Finite Group Databasebeta

← #13135215

Subgroups — #13135215

Order 1728.

300 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 43 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
1#11721864C211yesmaximal, characteristic
2#63663⋊C3C212yesmaximal, complemented
3#10589182—32nomaximal
4C2×Q8×H(3,𝔽3)C2211yescharacteristic
5C2×C4⋊#272C411yescharacteristic
6#88542—22no—
7C4×#1084C411yescharacteristic
8#3748⋊C3—22no—
9C3×#947—32no—
10C3×#947—61no—
11#63663D312yescomplemented
12C3×#836—32no—
13C22×#272—21no—
14Q8×H(3,𝔽3)—41no—
15C2×C4×H(3,𝔽3)C2×C411yescharacteristic
16C2×C4×H(3,𝔽3)D412yes—
17C2×#1084—21no—
18C4⋊#272—41no—
19(C3⋊(C3⋊C8))⋊C3—42no—
20#4542—91no—
21#4477—92no—
22C2×C32×Q8—21no—
23C2×C32×Q8C2×D312yes—
24C2×C3×D12—32no—
25C2×C3×D12—61no—
26#3748—22no—
27C3×(C2×D3)⋊C4—64no—
28C3×(C2×D3)⋊C4—62no—
29C3×(C2×D3)⋊C4—62no—
30C3×C4×Dic3—32no—
31C3×C4×Dic3—61no—
32C3×#208—62no—
33C22×H(3,𝔽3)C22⋊C411yescharacteristic
34C2×#272—42no—
35C2×#272—41no—
36C4×H(3,𝔽3)—21no—
37C4×H(3,𝔽3)—22no—
38#1084—41no—
39C3×#124—91no—
40#947—32no—
41#947—61no—
42C3×#101—92no—
43#836—32no—
44#836—92no—
45C22×C3×D3—62no—
46C22×C3×D3—62no—
47C32×Q8—42no—
48C32×Q8—42no—
49C2×C32×C4—21no—
50C2×C32×C4—22no—
51C2×C32×C4C4×D312yes—
52C2×C32×C4D1212yes—
53C2×C32×C4#6712yes—
54C2×C3×Dic3—62no—
55C2×C3×Dic3—62no—
56C3×D12—122no—
57C3×D12—122no—
58C3⋊(C3⋊C8)—42no—
59C3×C3⋊C8—122no—
60#355—271noSylow
61C2×H(3,𝔽3)—21no—
62C2×H(3,𝔽3)#9911yescharacteristic
63#272—81no—
64C2×C3×Q8—32no—
65C2×C3×Q8—61no—
66C2×C3×Q8D3211yescharacteristic
67C2×C3×D4—91no—
68C2×D12—32no—
69C2×D12—61no—
70C3×M16—182no—
71C3×C22⋊C4—182no—
72C3×C42—91no—
73(C2×D3)⋊C4—62no—
74(C2×D3)⋊C4—62no—
75(C2×D3)⋊C4—64no—
76C4×Dic3—32no—
77C4×Dic3—61no—
78#208—62no—
79#208—182no—
80C22×C32—21no—
81C22×C32(C2×D3)⋊C412yes—
82C2×C3×D3—122no—
83C2×C3×D3—124no—
84C2×C3×D3—124no—
85C2×C3×D3—122no—
86C32×C4—22no—
87C32×C4—22no—
88C32×C4—22no—
89C32×C4—22no—
90C32×C4—42no—
91C3×Dic3—122no—
92C3×Dic3—122no—
93#124—91no—
94#101—92no—
95H(3,𝔽3)#35511yescharacteristic, complemented, Sylow
96C23×C3—181no—
97C22×D3—62no—
98C22×D3—62no—
99C3×Q8—41no—
100C3×Q8—122no—
101C3×Q8—122no—
102C3×D4—361no—
103C2×C3×C4—32no—
104C2×C3×C4—32no—
105C2×C3×C4—32no—
106C2×C3×C4—61no—
107C2×C3×C4—62no—
108C2×C3×C4—181no—
109C2×C3×C4#62411yescharacteristic
110C2×C3×C4#62612yes—
111C2×Dic3—62no—
112C2×Dic3—62no—
113D12—122no—
114D12—122no—
115C3×C8—362no—
116C3⋊C8—122no—
117C3⋊C8—362no—
118C2×C32—22no—
119C2×C32—22no—
120C2×C32—21no—
121C2×C32#80612yes—
122C3×D3—242no—
123C3×D3—242no—
124C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
125C2×D4—91no—
126M16—182no—
127C22⋊C4—182no—
128C42—91no—
129C22×C3—32no—
130C22×C3—61no—
131C22×C3—362no—
132C22×C3—361no—
133C22×C3#372211yescharacteristic
134C2×D3—122no—
135C2×D3—122no—
136C2×D3—124no—
137C2×D3—124no—
138C3×C4—22no—
139C3×C4—21no—
140C3×C4—62no—
141C3×C4—62no—
142C3×C4—62no—
143C3×C4—62no—
144C3×C4—122no—
145C3×C4—361no—
146Dic3—122no—
147Dic3—122no—
148C32—21no—
149C32#447712yescomplemented
150C23—181no—
151Q8—41no—
152D4—361no—
153C2×C4—181no—
154C2×C4#619911yescharacteristic
155C2×C4#620012yes—
156C8—362no—
157C2×C3—21no—
158C2×C3—32no—
159C2×C3—62no—
160C2×C3—62no—
161C2×C3—61no—
162C2×C3—721no—
163C2×C3#6358111yescharacteristic
164D3—242no—
165D3—242no—
166C22—362no—
167C22—361no—
168C22#8845111yescharacteristic
169C4—22no—
170C4—21no—
171C4—361no—
172C3—32no—
173C3—61no—
174C3#1058918311yescharacteristic
175C2—21no—
176C2—721no—
177C2#1172033511yescharacteristic
subgroupG/H
centerC2#11720335
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC2×C3×Q8D32
Fitting, cosocleC2×Q8×H(3,𝔽3)C22
socleC2×C3#63581
prime radicalC2×C4⋊#272C4
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#355
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #63663⋊C3C2semidirect
11. #63663D3semidirect
95. H(3,𝔽3)#355semidirect
149. C32#4477semidirect

Maximal subgroups

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

subgroupindex
#117218642
#63663⋊C32
#105891823