Finite Group Databasebeta

← #13146694

Subgroups — #13146694

Order 1728.

389 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 45 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×#6188C211yesmaximal, characteristic, complemented
2C2×#88588C211yesmaximal, characteristic, complemented
3#11721318C212yesmaximal, complemented
4#11720705C212yesmaximal, complemented
5#11720704C211yesmaximal, characteristic, complemented
6C22×#1084C2211yescharacteristic
7C2×#638⋊C3—22no—
8C2×#6198C2212yes—
9C2×#6188—22no—
10C2×#6188C2211yescharacteristic
11C2×#6188C2211yescharacteristic
12#88588—22no—
13#88450—22no—
14#88449C2212yes—
15#63978—62no—
16C23×H(3,𝔽3)D411yescharacteristic
17C22×(C3⋊D3)⋊C3—22no—
18C2×#1084—21no—
19C2×#1084—21no—
20C2×#1084D411yescharacteristic
21C2×#1084C2311yescharacteristic
22#638⋊C3—44no—
23C2×(C3⋊Dic3)⋊C3—22no—
24#6198—22no—
25#6198—22no—
26#6188—21no—
27#6188D412yescomplemented
28#5115—91nomaximal
29C2×#638—22no—
30C2×C3×#67—62no—
31C22×C3×Dic3—62no—
32C2×#626—62no—
33Dic3⋊Dic3—62no—
34#3722—62no—
35(C2×D3)⋊Dic3—62no—
36C22×H(3,𝔽3)—21no—
37C22×H(3,𝔽3)—21no—
38C22×H(3,𝔽3)D812yescomplemented
39C22×H(3,𝔽3)C2×D411yescharacteristic
40C2×(C3⋊D3)⋊C3—42no—
41C2×(C3⋊D3)⋊C3—42no—
42C2×(C3⋊D3)⋊C3—42no—
43#1084—21no—
44#1084C2×D411yescharacteristic
45#1084#4111yescharacteristic
46(C3⋊Dic3)⋊C3—42no—
47C22×C3×C8—91no—
48#939—182no—
49#930—92no—
50C2×D24—91no—
51#821—92no—
52#818—91no—
53C23×C32—22no—
54C22×C3⋊D3—22no—
55C22×C3×D3—62no—
56#638—44no—
57C2×C3⋊Dic3—22no—
58C3×#67—124no—
59C2×C3×Dic3—62no—
60C2×C3×Dic3—62no—
61C2×C3×Dic3—62no—
62C2×C3×Dic3—122no—
63C2×C3×Dic3—122no—
64#626—122no—
65#626—122no—
66#466—271noSylow
67C2×H(3,𝔽3)—22no—
68C2×H(3,𝔽3)#12111yescharacteristic
69C2×H(3,𝔽3)C2×D811yescharacteristic
70C2×H(3,𝔽3)#13511yescharacteristic
71(C3⋊D3)⋊C3—82no—
72C2×C3×D4—182no—
73C22×C3×C4—91no—
74C2×#67—62no—
75C2×#67—182no—
76C2×#67—182no—
77C22×Dic3—62no—
78C2×D12—92no—
79C2×C3×C8—91no—
80C2×C3×C8—91no—
81C2×C3×C8—182no—
82#217—182no—
83(C2×D3)⋊C4—182no—
84(C2×D3)⋊C4—182no—
85C4⋊Dic3—92no—
86Dic3⋊C4—182no—
87D24—182no—
88C22×C32—22no—
89C22×C32—24no—
90C22×C32—42no—
91C22×C32—42no—
92C2×C3⋊D3—42no—
93C2×C3⋊D3—42no—
94C2×C3⋊D3—42no—
95C2×C3×D3—122no—
96C2×C3×D3—122no—
97C2×C3×D3—122no—
98C3⋊Dic3—42no—
99C3×Dic3—62no—
100C3×Dic3—62no—
101C3×Dic3—122no—
102C3×Dic3—122no—
103C2×D8—271no—
104C22×C8—91no—
105#121—272no—
106#107—271no—
107#102—272no—
108H(3,𝔽3)#46611yescharacteristic, complemented, Sylow
109C23×C3—62no—
