Finite Group Databasebeta

← C3×#10593540 #13167616

Subgroups — C3×#10593540 #13167616

Order 1728.

298 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C3×#64286C211yesmaximal, characteristic
2C3×#64283C211yesmaximal, characteristic
3C2×C3×#3787C211yesmaximal, characteristic
4C3×#63744C214yesmaximal, complemented
5#10593540C311yesmaximal, characteristic, direct product factor
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C3×C22⋊#157C412yes—
8C3×#3792—21no—
9C2×C3×#622—21no—
10C3×C4⋊#157C412yes—
11C3×C4×#157C2212yes—
12C3×#3788—21no—
13C3×#3787—21no—
14C3×#3787C2212yes—
15C3×#3787C2212yes—
16#64286C2×C311yescharacteristic
17#64283C2×C311yescharacteristic
18C2×#3787C2×C311yescharacteristic
19#63744C2×C314yescomplemented
20C22×C3×C3⋊D3C2×C411yescharacteristic
21C2×C3×#157C2×C414yes—
22C2×C33×C4Q811yescharacteristic
23C2×C3×C3⋊Dic3D411yescharacteristic
24C3×C4×C3⋊D3C2×C411yescharacteristic
25C3×C4×C3⋊D3D411yescharacteristic
26C3×C4×C3⋊D3Q811yescharacteristic
27C3×C4×C3⋊D3C2311yescharacteristic
28C3×#622—22no—
29C3×#622—22no—
30C3×#424—91nomaximal
31C2×C4×C3⋊D3C22×C311yescharacteristic
32C2×C3×C4×D3—121no—
33C22⋊#157C3×C412yes—
34#3792—21no—
35C2×#622—21no—
36C4⋊#157C3×C412yes—
37C4×#157C22×C312yes—
38#3788—21no—
39#3787—21no—
40#3787C22×C312yes—
41#3787C22×C312yes—
42C22×C33C4⋊C411yescharacteristic
43C2×C3×C3⋊D3—22no—
44C2×C3×C3⋊D3C4⋊C411yescharacteristic
45C2×C3×C3⋊D3C22×C411yescharacteristic
46C3×#157—24no—
47C33×C4C4⋊C411yescharacteristic
48C33×C4C2×Q811yescharacteristic
49C3×C3⋊Dic3C4⋊C411yescharacteristic
50C3×C3⋊Dic3C2×D411yescharacteristic
51C2×C3×M16—91no—
52C22×C3×C8—91no—
53C3×#117—91no—
54C3×C4⋊C8—94no—
55C22×C3⋊D3C2×C3×C411yescharacteristic
56C22×C3×D3—121no—
57C2×#157C2×C3×C414yes—
58C2×C32×C4—41no—
59C2×C32×C4—81no—
60C2×C32×C4C3×Q811yescharacteristic
61C2×C3⋊Dic3C3×D411yescharacteristic
62C4×C3⋊D3C2×C3×C411yescharacteristic
63C4×C3⋊D3C3×D411yescharacteristic
64C4×C3⋊D3C3×Q811yescharacteristic
65C4×C3⋊D3C23×C311yescharacteristic
66C2×C3×Dic3—121no—
67C3×C4×D3—121no—
68C3×C4×D3—121no—
69C3×C4×D3—121no—
70C3×C4×D3—121no—
71#622—22no—
72#622—22no—
73#424—91noSylow
74C2×C33—21no—
75C2×C33C2×C4⋊C411yescharacteristic
76C3×C3⋊D3—21no—
77C3×C3⋊D3#13112yes—
78C22×C3×C4—91no—
79C2×C4×D3—121no—
80C3×M16—181no—
81C3×M16—181no—
82C2×C3×C8—92no—
83C2×C3×C8—92no—
84C2×C3×C8—181no—
85C2×C3×C8—181no—
86C3×C4⋊C4—92no—
87C3×C22⋊C4—92no—
88C3×C42—92no—
89C22×C32—41no—
90C22×C32—81no—
91C22×C32C3×C4⋊C411yescharacteristic
92C2×C3⋊D3—22no—
93C2×C3⋊D3C3×C4⋊C411yescharacteristic
94C2×C3⋊D3C22×C3×C411yescharacteristic
95C2×C3×D3—121no—
96C2×C3×D3—121no—
97C2×C3×D3—124no—
98#157—24no—
99C32×C4—41no—
