Finite Group Databasebeta

← #13147397

Subgroups — #13147397

Order 1728.

799 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 799 conjugacy classes of subgroups — too many to draw. Only its 71 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×C4×#272C211yesmaximal, characteristic
2C2×C22⋊#1088C212yesmaximal
3#11720787C214yesmaximal, complemented
4C23×#272C2211yescharacteristic
5C22×#1088C2214yes—
6C22×C4×H(3,𝔽3)C411yescharacteristic, complemented
7C22×#1084C411yescharacteristic, complemented
8C2×C4×#272—24no—
9C2×C4×#272C412yescomplemented
10C2×C4×#272C2212yes—
11C22⋊#1088—24no—
12C22×C3×C4×D3—62no—
13C23×H(3,𝔽3)C2×C411yescharacteristic
14C22×#272—24no—
15C22×#272C2×C411yescharacteristic
16C22×#272D414yes—
17C22×#272C2311yescharacteristic
18C2×#1088—24no—
19C2×#1088—28no—
20C2×C4×H(3,𝔽3)—22no—
21C2×C4×H(3,𝔽3)C2×C412yes—
22C2×#1084—22no—
23C2×#1084C2×C412yes—
24C4×#272—24no—
25C4×#272—24no—
26C3×#381—91nomaximal
27C23×C3×D3—62no—
28C22×C32×C4—22no—
29C22×C3×Dic3—62no—
30C2×C3×C4×D3—64no—
31C2×C3×C4×D3—64no—
32C2×C3×C4×D3—616no—
33C22×H(3,𝔽3)—22no—
34C22×H(3,𝔽3)C22⋊C412yes—
35C22×H(3,𝔽3)C22×C411yescharacteristic
36C2×#272—22no—
37C2×#272—28no—
38C2×#272C22⋊C412yes—
39C2×#272C2×D412yes—
40C2×#272#4114yes—
41#1088—44no—
42C4×H(3,𝔽3)—22no—
43#1084—22no—
44C23×C3×C4—91no—
45C22×C4×D3—62no—
46C2×C3×C22⋊C4—92no—
47C3×#95—94no—
48C23×C32—22no—
49C22×C3×D3—62no—
50C22×C3×D3—68no—
51C22×C3×D3—62no—
52C22×C3×D3—616no—
53C2×C32×C4—28no—
54C2×C32×C4—24no—
55C2×C3×Dic3—64no—
56C2×C3×Dic3—68no—
57C3×C4×D3—616no—
58C3×C4×D3—616no—
59#381—91noSylow
60C2×H(3,𝔽3)—22no—
61C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
62C2×H(3,𝔽3)#11712yes—
63#272—22no—
64#272#12314yes—
65C24×C3—91no—
66C23×D3—62no—
67C22×C3×C4—62no—
68C22×C3×C4—91no—
69C22×C3×C4—92no—
70C22×C3×C4—92no—
71C22×C3×C4—94no—
72C22×C3×C4—184no—
73C22×C3×C4#15711yescharacteristic
74C22×Dic3—62no—
75C2×C4×D3—64no—
76C2×C4×D3—64no—
77C2×C4×D3—616no—
78C3×C22⋊C4—184no—
79C22×C32—28no—
80C22×C32—22no—
81C22×C32—24no—
82C2×C3×D3—64no—
83C2×C3×D3—64no—
84C2×C3×D3—68no—
85C2×C3×D3—632no—
86C2×C3×D3—68no—
87C32×C4—28no—
88C3×Dic3—68no—
89C23×C4—91no—
90C2×C22⋊C4—92no—
91#95—94no—
92H(3,𝔽3)#38111yescharacteristic, complemented, Sylow
93C23×C3—62no—
94C23×C3—91no—
95C23×C3—91no—
96C23×C3—94no—
97C23×C3—184no—
98C23×C3C2×#15711yescharacteristic
99C22×D3—62no—
100C22×D3—68no—
101C22×D3—62no—
102C22×D3—616no—
103C2×C3×C4—22no—
104C2×C3×C4—68no—
105C2×C3×C4—64no—
106C2×C3×C4—92no—
107C2×C3×C4—184no—
108C2×C3×C4—184no—
109C2×C3×C4—182no—
110C2×C3×C4—184no—
111C2×C3×C4—188no—
112C2×C3×C4C2×#15712yes—
113C2×Dic3—64no—
114C2×Dic3—68no—
115C4×D3—616no—
116C4×D3—616no—
117C2×C32—28no—
118C2×C32—24no—
119C2×C32—22no—
120C3×D3—68no—
121C3×D3—68no—
122C24—91no—
123C22×C4—91no—
124C22×C4—92no—
125C22×C4—92no—
126C22×C4—94no—
127C22×C4—184no—
128C22×C4#108811yescharacteristic, complemented
129C22⋊C4—184no—
130C22×C3—22no—
131C22×C3—68no—
132C22×C3—64no—
133C22×C3—62no—
134C22×C3—94no—
135C22×C3—92no—
136C22×C3—92no—
137C22×C3—188no—
138C22×C3—182no—
139C22×C3C22⋊#15712yes—
140C22×C3C22×#15711yescharacteristic
141C2×D3—64no—
142C2×D3—64no—
143C2×D3—68no—
144C2×D3—632no—
145C2×D3—68no—
146C3×C4—22no—
147C3×C4—68no—
148C3×C4—182no—
149C3×C4—364no—
150Dic3—68no—
151C32—22no—
152C23—91no—
153C23—94no—
154C23—91no—
155C23—184no—
156C23C2×#108811yescharacteristic
157C2×C4—22no—
158C2×C4—92no—
159C2×C4—184no—
160C2×C4—184no—
161C2×C4—182no—
162C2×C4—184no—
163C2×C4—188no—
164C2×C4C2×#108812yes—
165C2×C3—22no—
166C2×C3—68no—
167C2×C3—64no—
168C2×C3—62no—
169C2×C3—94no—
170C2×C3—182no—
171C2×C3#6428612yes—
172C2×C3C2×C22⋊#15711yescharacteristic
173D3—68no—
174D3—68no—
175C22—22no—
176C22—94no—
177C22—92no—
178C22—92no—
179C22—188no—
180C22—182no—
181C22C22⋊#108812yes—
182C22C22×#108811yescharacteristic
183C4—22no—
184C4—182no—
185C4—364no—
186C3—62no—
187C3#1059381111yescharacteristic
188C2—22no—
189C2—94no—
190C2—182no—
191C2#1172244812yes—
192C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC22×C4×H(3,𝔽3)C4
prime radicalC23×#272C22
cosocleC22×#272C23
hypercenterC22×C3×C4#157
hypocenterH(3,𝔽3)#381
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. #11720787C2semidirect
6. C22×C4×H(3,𝔽3)C4semidirect
7. C22×#1084C4semidirect
9. C2×C4×#272C4semidirect
92. H(3,𝔽3)#381semidirect
128. C22×C4#1088semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C2×C22⋊#10882
#117207872
C3×#3819