Finite Group Databasebeta

← #13157236

Subgroups — #13157236

Order 1728.

356 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 61 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×Q8×C3⋊Dic3C211yesmaximal, characteristic
2C4⋊(C3⋊#622)C211yesmaximal, characteristic
3C4×C3⋊#622C211yesmaximal, characteristic
4Q8⋊#622—31nomaximal
5C2×C33×Q8C411yescharacteristic
6C2×C3⋊#622—21no—
7C2×C3⋊#622—21no—
8C3×C4⋊(C3⋊Dic3)—21no—
9C3×C4⋊(C3⋊Dic3)C411yescharacteristic
10C3×C4×C3⋊Dic3—21no—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12Q8×C3⋊Dic3D311yescharacteristic
13C3×Q8×Dic3—62no—
14C4⋊#622—31no—
15C4×#622—31no—
16C33×Q8—21no—
17C33×Q8C812yescomplemented
18C2×C33×C4—21no—
19C2×C33×C4C2×C411yescharacteristic
20C2×C3×C3⋊Dic3—21no—
21C2×C3×C3⋊Dic3D411yescharacteristic
22C2×C3×C3⋊Dic3D411yescharacteristic
23C3⋊#622—22no—
24C3⋊#622—41no—
25#4535—91nomaximal
26C2×C32×Q8—22no—
27C2×C32×Q8—42no—
28C2×C32×Q8Dic311yescharacteristic
29C2×#622—61no—
30C2×#622—61no—
31C4⋊(C3⋊Dic3)—21no—
32C4⋊(C3⋊Dic3)Dic311yescharacteristic
33C4×C3⋊Dic3—21no—
34C4×C3⋊Dic3C2×D311yescharacteristic
35C3×C4⋊Dic3—62no—
36C3×C4⋊Dic3—64no—
37C3×C4×Dic3—62no—
38C3×C4×Dic3—64no—
39C22×C33C22⋊C411yescharacteristic
40C33×C4—22no—
41C33×C4C2×C811yescharacteristic
42C33×C4M1611yescharacteristic
43C3×C3⋊Dic3—21no—
44C3×C3⋊Dic3—41no—
45C3×C3⋊Dic3SD1611yescharacteristic
46C3×C3⋊Dic3Q1611yescharacteristic
47C3×C4×Q8—91no—
48Q8×Dic3—62no—
49C4⋊(C3⋊C8)—91no—
50C4×C3⋊C8—91no—
51C32×Q8—21no—
52C32×Q8—24no—
53C32×Q8—24no—
54C32×Q8—44no—
55C32×Q8—44no—
56C32×Q8C3⋊C812yescomplemented
57C2×C32×C4—21no—
58C2×C32×C4—24no—
59C2×C32×C4—22no—
60C2×C32×C4—44no—
61C2×C32×C4—42no—
62C2×C32×C4C2×Dic311yescharacteristic
63C2×C3⋊Dic3—21no—
64C2×C3⋊Dic3#6711yescharacteristic
65C2×C3⋊Dic3#6711yescharacteristic
66C2×C3×Dic3—62no—
67C2×C3×Dic3—62no—
68C2×C3×Dic3—64no—
69#622—62no—
70#622—121no—
71Q8⋊C8—271noSylow
72C2×C33C22⋊C811yescharacteristic
73C2×C33#10311yescharacteristic
74C2×C33#10411yescharacteristic
75C2×C3×Q8—22no—
76C2×C3×Q8—42no—
77C2×C3×Q8#15711yescharacteristic
78C3×C4⋊C4—91no—
79C3×C4⋊C4—181no—
80C3×C42—91no—
81C3×C42—181no—
82C4⋊Dic3—62no—
83C4⋊Dic3—64no—
84C4×Dic3—62no—
85C4×Dic3—64no—
86C2×C3⋊C8—181no—
87C2×C3⋊C8—181no—
88C22×C32—22no—
89C22×C32—42no—
90C22×C32#21711yescharacteristic
91C32×C4—22no—
92C32×C4—28no—
93C32×C4—22no—
94C32×C4—22no—
95C32×C4—48no—
96C32×C4—42no—
97C32×C4—42no—
98C32×C4C2×C3⋊C811yescharacteristic
99C32×C4#20811yescharacteristic
100C3⋊Dic3—21no—
101C3⋊Dic3—41no—
102C3⋊Dic3#21511yescharacteristic
