Finite Group Databasebeta

← C3×#10593413 #13166777

Subgroups — C3×#10593413 #13166777

Order 1728.

608 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 146 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
1C2×C3×C4×#157C211yesmaximal, characteristic
2C2×C3×#3773C214yesmaximal
3C3×#63900C212yesmaximal
4#10593413C311yesmaximal, characteristic, direct product factor
5C22×C3×#157C2212yes—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C3×#624C2214yes—
8C3×C4×#157C414yescomplemented
9C3×#3773—28no—
10C32×C4⋊Dic3—22no—
11C3×#3724—22no—
12C2×C4×#157C2×C311yescharacteristic
13C2×#3773C2×C314yes—
14C3×D3×C4⋊C4—62no—
15#63900C2×C312yes—
16C22×C3×C3⋊D3C2311yescharacteristic
17C2×C3×#157C2×C414yes—
18C2×C3×#157D414yes—
19C2×C3×#157Q814yes—
20C2×C33×C4D411yescharacteristic
21C2×C3×C3⋊Dic3D411yescharacteristic
22C3×C4×C3⋊D3C2×C412yes—
23C3×C4×C3⋊D3Q812yes—
24C2×C32×Dic3—24no—
25C3×#624—28no—
26C3×#384—91nomaximal
27C22×#157C22×C312yes—
28C2×C4×C3⋊D3C22×C311yescharacteristic
29C2×C3×C4×D3—62no—
30C2×C3×C4×D3—64no—
31C2×#624C22×C314yes—
32C4×#157C3×C414yescomplemented
33#3773—28no—
34C32×C4⋊C4—62no—
35C3×C4⋊Dic3—22no—
36C3×C4⋊Dic3—22no—
37C3×C4⋊Dic3—42no—
38C3×Dic3⋊C4—64no—
39#3724—22no—
40C22×C33C2×D411yescharacteristic
41C2×C3×C3⋊D3C22×C411yescharacteristic
42C2×C3×C3⋊D3C2×D412yes—
43C2×C3×C3⋊D3C2×Q812yes—
44C2×C3×C3⋊D3C2×Q811yescharacteristic
45C3×#157C4⋊C418yescomplemented
46C33×C4C4⋊C412yescomplemented
47C3×C3⋊Dic3C4⋊C412yescomplemented
48C32×Dic3—44no—
49C2×C3×C4⋊C4—92no—
50C2×C3×C4⋊C4—94no—
51C2×C3×C42—91no—
52D3×C4⋊C4—62no—
53C22×C3⋊D3C23×C311yescharacteristic
54C22×C3×D3—62no—
55C2×#157C2×C3×C414yes—
56C2×#157C3×D414yes—
57C2×#157C3×Q814yes—
58C2×C32×C4—22no—
59C2×C32×C4—42no—
60C2×C32×C4—64no—
61C2×C32×C4C3×D411yescharacteristic
62C2×C3⋊Dic3C3×D411yescharacteristic
63C4×C3⋊D3C2×C3×C412yes—
64C4×C3⋊D3C3×Q812yes—
65C2×C3×Dic3—24no—
66C2×C3×Dic3—24no—
67C2×C3×Dic3—44no—
68C2×C3×Dic3—62no—
69C3×C4×D3—64no—
70C3×C4×D3—64no—
71C3×C4×D3—128no—
72#624—28no—
73#384—91noSylow
74C2×C33C2×C4⋊C411yescharacteristic
75C2×C33#12711yescharacteristic
76C2×C33C4⋊Q811yescharacteristic
77C3×C3⋊D3C2×C4⋊C412yes—
78C3×C3⋊D3C4⋊Q812yes—
79C22×C3×C4—94no—
80C22×C3×C4—92no—
81C22×C3×C4—91no—
82C2×C4×D3—62no—
83C2×C4×D3—64no—
84C3×C4⋊C4—62no—
85C3×C4⋊C4—62no—
86C3×C4⋊C4—122no—
87C3×C4⋊C4—182no—
88C3×C4⋊C4—188no—
89C3×C42—94no—
90C4⋊Dic3—22no—
91Dic3⋊C4—64no—
92C22×C32—22no—
93C22×C32—42no—
94C22×C32C2×C3×D411yescharacteristic
95C2×C3⋊D3C22×C3×C411yescharacteristic
96C2×C3⋊D3C2×C3×D412yes—
97C2×C3⋊D3C2×C3×Q812yes—
98C2×C3⋊D3C2×C3×Q811yescharacteristic
99C2×C3×D3—62no—
100C2×C3×D3—64no—
101C2×C3×D3—62no—
102C2×C3×D3—64no—
103#157C3×C4⋊C418yescomplemented
104C32×C4—24no—
105C32×C4—44no—
106C32×C4—124no—
107C32×C4C3×C4⋊C412yescomplemented
108C3⋊Dic3C3×C4⋊C412yescomplemented
