Finite Group Databasebeta

← #10595445⋊C3 #13166157

Subgroups — #10595445⋊C3 #13166157

Order 1728.

376 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 46 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
1C3×#1010⋊C3C212yesmaximal, complemented
2C3⋊(C3⋊(#131⋊C3))C211yesmaximal, characteristic, complemented
3#10595445C311yesmaximal, characteristic, complemented
4#5838⋊C3—32nomaximal
5C32×#41⋊C3C2211yescharacteristic
6C3×D3×SL(2,𝔽3)—22no—
7C3×#3713—41nomaximal
8C3×#1010C2×C312yescomplemented
9C3×2−1+4⋊C3—32no—
10#1010⋊C3—22no—
11#1010⋊C3D312yescomplemented
12#64157C2×C311yescharacteristic, complemented
13C2×#3718—31no—
14#63827—32no—
15#63822—31no—
16C3⋊(#131⋊C3)—32no—
17C3⋊(#131⋊C3)—61no—
18C32×D12—42no—
19C32×SL(2,𝔽3)D411yescharacteristic
20C3×C3⋊(C3⋊C8)—41no—
21#5838—32no—
22#578⋊C3—91no—
23C32×#41C22×C311yescharacteristic
24C3×D3×Q8—22no—
25C3×#237—62no—
26C3×#235—32no—
27C2×C3×Dic6—32no—
28C3×#41⋊C3—22no—
29C3×#41⋊C3—21no—
30C3×#41⋊C3—41no—
31C3×#41⋊C3C2×D312yes—
32C2×C3×SL(2,𝔽3)—62no—
33D3×SL(2,𝔽3)—24no—
34D3×SL(2,𝔽3)—22no—
35#3748—31no—
36C2×C3⋊(C3⋊C8)—31no—
37C3×#213—122no—
38#3718—32no—
39#3718—31no—
40#3715—32no—
41#3713A411yescharacteristic
42C2×C32×D3—82no—
43C33×C4—41no—
44C3×2−1+4—32no—
45#1010C3×D312yescomplemented
462−1+4⋊C3—32no—
472−1+4⋊C3—62no—
48#951—92no—
49#950—92no—
50#948—32no—
51#948—61no—
52C2×#216—92no—
53#942—92no—
54#131⋊C3—91no—
55C32×Q8C3×D411yescharacteristic
56C32×D4—31no—
57C32×D4—122no—
58C2×C32×C4—31no—
59C3×#67—62no—
60C2×C3×Dic3—62no—
61C3×D12—42no—
62C3×D12—82no—
63C3×D12C2×A412yes—
64C3×C4×D3—62no—
65C3×Dic6—32no—
66C3×Dic6—62no—
67C3×SL(2,𝔽3)—22no—
68C3×SL(2,𝔽3)—21no—
69C3×SL(2,𝔽3)—41no—
70C3×SL(2,𝔽3)#6712yes—
71C3⋊(C3⋊C8)—31no—
72C3⋊(C3⋊C8)C2×A411yescharacteristic
73C3×C3⋊C8—122no—
74C3×C3⋊C8—241no—
75#578—91noSylow
76C2×C33—41no—
77C32×D3—162no—
78C3×#41—21no—
79C3×#41—92no—
80C3×#41—182no—
81C3×#41C2×C3×D312yes—
82C2×C3×Q8—62no—
83C2×C3×Q8—92no—
84D3×Q8—22no—
85#237—62no—
86#235—32no—
87C2×Dic6—32no—
88#41⋊C3—22no—
89#41⋊C3—41no—
90#41⋊C3D3211yescharacteristic
91C2×SL(2,𝔽3)—64no—
92C2×SL(2,𝔽3)—62no—
93C3×D8—361no—
94#216—92no—
95#216—94no—
96#215—92no—
97#214—92no—
98#213—32no—
99#208—92no—
100#208—181no—
101C2×C3⋊C8—92no—
102C2×C3⋊C8—181no—
103C22×C32—31no—
104C22×C32—242no—
105C2×C3×D3—22no—
106C2×C3×D3—82no—
107C2×C3×D3—84no—
108C32×C4—31no—
109C32×C4—42no—
110C32×C4—81no—
111C32×C4—82no—
112C32×C4—161no—
