Finite Group Databasebeta

← #13147292

Subgroups — #13147292

Order 1728.

451 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 62 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×#41⋊H(3,𝔽3)C211yesmaximal, characteristic
2#11720765C212yesmaximal, complemented
3#41×C3⋊Dic3C311yesmaximal, characteristic, complemented
4Dic3×#41⋊C3—31nomaximal
5#41⋊H(3,𝔽3)C412yescomplemented
6C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
7C4×(C3⋊Dic3)⋊C3—41nomaximal
8C2×C32×#41C2×C311yescharacteristic
9C2×C3×#41⋊C3—21no—
10C2×C3×#41⋊C3D311yescharacteristic
11Q8×C3⋊Dic3C2×C312yes—
12D4×C3⋊Dic3—32no—
13#64135—31no—
14C2×C4×C3⋊Dic3—31no—
15Dic3×SL(2,𝔽3)—32no—
16C2×C4×H(3,𝔽3)—41no—
17C2×(C3⋊Dic3)⋊C3—42no—
18Q8⋊H(3,𝔽3)C2×C411yescharacteristic, complemented
19Dic3×#41—31no—
20Dic3×#41—91no—
21C4×#41⋊C3—91no—
22C32×#41—32no—
23C32×#41C3×C412yescomplemented
24C2×C32×Q8C22×C311yescharacteristic
25C2×C32×D4—31no—
26C22×C32×C4—31no—
27C22×C3⋊Dic3—32no—
28C3×#41⋊C3—22no—
29C3×#41⋊C3Dic312yescomplemented
30C2×C3×SL(2,𝔽3)—21no—
31C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
32#3757—32no—
33C4⋊(C3⋊Dic3)—32no—
34C4×C3⋊Dic3—31no—
35C4×C3⋊Dic3—32no—
36C4×C3⋊Dic3A411yescharacteristic
37C3×C4×Dic3—121no—
38C22×H(3,𝔽3)—41no—
39C4×H(3,𝔽3)—42no—
40(C3⋊Dic3)⋊C3—44no—
41C2×C3×#41—31no—
42C2×C3×#41C3×D311yescharacteristic
43C2×#41⋊C3—31no—
44C2×#41⋊C3—61no—
45Q8×Dic3—32no—
46Q8×Dic3—92no—
47D4×Dic3—96no—
48D4×Dic3—92no—
49#926—93no—
50#926—91no—
51C2×C4×Dic3—91no—
52C2×C4×Dic3—93no—
53C4×SL(2,𝔽3)—92no—
54C23×C32—31no—
55C32×Q8—31no—
56C32×Q8C2×C3×C411yescharacteristic
57C32×D4—32no—
58C32×D4—32no—
59C2×C32×C4—31no—
60C2×C32×C4—32no—
61C2×C32×C4—32no—
62C2×C32×C4—41no—
63C2×C32×C4—81no—
64C2×C32×C4C2×A411yescharacteristic
65C2×C3⋊Dic3—32no—
66C2×C3⋊Dic3—64no—
67C2×C3⋊Dic3C2×A412yes—
68C2×C3×Dic3—122no—
69C3×SL(2,𝔽3)—21no—
70C3×SL(2,𝔽3)C2×Dic311yescharacteristic, complemented
71C4×#41—91noSylow
72C2×H(3,𝔽3)—41no—
73C2×H(3,𝔽3)—42no—
74C3×#41—36no—
75C3×#41—32no—
76C3×#41—32no—
77C3×#41C3×Dic312yes—
78C2×C3×Q8—31no—
79C2×C3×Q8C2×C3×D311yescharacteristic
80C2×C3×D4—31no—
81C2×C3×D4—33no—
82C22×C3×C4—33no—
83C22×C3×C4—31no—
84C22×Dic3—92no—
85C22×Dic3—96no—
86#41⋊C3—32no—
87#41⋊C3—62no—
88C2×SL(2,𝔽3)—31no—
89C2×SL(2,𝔽3)—61no—
90C3×C42—361no—
91#217—96no—
92#217—92no—
93C4⋊Dic3—96no—
94C4⋊Dic3—92no—
95C4×Dic3—31no—
96C4×Dic3—91no—
97C4×Dic3—92no—
98C4×Dic3—93no—
99C4×Dic3—96no—
100C4×Dic3—91no—
101C22×C32—32no—
102C22×C32—41no—
103C22×C32—62no—
104C22×C32—81no—
105C22×C32C22×A411yescharacteristic
106C32×C4—31no—
107C32×C4—31no—
108C32×C4—42no—
109C32×C4—82no—
110C32×C4C4×A412yes—
111C3⋊Dic3—62no—
112C3⋊Dic3#41⋊C314yescomplemented
113C3×Dic3—124no—
114C2×#41(C3⋊D3)⋊C311yescharacteristic
115C4×Q8—92no—
