Finite Group Databasebeta

← #10589358⋊C3 #13135415

Subgroups — #10589358⋊C3 #13135415

Order 1728.

239 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3746⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×#107C211yesmaximal, characteristic
3C2×(C3⋊(C3⋊C16))⋊C3C211yesmaximal, characteristic
4#10589358C311yesmaximal, characteristic, complemented
5C3×#4492—31nomaximal
6C2×#634⋊C3—21no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8C4⋊C4×H(3,𝔽3)—21no—
9#3746⋊C3—21no—
10#3746⋊C3C412yescomplemented
11(C3⋊(C3⋊C16))⋊C3—21no—
12C2×#3746C2×C311yescharacteristic
13C2×C3×Dic12—31no—
14C32×#107—21no—
15C32×#107D311yescharacteristic
16C32×#107C2×C311yescharacteristic
17C2×C3⋊(C3⋊C16)C2×C311yescharacteristic
18C2×C3×C3⋊C16—31no—
19C2×C4×H(3,𝔽3)—41no—
20C2×C4×H(3,𝔽3)D411yescharacteristic
21C2×(C3⋊Dic3)⋊C3—41no—
22#634⋊C3—22no—
23#634⋊C3—41no—
24C8×H(3,𝔽3)C2×C411yescharacteristic
25C8×H(3,𝔽3)D411yescharacteristic
26C3×#358—91no—
27#4492—31no—
28#4492—91no—
29C2×#634—21no—
30C2×C3×Dic6—61no—
31C2×C32×C8—21no—
32C2×C32×C8C2×D311yescharacteristic
33C2×C32×C8C22×C311yescharacteristic
34C32×C4⋊C4—22no—
35C32×C4⋊C4—21no—
36C32×C4⋊C4—21no—
37#3746—21no—
38#3746C3×C412yescomplemented
39C3×Dic12—32no—
40C3×Dic12—61no—
41C3⋊(C3⋊C16)—21no—
42C3×C3⋊C16—61no—
43C22×H(3,𝔽3)D811yescharacteristic
44C4×H(3,𝔽3)—81no—
45C4×H(3,𝔽3)C22⋊C411yescharacteristic
46C4×H(3,𝔽3)SD1611yescharacteristic
47(C3⋊Dic3)⋊C3—42no—
48C2×C3×Q16—91no—
49C2×Dic12—31no—
50C2×Dic12—91no—
51C2×C3×C16—91no—
52C3×#107—31no—
53C3×#107—31no—
54C3×#107—61no—
55C3×#107C3×D311yescharacteristic
56C2×C3⋊C16—31no—
57C2×C3⋊C16—91no—
58C2×C32×C4—21no—
59C2×C32×C4—42no—
60C2×C32×C4—41no—
61C2×C32×C4—41no—
62C2×C32×C4#6711yescharacteristic
63C2×C32×C4C3×D411yescharacteristic
64C2×C3⋊Dic3—41no—
65#634—22no—
66#634—41no—
67C2×C3×Dic3—121no—
68C3×Dic6—62no—
69C3×Dic6—121no—
70C32×C8—21no—
71C32×C8—21no—
72C32×C8C4×D311yescharacteristic
73C32×C8D1211yescharacteristic
74C32×C8C2×C3×C411yescharacteristic
75C32×C8C3×D411yescharacteristic
76#358—91noSylow
77C2×H(3,𝔽3)#10211yescharacteristic
78C2×H(3,𝔽3)SD3211yescharacteristic
79C2×H(3,𝔽3)Q3211yescharacteristic
80C2×C3×Q8—181no—
81C2×Dic6—61no—
82C2×Dic6—181no—
83C3×Q16—92no—
84C3×Q16—181no—
85C2×C3×C8—31no—
86C2×C3×C8—31no—
87C2×C3×C8—61no—
88C2×C3×C8C2×C3×D311yescharacteristic
89C3×C4⋊C4—21no—
90C3×C4⋊C4—61no—
91C3×C4⋊C4—61no—
92C3×C4⋊C4—62no—
93Dic12—32no—
94Dic12—61no—
95Dic12—92no—
96Dic12—181no—
97C3×C16—181no—
98C3⋊C16—61no—
99C3⋊C16—181no—
100C22×C32—21no—
101C22×C32#21311yescharacteristic
102C22×C32C3×D811yescharacteristic
103C32×C4—21no—
104C32×C4—21no—
105C32×C4—82no—
106C32×C4—81no—
107C32×C4—81no—
108C32×C4(C2×D3)⋊C411yescharacteristic
109C32×C4#21511yescharacteristic
110C32×C4C3×C22⋊C411yescharacteristic
111C32×C4C3×SD1611yescharacteristic
112C3⋊Dic3—42no—
113C3×Dic3—122no—
114C2×Q16—91no—