110C23×C3—182no—
111C23×C3GO+(4,𝔽2)11yescharacteristic
112C22×D3—62no—
113C22×D3—182no—
114C3×D4—364no—
115C2×C3×C4—91no—
116C2×C3×C4—91no—
117C2×C3×C4—181no—
118C2×C3×C4—181no—
119C2×C3×C4—182no—
120#67—124no—
121#67—364no—
122#67—362no—
123#67—362no—
124C2×Dic3—62no—
125C2×Dic3—62no—
126C2×Dic3—62no—
127C2×Dic3—122no—
128C2×Dic3—122no—
129C2×Dic3—182no—
130D12—182no—
131D12—182no—
132C3×C8—92no—
133C3×C8—181no—
134C2×C32—22no—
135C2×C32—22no—
136C2×C32—22no—
137C2×C32—44no—
138C3⋊D3—82no—
139C3×D3—242no—
140C2×D4—182no—
141C2×D4—272no—
142C22×C4—91no—
143D8—542no—
144C2×C8—91no—
145C2×C8—91no—
146C2×C8—182no—
147C4⋊C4—272no—
148C22⋊C4—542no—
149C22×C3—21no—
150C22×C3—21no—
151C22×C3—64no—
152C22×C3—62no—
153C22×C3—122no—
154C22×C3—122no—
155C22×C3—362no—
156C22×C3—362no—
157C22×C3—362no—
158C22×C3#377412yes—
159C22×C3C2×GO+(4,𝔽2)11yescharacteristic
160C2×D3—122no—
161C2×D3—122no—
162C2×D3—122no—
163C2×D3—362no—
164C2×D3—362no—
165C2×D3—362no—
166C3×C4—91no—
167C3×C4—91no—
168C3×C4—181no—
169C3×C4—362no—
170Dic3—62no—
171Dic3—62no—
172Dic3—122no—
173Dic3—122no—
174Dic3—362no—
175C32—22no—
176C23—182no—
177C23#1088⋊C211yescharacteristic
178D4—364no—
179D4—542no—
180D4—542no—
181C2×C4—91no—
182C2×C4—91no—
183C2×C4—181no—
184C2×C4—181no—
185C2×C4—182no—
186C8—92no—
187C8—181no—
188C2×C3—22no—
189C2×C3—62no—
190C2×C3—62no—
191C2×C3—62no—
192C2×C3—124no—
193C2×C3—722no—
194C2×C3C2×#377411yescharacteristic
195C2×C3#6423711yescharacteristic
196C2×C3#6424211yescharacteristic
197D3—242no—
198D3—722no—
199C22—21no—
200C22—21no—
201C22—362no—
202C22—362no—
203C22—362no—
204C22#8858812yescomplemented
205C22C2×#1088⋊C211yescharacteristic
206C4—91no—
207C4—91no—
208C4—181no—
209C4—362no—
210C3—62no—
211C3#1059352011yescharacteristic
212C2—22no—
213C2—722no—
214C2C2×#8858811yescharacteristic
215C2#1172231011yescharacteristic
216C2#1172231611yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator, cosocleC2×#1084C23
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC23×H(3,𝔽3)D4
hypercenterC23#1088⋊C2
hypocenterH(3,𝔽3)#466
LCSG ▷ C2×#1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×#6188C2semidirect
2. C2×#88588C2semidirect
3. #11721318C2semidirect
4. #11720705C2semidirect
5. #11720704C2semidirect
27. #6188D4semidirect
38. C22×H(3,𝔽3)D8semidirect
108. H(3,𝔽3)#466semidirect
204. C22#88588semidirect

Maximal subgroups

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

subgroupindex
C22×#61882
C2×#885882
#117213182
#117207052
#117207042
#51159