103C3⋊Dic3#21611yescharacteristic
104C3×Dic3—62no—
105C3×Dic3—62no—
106C3×Dic3—122no—
107C3×Dic3—124no—
108C4×Q8—91no—
109C4⋊C8—271no—
110C4×C8—271no—
111C33Q8⋊C811yescharacteristic, complemented, Sylow
112C3×Q8—24no—
113C3×Q8—21no—
114C3×Q8—24no—
115C3×Q8—44no—
116C3×Q8—44no—
117C3×Q8#62212yescomplemented
118C2×C3×C4—24no—
119C2×C3×C4—21no—
120C2×C3×C4—22no—
121C2×C3×C4—44no—
122C2×C3×C4—42no—
123C2×C3×C4—91no—
124C2×C3×C4—91no—
125C2×C3×C4—181no—
126C2×C3×C4C2×#15711yescharacteristic
127C2×Dic3—62no—
128C2×Dic3—62no—
129C2×Dic3—64no—
130C3⋊C8—182no—
131C3⋊C8—361no—
132C2×C32—22no—
133C2×C32—22no—
134C2×C32—22no—
135C2×C32—42no—
136C2×C32—42no—
137C2×C32—42no—
138C2×C32#83011yescharacteristic
139C2×C32#83511yescharacteristic
140C2×C32#83711yescharacteristic
141C2×Q8C3⋊#15711yescharacteristic
142C2×C8—541no—
143C2×C8—541no—
144C4⋊C4—91no—
145C4⋊C4—181no—
146C42—91no—
147C42—181no—
148C22×C3—22no—
149C22×C3—42no—
150C22×C3C22⋊#15711yescharacteristic
151C3×C4—28no—
152C3×C4—22no—
153C3×C4—22no—
154C3×C4—22no—
155C3×C4—48no—
156C3×C4—42no—
157C3×C4—42no—
158C3×C4—91no—
159C3×C4—91no—
160C3×C4—181no—
161C3×C4—361no—
162C3×C4C2×#62211yescharacteristic
163C3×C4#379211yescharacteristic
164Dic3—62no—
165Dic3—62no—
166Dic3—122no—
167Dic3—124no—
168C32—22no—
169C32—42no—
170C32#453511yescharacteristic, complemented
171Q8—21no—
172Q8C3⋊#62212yescomplemented
173C2×C4—21no—
174C2×C4—91no—
175C2×C4—91no—
176C2×C4—181no—
177C2×C4C2×C3⋊#15711yescharacteristic
178C8—542no—
179C8—1081no—
180C2×C3—22no—
181C2×C3—22no—
182C2×C3—22no—
183C2×C3—42no—
184C2×C3—42no—
185C2×C3—42no—
186C2×C3#6378411yescharacteristic
187C2×C3#6378511yescharacteristic
188C2×C3C22⋊#62211yescharacteristic
189C22#8899411yescharacteristic
190C4—22no—
191C4—91no—
192C4—91no—
193C4—181no—
194C4—361no—
195C4C2×C3⋊#62211yescharacteristic
196C4#8899311yescharacteristic
197C3—22no—
198C3—42no—
199C3Q8⋊#62211yescharacteristic, complemented
200C2#1172310911yescharacteristic
201C2#1172311011yescharacteristic
202C2#1172311211yescharacteristic
subgroupG/H
centerC22#88994
commutatorC33×C4C2×C8
FrattiniC2×C4C2×C3⋊#157
FittingC2×C33×Q8C4
socle, prime radicalC22×C33C22⋊C4
cosocleC3×C4×C3⋊Dic3C22
hypercenterC2×Q8C3⋊#157
hypocenterC33Q8⋊C8
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
17. C33×Q8C8semidirect
56. C32×Q8C3⋊C8semidirect
111. C33Q8⋊C8semidirect
117. C3×Q8#622semidirect
170. C32#4535semidirect
172. Q8C3⋊#622semidirect
199. C3Q8⋊#622semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C3⋊Dic32
C4⋊(C3⋊#622)2
C4×C3⋊#6222
Q8⋊#6223
#45359