Finite Group Databasebeta

← C3×#10590208 #13157133

Subgroups — C3×#10590208 #13157133

Order 1728.

488 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 76 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×#3792C212yesmaximal
2C2×C3×#3750C211yesmaximal, characteristic
3#10590208C311yesmaximal, characteristic, direct product factor
4C22×C33×C4C411yescharacteristic
5C22×C3×C3⋊Dic3C411yescharacteristic
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C3×#3792—22no—
8C3×#3792C414yescomplemented
9C2×C3×#622—22no—
10C3×#3750—22no—
11C2×#3792C2×C312yes—
12C2×#3750C2×C311yescharacteristic
13C2×C3×Dic3⋊C4—62no—
14C23×C33C2×C411yescharacteristic
15C2×C33×C4—21no—
16C2×C33×C4—41no—
17C2×C3×C3⋊Dic3—21no—
18C2×C3×C3⋊Dic3—41no—
19C2×C3×C3⋊Dic3C2×C412yes—
20C2×C3×C3⋊Dic3D411yescharacteristic
21C2×C3×C3⋊Dic3D412yes—
22C2×C3×C3⋊Dic3Q811yescharacteristic
23C3×#622—24no—
24C3×#343—91nomaximal
25C22×C32×C4—22no—
26C22×C32×C4—42no—
27C22×C32×C4C3×C411yescharacteristic
28C22×C3⋊Dic3C3×C411yescharacteristic
29C22×C3⋊Dic3C22×C311yescharacteristic
30C22×C3×Dic3—62no—
31C22×C3×Dic3—62no—
32#3792—22no—
33#3792C3×C414yescomplemented
34C2×#622—22no—
35#3750—22no—
36C3×Dic3⋊C4—68no—
37C22×C33—21no—
38C22×C33—21no—
39C22×C33C4211yescharacteristic
40C22×C33C22⋊C411yescharacteristic
41C22×C33C4⋊C411yescharacteristic
42C33×C4—41no—
43C3×C3⋊Dic3—41no—
44C3×C3⋊Dic3C22⋊C412yes—
45C3×C3⋊Dic3C4⋊C412yes—
46C2×C3×M16—92no—
47C2×C3×C4⋊C4—91no—
48C2×Dic3⋊C4—62no—
49C23×C32—22no—
50C23×C32—42no—
51C23×C32C2×C3×C411yescharacteristic
52C2×C32×C4—21no—
53C2×C32×C4—24no—
54C2×C32×C4—44no—
55C2×C32×C4—44no—
56C2×C32×C4—48no—
57C2×C32×C4—41no—
58C2×C3⋊Dic3—21no—
59C2×C3⋊Dic3—41no—
60C2×C3⋊Dic3C2×C3×C412yes—
61C2×C3⋊Dic3C3×D411yescharacteristic
62C2×C3⋊Dic3C3×D412yes—
63C2×C3⋊Dic3C3×Q811yescharacteristic
64C2×C3×Dic3—64no—
65C2×C3×Dic3—64no—
66C2×C3×Dic3—62no—
67C2×C3×Dic3—62no—
68C2×C3×Dic3—64no—
69C2×C3×Dic3—124no—
70#622—24no—
71#343—91noSylow
72C2×C33—21no—
73C2×C33—21no—
74C2×C33#9511yescharacteristic
75C2×C33#10011yescharacteristic
76C2×C33#10111yescharacteristic
77C22×C3×C4—22no—
78C22×C3×C4—42no—
79C22×C3×C4—91no—
80C22×C3×C4—91no—
81C22×C3×C4#15711yescharacteristic
82C22×Dic3—62no—
83C22×Dic3—62no—
84C3×M16—94no—
85C3×M16—182no—
86C2×C3×C8—182no—
87C3×C4⋊C4—182no—
88Dic3⋊C4—68no—
89C22×C32—21no—
90C22×C32—21no—
91C22×C32—24no—
92C22×C32—24no—
93C22×C32—22no—
94C22×C32—22no—
95C22×C32—22no—
96C22×C32—44no—
97C22×C32—44no—
98C22×C32—42no—
99C22×C32—42no—
100C22×C32—42no—
101C22×C32C3×C4211yescharacteristic
102C22×C32C3×C22⋊C411yescharacteristic
103C22×C32C3×C4⋊C411yescharacteristic
104C32×C4—44no—
105C32×C4—48no—
106C32×C4—41no—
107C3⋊Dic3—41no—
108C3⋊Dic3C3×C22⋊C412yes—
109C3⋊Dic3C3×C4⋊C412yes—
110C3×Dic3—64no—
111C3×Dic3—64no—
112C3×Dic3—124no—
113C2×M16—92no—
