Finite Group Databasebeta

← #13147003

Subgroups — #13147003

Order 1728.

289 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 42 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
1C4×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic
2#11720738C211yesmaximal, characteristic
3#11720738C211yesmaximal, characteristic
4C3×#5389—33nomaximal
5C3×#5022—31nomaximal
6C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
7#88540—41nomaximal
8C32×C4×Q8D311yescharacteristic
9C3×#870—31no—
10C3×C4×SL(2,𝔽3)D313yes—
11C3×#859—33no—
12C3×#859—33no—
13C3×#835—31no—
14C3×#835—31no—
15C3×#808—31no—
16C3×#807—31no—
17C3×C4⋊(C3⋊C8)—31no—
18C2×C4×H(3,𝔽3)—41no—
19C2×#1084—41no—
20C2×#1084—41no—
21Q8⋊H(3,𝔽3)Q811yescharacteristic
22#5389—33no—
23C3×#479—91no—
24#5022—31no—
25C2×C32×Q8C2×D311yescharacteristic
26C2×C3×SL(2,𝔽3)C2×D313yes—
27C32×C4⋊C4—31no—
28C32×C42—31no—
29C3×C4⋊Dic3—31no—
30C3×C4⋊Dic3—31no—
31C3×Dic3⋊C4—61no—
32C3×Dic3⋊C4—61no—
33C3×Dic3⋊C4—123no—
34C2×C3×C3⋊C8—31no—
35C2×C3×C3⋊C8—31no—
36C22×H(3,𝔽3)—41no—
37C4×H(3,𝔽3)—81no—
38#1084—42no—
39#1084—81no—
40C3×#125—91no—
41C3×C4×Q8—31no—
42C3×C4×Q8C3⋊D311yescharacteristic
43#870—31no—
44C4×SL(2,𝔽3)—33no—
45#859—33no—
46#859—33no—
47C3×#107—91no—
48C3×#106—91no—
49C3×C4⋊C8—91no—
50C3×#103—91no—
51C3×#103—91no—
52#835—31no—
53#835—31no—
54#808—31no—
55#807—31no—
56C4⋊(C3⋊C8)—31no—
57C32×Q8—31no—
58C32×Q8Dic611yescharacteristic
59C2×C32×C4—31no—
60C2×C32×C4—31no—
61C2×C32×C4—43no—
62C2×C32×C4S411yescharacteristic
63C2×C3×Dic3—61no—
64C2×C3×Dic3—61no—
65C2×C3×Dic3—123no—
66C2×C3×Dic3—123no—
67C3×SL(2,𝔽3)Dic613yes—
68C3×C3⋊C8—61no—
69C3×C3⋊C8—61no—
70#479—91noSylow
71C2×H(3,𝔽3)—41no—
72C2×H(3,𝔽3)—41no—
73C2×H(3,𝔽3)—41no—
74C2×C3×Q8—31no—
75C2×C3×Q8C2×C3⋊D311yescharacteristic
76C2×SL(2,𝔽3)—33no—
77C2×C3×C8—91no—
78C2×C3×C8—91no—
79C3×C4⋊C4—31no—
80C3×C4⋊C4—31no—
81C3×C4⋊C4—61no—
82C3×C4⋊C4—91no—
83C3×C4⋊C4—91no—
84C3×C4⋊C4—181no—
85C3×C4⋊C4—181no—
86C3×C42—31no—
87C3×C42—31no—
88C3×C42—61no—
89C4⋊Dic3—31no—
90C4⋊Dic3—31no—
91Dic3⋊C4—61no—
92Dic3⋊C4—61no—
93Dic3⋊C4—123no—
94C2×C3⋊C8—31no—
95C2×C3⋊C8—31no—
96C22×C32—43no—
97C22×C32C2×S411yescharacteristic
98C32×C4—21no—
99C32×C4—31no—
100C32×C4—31no—
101C32×C4—61no—
102C32×C4—83no—
103C3×Dic3—126no—
104C3×Dic3—121no—
105C3×Dic3—121no—
106C3×Dic3—243no—
107#125—91no—
