Finite Group Databasebeta

← #13136898

Subgroups — #13136898

Order 1728.

355 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 55 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×C4⋊#1088C211yesmaximal, characteristic
2C2×#88626C211yesmaximal, characteristic
3C4⋊C4×#272C211yesmaximal, characteristic
4C22×#1088—21no—
5C2×C4×#272—21no—
6C2×C4×#272C2211yescharacteristic
7C2×#6188—21no—
8C4⋊#1088—21no—
9C4⋊#1088C412yescomplemented
10#88626—21no—
11#88626C412yescomplemented
12C4⋊C4×H(3,𝔽3)C411yescharacteristic, complemented
13C4⋊#1084C411yescharacteristic, complemented
14#88540—21no—
15C3×D3×C4⋊C4—62no—
16C22×#272D411yescharacteristic
17C2×#1088—22no—
18C2×#1088—41no—
19C2×#1088—41no—
20C2×C4×H(3,𝔽3)—21no—
21C2×C4×H(3,𝔽3)C2×C411yescharacteristic
22C2×#1084—21no—
23C2×#1084D411yescharacteristic
24C4×#272—41no—
25C4×#272—41no—
26C4×#272C2×C411yescharacteristic
27C4×#272C2×C411yescharacteristic
28C4×#272D411yescharacteristic
29C4×#272Q811yescharacteristic
30#6188—22no—
31C3×#340—91nomaximal
32C2×C3×C4×D3—62no—
33C2×C3×C4×D3—64no—
34C32×C4⋊C4—22no—
35C3×C4⋊Dic3—62no—
36C3×Dic3⋊C4—64no—
37C22×H(3,𝔽3)C22⋊C411yescharacteristic
38C2×#272C22⋊C411yescharacteristic
39C2×#272C4⋊C411yescharacteristic
40C2×#272D811yescharacteristic
41C2×#272SD1611yescharacteristic
42C2×#272SD1611yescharacteristic
43C2×#272Q1611yescharacteristic
44#1088—42no—
45C4×H(3,𝔽3)—41no—
46C4×H(3,𝔽3)C4211yescharacteristic
47C4×H(3,𝔽3)C4⋊C411yescharacteristic
48#1084—41no—
49#1084C22⋊C411yescharacteristic
50#1084C4⋊C411yescharacteristic
51C22×C3×C8—91no—
52C2×C3×C4⋊C4—91no—
53C2×C3×C4⋊C4—91no—
54D3×C4⋊C4—62no—
55C22×C3×D3—62no—
56C2×C32×C4—24no—
57C2×C32×C4—22no—
58C2×C3×Dic3—62no—
59C2×C3×Dic3—64no—
60C3×C4×D3—62no—
61C3×C4×D3—62no—
62C3×C4×D3—62no—
63C3×C4×D3—62no—
64C3×C4×D3—124no—
65C3×C4×D3—124no—
66#340—91noSylow
67C2×H(3,𝔽3)#9511yescharacteristic
68C2×H(3,𝔽3)#10211yescharacteristic
69C2×H(3,𝔽3)#10311yescharacteristic
70#272#10211yescharacteristic
71#272#10311yescharacteristic
72#272#10611yescharacteristic
73#272#10711yescharacteristic
74C22×C3×C4—91no—
75C22×C3×C4—181no—
76C22×C3×C4—181no—
77C2×C4×D3—62no—
78C2×C4×D3—64no—
79C2×C3×C8—92no—
80C2×C3×C8—181no—
81C2×C3×C8—181no—
82C3×C4⋊C4—62no—
83C3×C4⋊C4—92no—
84C3×C4⋊C4—91no—
85C3×C4⋊C4—181no—
86C3×C4⋊C4—181no—
87C3×C4⋊C4#15711yescharacteristic
88C4⋊Dic3—62no—
89Dic3⋊C4—64no—
90C22×C32—22no—
91C2×C3×D3—62no—
92C2×C3×D3—62no—
93C2×C3×D3—62no—
94C2×C3×D3—62no—
95C2×C3×D3—62no—
96C2×C3×D3—62no—
97C32×C4—22no—
98C32×C4—22no—
99C32×C4—44no—
100C3×Dic3—62no—
101C3×Dic3—62no—
102C3×Dic3—124no—
103C22×C8—91no—
104C2×C4⋊C4—91no—
105C2×C4⋊C4—91no—
106H(3,𝔽3)#34011yescharacteristic, complemented, Sylow
107C23×C3—91no—
