Finite Group Databasebeta

← #101×#272 #13143890

Subgroups — #101×#272 #13143890

Order 1728.

803 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 803 conjugacy classes of subgroups — too many to draw. Only its 113 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×Q8×#272C211yesmaximal, characteristic
2M16×#272C212yesmaximal
3H(3,𝔽3)×#101C211yesmaximal, characteristic, complemented
4#11720584C211yesmaximal, characteristic, complemented
5#11720561C212yesmaximal, complemented
6C3×D3×#101—34nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8Q8×#272—22no—
9Q8×#272—22no—
10C2×C4×#272C412yes—
11C2×C4×#272C2211yescharacteristic
12C2×#6229C412yes—
13C2×#6229C2211yescharacteristic
14M16×H(3,𝔽3)C2212yes—
15#88538C2212yes—
16#88527—22no—
17C8×#272—22no—
18C2×C3×D3×Q8—34no—
19C3×D3×M16—38no—
20C32×#101D314yescomplemented
21C3×#836—34no—
22C3×#824—38no—
23C22×#272C2×C411yescharacteristic
24Q8×H(3,𝔽3)—22no—
25C2×C4×H(3,𝔽3)C2×C412yes—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×#1084C2×C412yes—
28C2×#1084C2×C411yescharacteristic
29C4×#272—22no—
30C4×#272—22no—
31C4×#272D412yes—
32C4×#272D412yes—
33#6229—22no—
34#6229—24no—
35C8×H(3,𝔽3)—22no—
36#6180—22no—
37C2×C3×#101—91no—
38D3×#101—34no—
39C2×C32×Q8C2×D314yes—
40C3×D3×Q8—68no—
41C3×D3×Q8—68no—
42C2×C3×C4×D3—34no—
43C2×C3×C4×D3—38no—
44C2×C3×Dic6—38no—
45C2×C3×Dic6—34no—
46C32×M16C2×D318yes—
47C3×#208—38no—
48C3×ΓL(1,𝔽25)—68no—
49C3×D3×C8—68no—
50C22×H(3,𝔽3)C22×C411yescharacteristic
51C2×#272—22no—
52C2×#272C22⋊C411yescharacteristic
53C2×#272C22⋊C411yescharacteristic
54C4×H(3,𝔽3)—22no—
55C4×H(3,𝔽3)C2×D412yes—
56#1084—22no—
57#1084C22⋊C412yes—
58C22×C3×Q8—91no—
59C2×D3×Q8—34no—
60C2×C3×M16—92no—
61D3×M16—38no—
62C3×#101—34no—
63C3×#101—91no—
64C3×#101—92no—
65C3×#101C3⋊D311yescharacteristic
66#836—34no—
67#824—38no—
68C22×C3×D3—34no—
69C32×Q8—28no—
70C2×C32×C4C4×D318yes—
71C2×C32×C4C22×D314yes—
72C2×C3×Dic3—34no—
73C2×C3×Dic3—38no—
74C3×C4×D3—38no—
75C3×C4×D3—38no—
76C3×C4×D3—68no—
77C3×C4×D3—68no—
78C3×Dic6—616no—
79C3×Dic6—68no—
80C32×C8—28no—
81C3×C3⋊C8—68no—
82C2×#101—91noSylow
83C2×H(3,𝔽3)—21no—
84C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
85#272—21no—
86#272#10112yesdirect product factor
87C2×C3×Q8—34no—
88C2×C3×Q8—92no—
89C2×C3×Q8—91no—
90C2×C3×Q8—182no—
91C2×C3×Q8—182no—
92C2×C3×Q8C2×C3⋊D311yescharacteristic
93C22×C3×C4—91no—
94C22×C3×C4—92no—
95D3×Q8—68no—
96D3×Q8—68no—
97C2×C4×D3—34no—
98C2×C4×D3—38no—
99C2×Dic6—34no—
100C2×Dic6—38no—
101C3×M16—38no—
102C3×M16—92no—
103C3×M16—182no—
104C3×M16C2×C3⋊D312yes—
105C2×C3×C8—182no—
106#208—38no—
107ΓL(1,𝔽25)—68no—
108D3×C8—68no—
109C22×C32C2×C4×D314yes—
110C2×C3×D3—34no—
