Finite Group Databasebeta

← #13143879

Subgroups — #13143879

Order 1728.

825 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 825 conjugacy classes of subgroups — too many to draw. Only its 111 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×#88744C211yesmaximal, characteristic
2M16×#272C212yesmaximal, complemented
3H(3,𝔽3)×#100C211yesmaximal, characteristic, complemented
4#11720579C211yesmaximal, characteristic, complemented
5#11720561C212yesmaximal, complemented
6C3×#4745—34nomaximal
7C2×D4×H(3,𝔽3)C2211yescharacteristic
8C2×#6235C412yes—
9C22×#1084C412yes—
10#88744—22no—
11#88744—22no—
12C2×C4×#272C2211yescharacteristic
13C2×#6229C2211yescharacteristic
14M16×H(3,𝔽3)C2212yes—
15#88538C2212yes—
16#88527—22no—
17C8×#272—22no—
18C2×C3×#237—34no—
19C3×D3×M16—38no—
20C32×#100D314yescomplemented
21C3×#833—34no—
22C3×#824—38no—
23C23×H(3,𝔽3)C2×C412yes—
24C22×#272C2×C411yescharacteristic
25D4×H(3,𝔽3)—22no—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27#6235—24no—
28C2×#1084—22no—
29C2×#1084—22no—
30C2×#1084C2×C412yes—
31C2×#1084C2×C411yescharacteristic
32C4×#272D412yes—
33C4×#272D412yes—
34#6229—22no—
35C8×H(3,𝔽3)—22no—
36#6180—22no—
37C3×#413—91no—
38#4745—34no—
39C2×C32×D4C2×D314yes—
40C2×C3×#67—38no—
41C22×C3×Dic3—38no—
42C3×#237—68no—
43C3×#237—68no—
44C2×C3×C4×D3—34no—
45C2×C3×Dic6—34no—
46C32×M16C2×D318yes—
47C3×#208—38no—
48C3×ΓL(1,𝔽25)—68no—
49C3×D3×C8—68no—
50C22×H(3,𝔽3)—22no—
51C22×H(3,𝔽3)—42no—
52C22×H(3,𝔽3)C22×C411yescharacteristic
53C2×#272—41no—
54C2×#272C22⋊C412yes—
55C4×H(3,𝔽3)C2×D412yes—
56#1084—22no—
57#1084C22⋊C411yescharacteristic
58#1084C22⋊C411yescharacteristic
59C2×C3×#41—91no—
60C2×#237—34no—
61C2×C3×M16—92no—
62D3×M16—38no—
63C3×#101—92no—
64C3×#100—34no—
65C3×#100—91no—
66C3×#100C3⋊D311yescharacteristic
67#833—34no—
68#824—38no—
69C23×C32C4×D318yes—
70C22×C3×D3—34no—
71C32×D4—28no—
72C2×C32×C4C22×D314yes—
73C3×#67—616no—
74C2×C3×Dic3—34no—
75C2×C3×Dic3—38no—
76C2×C3×Dic3—68no—
77C2×C3×Dic3—68no—
78C3×C4×D3—38no—
79C3×C4×D3—38no—
80C3×Dic6—68no—
81C32×C8—28no—
82C3×C3⋊C8—68no—
83#413—91noSylow
84C2×H(3,𝔽3)—21no—
85C2×H(3,𝔽3)—42no—
86C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
87#272—22no—
88C3×#41—182no—
89C3×#41—182no—
90C2×C3×Q8—91no—
91C2×C3×D4—34no—
92C2×C3×D4—92no—
93C2×C3×D4C2×C3⋊D311yescharacteristic
94C22×C3×C4—91no—
95C22×C3×C4—92no—
96C2×#67—38no—
97C22×Dic3—38no—
98#237—68no—
99#237—68no—
100C2×C4×D3—34no—
101C2×Dic6—34no—
102C3×M16—38no—
103C3×M16—92no—
104C3×M16—182no—
105C3×M16C2×C3⋊D312yes—
106C2×C3×C8—182no—
107#208—38no—
108ΓL(1,𝔽25)—68no—
109D3×C8—68no—
110C22×C32—28no—
111C22×C32—48no—
112C22×C32C2×C4×D314yes—
113C2×C3×D3—38no—
114C2×C3×D3—124no—