115C2×C16—91no—
116#107(C3⋊D3)⋊C311yescharacteristic
117H(3,𝔽3)#35811yescharacteristic, complemented, Sylow
118C3×Q8—182no—
119C3×Q8—361no—
120C2×C3×C4—31no—
121C2×C3×C4—31no—
122C2×C3×C4—41no—
123C2×C3×C4—61no—
124C2×C3×C4—121no—
125C2×C3×C4—121no—
126C2×C3×C4—122no—
127C2×C3×C4—361no—
128C2×C3×C4C3×#6711yescharacteristic
129C2×Dic3—121no—
130C2×Dic3—361no—
131Dic6—62no—
132Dic6—121no—
133Dic6—182no—
134Dic6—361no—
135C3×C8—31no—
136C3×C8—31no—
137C3×C8—31no—
138C3×C8—31no—
139C3×C8—61no—
140C3×C8—61no—
141C3×C8C3×C4×D311yescharacteristic
142C3×C8C3×D1211yescharacteristic
143C2×C32—21no—
144C2×C32—21no—
145C2×C32—21no—
146C2×C32#80911yescharacteristic
147C2×C32#82711yescharacteristic
148C2×C32#82911yescharacteristic
149C2×C32C3×#10211yescharacteristic
150C2×C32C3×SD3211yescharacteristic
151C2×C32C3×Q3211yescharacteristic
152C2×Q8—181no—
153Q16—92no—
154Q16—181no—
155C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
156C4⋊C4—21no—
157C16—181no—
158C22×C3—31no—
159C22×C3—31no—
160C22×C3—61no—
161C22×C3C3×#21311yescharacteristic
162C3×C4—31no—
163C3×C4—31no—
164C3×C4—31no—
165C3×C4—31no—
166C3×C4—61no—
167C3×C4—61no—
168C3×C4—81no—
169C3×C4—241no—
170C3×C4—241no—
171C3×C4—242no—
172C3×C4—362no—
173C3×C4C3×(C2×D3)⋊C411yescharacteristic
174C3×C4C3×#21511yescharacteristic
175Dic3—122no—
176Dic3—362no—
177C32—21no—
178C32#449211yescharacteristic, complemented
179C32C3×#35811yescharacteristic, complemented
180Q8—182no—
181Q8—361no—
182C2×C4—41no—
183C2×C4—361no—
184C2×C4#638⋊C311yescharacteristic
185C8C4×(C3⋊D3)⋊C311yescharacteristic
186C8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
187C2×C3—31no—
188C2×C3—31no—
189C2×C3—31no—
190C2×C3—31no—
191C2×C3—31no—
192C2×C3—31no—
193C2×C3—61no—
194C2×C3—61no—
195C2×C3—61no—
196C2×C3C3×#80911yescharacteristic
197C2×C3C3×#82711yescharacteristic
198C2×C3C3×#82911yescharacteristic
199C22#3753⋊C311yescharacteristic
200C4—81no—
201C4—362no—
202C4#3752⋊C311yescharacteristic
203C4#3755⋊C311yescharacteristic
204C3—31no—
205C3—31no—
206C3—61no—
207C3C3×#449211yescharacteristic
208C2#63636⋊C311yescharacteristic
209C2#63654⋊C311yescharacteristic
210C2#63656⋊C311yescharacteristic
subgroupG/H
centerC22#3753⋊C3
commutatorC32×C8C2×C3×C4
FrattiniC2×C3×C8C2×C3×D3
FittingH(3,𝔽3)×#107C2
socleC22×C3C3×#213
prime radicalC22×H(3,𝔽3)D8
cosocleC2×C32×C8C22×C3
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×#358
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ #107

Complements

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

NH ≅ G/Ntype
4. #10589358C3semidirect
10. #3746⋊C3C4semidirect
38. #3746C3×C4semidirect
117. H(3,𝔽3)#358semidirect
178. C32#4492semidirect
179. C32C3×#358semidirect

Maximal subgroups

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

subgroupindex
C2×#3746⋊C32
H(3,𝔽3)×#1072
C2×(C3⋊(C3⋊C16))⋊C32
#105893583
C3×#44923