Finite Group Databasebeta

← #13146617

Subgroups — #13146617

Order 1728.

409 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 51 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×#88590C211yesmaximal, characteristic
3#11721147C212yesmaximal
4#11720707C212yesmaximal
5#11720707C211yesmaximal, characteristic
6C22×#1088C2211yescharacteristic
7C22×#1088C2211yescharacteristic
8C2×C4×#272C2211yescharacteristic
9C2×#6199C2212yes—
10C2×#6199C2212yes—
11C4⋊#1088—22no—
12#88590—22no—
13#3750⋊C3—22no—
14#88448—22no—
15D3×Dic3⋊C4—62no—
16C22×#272C2311yescharacteristic
17C2×#1088—22no—
18C2×#1088—21no—
19C2×#1088—22no—
20C2×#1088D412yes—
21C2×C4×H(3,𝔽3)D411yescharacteristic
22C2×#1084D411yescharacteristic
23C4×#272—21no—
24C4×#272—21no—
25C2×(C3⋊Dic3)⋊C3—22no—
26C2×(C3⋊Dic3)⋊C3—22no—
27#6199—22no—
28#6199—22no—
29#6199—22no—
30#6199Q814yes—
31#4981—91nomaximal
32C2×C3×C4×D3—62no—
33C2×D3×Dic3—62no—
34C2×D3×Dic3—62no—
35#3750—22no—
36C3×Dic3⋊C4—62no—
37#3724—62no—
38Dic3⋊Dic3—62no—
39C22×H(3,𝔽3)C2×D411yescharacteristic
40C2×#272C2×D411yescharacteristic
41C2×#272C2×Q812yes—
42C2×#272#4112yes—
43C2×#272#4111yescharacteristic
44#1088—22no—
45#1088—41no—
46C4×H(3,𝔽3)—21no—
47#1084—21no—
48(C3⋊Dic3)⋊C3—24no—
49(C3⋊Dic3)⋊C3—42no—
50C2×C3×C4⋊C4—91no—
51C2×C4⋊Dic3—91no—
52C2×Dic3⋊C4—92no—
53D3×C4⋊C4—182no—
54C3⋊#95—91no—
55C3⋊#95—92no—
56C22×C3×D3—62no—
57C2×C32×C4—22no—
58C2×C3⋊Dic3—22no—
59C2×C3⋊Dic3—22no—
60C2×C3×Dic3—62no—
61C2×C3×Dic3—62no—
62C2×C3×Dic3—62no—
63C3×C4×D3—122no—
64C3×C4×D3—122no—
65D3×Dic3—64no—
66D3×Dic3—64no—
67D3×Dic3—122no—
68D3×Dic3—122no—
69#398—271noSylow
70C2×H(3,𝔽3)#12111yescharacteristic
71C2×H(3,𝔽3)C22⋊Q812yes—
72#272#12311yescharacteristic
73#272#12512yes—
74#272C4⋊Q811yescharacteristic
75C22×C3×C4—91no—
76C22×C3×C4—91no—
77C22×C3×C4—91no—
78C22×Dic3—92no—
79C22×Dic3—92no—
80C2×C4×D3—62no—
81C2×C4×D3—182no—
82C2×C4×D3—182no—
83C3×C4⋊C4—182no—
84C3×C4⋊C4—182no—
85C4⋊Dic3—182no—
86C4⋊Dic3—182no—
87Dic3⋊C4—62no—
88Dic3⋊C4—182no—
89Dic3⋊C4—182no—
90Dic3⋊C4—182no—
91C22×C32—22no—
92C2×C3×D3—62no—
93C2×C3×D3—62no—
94C2×C3×D3—62no—
95C2×C3×D3—62no—
96C2×C3×D3—62no—
97C2×C3×D3—62no—
98C32×C4—42no—
99C3⋊Dic3—24no—
100C3⋊Dic3—42no—
101C3×Dic3—64no—
102C3×Dic3—122no—
103C3×Dic3—122no—
104C2×C4⋊C4—91no—
105C2×C4⋊C4—272no—
106C2×C4⋊C4—271no—
107#95—271no—
108#95—272no—