116C4×D4—272no—
117#117—271no—
118C2×C42—271no—
119H(3,𝔽3)—41noSylow
120C23×C3—33no—
121C23×C3—31no—
122C3×Q8—31no—
123C3×Q8—33no—
124C3×Q8—31no—
125C3×Q8C2×C3×Dic311yescharacteristic
126C3×D4—32no—
127C3×D4—36no—
128C3×D4—36no—
129C3×D4—32no—
130C2×C3×C4—33no—
131C2×C3×C4—36no—
132C2×C3×C4—36no—
133C2×C3×C4—31no—
134C2×C3×C4—32no—
135C2×C3×C4—32no—
136C2×C3×C4—31no—
137C2×C3×C4—121no—
138C2×C3×C4—241no—
139C2×C3×C4—362no—
140C2×C3×C4D3×A411yescharacteristic
141C2×Dic3—32no—
142C2×Dic3—92no—
143C2×Dic3—96no—
144C2×Dic3—92no—
145C2×Dic3—184no—
146C2×Dic3—1812no—
147SL(2,𝔽3)—31no—
148SL(2,𝔽3)—61no—
149C2×C32—42no—
150C2×C32—41no—
151C2×C32—62no—
152C2×C32—82no—
153C2×C32—81no—
154C2×C32C2×C4×A411yescharacteristic
155C2×C32C2×#41⋊C312yes—
156#41—32no—
157#41(C3⋊Dic3)⋊C312yescomplemented
158C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
159C2×D4—31no—
160C22×C4—31no—
161C22×C4—272no—
162C4⋊C4—272no—
163C22⋊C4—272no—
164C42—91no—
165C42—272no—
166C42—271no—
167C22×C3—36no—
168C22×C3—32no—
169C22×C3—31no—
170C22×C3—66no—
171C22×C3—62no—
172C22×C3—121no—
173C22×C3—241no—
174C22×C3C2×D3×A411yescharacteristic
175C3×C4—33no—
176C3×C4—33no—
177C3×C4—31no—
178C3×C4—31no—
179C3×C4—32no—
180C3×C4—122no—
181C3×C4—242no—
182C3×C4—364no—
183C3×C4Dic3×A412yes—
184Dic3—34no—
185Dic3—94no—
186Dic3—186no—
187Dic3—182no—
188C32—41no—
189C32—81no—
190C32C4×#41⋊C311yescharacteristic, complemented
191C23—31no—
192Q8—31no—
193Q8C2×(C3⋊Dic3)⋊C311yescharacteristic
194D4—32no—
195D4—32no—
196C2×C4—31no—
197C2×C4—32no—
198C2×C4—32no—
199C2×C4—92no—
200C2×C4—272no—
201C2×C4—544no—
202C2×C4(C3⋊D3)⋊A411yescharacteristic
203C2×C3—32no—
204C2×C3—31no—
205C2×C3—66no—
206C2×C3—62no—
207C2×C3—122no—
208C2×C3—121no—
209C2×C3—242no—
210C2×C3—241no—
211C2×C3D3×#41⋊C312yes—
212C2×C3C2×Dic3×A411yescharacteristic
213C22—32no—
214C22—62no—
215C22C2×(C3⋊D3)⋊A411yescharacteristic
216C4—31no—
217C4—31no—
218C4—94no—
219C4—542no—
220C4(C3⋊Dic3)⋊A412yes—
221C3—31no—
222C3—121no—
223C3—241no—
224C3Dic3×#41⋊C311yescharacteristic
225C2—62no—
226C2#1172243712yes—
227C2C2×(C3⋊Dic3)⋊A411yescharacteristic
subgroupG/H
center, hypercenterC2×C4(C3⋊D3)⋊A4
commutator, hypocenterC32×Q8C2×C3×C4
Frattini, socleC22×C3C2×D3×A4
FittingC2×C32×#41C2×C3
prime radicalC2×#41⋊H(3,𝔽3)C2
cosocleC2×C32×Q8C22×C3
LCSG ▷ C32×Q8
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11720765C2semidirect
3. #41×C3⋊Dic3C3semidirect
5. #41⋊H(3,𝔽3)C4semidirect
18. Q8⋊H(3,𝔽3)C2×C4semidirect
23. C32×#41C3×C4semidirect
29. C3×#41⋊C3Dic3semidirect
70. C3×SL(2,𝔽3)C2×Dic3semidirect
112. C3⋊Dic3#41⋊C3semidirect
157. #41(C3⋊Dic3)⋊C3semidirect
190. C32C4×#41⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
#117207652
#41×C3⋊Dic33
Dic3×#41⋊C33
C4×(C3⋊Dic3)⋊C34