Finite Group Databasebeta

← C3×#10590098 #13155142

Subgroups — C3×#10590098 #13155142

Order 1728.

450 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 70 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×C22⋊#157C212yesmaximal
2C2×C3×C22⋊#157C211yesmaximal, characteristic
3#10590098C311yesmaximal, characteristic, direct product factor
4C23×C3×C3⋊D3C2211yescharacteristic
5C22×C3×#157—22no—
6C22×C3×#157C412yescomplemented
7C3×C22⋊#157—22no—
8C3×C22⋊#157—22no—
9C3×C22⋊#157C414yescomplemented
10C2×C22⋊#157C2×C312yes—
11C2×C22⋊#157C2×C311yescharacteristic
12C23×C33Q811yescharacteristic
13C22×C3×C3⋊D3—42no—
14C22×C3×C3⋊D3C2×C411yescharacteristic
15C22×C3×C3⋊D3C2×C412yes—
16C22×C3×C3⋊D3D412yes—
17C22×C3×C3⋊D3D411yescharacteristic
18C2×C3×#157—24no—
19C2×C3×#157—22no—
20C2×C3×#157—44no—
21C2×C3×#157—42no—
22C3×#342—91nomaximal
23C23×C3⋊D3C22×C311yescharacteristic
24C23×C3×D3—121no—
25C22×#157—22no—
26C22×#157C3×C412yescomplemented
27C22⋊#157—22no—
28C22⋊#157—22no—
29C22⋊#157C3×C414yescomplemented
30C22×C33—41no—
31C22×C33C4⋊C411yescharacteristic
32C22×C33C4⋊C412yes—
33C2×C3×C3⋊D3—22no—
34C2×C3×C3⋊D3—24no—
35C2×C3×C3⋊D3—42no—
36C2×C3×C3⋊D3—41no—
37C2×C3×C3⋊D3C4211yescharacteristic
38C2×C3×C3⋊D3C22⋊C412yes—
39C2×C3×C3⋊D3C22⋊C411yescharacteristic
40C3×#157—44no—
41C3×#157—42no—
42C2×C3×C22⋊C4—92no—
43C2×C3×C22⋊C4—91no—
44C23×C32—41no—
45C23×C32—81no—
46C23×C32C3×Q811yescharacteristic
47C22×C3⋊D3—42no—
48C22×C3⋊D3C2×C3×C411yescharacteristic
49C22×C3⋊D3C2×C3×C412yes—
50C22×C3⋊D3C3×D412yes—
51C22×C3⋊D3C3×D411yescharacteristic
52C22×C3×D3—121no—
53C22×C3×D3—122no—
54C22×C3×D3—122no—
55C22×C3×D3—121no—
56C22×C3×D3—128no—
57C2×#157—24no—
58C2×#157—22no—
59C2×#157—44no—
60C2×#157—42no—
61#342—91noSylow
62C2×C33—21no—
63C2×C33—22no—
64C2×C33#9511yescharacteristic
65C3×C3⋊D3—22no—
66C3×C3⋊D3—21no—
67C3×C3⋊D3#9912yes—
68C24×C3—91no—
69C23×D3—121no—
70C22×C3×C4—92no—
71C22×C3×C4—182no—
72C3×C22⋊C4—94no—
73C3×C22⋊C4—182no—
74C3×C22⋊C4—182no—
75C22×C32—41no—
76C22×C32—44no—
77C22×C32—42no—
78C22×C32—41no—
79C22×C32—84no—
80C22×C32—81no—
81C22×C32—82no—
82C22×C32C3×C4⋊C411yescharacteristic
83C22×C32C3×C4⋊C412yes—
84C2×C3⋊D3—22no—
85C2×C3⋊D3—24no—
86C2×C3⋊D3—42no—
87C2×C3⋊D3—41no—
88C2×C3⋊D3C3×C4211yescharacteristic
89C2×C3⋊D3C3×C22⋊C412yes—
90C2×C3⋊D3C3×C22⋊C411yescharacteristic
91C2×C3×D3—121no—
92C2×C3×D3—121no—
93C2×C3×D3—122no—
94C2×C3×D3—124no—
95C2×C3×D3—128no—
96C2×C3×D3—128no—
97C2×C3×D3—124no—
98#157—44no—
99#157—42no—
100C2×C22⋊C4—92no—
101C2×C22⋊C4—91no—
102C33#34211yescharacteristic, complemented, Sylow