109H(3,𝔽3)#39811yescharacteristic, complemented, Sylow
110C23×C3—91no—
111C22×D3—62no—
112C2×C3×C4—62no—
113C2×C3×C4—92no—
114C2×C3×C4—91no—
115C2×C3×C4—181no—
116C2×C3×C4—181no—
117C2×C3×C4—182no—
118C2×C3×C4—182no—
119C2×C3×C4—182no—
120C2×C3×C4—181no—
121C2×C3×C4—182no—
122C2×C3×C4GO+(4,𝔽2)11yescharacteristic
123C2×Dic3—62no—
124C2×Dic3—62no—
125C2×Dic3—62no—
126C2×Dic3—94no—
127C2×Dic3—182no—
128C2×Dic3—182no—
129C2×Dic3—182no—
130C2×Dic3—182no—
131C2×Dic3—182no—
132C4×D3—122no—
133C4×D3—122no—
134C4×D3—184no—
135C4×D3—184no—
136C4×D3—362no—
137C4×D3—362no—
138C2×C32—22no—
139C2×C32—22no—
140C2×C32—22no—
141C3×D3—62no—
142C3×D3—62no—
143C3×D3—62no—
144C3×D3—62no—
145C22×C4—91no—
146C22×C4—91no—
147C22×C4—91no—
148C22×C4—272no—
149C22×C4—272no—
150C4⋊C4—182no—
151C4⋊C4—182no—
152C4⋊C4—542no—
153C4⋊C4—542no—
154C22×C3—62no—
155C22×C3—92no—
156C22×C3—92no—
157C22×C3—91no—
158C22×C3—91no—
159C22×C3C2×GO+(4,𝔽2)11yescharacteristic
160C2×D3—62no—
161C2×D3—62no—
162C2×D3—62no—
163C2×D3—62no—
164C2×D3—62no—
165C2×D3—62no—
166C3×C4—21no—
167C3×C4—122no—
168C3×C4—184no—
169C3×C4—182no—
170C3×C4—181no—
171C3×C4—362no—
172C3×C4—361no—
173Dic3—64no—
174Dic3—122no—
175Dic3—122no—
176Dic3—184no—
177Dic3—362no—
178C32—22no—
179C23—91no—
180C2×C4—92no—
181C2×C4—91no—
182C2×C4—181no—
183C2×C4—181no—
184C2×C4—182no—
185C2×C4—182no—
186C2×C4—181no—
187C2×C4—182no—
188C2×C4—182no—
189C2×C4—274no—
190C2×C4—542no—
191C2×C4—542no—
192C2×C4—542no—
193C2×C4#1088⋊C211yescharacteristic
194C2×C3—62no—
195C2×C3—62no—
196C2×C3—62no—
197C2×C3—91no—
198C2×C3—91no—
199C2×C3—92no—
200C2×C3D32⋊Q812yes—
201C2×C3#6424211yescharacteristic
202D3—62no—
203D3—62no—
204D3—62no—
205D3—62no—
206C22—92no—
207C22—92no—
208C22—91no—
209C22—91no—
210C22C2×#1088⋊C211yescharacteristic
211C4—21no—
212C4—182no—
213C4—184no—
214C4—181no—
215C4—361no—
216C4—362no—
217C3—62no—
218C3#1059334011yescharacteristic
219C2—91no—
220C2—92no—
221C2—91no—
222C2(((C3⋊D3)⋊C3)⋊C2)⋊Q812yes—
223C2#1172231611yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator, prime radical, cosocleC22×#272C23
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#398
LCSG ▷ C22×#272 ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
109. H(3,𝔽3)#398semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10882
C2×#885902
#117211472
#117207072
#49819