Finite Group Databasebeta

← #13157178

Subgroups — #13157178

Order 1728.

316 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 49 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×#64097C211yesmaximal, characteristic
2#11723104C211yesmaximal, characteristic
3#11723104C211yesmaximal, characteristic
4#10590162—31nomaximal
5C2×C3×#634—21no—
6C2×C3⋊#622—21no—
7C2×C3⋊#622—21no—
8C33×C4⋊C4C411yescharacteristic
9C3×C4⋊(C3⋊Dic3)C411yescharacteristic
10C3×#3750—21no—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12#64097D311yescharacteristic
13C3×#888—62no—
14#63779—31no—
15#63779—31no—
16C2×C33×C4—21no—
17C2×C33×C4C2×C411yescharacteristic
18C2×C3×C3⋊Dic3—21no—
19C2×C3×C3⋊Dic3D411yescharacteristic
20C2×C3×C3⋊Dic3D411yescharacteristic
21C3×#634—41no—
22C3×#634—41no—
23C3⋊#622—41no—
24C3⋊#622—41no—
25#4547—91nomaximal
26C2×#634—21no—
27C2×C3×Dic6—64no—
28C2×#622—61no—
29C2×#622—61no—
30C32×C4⋊C4—22no—
31C32×C4⋊C4—42no—
32C32×C4⋊C4Dic311yescharacteristic
33C4⋊(C3⋊Dic3)Dic311yescharacteristic
34#3750—21no—
35C4×C3⋊Dic3C2×D311yescharacteristic
36C3×C4⋊Dic3—62no—
37C3×Dic3⋊C4—64no—
38C3×C4×Dic3—62no—
39C22×C33C22⋊C411yescharacteristic
40C33×C4—21no—
41C33×C4—41no—
42C3×C3⋊Dic3—41no—
43C3×C3⋊Dic3D811yescharacteristic
44C3×C3⋊Dic3SD1611yescharacteristic
45C3×C3⋊Dic3SD1611yescharacteristic
46C3×C3⋊Dic3Q1611yescharacteristic
47C3×C4⋊Q8—91no—
48#888—62no—
49C4⋊(C3⋊C8)—91no—
50C4⋊(C3⋊C8)—91no—
51C2×C32×C4—21no—
52C2×C32×C4—24no—
53C2×C32×C4—22no—
54C2×C32×C4—44no—
55C2×C32×C4—42no—
56C2×C32×C4C2×Dic311yescharacteristic
57C2×C3⋊Dic3—21no—
58C2×C3⋊Dic3#6711yescharacteristic
59C2×C3⋊Dic3#6711yescharacteristic
60#634—41no—
61#634—41no—
62C2×C3×Dic3—62no—
63C2×C3×Dic3—62no—
64C2×C3×Dic3—64no—
65C3×Dic6—124no—
66C3×Dic6—124no—
67#622—121no—
68#622—121no—
69#332—271noSylow
70C2×C33#10111yescharacteristic
71C2×C33#10211yescharacteristic
72C2×C33#10311yescharacteristic
73C2×C3×Q8—181no—
74C2×Dic6—64no—
75C3×C4⋊C4—22no—
76C3×C4⋊C4—42no—
77C3×C4⋊C4—91no—
78C3×C4⋊C4—181no—
79C3×C4⋊C4#15711yescharacteristic
80C3×C42—91no—
81C4⋊Dic3—62no—
82Dic3⋊C4—64no—
83C4×Dic3—62no—
84C2×C3⋊C8—181no—
85C2×C3⋊C8—181no—
86C22×C32—22no—
87C22×C32—42no—
88C22×C32#21711yescharacteristic
89C32×C4—21no—
90C32×C4—24no—
91C32×C4—44no—
92C32×C4—41no—
93C32×C4—44no—
94C32×C4—84no—
95C3⋊Dic3—41no—
96C3⋊Dic3#21311yescharacteristic
97C3⋊Dic3#21411yescharacteristic
98C3⋊Dic3#21511yescharacteristic
99C3⋊Dic3#21611yescharacteristic
100C3×Dic3—62no—
101C3×Dic3—62no—
102C3×Dic3—62no—
103C3×Dic3—62no—
104C3×Dic3—124no—
105C4⋊Q8—91no—
106C4⋊C8—271no—
107C4⋊C8—271no—
108C33#33211yescharacteristic, complemented, Sylow
109C3×Q8—361no—
110C3×Q8—361no—
111C2×C3×C4—24no—
112C2×C3×C4—21no—
113C2×C3×C4—22no—
114C2×C3×C4—44no—
115C2×C3×C4—42no—
116C2×C3×C4—91no—
117C2×C3×C4—91no—
118C2×C3×C4—181no—
119C2×C3×C4C2×#15711yescharacteristic
120C2×Dic3—62no—
121C2×Dic3—62no—
122C2×Dic3—64no—
123Dic6—124no—
124Dic6—124no—
125C3⋊C8—361no—
126C3⋊C8—361no—
127C2×C32—22no—
128C2×C32—22no—
129C2×C32—22no—
130C2×C32—42no—
131C2×C32—42no—
132C2×C32—42no—
133C2×C32#83211yescharacteristic
134C2×C32#83511yescharacteristic
135C2×C32#83611yescharacteristic
136C2×Q8—181no—
137C2×C8—541no—
138C2×C8—541no—
139C4⋊C4—91no—
140C4⋊C4—181no—
141C4⋊C4C3⋊#15711yescharacteristic
142C42—91no—
143C22×C3—22no—
144C22×C3—42no—
145C22×C3C22⋊#15711yescharacteristic
146C3×C4—24no—
147C3×C4—21no—
148C3×C4—44no—
149C3×C4—44no—
150C3×C4—41no—
151C3×C4—84no—
152C3×C4—91no—
153C3×C4—91no—
154C3×C4—91no—
155C3×C4—91no—
156C3×C4—361no—
157Dic3—62no—
158Dic3—62no—
159Dic3—62no—
160Dic3—62no—
161Dic3—124no—
162C32—22no—
163C32—42no—
164C32#454711yescharacteristic, complemented
165Q8—361no—
166Q8—361no—
167C2×C4—21no—
168C2×C4—91no—
169C2×C4—91no—
170C2×C4—181no—
171C2×C4C2×C3⋊#15711yescharacteristic
172C8—1081no—
173C8—1081no—
174C2×C3—22no—
175C2×C3—22no—
176C2×C3—22no—
177C2×C3—42no—
178C2×C3—42no—
179C2×C3—42no—
180C2×C3#6378011yescharacteristic
181C2×C3#6378211yescharacteristic
182C2×C3#6378411yescharacteristic
183C22#8899411yescharacteristic
184C4—21no—
185C4—41no—
186C4—91no—
187C4—91no—
188C4—91no—
189C4—91no—
190C4—361no—
191C3—22no—
192C3—42no—
193C3#1059016211yescharacteristic, complemented
194C2#1172310511yescharacteristic
195C2#1172310711yescharacteristic
196C2#1172310911yescharacteristic
subgroupG/H
centerC22#88994
commutatorC2×C33×C4C2×C4
FrattiniC2×C4C2×C3⋊#157
FittingC33×C4⋊C4C4
socle, prime radicalC22×C33C22⋊C4
cosocleC3×C4×C3⋊Dic3C22
hypercenterC4⋊C4C3⋊#157
hypocenterC33#332
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
108. C33#332semidirect
164. C32#4547semidirect
193. C3#10590162semidirect

Maximal subgroups

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

subgroupindex
C3×#640972
#117231042
#105901623
#45479