Finite Group Databasebeta

← C22×#88940 #13181163

Subgroups — C22×#88940 #13181163

Order 1728.

1416 conjugacy classes of subgroups; only the 319 normal subgroups are shown.

normal subgroups only

This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 319 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×#6294C212yesmaximal
3C2×#88940C2112yesmaximal, direct product factor
4C22×#3773—31nomaximal
5C23×C3×C3⋊D3C2211yescharacteristic
6C22×C3×#157C418yescomplemented
7C22×C3×#157C2216yes—
8C22×C3×C3⋊Dic3—22no—
9C2×#6294—26no—
10C2×#6294C22112yes—
11#88940C22116yesdirect product factor
12C23×#157D311yescharacteristic
13C22×#624—32no—
14C22×D3×Dic3—62no—
15C2×#3773—312no—
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×C3×C3⋊Dic3—212no—
24#6294—212no—
25#6294C2318yes—
26C22×C4⋊Dic3—91nomaximal
27C23×C3⋊D3C2×D311yescharacteristic
28C23×C3×D3—62no—
29C22×#157Dic318yescomplemented
30C22×#157C2×D316yes—
31C22×C3⋊Dic3—22no—
32C22×C3×Dic3—62no—
33C22×C3×Dic3—62no—
34C22×C3×Dic3—122no—
35C2×#624—312no—
36C2×#624—66no—
37C2×D3×Dic3—612no—
38C2×D3×Dic3—612no—
39#3773—316no—
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
48C3×C3⋊Dic3—28no—
49C23×C3×C4—91no—
50C23×Dic3—92no—
51C22×C4×D3—182no—
52C2×C4⋊Dic3—912no—
53C23×C32—22no—
54C23×C32—42no—
55C23×C32D1211yescharacteristic
56C22×C3⋊D3Dic614yes—
57C22×C3⋊D3D1213yes—
58C22×C3⋊D3C2×Dic314yes—
59C22×C3⋊D3C22×D313yes—
60C22×C3×D3—68no—
61C22×C3×D3—66no—
62C22×C3×D3—68no—
63C22×C3×D3—66no—
64C2×#157C2×Dic3124yescomplemented
65C2×#157C22×D314yes—
66C2×C3⋊Dic3—212no—
67C2×C3×Dic3—612no—
68C2×C3×Dic3—612no—
69C2×C3×Dic3—1212no—
70#624—38no—
71#624—612no—
72D3×Dic3—68no—
73D3×Dic3—624no—
74C22×C4⋊C4—271noSylow
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—98no—
83C22×C3×C4—96no—
84C22×C3×C4—182no—
85C22×Dic3—62no—
86C22×Dic3—62no—
87C22×Dic3—912no—
88C22×Dic3—122no—
89C22×Dic3—186no—
90C2×C4×D3—1812no—
91C2×C4×D3—1812no—
92C4⋊Dic3—916no—
93C22×C32—26no—
94C22×C32—28no—
95C22×C32—48no—
96C22×C32—46no—
97C22×C32C4⋊Dic314yescomplemented
98C22×C32C2×D1213yes—
99C2×C3⋊D3C4⋊Dic3112yescomplemented
100C2×C3⋊D3C2×Dic616yes—
101C2×C3⋊D3C2×D1213yes—
102C2×C3⋊D3C22×Dic316yes—
103C2×C3⋊D3C23×D311yescharacteristic
104C2×C3×D3—624no—
105C2×C3×D3—612no—
106C2×C3×D3—66no—
107C2×C3×D3—612no—
108C2×C3×D3—62no—
109#157C22×Dic318yescomplemented
110C3⋊Dic3—28no—
111C3×Dic3—68no—
112C3×Dic3—68no—
113C3×Dic3—128no—
114C23×C4—91no—
115C23×C4—272no—
116C2×C4⋊C4—2712no—
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—924no—
130C2×C3×C4—94no—
131C2×C3×C4—1812no—
132C2×Dic3—612no—
