Finite Group Databasebeta

← #13144346

Subgroups — #13144346

Order 1728.

1023 conjugacy classes of subgroups; only the 233 normal subgroups are shown.

normal subgroups only

This group has 1023 conjugacy classes of subgroups — too many to draw. Only its 233 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
1C22×#6229C211yesmaximal, characteristic
2C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
3C2×C4×#1084C211yesmaximal, characteristic
4#11720574C214yesmaximal
5C3×#4973—34nomaximal
6C22×C4×H(3,𝔽3)C2212yes—
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2212yes—
9C22×#1084C2212yes—
10C2×#6229—22no—
11C2×#6229—22no—
12C2×#6229C414yescomplemented
13C4⋊C4×H(3,𝔽3)—22no—
14C4×#1084—22no—
15C22×C3×Dic6—34no—
16C2×C32×C4⋊C4D314yes—
17C2×C3×C4×Dic3—34no—
18C3×C3⋊#95—316no—
19C23×H(3,𝔽3)C2311yescharacteristic
20C2×C4×H(3,𝔽3)—22no—
21C2×C4×H(3,𝔽3)—24no—
22C2×C4×H(3,𝔽3)C2×C412yes—
23C2×C4×H(3,𝔽3)D412yes—
24C2×C4×H(3,𝔽3)D412yes—
25C2×#1084—22no—
26C2×#1084—22no—
27C2×#1084—22no—
28C2×#1084—22no—
29C2×#1084C2×C414yes—
30C2×#1084Q814yescomplemented
31#6229—28no—
32C3×#391—91no—
33#4973—34no—
34C22×C32×C4C2×D318yes—
35C22×C32×C4C2×D314yes—
36C22×C3×Dic3—38no—
37C22×C3×Dic3—38no—
38C2×C3×Dic6—316no—
39C2×C3×Dic6—68no—
40C2×C3×Dic6—68no—
41C32×C4⋊C4—28no—
42C3×C4×Dic3—68no—
43C22×H(3,𝔽3)C22×C411yescharacteristic
44C22×H(3,𝔽3)C2×D411yescharacteristic
45C22×H(3,𝔽3)C2×D411yescharacteristic
46C22×H(3,𝔽3)C2×Q812yes—
47C22×H(3,𝔽3)#4112yes—
48C4×H(3,𝔽3)—42no—
49C4×H(3,𝔽3)C22⋊C414yes—
50#1084—24no—
51#1084—24no—
52C22×C3×Q8—91no—
53C22×Dic6—34no—
54C2×C3×C4⋊C4—34no—
55C2×C3×C4⋊C4C3⋊D311yescharacteristic
56C2×C3×C42—91no—
57C2×C4×Dic3—34no—
58C3×#95—94no—
59C3⋊#95—316no—
60C23×C32C22×D314yes—
61C2×C32×C4—216no—
62C2×C32×C4—28no—
63C2×C32×C4C4×D318yes—
64C2×C32×C4D1218yes—
65C2×C32×C4#6718yes—
66C2×C3×Dic3—316no—
67C2×C3×Dic3—316no—
68C2×C3×Dic3—68no—
69C2×C3×Dic3—68no—
70C2×C3×Dic3—68no—
71C2×C3×Dic3—68no—
72C3×Dic6—632no—
73#391—91noSylow
74C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
75C2×H(3,𝔽3)C4×Q812yes—
76C2×H(3,𝔽3)C22⋊Q812yes—
77C2×H(3,𝔽3)#12411yescharacteristic
78C2×H(3,𝔽3)C4⋊Q811yescharacteristic
79C2×C3×Q8—94no—
80C2×C3×Q8—182no—
81C2×C3×Q8—182no—
82C22×C3×C4—34no—
83C22×C3×C4—38no—
84C22×C3×C4—92no—
85C22×C3×C4—92no—
86C22×C3×C4C2×C3⋊D312yes—
87C22×C3×C4C2×C3⋊D311yescharacteristic
88C22×Dic3—38no—
89C22×Dic3—38no—
90C2×Dic6—316no—
91C2×Dic6—68no—
92C2×Dic6—68no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—68no—
95C3×C42—182no—
96C4×Dic3—68no—
97C22×C32C2×C4×D314yes—
98C22×C32C2×D1214yes—
99C22×C32#23718yes—
100C22×C32D3×Q818yes—
101C22×C32C2×#6714yes—
102C32×C4—48no—
103C32×C4(C2×D3)⋊C4116yes—
