Finite Group Databasebeta

← #13144353

Subgroups — #13144353

Order 1728.

1001 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1001 conjugacy classes of subgroups — too many to draw. Only its 187 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×#6229C211yesmaximal, characteristic
2C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
3C2×#88540C212yesmaximal
4#11720574C212yesmaximal
5#11720574C211yesmaximal, characteristic
6C3×#4979—34nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2212yes—
11C22×#1084C2212yes—
12C2×#6229—22no—
13C2×#6229—24no—
14C4⋊C4×H(3,𝔽3)—22no—
15#88540—24no—
16C22×C3×Dic6—34no—
17C2×C32×C4⋊C4D314yes—
18C2×C3×Dic3⋊C4—38no—
19C3×C3⋊#95—38no—
20C3×C3⋊#95—34no—
21C23×H(3,𝔽3)C2311yescharacteristic
22C2×C4×H(3,𝔽3)—22no—
23C2×C4×H(3,𝔽3)—21no—
24C2×C4×H(3,𝔽3)—21no—
25C2×C4×H(3,𝔽3)—22no—
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)D412yes—
28C2×C4×H(3,𝔽3)Q812yescomplemented
29C2×#1084—22no—
30C2×#1084—22no—
31C2×#1084—22no—
32C2×#1084—22no—
33C2×#1084D414yes—
34C2×#1084Q814yescomplemented
35#6229—28no—
36C3×#395—91no—
37#4979—34no—
38C22×C32×C4C2×D314yes—
39C22×C32×C4C2×D314yes—
40C22×C32×C4C2×D314yes—
41C22×C3×Dic3—38no—
42C22×C3×Dic3—38no—
43C2×C3×Dic6—616no—
44C2×C3×Dic6—68no—
45C32×C4⋊C4—28no—
46C3×Dic3⋊C4—616no—
47C22×H(3,𝔽3)C2×D412yes—
48C22×H(3,𝔽3)C2×D411yescharacteristic
49C22×H(3,𝔽3)C2×Q811yescharacteristic
50C22×H(3,𝔽3)C2×Q812yes—
51C22×H(3,𝔽3)#4111yescharacteristic
52C4×H(3,𝔽3)—22no—
53C4×H(3,𝔽3)—22no—
54C4×H(3,𝔽3)—41no—
55#1084—24no—
56#1084—24no—
57C22×C3×Q8—91no—
58C22×Dic6—34no—
59C2×C3×C4⋊C4—34no—
60C2×C3×C4⋊C4—92no—
61C2×C3×C4⋊C4C3⋊D311yescharacteristic
62C2×Dic3⋊C4—38no—
63C3×#95—91no—
64C3×#95—92no—
65C3⋊#95—34no—
66C3⋊#95—38no—
67C23×C32C22×D314yes—
68C2×C32×C4—24no—
69C2×C32×C4—28no—
70C2×C32×C4—24no—
71C2×C32×C4—28no—
72C2×C32×C4—24no—
73C2×C32×C4Dic618yescomplemented
74C2×C32×C4#6718yes—
75C2×C3×Dic3—316no—
76C2×C3×Dic3—316no—
77C2×C3×Dic3—68no—
78C2×C3×Dic3—68no—
79C2×C3×Dic3—68no—
80C2×C3×Dic3—68no—
81C3×Dic6—632no—
82#395—91noSylow
83C2×H(3,𝔽3)#12011yescharacteristic
84C2×H(3,𝔽3)C22⋊Q812yes—
85C2×H(3,𝔽3)C22⋊Q811yescharacteristic
86C2×H(3,𝔽3)C4⋊Q812yes—
87C2×H(3,𝔽3)C4⋊Q811yescharacteristic
88C2×C3×Q8—184no—
89C2×C3×Q8—182no—
90C22×C3×C4—34no—
91C22×C3×C4—34no—
92C22×C3×C4—34no—
93C22×C3×C4—92no—
94C22×C3×C4—92no—
95C22×C3×C4C2×C3⋊D311yescharacteristic
96C22×C3×C4C2×C3⋊D311yescharacteristic
97C22×C3×C4C2×C3⋊D311yescharacteristic
98C22×Dic3—38no—
99C22×Dic3—38no—
100C2×Dic6—616no—
101C2×Dic6—68no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—68no—
104C3×C4⋊C4—184no—
105Dic3⋊C4—616no—
106C22×C32C2×Dic614yes—
107C22×C32#23514yes—
108C22×C32D3×D418yes—
109C22×C32D3×Q818yes—
110C22×C32C2×#6714yes—
111C32×C4—28no—
112C32×C4—28no—
113C32×C4—44no—
114C3×Dic3—616no—
115C3×Dic3—616no—
