Finite Group Databasebeta

← #13144342

Subgroups — #13144342

Order 1728.

572 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 123 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×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
2#11721824C211yesmaximal, characteristic
3#11720573C211yesmaximal, characteristic
4#11720532C212yesmaximal, complemented
5#11720530C212yesmaximal, complemented
6C3×#4971—34nomaximal
7C22×C4×H(3,𝔽3)—21no—
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#6229C2211yescharacteristic
10C4⋊C4×H(3,𝔽3)—21no—
11C4⋊C4×H(3,𝔽3)C2212yes—
12#88552—21no—
13C4⋊#1084C2211yescharacteristic
14#88540—21no—
15C2×#6180C2212yes—
16C2×C32×C4⋊C4D314yes—
17C3×#924—34no—
18C3×#830—34no—
19C3×#810—38no—
20C3×#808—38no—
21C23×H(3,𝔽3)D411yescharacteristic
22C2×C4×H(3,𝔽3)—22no—
23C2×C4×H(3,𝔽3)—22no—
24C2×C4×H(3,𝔽3)—42no—
25C2×C4×H(3,𝔽3)D411yescharacteristic
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×#1084—21no—
28C2×#1084—21no—
29#6229—22no—
30#6180—22no—
31C3×#483—91no—
32#4971—34no—
33C22×C32×C4—24no—
34C22×C32×C4C2×D314yes—
35C2×C3×Dic6—34no—
36C32×C4⋊C4—24no—
37C32×C4⋊C4C2×D318yes—
38C3×#217—64no—
39C3×C4⋊Dic3—34no—
40C3×Dic3⋊C4—64no—
41C2×C3×C3⋊C8—38no—
42C22×H(3,𝔽3)—21no—
43C22×H(3,𝔽3)—21no—
44C22×H(3,𝔽3)SD1612yes—
45C22×H(3,𝔽3)C2×D411yescharacteristic
46C4×H(3,𝔽3)—21no—
47C4×H(3,𝔽3)—42no—
48C4×H(3,𝔽3)#4112yes—
49#1084—41no—
50#1084—41no—
51C3×C22⋊Q8—91no—
52C2×C3×C4⋊C4—34no—
53C2×C3×C4⋊C4C3⋊D311yescharacteristic
54#924—34no—
55C3×#106—92no—
56C3×#103—92no—
57C3×C22⋊C8—91no—
58#830—34no—
59#810—38no—
60#808—38no—
61C23×C32#6714yes—
62C2×C32×C4—28no—
63C2×C32×C4—28no—
64C2×C32×C4—48no—
65C2×C32×C4#6714yes—
66C2×C32×C4C22×D314yes—
67C2×C3×Dic3—64no—
68C2×C3×Dic3—64no—
69C3×Dic6—68no—
70C3×C3⋊C8—68no—
71#483—91noSylow
72C2×H(3,𝔽3)—22no—
73C2×H(3,𝔽3)#12311yescharacteristic
74C2×H(3,𝔽3)C2×SD1611yescharacteristic
75C2×H(3,𝔽3)#13711yescharacteristic
76C2×C3×Q8—91no—
77C22×C3×C4—21no—
78C22×C3×C4—34no—
79C22×C3×C4—64no—
80C22×C3×C4C2×C3⋊D311yescharacteristic
81C2×Dic6—34no—
82C2×C3×C8—92no—
83C3×C4⋊C4—21no—
84C3×C4⋊C4—38no—
85C3×C4⋊C4—64no—
86C3×C4⋊C4—91no—
87C3×C4⋊C4—181no—
88C3×C4⋊C4C2×C3⋊D312yes—
89C3×C22⋊C4—181no—
90#217—64no—
91C4⋊Dic3—34no—
92Dic3⋊C4—64no—
93C2×C3⋊C8—38no—
94C22×C32—24no—
95C22×C32—24no—
96C22×C32#21418yes—
97C22×C32C2×#6714yes—
98C32×C4—24no—
99C32×C4—48no—
100C32×C4#23518yes—
101C3×Dic3—124no—
102C3×Dic3—124no—
103C22⋊Q8—91no—
104C2×C4⋊C4#27211yescharacteristic
105#106—92no—
106#103—92no—
107C22⋊C8—91no—
108H(3,𝔽3)#48311yescharacteristic, complemented, Sylow
109C23×C3—34no—
110C23×C3#63811yescharacteristic
111C3×Q8—182no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—34no—
115C2×C3×C4—34no—
116C2×C3×C4—42no—
117C2×C3×C4—68no—
118C2×C3×C4—68no—
119C2×C3×C4—128no—
120C2×C3×C4—181no—
121C2×C3×C4—181no—
122C2×C3×C4#63811yescharacteristic
123C2×C3×C4C22×C3⋊D311yescharacteristic
124C2×Dic3—64no—
125C2×Dic3—64no—
126Dic6—68no—
127C3×C8—182no—
128C3⋊C8—68no—
129C2×C32—28no—
130C2×C32#92914yes—
131C2×C32C2×#21414yes—
132C2×C32#94214yes—
133C2×Q8—91no—
134C22×C4—21no—
135C22×C4C2×#27211yescharacteristic
136C2×C8—92no—
137C4⋊C4—21no—
138C4⋊C4—91no—
139C4⋊C4—181no—
140C4⋊C4C2×#27212yes—
141C22⋊C4—181no—
142C22×C3—21no—
143C22×C3—21no—
144C22×C3—34no—
145C22×C3—38no—
146C22×C3—64no—
147C22×C3—64no—
148C22×C3#375412yes—
149C22×C3C2×#63811yescharacteristic
150C3×C4—21no—
151C3×C4—38no—
152C3×C4—42no—
153C3×C4—64no—
154C3×C4—128no—
155C3×C4—361no—
156C3×C4—361no—
157C3×C4#382812yes—
158Dic3—124no—
159Dic3—124no—
160C32#497114yescomplemented
161C23#623511yescharacteristic
162Q8—182no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—42no—
166C2×C4—181no—
167C2×C4—181no—
168C2×C4#623511yescharacteristic
169C2×C4C22×#27211yescharacteristic
170C8—182no—
171C2×C3—22no—
172C2×C3—34no—
173C2×C3—34no—
174C2×C3—34no—
175C2×C3—68no—
176C2×C3#6413811yescharacteristic
177C2×C3C2×#375411yescharacteristic
178C2×C3#6415111yescharacteristic
179C22—21no—
180C22—21no—
181C22#8854612yes—
182C22C2×#623511yescharacteristic
183C4—21no—
184C4—42no—
185C4—361no—
186C4—361no—
187C4#8874012yes—
188C3—34no—
189C3#1059267011yescharacteristic
190C2—22no—
191C2#1172182911yescharacteristic
192C2C2×#8854611yescharacteristic
193C2#1172185911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radicalC23×H(3,𝔽3)D4
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)#483
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #11720532C2semidirect
5. #11720530C2semidirect
108. H(3,𝔽3)#483semidirect
160. C32#4971semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×H(3,𝔽3)2
#117218242
#117205732
#117205322
#117205302
C3×#49713