111C2×C3×D3—34no—
112C2×C3×D3—68no—
113C32×C4—28no—
114C32×C4D3×D418yes—
115C3×Dic3—38no—
116C3×Dic3—68no—
117C22×Q8—91no—
118C2×M16—92no—
119#101—91no—
120#101—92no—
121#101#27211yescharacteristic, direct product factor
122H(3,𝔽3)C2×#10111yescharacteristic, complemented, Sylow
123C23×C3—91no—
124C22×D3—34no—
125C3×Q8—22no—
126C3×Q8—68no—
127C3×Q8—184no—
128C3×Q8—182no—
129C2×C3×C4—38no—
130C2×C3×C4—34no—
131C2×C3×C4—91no—
132C2×C3×C4—92no—
133C2×C3×C4—92no—
134C2×C3×C4—92no—
135C2×C3×C4—182no—
136C2×C3×C4—182no—
137C2×C3×C4C4×C3⋊D312yes—
138C2×C3×C4C22×C3⋊D311yescharacteristic
139C2×Dic3—34no—
140C2×Dic3—38no—
141C4×D3—38no—
142C4×D3—38no—
143C4×D3—68no—
144C4×D3—68no—
145Dic6—616no—
146Dic6—68no—
147C3×C8—22no—
148C3×C8—68no—
149C3×C8—182no—
150C3⋊C8—68no—
151C2×C32—24no—
152C2×C32D3×C22⋊C414yes—
153C3×D3—38no—
154C3×D3—64no—
155C2×Q8—92no—
156C2×Q8—91no—
157C2×Q8—182no—
158C2×Q8—182no—
159C2×Q8C2×#27211yescharacteristic
160C22×C4—91no—
161C22×C4—92no—
162M16—92no—
163M16—182no—
164M16C2×#27212yes—
165C2×C8—182no—
166C22×C3—34no—
167C22×C3—91no—
168C22×C3—91no—
169C22×C3—182no—
170C22×C3C2×C4×C3⋊D311yescharacteristic
171C2×D3—34no—
172C2×D3—34no—
173C2×D3—68no—
174C3×C4—22no—
175C3×C4—38no—
176C3×C4—68no—
177C3×C4—92no—
178C3×C4—182no—
179C3×C4D4×C3⋊D312yes—
180Dic3—38no—
181Dic3—68no—
182C32D3×#10114yescomplemented
183C23—91no—
184Q8—22no—
185Q8—184no—
186Q8—182no—
187C2×C4—91no—
188C2×C4—92no—
189C2×C4—92no—
190C2×C4—92no—
191C2×C4—182no—
192C2×C4—182no—
193C2×C4C4×#27212yes—
194C2×C4C22×#27211yescharacteristic
195C8—22no—
196C8—182no—
197C2×C3—21no—
198C2×C3—34no—
199C2×C3—64no—
200C2×C3—92no—
201C2×C3—181no—
202C2×C3C22⋊C4×C3⋊D311yescharacteristic
203D3—38no—
204D3—64no—
205C22—91no—
206C22—91no—
207C22—182no—
208C22C2×C4×#27211yescharacteristic
209C4—22no—
210C4—92no—
211C4—182no—
212C4D4×#27212yes—
213C3—34no—
214C3C3⋊D3×#10111yescharacteristic
215C2—21no—
216C2—92no—
217C2—181no—
218C2C22⋊C4×#27211yescharacteristic
subgroupG/H
center, socleC2×C3C22⋊C4×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#101C2
prime radicalC22×#272C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#101C3⋊D3
hypocenterH(3,𝔽3)C2×#101
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#101

Complements

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

NH ≅ G/Ntype
3. H(3,𝔽3)×#101C2semidirect
4. #11720584C2semidirect
5. #11720561C2semidirect
20. C32×#101D3semidirect
86. #272#101direct (N × H)
121. #101#272direct (N × H)
122. H(3,𝔽3)C2×#101semidirect
182. C32D3×#101semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#2722
M16×#2722
H(3,𝔽3)×#1012
#117205842
#117205612
C3×D3×#1013