103C23×C3—41no—
104C23×C3—81no—
105C23×C3—92no—
106C23×C3—91no—
107C23×C3—92no—
108C23×C3—91no—
109C23×C3—362no—
110C23×C3PSU(3,𝔽2)11yescharacteristic
111C22×D3—121no—
112C22×D3—122no—
113C22×D3—122no—
114C22×D3—121no—
115C22×D3—128no—
116C2×C3×C4—184no—
117C2×C3×C4—182no—
118C2×C3×C4—364no—
119C2×C3×C4—362no—
120C2×C32—21no—
121C2×C32—22no—
122C2×C32—42no—
123C2×C32—44no—
124C2×C32—41no—
125C2×C32—82no—
126C2×C32—84no—
127C2×C32—81no—
128C2×C32C3×#9511yescharacteristic
129C3⋊D3—22no—
130C3⋊D3—21no—
131C3⋊D3C3×#9912yes—
132C3×D3—122no—
133C3×D3—124no—
134C3×D3—122no—
135C24—91no—
136C22×C4—92no—
137C22×C4—182no—
138C22⋊C4—94no—
139C22⋊C4—182no—
140C22⋊C4—182no—
141C22×C3—41no—
142C22×C3—44no—
143C22×C3—42no—
144C22×C3—41no—
145C22×C3—84no—
146C22×C3—82no—
147C22×C3—81no—
148C22×C3—91no—
149C22×C3—92no—
150C22×C3—91no—
151C22×C3—184no—
152C22×C3—182no—
153C22×C3—361no—
154C22×C3—362no—
155C22×C3#377712yes—
156C22×C3#377711yescharacteristic
157C2×D3—121no—
158C2×D3—121no—
159C2×D3—122no—
160C2×D3—124no—
161C2×D3—128no—
162C2×D3—128no—
163C2×D3—124no—
164C3×C4—362no—
165C3×C4—364no—
166C32—41no—
167C32—81no—
168C32C3×#34211yescharacteristic, complemented
169C23—92no—
170C23—91no—
171C23—92no—
172C23—91no—
173C23—362no—
174C23C3×PSU(3,𝔽2)11yescharacteristic
175C2×C4—184no—
176C2×C4—182no—
177C2×C4—364no—
178C2×C4—362no—
179C2×C3—22no—
180C2×C3—21no—
181C2×C3—42no—
182C2×C3—44no—
183C2×C3—41no—
184C2×C3—84no—
185C2×C3—82no—
186C2×C3—81no—
187C2×C3—92no—
188C2×C3—182no—
189C2×C3—181no—
190C2×C3#6374711yescharacteristic
191D3—122no—
192D3—124no—
193D3—122no—
194C22—41no—
195C22—91no—
196C22—92no—
197C22—91no—
198C22—184no—
199C22—182no—
200C22—361no—
201C22—362no—
202C22C3×#377712yes—
203C22C3×#377711yescharacteristic
204C4—362no—
205C4—364no—
206C3—41no—
207C3—81no—
208C3#1059009811yescharacteristic, direct product factor
209C2—22no—
210C2—21no—
211C2—92no—
212C2—182no—
213C2—181no—
214C2C3×#6374711yescharacteristic
subgroupG/H
centerC2×C3#63747
commutatorC2×C3⋊D3C3×C42
FrattiniC23C3×PSU(3,𝔽2)
FittingC23×C33Q8
socleC2×C33#95
prime radicalC23×C3×C3⋊D3C22
cosocleC23×C3⋊D3C22×C3
hypercenterC23×C3PSU(3,𝔽2)
hypocenterC32C3×#342
LCSG ▷ C2×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
3. #10590098C3direct (N × H)
6. C22×C3×#157C4semidirect
9. C3×C22⋊#157C4semidirect
26. C22×#157C3×C4semidirect
29. C22⋊#157C3×C4semidirect
102. C33#342semidirect
168. C32C3×#342semidirect
208. C3#10590098direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊#1572
#105900983
C3×#3429