100C32×C4—41no—
101C32×C4—81no—
102C32×C4—81no—
103C32×C4C3×C4⋊C411yescharacteristic
104C32×C4C2×C3×Q811yescharacteristic
105C3⋊Dic3C3×C4⋊C411yescharacteristic
106C3⋊Dic3C2×C3×D411yescharacteristic
107C3×Dic3—121no—
108C3×Dic3—121no—
109C2×M16—91no—
110C22×C8—91no—
111#117—91no—
112C4⋊C8—94no—
113C33#42411yescharacteristic, complemented, Sylow
114C23×C3—91no—
115C22×D3—121no—
116C2×C3×C4—41no—
117C2×C3×C4—81no—
118C2×C3×C4—91no—
119C2×C3×C4—91no—
120C2×C3×C4—91no—
121C2×C3×C4—91no—
122C2×C3×C4—91no—
123C2×C3×C4—94no—
124C2×C3×C4PSU(3,𝔽2)11yescharacteristic
125C2×Dic3—121no—
126C4×D3—121no—
127C4×D3—121no—
128C4×D3—121no—
129C4×D3—121no—
130C3×C8—182no—
131C3×C8—182no—
132C2×C32—21no—
133C2×C32—42no—
134C2×C32—41no—
135C2×C32—82no—
136C2×C32—81no—
137C2×C32C2×C3×C4⋊C411yescharacteristic
138C3⋊D3—21no—
139C3⋊D3C3×#13112yes—
140C3×D3—122no—
141C3×D3—122no—
142C22×C4—91no—
143M16—181no—
144M16—181no—
145C2×C8—92no—
146C2×C8—92no—
147C2×C8—181no—
148C2×C8—181no—
149C4⋊C4—92no—
150C22⋊C4—92no—
151C42—92no—
152C22×C3—41no—
153C22×C3—81no—
154C22×C3—91no—
155C22×C3—91no—
156C22×C3—182no—
157C22×C3#377711yescharacteristic
158C2×D3—121no—
159C2×D3—121no—
160C2×D3—124no—
161C3×C4—41no—
162C3×C4—41no—
163C3×C4—81no—
164C3×C4—81no—
165C3×C4—91no—
166C3×C4—91no—
167C3×C4—184no—
168C3×C4#377711yescharacteristic
169C3×C4C2×PSU(3,𝔽2)11yescharacteristic
170Dic3—121no—
171Dic3—121no—
172C32—41no—
173C32—81no—
174C32C3×#42411yescharacteristic, complemented
175C23—91no—
176C2×C4—91no—
177C2×C4—91no—
178C2×C4—91no—
179C2×C4—91no—
180C2×C4—91no—
181C2×C4—94no—
182C2×C4C3×PSU(3,𝔽2)11yescharacteristic
183C8—182no—
184C8—182no—
185C2×C3—21no—
186C2×C3—42no—
187C2×C3—41no—
188C2×C3—82no—
189C2×C3—81no—
190C2×C3—92no—
191C2×C3—181no—
192C2×C3C2×#377711yescharacteristic
193D3—122no—
194D3—122no—
195C22—91no—
196C22—91no—
197C22—182no—
198C22C3×#377711yescharacteristic
199C4—91no—
200C4—91no—
201C4—184no—
202C4C3×#377711yescharacteristic
203C4C2×C3×PSU(3,𝔽2)11yescharacteristic
204C3—41no—
205C3—81no—
206C3#1059354011yescharacteristic, direct product factor
207C2—21no—
208C2—92no—
209C2—181no—
210C2C2×C3×#377711yescharacteristic
subgroupG/H
centerC3×C4C2×PSU(3,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC4C2×C3×PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC4×C3⋊D3C23×C3
hypercenterC2×C3×C4PSU(3,𝔽2)
hypocenterC32C3×#424
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63744C2semidirect
5. #10593540C3direct (N × H)
19. #63744C2×C3semidirect
113. C33#424semidirect
174. C32C3×#424semidirect
206. C3#10593540direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642862
C3×#642832
C2×C3×#37872
C3×#637442
#105935403
C3×#4249