Finite Group Databasebeta

← #13143713

Subgroups — #13143713

Order 1728.

847 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 847 conjugacy classes of subgroups — too many to draw. Only its 181 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×#88540C212yesmaximal
2C2×C4×#1084C211yesmaximal, characteristic
3H(3,𝔽3)×#95C211yesmaximal, characteristic
4#11720574C211yesmaximal, characteristic
5#11720574C212yesmaximal
6C3×#4650—34nomaximal
7C22×C4×H(3,𝔽3)C2212yes—
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2212yes—
10C22×#1084C2211yescharacteristic
11C22×#1084C2211yescharacteristic
12#88540—24no—
13C4×#1084C414yescomplemented
14C2×C3×Dic3⋊C4—38no—
15C2×C3×C4×Dic3—34no—
16C32×#95D314yes—
17C3×C3⋊#95—38no—
18C3×C3⋊#95—34no—
19C23×H(3,𝔽3)C2311yescharacteristic
20C2×C4×H(3,𝔽3)—22no—
21C2×C4×H(3,𝔽3)—22no—
22C2×C4×H(3,𝔽3)—22no—
23C2×C4×H(3,𝔽3)—22no—
24C2×C4×H(3,𝔽3)C2×C412yes—
25C2×#1084—22no—
26C2×#1084—22no—
27C2×#1084—22no—
28C2×#1084—22no—
29C2×#1084C2×C412yes—
30C2×#1084C2×C412yes—
31C2×#1084D412yes—
32C2×#1084Q812yes—
33C3×#382—91no—
34#4650—34no—
35C22×C32×C4C2×D318yes—
36C22×C32×C4C2×D314yes—
37C22×C3×Dic3—34no—
38C22×C3×Dic3—34no—
39C22×C3×Dic3—38no—
40C3×Dic3⋊C4—616no—
41C3×C4×Dic3—316no—
42C22×H(3,𝔽3)C22×C411yescharacteristic
43C22×H(3,𝔽3)C2×D411yescharacteristic
44C22×H(3,𝔽3)C2×Q811yescharacteristic
45C22×H(3,𝔽3)#4112yes—
46C22×H(3,𝔽3)#4112yes—
47C4×H(3,𝔽3)—22no—
48C4×H(3,𝔽3)—42no—
49#1084—22no—
50#1084—42no—
51#1084C4⋊C414yes—
52C2×C3×C4⋊C4—92no—
53C2×C3×C42—91no—
54C2×Dic3⋊C4—38no—
55C2×C4×Dic3—34no—
56C3×#95—34no—
57C3×#95—92no—
58C3×#95—91no—
59C3×#95C3⋊D311yescharacteristic
60C3⋊#95—38no—
61C3⋊#95—34no—
62C23×C32C22×D314yes—
63C2×C32×C4—28no—
64C2×C32×C4—28no—
65C2×C32×C4—28no—
66C2×C32×C4—28no—
67C2×C32×C4C4×D318yes—
68C2×C3×Dic3—38no—
69C2×C3×Dic3—38no—
70C2×C3×Dic3—38no—
71C2×C3×Dic3—38no—
72C2×C3×Dic3—68no—
73C2×C3×Dic3—68no—
74C2×C3×Dic3—68no—
75C2×C3×Dic3—68no—
76#382—91noSylow
77C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
78C2×H(3,𝔽3)#11711yescharacteristic
79C2×H(3,𝔽3)#11711yescharacteristic
80C2×H(3,𝔽3)#12412yes—
81C2×H(3,𝔽3)#12512yes—
82C22×C3×C4—34no—
83C22×C3×C4—38no—
84C22×C3×C4—91no—
85C22×C3×C4—91no—
86C22×C3×C4—92no—
87C22×C3×C4C2×C3⋊D312yes—
88C22×C3×C4C2×C3⋊D311yescharacteristic
89C22×Dic3—34no—
90C22×Dic3—34no—
91C22×Dic3—38no—
92C3×C4⋊C4—184no—
93C3×C42—94no—
94Dic3⋊C4—616no—
95C4×Dic3—316no—
96C22×C32C2×C4×D314yes—
97C22×C32#23518yes—
98C22×C32D3×D414yes—
99C22×C32#23718yes—
100C22×C32D3×Q814yes—
101C32×C4—28no—
102C32×C4—48no—
103C3×Dic3—316no—
104C3×Dic3—68no—