115C32×C4D3×D418yes—
116C3×Dic3—34no—
117C3×Dic3—34no—
118C3×Dic3—68no—
119C2×#41—91no—
120C2×M16—92no—
121#101—92no—
122#100—91no—
123#100#27211yescharacteristic, complemented
124H(3,𝔽3)#41311yescharacteristic, complemented, Sylow
125C23×C3—38no—
126C23×C3—91no—
127C23×C3C4×C3⋊D312yes—
128C22×D3—34no—
129C3×Q8—182no—
130C3×D4—22no—
131C3×D4—68no—
132C3×D4—184no—
133C2×C3×C4—34no—
134C2×C3×C4—91no—
135C2×C3×C4—92no—
136C2×C3×C4—92no—
137C2×C3×C4—92no—
138C2×C3×C4—182no—
139C2×C3×C4—182no—
140C2×C3×C4C22×C3⋊D311yescharacteristic
141#67—616no—
142C2×Dic3—34no—
143C2×Dic3—38no—
144C2×Dic3—68no—
145C2×Dic3—68no—
146C4×D3—38no—
147C4×D3—38no—
148Dic6—68no—
149C3×C8—22no—
150C3×C8—68no—
151C3×C8—182no—
152C3⋊C8—68no—
153C2×C32—24no—
154C2×C32—48no—
155C2×C32D3×C22⋊C414yes—
156C3×D3—68no—
157#41—182no—
158#41—182no—
159C2×Q8—91no—
160C2×D4—92no—
161C2×D4C2×#27211yescharacteristic
162C22×C4—91no—
163C22×C4—92no—
164M16—92no—
165M16—182no—
166M16C2×#27212yes—
167C2×C8—182no—
168C22×C3—22no—
169C22×C3—34no—
170C22×C3—42no—
171C22×C3—68no—
172C22×C3—92no—
173C22×C3—128no—
174C22×C3—361no—
175C22×C3C2×C4×C3⋊D311yescharacteristic
176C2×D3—38no—
177C2×D3—124no—
178C3×C4—38no—
179C3×C4—91no—
180C3×C4—91no—
181C3×C4—182no—
182C3×C4D4×C3⋊D312yes—
183Dic3—34no—
184Dic3—34no—
185Dic3—68no—
186C32#474514yescomplemented
187C23—91no—
188C23C4×#27212yes—
189Q8—182no—
190D4—22no—
191D4—184no—
192C2×C4—91no—
193C2×C4—92no—
194C2×C4—92no—
195C2×C4—92no—
196C2×C4—182no—
197C2×C4—182no—
198C2×C4C22×#27211yescharacteristic
199C8—22no—
200C8—182no—
201C2×C3—21no—
202C2×C3—34no—
203C2×C3—42no—
204C2×C3—64no—
205C2×C3—128no—
206C2×C3—182no—
207C2×C3C22⋊C4×C3⋊D311yescharacteristic
208D3—68no—
209C22—22no—
210C22—42no—
211C22—92no—
212C22—361no—
213C22C2×C4×#27211yescharacteristic
214C4—91no—
215C4—91no—
216C4—182no—
217C4D4×#27212yes—
218C3—34no—
219C3#1059244411yescharacteristic
220C2—21no—
221C2—42no—
222C2—182no—
223C2C22⋊C4×#27211yescharacteristic
subgroupG/H
center, socleC2×C3C22⋊C4×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#100C2
prime radicalC2×#88744C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#100C3⋊D3
hypocenterH(3,𝔽3)#413
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#100

Complements

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

NH ≅ G/Ntype
2. M16×#272C2semidirect
3. H(3,𝔽3)×#100C2semidirect
4. #11720579C2semidirect
5. #11720561C2semidirect
20. C32×#100D3semidirect
123. #100#272semidirect
124. H(3,𝔽3)#413semidirect
186. C32#4745semidirect

Maximal subgroups

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

subgroupindex
C2×#887442
M16×#2722
H(3,𝔽3)×#1002
#117205792
#117205612
C3×#47453