109C3×Dic3—44no—
110C3×Dic3—44no—
111C3×Dic3—64no—
112C3×Dic3—84no—
113C2×C4⋊C4—92no—
114C2×C4⋊C4—94no—
115C2×C42—91no—
116C33#38411yescharacteristic, complemented, Sylow
117C23×C3—91no—
118C22×D3—62no—
119C2×C3×C4—22no—
120C2×C3×C4—42no—
121C2×C3×C4—64no—
122C2×C3×C4—64no—
123C2×C3×C4—94no—
124C2×C3×C4—94no—
125C2×C3×C4—94no—
126C2×C3×C4—91no—
127C2×C3×C4—92no—
128C2×C3×C4—92no—
129C2×C3×C4—124no—
130C2×C3×C4—188no—
131C2×C3×C4GO+(4,𝔽2)11yescharacteristic
132C2×Dic3—24no—
133C2×Dic3—62no—
134C4×D3—64no—
135C4×D3—64no—
136C4×D3—128no—
137C2×C32—22no—
138C2×C32—22no—
139C2×C32—22no—
140C2×C32—42no—
141C2×C32—42no—
142C2×C32—42no—
143C2×C32C2×C3×C4⋊C411yescharacteristic
144C2×C32C3×#12711yescharacteristic
145C2×C32C3×C4⋊Q811yescharacteristic
146C3⋊D3C2×C3×C4⋊C412yes—
147C3⋊D3C3×C4⋊Q812yes—
148C3×D3—64no—
149C3×D3—64no—
150C22×C4—94no—
151C22×C4—92no—
152C22×C4—91no—
153C4⋊C4—62no—
154C4⋊C4—182no—
155C4⋊C4—188no—
156C42—94no—
157C22×C3—22no—
158C22×C3—42no—
159C22×C3—92no—
160C22×C3—92no—
161C22×C3—91no—
162C22×C3—91no—
163C22×C3C2×GO+(4,𝔽2)11yescharacteristic
164C2×D3—62no—
165C2×D3—62no—
166C2×D3—64no—
167C2×D3—64no—
168C3×C4—24no—
169C3×C4—44no—
170C3×C4—98no—
171C3×C4—92no—
172C3×C4—124no—
173C3×C4—124no—
174C3×C4—244no—
175C3×C4#377312yescomplemented
176Dic3—44no—
177Dic3—64no—
178C32—22no—
179C32—42no—
180C32C3×#38411yescharacteristic, complemented
181C23—91no—
182C2×C4—64no—
183C2×C4—94no—
184C2×C4—94no—
185C2×C4—94no—
186C2×C4—91no—
187C2×C4—92no—
188C2×C4—92no—
189C2×C4—188no—
190C2×C4C3×GO+(4,𝔽2)11yescharacteristic
191C2×C3—22no—
192C2×C3—22no—
193C2×C3—22no—
194C2×C3—42no—
195C2×C3—42no—
196C2×C3—42no—
197C2×C3—92no—
198C2×C3—92no—
199C2×C3#6422211yescharacteristic
200C2×C3#6423111yescharacteristic
201C2×C3C2×#377311yescharacteristic
202D3—64no—
203D3—64no—
204C22—91no—
205C22—91no—
206C22—92no—
207C22—92no—
208C22C2×C3×GO+(4,𝔽2)11yescharacteristic
209C4—98no—
210C4—92no—
211C4—124no—
212C4C3×#377312yescomplemented
213C3—22no—
214C3—42no—
215C3#1059341311yescharacteristic, direct product factor
216C2—92no—
217C2—92no—
218C2C3×#6422211yescharacteristic
219C2C3×#6423111yescharacteristic
220C2C2×C3×#377311yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3C23
cosocleC22×C3⋊D3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#384
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
4. #10593413C3direct (N × H)
8. C3×C4×#157C4semidirect
32. C4×#157C3×C4semidirect
45. C3×#157C4⋊C4semidirect
46. C33×C4C4⋊C4semidirect
47. C3×C3⋊Dic3C4⋊C4semidirect
103. #157C3×C4⋊C4semidirect
107. C32×C4C3×C4⋊C4semidirect
108. C3⋊Dic3C3×C4⋊C4semidirect
116. C33#384semidirect
175. C3×C4#3773semidirect
180. C32C3×#384semidirect
212. C4C3×#3773semidirect
215. C3#10593413direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×#1572
C2×C3×#37732
C3×#639002
#105934133
C3×#3849