133C2×Dic3—612no—
134C2×Dic3—98no—
135C2×Dic3—1212no—
136C2×Dic3—1812no—
137C4×D3—188no—
138C4×D3—1824no—
139C2×C32—22no—
140C2×C32—212no—
141C2×C32—412no—
142C2×C32—42no—
143C2×C32C2×C4⋊Dic316yescomplemented
144C2×C32C22×D1211yescharacteristic
145C3⋊D3C2×C4⋊Dic316yescomplemented
146C3⋊D3C22×Dic611yescharacteristic
147C3⋊D3C23×Dic311yescharacteristic
148C3×D3—612no—
149C3×D3—62no—
150C3×D3—62no—
151C24—91no—
152C22×C4—98no—
153C22×C4—96no—
154C22×C4—182no—
155C22×C4—2712no—
156C22×C4—546no—
157C4⋊C4—2716no—
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#377314yescomplemented
168C22×C3C2×GO+(4,𝔽2)13yes—
169C2×D3—66no—
170C2×D3—62no—
171C2×D3—624no—
172C2×D3—612no—
173C2×D3—612no—
174C3×C4—98no—
175C3×C4—188no—
176Dic3—68no—
177Dic3—68no—
178Dic3—128no—
179C32—22no—
180C32—42no—
181C32C22×C4⋊Dic311yescharacteristic, complemented
182C23—93no—
183C23—93no—
184C23—94no—
185C23—94no—
186C23#157⋊D311yescharacteristic
187C2×C4—94no—
188C2×C4—924no—
189C2×C4—1812no—
190C2×C4—278no—
191C2×C4—5412no—
192C2×C3—212no—
193C2×C3—22no—
194C2×C3—412no—
195C2×C3—42no—
196C2×C3—91no—
197C2×C3—91no—
198C2×C3—96no—
199C2×C3C2×#377316yescomplemented
200C2×C3C22×GO+(4,𝔽2)11yescharacteristic
201D3—612no—
202D3—62no—
203D3—62no—
204C22—96no—
205C22—96no—
206C22—912no—
207C22—91no—
208C22—93no—
209C22#8894014yesdirect product factor
210C22C2×#157⋊D313yes—
211C4—98no—
212C4—188no—
213C3—22no—
214C3—42no—
215C3C22×#377311yescharacteristic, complemented
216C2—91no—
217C2—96no—
218C2—91no—
219C2C2×#8894016yesdirect product factor
220C2C22×#157⋊D311yescharacteristic
subgroupG/H
center, hypercenterC23#157⋊D3
commutatorC3×C3⋊D3C23×C4
FrattiniC2C22×#157⋊D3
Fitting, socleC23×C33D4
prime radicalC23×C3×C3⋊D3C22
cosocleC2×C3×C3⋊D3C24
hypocenterC33C22×C4⋊C4
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
3. C2×#88940C2direct (N × H)
6. C22×C3×#157C4semidirect
11. #88940C22direct (N × H)
21. C2×C3×#157C2×C4semidirect
29. C22×#157Dic3semidirect
40. C22×C33C4⋊C4semidirect
42. C2×C3×C3⋊D3C4⋊C4semidirect
47. C3×#157C22×C4semidirect
64. C2×#157C2×Dic3semidirect
75. C2×C33C2×C4⋊C4semidirect
77. C3×C3⋊D3C2×C4⋊C4semidirect
97. C22×C32C4⋊Dic3semidirect
99. C2×C3⋊D3C4⋊Dic3semidirect
109. #157C22×Dic3semidirect
117. C33C22×C4⋊C4semidirect
143. C2×C32C2×C4⋊Dic3semidirect
145. C3⋊D3C2×C4⋊Dic3semidirect
167. C22×C3#3773semidirect
181. C32C22×C4⋊Dic3semidirect
199. C2×C3C2×#3773semidirect
209. C22#88940direct (N × H)
215. C3C22×#3773semidirect
219. C2C2×#88940direct (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×#62942
C2×#889402
C22×#37733
C22×C4⋊Dic39