114C2×C4⋊C4—91no—
115C33#34311yescharacteristic, complemented, Sylow
116C23×C3—22no—
117C23×C3—42no—
118C23×C3C2×#15711yescharacteristic
119C2×C3×C4—24no—
120C2×C3×C4—21no—
121C2×C3×C4—44no—
122C2×C3×C4—44no—
123C2×C3×C4—48no—
124C2×C3×C4—41no—
125C2×C3×C4—92no—
126C2×C3×C4—91no—
127C2×C3×C4—91no—
128C2×C3×C4—92no—
129C2×C3×C4—181no—
130C2×C3×C4—361no—
131C2×Dic3—64no—
132C2×Dic3—64no—
133C2×Dic3—62no—
134C2×Dic3—62no—
135C2×Dic3—64no—
136C2×Dic3—124no—
137C3×C8—184no—
138C2×C32—21no—
139C2×C32—21no—
140C2×C32—24no—
141C2×C32—24no—
142C2×C32—22no—
143C2×C32—22no—
144C2×C32—22no—
145C2×C32—44no—
146C2×C32—44no—
147C2×C32—42no—
148C2×C32—42no—
149C2×C32—42no—
150C2×C32C3×#9511yescharacteristic
151C2×C32C3×#10011yescharacteristic
152C2×C32C3×#10111yescharacteristic
153C22×C4—91no—
154C22×C4—91no—
155C22×C4C3×#15711yescharacteristic
156M16—94no—
157M16—182no—
158C2×C8—182no—
159C4⋊C4—182no—
160C22×C3—24no—
161C22×C3—24no—
162C22×C3—21no—
163C22×C3—21no—
164C22×C3—22no—
165C22×C3—22no—
166C22×C3—22no—
167C22×C3—44no—
168C22×C3—44no—
169C22×C3—42no—
170C22×C3—42no—
171C22×C3—42no—
172C22×C3C4×#15711yescharacteristic
173C22×C3C4⋊#15711yescharacteristic
174C22×C3C22⋊#15711yescharacteristic
175C3×C4—44no—
176C3×C4—48no—
177C3×C4—41no—
178C3×C4—92no—
179C3×C4—92no—
180C3×C4—361no—
181Dic3—64no—
182Dic3—64no—
183Dic3—124no—
184C32—22no—
185C32—42no—
186C32C3×#34311yescharacteristic, complemented
187C23C2×C3×#15711yescharacteristic
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—92no—
191C2×C4—91no—
192C2×C4—91no—
193C2×C4—92no—
194C2×C4—181no—
195C2×C4—361no—
196C8—184no—
197C2×C3—24no—
198C2×C3—24no—
199C2×C3—21no—
200C2×C3—21no—
201C2×C3—22no—
202C2×C3—22no—
203C2×C3—22no—
204C2×C3—44no—
205C2×C3—44no—
206C2×C3—42no—
207C2×C3—42no—
208C2×C3—42no—
209C2×C3#6378011yescharacteristic
210C2×C3#6378111yescharacteristic
211C2×C3#6378811yescharacteristic
212C22—21no—
213C22—21no—
214C22C3×C4×#15711yescharacteristic
215C22C3×C4⋊#15711yescharacteristic
216C22C3×C22⋊#15711yescharacteristic
217C4—41no—
218C4—92no—
219C4—92no—
220C4—361no—
221C3—22no—
222C3—42no—
223C3#1059020811yescharacteristic, direct product factor
224C2—21no—
225C2—21no—
226C2C3×#6378011yescharacteristic
227C2C3×#6378111yescharacteristic
228C2C3×#6378811yescharacteristic
subgroupG/H
centerC22×C3C22⋊#157
commutatorC22×C32C3×C42
FrattiniC23C2×C3×#157
FittingC22×C33×C4C4
socleC22×C33C22⋊C4
prime radicalC23×C33C2×C4
cosocleC22×C3⋊Dic3C22×C3
hypercenterC22×C3×C4#157
hypocenterC32C3×#343
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. #10590208C3direct (N × H)
8. C3×#3792C4semidirect
33. #3792C3×C4semidirect
115. C33#343semidirect
186. C32C3×#343semidirect
223. C3#10590208direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37922
C2×C3×#37502
#105902083
C3×#3439