108C22×D3—62no—
109C2×C3×C4—21no—
110C2×C3×C4—64no—
111C2×C3×C4—62no—
112C2×C3×C4—91no—
113C2×C3×C4—91no—
114C2×C3×C4—91no—
115C2×C3×C4—91no—
116C2×C3×C4—91no—
117C2×C3×C4—182no—
118C2×C3×C4—181no—
119C2×C3×C4—361no—
120C2×C3×C4—361no—
121C2×C3×C4—361no—
122C2×C3×C4—361no—
123C2×C3×C4C2×#15711yescharacteristic
124C2×Dic3—62no—
125C2×Dic3—64no—
126C4×D3—62no—
127C4×D3—62no—
128C4×D3—62no—
129C4×D3—62no—
130C4×D3—124no—
131C4×D3—124no—
132C3×C8—182no—
133C2×C32—22no—
134C2×C32—22no—
135C2×C32—22no—
136C3×D3—62no—
137C3×D3—62no—
138C3×D3—62no—
139C3×D3—62no—
140C22×C4—91no—
141C22×C4—181no—
142C22×C4—181no—
143C2×C8—92no—
144C2×C8—181no—
145C2×C8—181no—
146C4⋊C4—92no—
147C4⋊C4—91no—
148C4⋊C4—181no—
149C4⋊C4—181no—
150C4⋊C4#108811yescharacteristic, complemented
151C22×C3—62no—
152C22×C3—91no—
153C22×C3—91no—
154C22×C3—91no—
155C22×C3—91no—
156C22×C3—91no—
157C22×C3—91no—
158C22×C3C22⋊#15711yescharacteristic
159C2×D3—62no—
160C2×D3—62no—
161C2×D3—62no—
162C2×D3—62no—
163C2×D3—62no—
164C2×D3—62no—
165C3×C4—41no—
166C3×C4—62no—
167C3×C4—62no—
168C3×C4—91no—
169C3×C4—91no—
170C3×C4—124no—
171C3×C4—362no—
172C3×C4—361no—
173C3×C4C4×#15711yescharacteristic
174C3×C4C4⋊#15711yescharacteristic
175Dic3—62no—
176Dic3—62no—
177Dic3—124no—
178C32—22no—
179C23—91no—
180C2×C4—21no—
181C2×C4—91no—
182C2×C4—91no—
183C2×C4—91no—
184C2×C4—91no—
185C2×C4—91no—
186C2×C4—182no—
187C2×C4—181no—
188C2×C4—361no—
189C2×C4—361no—
190C2×C4—361no—
191C2×C4—361no—
192C2×C4C2×#108811yescharacteristic
193C8—182no—
194C2×C3—62no—
195C2×C3—62no—
196C2×C3—62no—
197C2×C3—91no—
198C2×C3—91no—
199C2×C3—91no—
200C2×C3—91no—
201C2×C3#6378111yescharacteristic
202C2×C3#6378211yescharacteristic
203C2×C3#6378411yescharacteristic
204D3—62no—
205D3—62no—
206D3—62no—
207D3—62no—
208C22—91no—
209C22—91no—
210C22—91no—
211C22—91no—
212C22—91no—
213C22—91no—
214C22C22⋊#108811yescharacteristic
215C4—41no—
216C4—91no—
217C4—91no—
218C4—362no—
219C4—361no—
220C4C4×#108811yescharacteristic
221C4C4⋊#108811yescharacteristic
222C3—62no—
223C3#1059016311yescharacteristic
224C2—91no—
225C2—91no—
226C2—91no—
227C2—91no—
228C2#1172078711yescharacteristic
229C2#1172078811yescharacteristic
230C2#1172079011yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#157
commutatorC4×H(3,𝔽3)C42
FrattiniC2×C3×C4C2×#157
FittingC4⋊C4×H(3,𝔽3)C4
prime radicalC22×#272D4
cosocleC2×C4×#272C22
hypercenterC3×C4⋊C4#157
hypocenterH(3,𝔽3)#340
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
9. C4⋊#1088C4semidirect
11. #88626C4semidirect
12. C4⋊C4×H(3,𝔽3)C4semidirect
13. C4⋊#1084C4semidirect
106. H(3,𝔽3)#340semidirect
150. C4⋊C4#1088semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10882
C2×#886262
C4⋊C4×#2722
C3×#3409