104C3×Dic3—616no—
105C3×Dic3—616no—
106C22×Q8—91no—
107C2×C4⋊C4#27211yescharacteristic
108C2×C42—91no—
109#95—94no—
110H(3,𝔽3)#39111yescharacteristic, complemented, Sylow
111C23×C3—34no—
112C23×C3C22×C3⋊D311yescharacteristic
113C3×Q8—188no—
114C2×C3×C4—24no—
115C2×C3×C4—22no—
116C2×C3×C4—38no—
117C2×C3×C4—38no—
118C2×C3×C4—38no—
119C2×C3×C4—68no—
120C2×C3×C4—616no—
121C2×C3×C4—94no—
122C2×C3×C4—94no—
123C2×C3×C4—182no—
124C2×C3×C4—182no—
125C2×C3×C4—182no—
126C2×C3×C4—182no—
127C2×C3×C4C4×C3⋊D312yes—
128C2×C3×C4C4⋊(C3⋊D3)12yes—
129C2×C3×C4#63812yes—
130C2×Dic3—316no—
131C2×Dic3—316no—
132C2×Dic3—68no—
133C2×Dic3—68no—
134C2×Dic3—68no—
135C2×Dic3—68no—
136Dic6—632no—
137C2×C32#88718yes—
138C2×C32#89718yes—
139C2×C32C2×(C2×D3)⋊C414yes—
140C2×C32#93614yes—
141C2×C32#94414yes—
142C2×Q8—94no—
143C2×Q8—182no—
144C2×Q8—182no—
145C22×C4—92no—
146C22×C4—92no—
147C22×C4C2×#27212yes—
148C22×C4C2×#27211yescharacteristic
149C4⋊C4—22no—
150C42—182no—
151C22×C3—34no—
152C22×C3—34no—
153C22×C3—38no—
154C22×C3—38no—
155C22×C3—34no—
156C22×C3C2×C4×C3⋊D311yescharacteristic
157C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
158C22×C3#383012yes—
159C22×C3Q8×C3⋊D312yes—
160C22×C3C2×#63811yescharacteristic
161C3×C4—316no—
162C3×C4—42no—
163C3×C4—128no—
164C3×C4—184no—
165C3×C4—184no—
166C3×C4#375214yes—
167Dic3—616no—
168Dic3—616no—
169C32#497314yescomplemented
170C23C22×#27211yescharacteristic
171Q8—188no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—94no—
175C2×C4—94no—
176C2×C4—182no—
177C2×C4—182no—
178C2×C4—182no—
179C2×C4—182no—
180C2×C4C4×#27212yes—
181C2×C4C4⋊#27212yes—
182C2×C4#623512yes—
183C2×C3—38no—
184C2×C3—38no—
185C2×C3—34no—
186C2×C3—34no—
187C2×C3—34no—
188C2×C3#6409612yes—
189C2×C3#6410612yes—
190C2×C3C2×#375211yescharacteristic
191C2×C3#6414511yescharacteristic
192C2×C3#6415311yescharacteristic
193C22C2×C4×#27211yescharacteristic
194C22C2×C4⋊#27211yescharacteristic
195C22#8874412yes—
196C22Q8×#27212yes—
197C22C2×#623511yescharacteristic
198C4—42no—
199C4—184no—
200C4—184no—
201C4#8854214yes—
202C3—34no—
203C3#1059267211yescharacteristic
204C2#1172175512yes—
205C2#1172176512yes—
206C2C2×#8854211yescharacteristic
207C2#1172184611yescharacteristic
208C2#1172186111yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radical, cosocleC23×H(3,𝔽3)C23
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)#391
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
12. C2×#6229C4semidirect
30. C2×#1084Q8semidirect
110. H(3,𝔽3)#391semidirect
169. C32#4973semidirect

Maximal subgroups

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

subgroupindex
C22×#62292
C2×C4⋊C4×H(3,𝔽3)2
C2×C4×#10842
#117205742
C3×#49733