105C3×Dic3—128no—
106C2×C4⋊C4—92no—
107C2×C42—91no—
108#95—92no—
109#95—91no—
110#95#27211yescharacteristic
111H(3,𝔽3)#38211yescharacteristic, complemented, Sylow
112C23×C3—34no—
113C23×C3C22×C3⋊D311yescharacteristic
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—38no—
119C2×C3×C4—68no—
120C2×C3×C4—68no—
121C2×C3×C4—68no—
122C2×C3×C4—68no—
123C2×C3×C4—92no—
124C2×C3×C4—92no—
125C2×C3×C4—92no—
126C2×C3×C4—92no—
127C2×C3×C4—182no—
128C2×C3×C4—182no—
129C2×C3×C4—182no—
130C2×C3×C4—182no—
131C2×C3×C4C4×C3⋊D312yes—
132C2×Dic3—38no—
133C2×Dic3—38no—
134C2×Dic3—38no—
135C2×Dic3—38no—
136C2×Dic3—68no—
137C2×Dic3—68no—
138C2×Dic3—68no—
139C2×Dic3—68no—
140C2×C32#87214yes—
141C2×C32#87714yes—
142C2×C32#88518yes—
143C2×C32#88918yes—
144C2×C32D3×C4⋊C414yes—
145C22×C4—91no—
146C22×C4—91no—
147C22×C4—92no—
148C22×C4C2×#27212yes—
149C22×C4C2×#27211yescharacteristic
150C4⋊C4—184no—
151C42—94no—
152C22×C3—34no—
153C22×C3—38no—
154C22×C3—34no—
155C22×C3—38no—
156C22×C3—34no—
157C22×C3C2×C4×C3⋊D311yescharacteristic
158C22×C3#382812yes—
159C22×C3D4×C3⋊D311yescharacteristic
160C22×C3#383012yes—
161C22×C3Q8×C3⋊D311yescharacteristic
162C3×C4—22no—
163C3×C4—42no—
164C3×C4—68no—
165C3×C4—94no—
166C3×C4—128no—
167C3×C4—182no—
168C3×C4—362no—
169Dic3—316no—
170Dic3—68no—
171Dic3—128no—
172C32#465014yescomplemented
173C23C22×#27211yescharacteristic
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—92no—
179C2×C4—92no—
180C2×C4—92no—
181C2×C4—92no—
182C2×C4—182no—
183C2×C4—182no—
184C2×C4—182no—
185C2×C4—182no—
186C2×C4C4×#27212yes—
187C2×C3—34no—
188C2×C3—38no—
189C2×C3—38no—
190C2×C3—34no—
191C2×C3—34no—
192C2×C3#6408111yescharacteristic
193C2×C3#6408611yescharacteristic
194C2×C3#6409412yes—
195C2×C3#6409812yes—
196C2×C3C4⋊C4×C3⋊D311yescharacteristic
197C22C2×C4×#27211yescharacteristic
198C22#8874012yes—
199C22D4×#27211yescharacteristic
200C22#8874412yes—
201C22Q8×#27211yescharacteristic
202C4—22no—
203C4—42no—
204C4—94no—
205C4—182no—
206C4—362no—
207C3—34no—
208C3#1059234911yescharacteristic
209C2#1172171811yescharacteristic
210C2#1172173311yescharacteristic
211C2#1172174112yes—
212C2#1172175712yes—
213C2C4⋊C4×#27211yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingH(3,𝔽3)×#95C2
prime radical, cosocleC23×H(3,𝔽3)C23
hypercenterC3×#95C3⋊D3
hypocenterH(3,𝔽3)#382
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
13. C4×#1084C4semidirect
111. H(3,𝔽3)#382semidirect
172. C32#4650semidirect

Maximal subgroups

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

subgroupindex
C2×#885402
C2×C4×#10842
H(3,𝔽3)×#952
#117205742
C3×#46503