116C22×Q8—91no—
117C2×C4⋊C4—92no—
118C2×C4⋊C4#27211yescharacteristic
119#95—91no—
120#95—92no—
121H(3,𝔽3)#39511yescharacteristic, complemented, Sylow
122C23×C3—34no—
123C23×C3C22×C3⋊D311yescharacteristic
124C3×Q8—188no—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—22no—
129C2×C3×C4—21no—
130C2×C3×C4—38no—
131C2×C3×C4—38no—
132C2×C3×C4—64no—
133C2×C3×C4—68no—
134C2×C3×C4—64no—
135C2×C3×C4—68no—
136C2×C3×C4—64no—
137C2×C3×C4—94no—
138C2×C3×C4—94no—
139C2×C3×C4—182no—
140C2×C3×C4—182no—
141C2×C3×C4—182no—
142C2×C3×C4—182no—
143C2×C3×C4#63412yes—
144C2×C3×C4#63812yes—
145C2×Dic3—316no—
146C2×Dic3—316no—
147C2×Dic3—68no—
148C2×Dic3—68no—
149C2×Dic3—68no—
150C2×Dic3—68no—
151Dic6—632no—
152C2×C32#88818yes—
153C2×C32#89618yes—
154C2×C32#92414yes—
155C2×C32#93714yes—
156C2×C32#94414yes—
157C2×Q8—184no—
158C2×Q8—182no—
159C22×C4—92no—
160C22×C4—92no—
161C22×C4C2×#27211yescharacteristic
162C22×C4C2×#27211yescharacteristic
163C22×C4C2×#27211yescharacteristic
164C4⋊C4—22no—
165C4⋊C4—184no—
166C22×C3—34no—
167C22×C3—34no—
168C22×C3—38no—
169C22×C3—38no—
170C22×C3—34no—
171C22×C3C2×#63411yescharacteristic
172C22×C3#382811yescharacteristic
173C22×C3D4×C3⋊D312yes—
174C22×C3Q8×C3⋊D312yes—
175C22×C3C2×#63811yescharacteristic
176C3×C4—22no—
177C3×C4—22no—
178C3×C4—41no—
179C3×C4—68no—
180C3×C4—68no—
181C3×C4—124no—
182C3×C4—184no—
183C3×C4—184no—
184Dic3—616no—
185Dic3—616no—
186C32#497914yescomplemented
187C23C22×#27211yescharacteristic
188Q8—188no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—94no—
195C2×C4—94no—
196C2×C4—182no—
197C2×C4—182no—
198C2×C4—182no—
199C2×C4—182no—
200C2×C4#622912yescomplemented
201C2×C4#623512yes—
202C2×C3—38no—
203C2×C3—38no—
204C2×C3—34no—
205C2×C3—34no—
206C2×C3—34no—
207C2×C3#6409712yes—
208C2×C3#6410512yes—
209C2×C3#6413311yescharacteristic
210C2×C3#6414611yescharacteristic
211C2×C3#6415311yescharacteristic
212C22C2×#622911yescharacteristic
213C22#8874011yescharacteristic
214C22D4×#27212yes—
215C22Q8×#27212yes—
216C22C2×#623511yescharacteristic
217C4—22no—
218C4—22no—
219C4—41no—
220C4—184no—
221C4—184no—
222C3—34no—
223C3#1059267811yescharacteristic
224C2#1172175612yes—
225C2#1172176412yes—
226C2#1172182411yescharacteristic
227C2#1172184711yescharacteristic
228C2#1172186111yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, prime radical, cosocleC23×H(3,𝔽3)C23
FittingC2×C4⋊C4×H(3,𝔽3)C2
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)#395
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
28. C2×C4×H(3,𝔽3)Q8semidirect
34. C2×#1084Q8semidirect
73. C2×C32×C4Dic6semidirect
121. H(3,𝔽3)#395semidirect
186. C32#4979semidirect
200. C2×C4#6229semidirect

Maximal subgroups

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

subgroupindex
C22×#62292
C2×C4⋊C4×H(3,𝔽3)2
C2×#885402
#117205742
C3×#49793