108C4×Q8#27211yescharacteristic
109#107—91no—
110#106—91no—
111C4⋊C8—91no—
112#103—91no—
113#103—91no—
114H(3,𝔽3)—41noSylow
115C3×Q8—31no—
116C3×Q8—31no—
117C3×Q8—31no—
118C3×Q8—61no—
119C3×Q8#63411yescharacteristic
120C2×C3×C4—31no—
121C2×C3×C4—31no—
122C2×C3×C4—31no—
123C2×C3×C4—31no—
124C2×C3×C4—31no—
125C2×C3×C4—61no—
126C2×C3×C4—61no—
127C2×C3×C4—123no—
128C2×C3×C4—181no—
129C2×C3×C4—181no—
130C2×C3×C4C3⋊S411yescharacteristic
131C2×Dic3—61no—
132C2×Dic3—61no—
133C2×Dic3—123no—
134C2×Dic3—123no—
135SL(2,𝔽3)—33no—
136C3×C8—181no—
137C3×C8—181no—
138C3⋊C8—61no—
139C3⋊C8—61no—
140C2×C32—43no—
141C2×C32—43no—
142C2×C32—43no—
143C2×C32A4⋊Q811yescharacteristic
144C2×C32#98311yescharacteristic
145C2×C32#98511yescharacteristic
146C2×Q8C2×#27211yescharacteristic
147C2×C8—91no—
148C2×C8—91no—
149C4⋊C4—31no—
150C4⋊C4—91no—
151C4⋊C4—91no—
152C4⋊C4—181no—
153C4⋊C4—181no—
154C42—31no—
155C22×C3—31no—
156C22×C3—123no—
157C22×C3C2×C3⋊S411yescharacteristic
158C3×C4—21no—
159C3×C4—31no—
160C3×C4—31no—
161C3×C4—31no—
162C3×C4—31no—
163C3×C4—61no—
164C3×C4—61no—
165C3×C4—61no—
166C3×C4—61no—
167C3×C4—61no—
168C3×C4—121no—
169C3×C4—243no—
170C3×C4—361no—
171C3×C4—361no—
172Dic3—121no—
173Dic3—121no—
174Dic3—126no—
175Dic3—243no—
176C32—43no—
177C32#538911yescharacteristic, complemented
178Q8—31no—
179Q8#622911yescharacteristic
180C2×C4—31no—
181C2×C4—31no—
182C2×C4—181no—
183C2×C4—181no—
184C2×C4#625811yescharacteristic
185C8—181no—
186C8—181no—
187C2×C3—31no—
188C2×C3—31no—
189C2×C3—31no—
190C2×C3—123no—
191C2×C3—123no—
192C2×C3—123no—
193C2×C3#6425911yescharacteristic
194C2×C3#6426411yescharacteristic
195C2×C3#6426611yescharacteristic
196C22C2×#625811yescharacteristic
197C4—21no—
198C4—31no—
199C4—31no—
200C4—61no—
201C4—361no—
202C4—361no—
203C3—31no—
204C3—123no—
205C3#1059362311yescharacteristic
206C2#1172238011yescharacteristic
207C2#1172238811yescharacteristic
208C2#1172239311yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C2×C3⋊S4
commutator, prime radical, cosocleC2×Q8⋊H(3,𝔽3)C22
FittingC32×C4×Q8D3
hypercenterC2×C3×C4C3⋊S4
hypocenterQ8⋊H(3,𝔽3)Q8
LCSG ▷ C2×Q8⋊H(3,𝔽3) ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
177. C32#5389semidirect

Maximal subgroups

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

subgroupindex
C4×Q8⋊H(3,𝔽3)2
#117207382
C3×#53893
C3×#50223
#885404