Finite Group Databasebeta

← C2×C3×#63770 #13173240

Subgroups — C2×C3×#63770 #13173240

Order 1728.

408 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 104 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×#3788C212yesmaximal
2C2×C3×C8×C3⋊D3C211yesmaximal, characteristic
3C3×#63770C214yesmaximal, direct product factor
4C2×#63770C311yesmaximal, characteristic, direct product factor
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6C2×C3×#622—22no—
7C3×#3788—22no—
8C3×#3788C2214yes—
9C2×C33×C8C411yescharacteristic
10C2×C3×C3⋊(C3⋊C8)C411yescharacteristic
11C3×C8×C3⋊D3C412yes—
12C3×C8×C3⋊D3C2212yes—
13C2×#3788C2×C312yes—
14C2×C8×C3⋊D3C2×C311yescharacteristic
15C2×C3×D3×C8—62no—
16#63770C2×C314yesdirect product factor
17C22×C3×C3⋊D3Q811yescharacteristic
18C2×C33×C4C2×C411yescharacteristic
19C2×C3×C3⋊Dic3D411yescharacteristic
20C3×C4×C3⋊D3C2×C411yescharacteristic
21C3×C4×C3⋊D3D411yescharacteristic
22C3×C4×C3⋊D3Q811yescharacteristic
23C3×C4×C3⋊D3C2311yescharacteristic
24C3×#622—24no—
25C33×C8C2×C412yes—
26C3×C3⋊(C3⋊C8)C2×C412yes—
27C2×C3×#108—91nomaximal
28C2×C4×C3⋊D3C22×C311yescharacteristic
29C2×C3×C4×D3—62no—
30C2×#622—22no—
31#3788—22no—
32#3788C22×C314yes—
33C2×C32×C8—22no—
34C2×C32×C8—42no—
35C2×C32×C8C3×C411yescharacteristic
36C2×C3⋊(C3⋊C8)C3×C411yescharacteristic
37C8×C3⋊D3C3×C412yes—
38C8×C3⋊D3C22×C312yes—
39C2×C3×C3⋊C8—62no—
40C3×D3×C8—64no—
41C3×D3×C8—64no—
42C22×C33C4⋊C411yescharacteristic
43C2×C3×C3⋊D3—22no—
44C2×C3×C3⋊D3C4⋊C411yescharacteristic
45C2×C3×C3⋊D3C2×Q811yescharacteristic
46C33×C4C4⋊C411yescharacteristic
47C33×C4C22×C411yescharacteristic
48C3×C3⋊Dic3C4⋊C411yescharacteristic
49C3×C3⋊Dic3C2×D411yescharacteristic
50C2×C3×M16—92no—
51C22×C3×C8—91no—
52C2×D3×C8—62no—
53C3×#108—94no—
54C22×C3⋊D3C3×Q811yescharacteristic
55C22×C3×D3—62no—
56C2×C32×C4—22no—
57C2×C32×C4—42no—
58C2×C32×C4C2×C3×C411yescharacteristic
59C2×C3⋊Dic3C3×D411yescharacteristic
60C4×C3⋊D3C2×C3×C411yescharacteristic
61C4×C3⋊D3C3×D411yescharacteristic
62C4×C3⋊D3C3×Q811yescharacteristic
63C4×C3⋊D3C23×C311yescharacteristic
64C2×C3×Dic3—62no—
65C3×C4×D3—62no—
66C3×C4×D3—62no—
67C3×C4×D3—62no—
68C3×C4×D3—62no—
69#622—24no—
70C32×C8—24no—
71C32×C8—44no—
72C32×C8C2×C3×C412yes—
73C3⋊(C3⋊C8)C2×C3×C412yes—
74C3×C3⋊C8—64no—
75C2×#108—91noSylow
76C2×C33#10812yescomplemented
77C2×C33C2×C4⋊C411yescharacteristic
78C3×C3⋊D3—21no—
79C3×C3⋊D3—21no—
80C22×C3×C4—91no—
81C2×C4×D3—62no—
82C3×M16—94no—
83C3×M16—182no—
84C2×C3×C8—22no—
85C2×C3×C8—42no—
86C2×C3×C8—91no—
87C2×C3×C8—92no—
88C2×C3×C8—92no—
89C2×C3×C8—182no—
90C2×C3×C8#15711yescharacteristic
91C2×C3⋊C8—62no—
92D3×C8—64no—
93D3×C8—64no—
94C22×C32—22no—
95C22×C32—42no—
96C22×C32C3×C4⋊C411yescharacteristic
97C2×C3⋊D3—22no—
98C2×C3⋊D3C3×C4⋊C411yescharacteristic
99C2×C3⋊D3C2×C3×Q811yescharacteristic
100C2×C3×D3—62no—
101C2×C3×D3—62no—
102C2×C3×D3—68no—
103C32×C4—22no—
