Finite Group Databasebeta

← #13147316

Subgroups — #13147316

Order 1728.

381 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C42×#272C211yesmaximal, characteristic
2#11720784C212yesmaximal
3#11720783C212yesmaximal, complemented
4C4⋊#6188C212yesmaximal, complemented
5C2×C4×#272C411yescharacteristic
6C2×C4×#272C411yescharacteristic
7C2×C4×#272C2211yescharacteristic
8C2×#6188C2214yes—
9C42×H(3,𝔽3)C411yescharacteristic
10C4×#1084C411yescharacteristic
11C4×#1084C2211yescharacteristic
12C4×#1084C2211yescharacteristic
13C3×C42×D3—62no—
14C22×#272C2×C411yescharacteristic
15C2×C4×H(3,𝔽3)C2×C411yescharacteristic
16C2×C4×H(3,𝔽3)C2×C411yescharacteristic
17C2×C4×H(3,𝔽3)C2×C411yescharacteristic
18C2×#1084C2×C411yescharacteristic
19C2×#1084C2×C411yescharacteristic
20C2×#1084C2311yescharacteristic
21C4×#272—22no—
22C4×#272—22no—
23C4×#272—22no—
24#6188—24no—
25C3×#432—91nomaximal
26C2×C3×C4×D3—62no—
27C2×C3×C4×D3—62no—
28C2×C3×C4×D3—62no—
29C32×C42—22no—
30C3×C4×Dic3—62no—
31C3×C4×Dic3—62no—
32C3×C4×Dic3—62no—
33C22×H(3,𝔽3)C22×C411yescharacteristic
34C2×#272—21no—
35C2×#272—21no—
36C2×#272M1612yes—
37C4×H(3,𝔽3)—21no—
38C4×H(3,𝔽3)—21no—
39C4×H(3,𝔽3)M1612yes—
40#1084—21no—
41#1084—21no—
42#1084#4112yes—
43C2×C3×C42—91no—
44C42×D3—62no—
45C3×C4⋊C8—92no—
46C3×C22⋊C8—92no—
47C3×#97—92no—
48C22×C3×D3—62no—
49C2×C32×C4—22no—
50C2×C32×C4—22no—
51C2×C32×C4—22no—
52C2×C3×Dic3—62no—
53C2×C3×Dic3—62no—
54C2×C3×Dic3—62no—
55C3×C4×D3—68no—
56C3×C4×D3—68no—
57C3×C4×D3—68no—
58#432—91noSylow
59C2×H(3,𝔽3)#11711yescharacteristic
60C2×H(3,𝔽3)C2×M1611yescharacteristic
61C2×H(3,𝔽3)C2×M1611yescharacteristic
62#272—22no—
63C22×C3×C4—91no—
64C22×C3×C4—91no—
65C22×C3×C4—91no—
66C2×C4×D3—62no—
67C2×C4×D3—62no—
68C2×C4×D3—62no—
69C2×C3×C8—94no—
70C3×C42—62no—
71C3×C42—91no—
72C3×C42—91no—
73C3×C42—91no—
74C3×C42#15711yescharacteristic
75C4×Dic3—62no—
76C4×Dic3—62no—
77C4×Dic3—62no—
78C22×C32—22no—
79C2×C3×D3—64no—
80C2×C3×D3—64no—
81C2×C3×D3—64no—
82C32×C4—24no—
83C32×C4—24no—
84C32×C4—24no—
85C3×Dic3—64no—
86C3×Dic3—64no—
87C3×Dic3—64no—
88C2×C42—91no—
89C4⋊C8—92no—
90C22⋊C8—92no—
91#97—92no—
92H(3,𝔽3)#43211yescharacteristic, complemented, Sylow
93C23×C3—91no—
94C22×D3—62no—
95C2×C3×C4—62no—
96C2×C3×C4—62no—
97C2×C3×C4—62no—
98C2×C3×C4—91no—
99C2×C3×C4—91no—
100C2×C3×C4—91no—
101C2×C3×C4—182no—
102C2×C3×C4—182no—
103C2×C3×C4—182no—
104C2×C3×C4C2×#15711yescharacteristic
105C2×C3×C4C2×#15711yescharacteristic
106C2×C3×C4C2×#15711yescharacteristic
107C2×Dic3—62no—
108C2×Dic3—62no—
109C2×Dic3—62no—
110C4×D3—68no—
111C4×D3—68no—
112C4×D3—68no—
113C3×C8—184no—
114C2×C32—22no—
115C2×C32—22no—
116C2×C32—22no—
117C3×D3—68no—
118C22×C4—91no—
119C22×C4—91no—
120C22×C4—91no—
121C2×C8—94no—
122C42—91no—
123C42—91no—
124C42—91no—
125C42#108811yescharacteristic
126C22×C3—62no—
127C22×C3—92no—
128C22×C3—181no—
129C22×C3—181no—
130C22×C3C22×#15711yescharacteristic
131C2×D3—64no—
132C2×D3—64no—
133C2×D3—64no—
134C3×C4—21no—
135C3×C4—21no—
136C3×C4—64no—
137C3×C4—64no—
138C3×C4—64no—
139C3×C4—92no—
140C3×C4—181no—
141C3×C4—181no—
142C3×C4#378812yes—
143Dic3—64no—
144Dic3—64no—
145Dic3—64no—
146C32—22no—
147C23—91no—
148C2×C4—91no—
149C2×C4—91no—
150C2×C4—91no—
151C2×C4—182no—
152C2×C4—182no—
153C2×C4—182no—
154C2×C4C2×#108811yescharacteristic
155C2×C4C2×#108811yescharacteristic
156C2×C4C2×#108811yescharacteristic
157C8—184no—
158C2×C3—62no—
159C2×C3—62no—
160C2×C3—62no—
161C2×C3—182no—
162C2×C3C2×#378811yescharacteristic
163C2×C3#6428311yescharacteristic
164C2×C3#6428611yescharacteristic
165D3—68no—
166C22—92no—
167C22—181no—
168C22—181no—
169C22C22×#108811yescharacteristic
170C4—21no—
171C4—21no—
172C4—92no—
173C4—181no—
174C4—181no—
175C4#8862712yes—
176C3—62no—
177C3#1059373011yescharacteristic
178C2—182no—
179C2C2×#8862711yescharacteristic
180C2#1172244511yescharacteristic
181C2#1172244811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC42×H(3,𝔽3)C4
prime radicalC22×#272C2×C4
cosocleC2×#1084C23
hypercenterC3×C42#157
hypocenterH(3,𝔽3)#432
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
3. #11720783C2semidirect
4. C4⋊#6188C2semidirect
92. H(3,𝔽3)#432semidirect

Maximal subgroups

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

subgroupindex
C42×#2722
#117207842
#117207832
C4⋊#61882
C3×#4329