113C32×C4C22×A411yescharacteristic
114C3×Dic3—62no—
1152−1+4—32no—
116#137—272no—
117#135—271no—
118C2×Q16—271no—
119#131—91no—
120C33—41noSylow
121C3×Q8—21no—
122C3×Q8—92no—
123C3×Q8—182no—
124C3×Q8C3×#6712yes—
125C3×D4—32no—
126C3×D4—32no—
127C3×D4—61no—
128C3×D4—122no—
129C3×D4—182no—
130C3×D4—242no—
131C2×C3×C4—32no—
132C2×C3×C4—61no—
133C2×C3×C4—182no—
134C2×C3×C4—182no—
135#67—62no—
136C2×Dic3—62no—
137D12D3×A412yes—
138C4×D3—62no—
139Dic6—32no—
140Dic6—62no—
141SL(2,𝔽3)—22no—
142SL(2,𝔽3)—41no—
143SL(2,𝔽3)#62511yescharacteristic
144C3×C8—361no—
145C3⋊C8—32no—
146C3⋊C8—61no—
147C3⋊C8—92no—
148C3⋊C8—181no—
149C2×C32—42no—
150C2×C32—61no—
151C2×C32—81no—
152C2×C32—82no—
153C2×C32—161no—
154C2×C32—482no—
155C2×C32D4×A411yescharacteristic
156C3×D3—42no—
157C3×D3—162no—
158C3×D3—164no—
159#41—92no—
160#41—182no—
161#41C3×D3211yescharacteristic
162C2×Q8—62no—
163C2×Q8—92no—
164Q16—271no—
165Q16—272no—
166SD16—272no—
167D8—91no—
168M16—271no—
169C2×C8—271no—
170C22×C3—32no—
171C22×C3—61no—
172C22×C3—62no—
173C22×C3—242no—
174C22×C3—244no—
175C2×D3—22no—
176C3×C4—21no—
177C3×C4—32no—
178C3×C4—41no—
179C3×C4—61no—
180C3×C4—82no—
181C3×C4—161no—
182C3×C4—182no—
183C3×C4C2×D3×A412yes—
184Dic3—62no—
185C32—42no—
186C32—81no—
187C32—82no—
188C32—161no—
189C32#578⋊C311yescharacteristic, complemented
190Q8—92no—
191Q8—182no—
192Q8C3×#62511yescharacteristic
193D4—31no—
194D4—32no—
195D4—182no—
196C2×C4—31no—
197C2×C4—182no—
198C2×C4—182no—
199C8—91no—
200C8—271no—
201C2×C3—21no—
202C2×C3—41no—
203C2×C3—62no—
204C2×C3—82no—
205C2×C3—121no—
206C2×C3—122no—
207C2×C3—161no—
208C2×C3—482no—
209C2×C3—484no—
210C2×C3A4×#6712yes—
211D3—42no—
212C22—31no—
213C22—62no—
214C4—31no—
215C4—182no—
216C4D32×A411yescharacteristic
217C3—21no—
218C3—41no—
219C3—82no—
220C3—161no—
221C3#5838⋊C312yescomplemented
222C2—61no—
223C2—122no—
224C2A4×#62511yescharacteristic
subgroupG/H
centerC2A4×#625
commutator, Fitting, cosocleC32×#41C22×C3
Frattini, hypercenterC4D32×A4
socleC2×C32D4×A4
hypocenterC32×Q8C3×D4
LCSG ▷ C32×#41 ▷ C32×Q8
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C3×#1010⋊C3C2semidirect
2. C3⋊(C3⋊(#131⋊C3))C2semidirect
3. #10595445C3semidirect
8. C3×#1010C2×C3semidirect
11. #1010⋊C3D3semidirect
12. #64157C2×C3semidirect
45. #1010C3×D3semidirect
189. C32#578⋊C3semidirect
221. C3#5838⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#1010⋊C32
C3⋊(C3⋊(#131⋊C3))2
#105954453
#5838⋊C33
C3×#37134