104C32×C4—22no—
105C32×C4—42no—
106C32×C4—42no—
107C32×C4C3×C4⋊C411yescharacteristic
108C32×C4C22×C3×C411yescharacteristic
109C3⋊Dic3C3×C4⋊C411yescharacteristic
110C3⋊Dic3C2×C3×D411yescharacteristic
111C3×Dic3—62no—
112C3×Dic3—62no—
113C2×M16—92no—
114C22×C8—91no—
115#108—94no—
116C33C2×#10811yescharacteristic, complemented, Sylow
117C23×C3—91no—
118C22×D3—62no—
119C2×C3×C4—22no—
120C2×C3×C4—42no—
121C2×C3×C4—91no—
122C2×C3×C4—91no—
123C2×C3×C4—91no—
124C2×C3×C4—91no—
125C2×C3×C4—91no—
126C2×C3×C4C2×#15711yescharacteristic
127C2×Dic3—62no—
128C4×D3—62no—
129C4×D3—62no—
130C4×D3—62no—
131C4×D3—62no—
132C3×C8—24no—
133C3×C8—44no—
134C3×C8—92no—
135C3×C8—184no—
136C3×C8C2×#15712yes—
137C3⋊C8—64no—
138C2×C32—24no—
139C2×C32—22no—
140C2×C32—44no—
141C2×C32—42no—
142C2×C32C3×#10812yescomplemented
143C2×C32C2×C3×C4⋊C411yescharacteristic
144C3⋊D3—21no—
145C3⋊D3—21no—
146C3×D3—64no—
147C3×D3—64no—
148C22×C4—91no—
149M16—94no—
150M16—182no—
151C2×C8—91no—
152C2×C8—92no—
153C2×C8—92no—
154C2×C8—182no—
155C2×C8C3×#15711yescharacteristic
156C22×C3—22no—
157C22×C3—42no—
158C22×C3—91no—
159C22×C3—91no—
160C22×C3—182no—
161C22×C3C4⋊#15711yescharacteristic
162C2×D3—62no—
163C2×D3—62no—
164C2×D3—68no—
165C3×C4—22no—
166C3×C4—22no—
167C3×C4—42no—
168C3×C4—42no—
169C3×C4—91no—
170C3×C4—91no—
171C3×C4C4⋊#15711yescharacteristic
172C3×C4C22×#15711yescharacteristic
173Dic3—62no—
174Dic3—62no—
175C32—22no—
176C32—42no—
177C32C2×C3×#10811yescharacteristic, complemented
178C23—91no—
179C2×C4—91no—
180C2×C4—91no—
181C2×C4—91no—
182C2×C4—91no—
183C2×C4—91no—
184C2×C4C2×C3×#15711yescharacteristic
185C8—92no—
186C8—184no—
187C8C2×C3×#15712yes—
188C2×C3—24no—
189C2×C3—22no—
190C2×C3—44no—
191C2×C3—42no—
192C2×C3—181no—
193C2×C3—181no—
194C2×C3#6377012yesdirect product factor
195C2×C3C2×C4⋊#15711yescharacteristic
196D3—64no—
197D3—64no—
198C22—91no—
199C22—91no—
200C22—182no—
201C22C3×C4⋊#15711yescharacteristic
202C4—91no—
203C4—91no—
204C4C3×C4⋊#15711yescharacteristic
205C4C22×C3×#15711yescharacteristic
206C3—22no—
207C3—42no—
208C3C2×#6377011yescharacteristic, direct product factor
209C2—181no—
210C2—181no—
211C2C3×#6377012yesdirect product factor
212C2C2×C3×C4⋊#15711yescharacteristic
subgroupG/H
centerC22×C3C4⋊#157
commutatorC32×C4C22×C3×C4
FrattiniC4C22×C3×#157
FittingC2×C33×C8C4
socleC22×C33C4⋊C4
prime radicalC22×C3×C3⋊D3Q8
cosocleC4×C3⋊D3C23×C3
hypercenterC2×C3×C8#157
hypocenterC32C2×C3×#108
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
3. C3×#63770C2direct (N × H)
4. C2×#63770C3direct (N × H)
16. #63770C2×C3direct (N × H)
76. C2×C33#108semidirect
116. C33C2×#108semidirect
142. C2×C32C3×#108semidirect
177. C32C2×C3×#108semidirect
194. C2×C3#63770direct (N × H)
208. C3C2×#63770direct (N × H)
211. C2C3×#63770direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37882
C2×C3×C8×C3⋊D32
C3×#637702
C2×#637703
C2×C3×#1089