3GO+(4,𝔽2)
commutatorC3⋊D3C23×C3×C4
FrattiniC2C22×C3×GO+(4,𝔽2)
Fitting, socleC23×C33D4
prime radicalC23×C3×C3⋊D3C22
cosocleC2×C3⋊D3C24×C3
hypocenterC32C22×C3×C4⋊C4
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
3. C2×C3×#3773C2direct (N × H)
4. C22×#3773C3direct (N × H)
6. C22×C3×#157C4semidirect
11. C3×#3773C22direct (N × H)
15. C2×#3773C2×C3direct (N × H)
21. C2×C3×#157C2×C4semidirect
29. C22×#157C3×C4semidirect
39. #3773C22×C3direct (N × H)
40. C22×C33C4⋊C4semidirect
42. C2×C3×C3⋊D3C4⋊C4semidirect
47. C3×#157C22×C4semidirect
64. C2×#157C2×C3×C4semidirect
75. C2×C33C2×C4⋊C4semidirect
77. C3×C3⋊D3C2×C4⋊C4semidirect
97. C22×C32C3×C4⋊C4semidirect
99. C2×C3⋊D3C3×C4⋊C4semidirect
109. #157C22×C3×C4semidirect
117. C33C22×C4⋊C4semidirect
143. C2×C32C2×C3×C4⋊C4semidirect
145. C3⋊D3C2×C3×C4⋊C4semidirect
167. C22×C3#3773direct (N × H)
181. C32C22×C3×C4⋊C4semidirect
199. C2×C3C2×#3773direct (N × H)
209. C22C3×#3773direct (N × H)
215. C3C22×#3773direct (N × H)
219. C2C2×C3×#3773direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×#1572
C22×C3×#6242
C2×C3×#37732
C22×#37733
C22×C3×C4⋊C49