Finite Group Databasebeta

← #13164884

Subgroups — #13164884

Order 1728.

1034 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 56 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
1#11722587C211yesmaximal, characteristic, complemented
2C22×C32×#41C311yesmaximal, characteristic, complemented
3C3×#5948C313yesmaximal, complemented
4#41⋊H(3,𝔽3)—44nomaximal
5C4×C22⋊H(3,𝔽3)—41nomaximal
6C2×C32×#41—34no—
7C2×C32×#41—34no—
8C22×C32×Q8C2×C311yescharacteristic
9C22×C32×D4—31no—
10C23×C32×C4—31no—
11C3×C22⋊SL(2,𝔽3)C2×C313yescomplemented
12C2×C22⋊H(3,𝔽3)—41no—
13Q8⋊H(3,𝔽3)—44no—
14C22×C3×#41—31no—
15C22×C3×#41C3211yescharacteristic
16#5948—33no—
17C24×C32—31no—
18C32×#41—312no—
19C32×#41—34no—
20C32×#41—34no—
21C32×#41A414yescomplemented
22C2×C32×Q8—34no—
23C2×C32×D4—312no—
24C22×C32×C4—36no—
25C22×C32×C4—36no—
26C22×C32×C4—31no—
27C22×C32×C4A411yescharacteristic
28C3×#41⋊C3—412no—
29C3×C4×A4—43no—
30C22⋊H(3,𝔽3)—41no—
31C4×H(3,𝔽3)—161no—
32C2×C3×#41—34no—
33C2×C3×#41—312no—
34C2×C3×#41—312no—
35C2×C3×#41—34no—
36C22×C3×Q8—31no—
37C22×C3×Q8C2×C3211yescharacteristic
38C22×C3×D4—31no—
39C22×C3×D4—33no—
40C23×C3×C4—33no—
41C23×C3×C4—31no—
42C22⋊SL(2,𝔽3)—33no—
43C23×C32—36no—
44C23×C32—63no—
45C23×C32—61no—
46C23×C32C2×A411yescharacteristic
47C2×C3×A4—43no—
48C32×Q8—34no—
49C32×Q8C2×A414yes—
50C32×D4—34no—
51C32×D4—312no—
52C2×C32×C4—36no—
53C2×C32×C4—312no—
54C2×C32×C4—31no—
55C2×C32×C4—34no—
56C2×C32×C4—31no—
57C3×SL(2,𝔽3)—412no—
58C22×#41H(3,𝔽3)11yescharacteristic, complemented, Sylow
59C2×H(3,𝔽3)—161no—
60C24×C3—33no—
61C24×C3—31no—
62C3×#41—312no—
63C3×#41—336no—
64C3×#41—312no—
65C3×#41—34no—
66C3×#41—312no—
67C3×#41—34no—
68C3×#41—34no—
69C3×#41C3×A414yes—
70C2×C3×Q8—312no—
71C2×C3×Q8—34no—
72C2×C3×D4—312no—
73C2×C3×D4—336no—
74C22×C3×C4—33no—
75C22×C3×C4—318no—
76C22×C3×C4—318no—
77C22×C3×C4—36no—
78C22×C3×C4—36no—
79C22×C3×C4—31no—
80C22×C3×C4—31no—
81C22×C3×C4C3×A411yescharacteristic
82#41⋊C3—1212no—
83C4×A4—123no—
84C22×C32—31no—
85C22×C32—31no—
86C22×C32—34no—
87C22×C32—66no—
88C22×C32—66no—
89C22×C32#41⋊C311yescharacteristic, complemented
90C3×A4—43no—
91C32×C4—31no—
92C32×C4—34no—
93C32×C4—163no—
94C32×C4C22⋊A411yescharacteristic
95C2×#41—34no—
96C2×#41—34no—
97C22×Q8C2×H(3,𝔽3)11yescharacteristic
98C22×D4—31no—
99C23×C4—31no—
100H(3,𝔽3)—161noSylow
101C23×C3—318no—
102C23×C3—36no—
103C23×C3—31no—
104C23×C3—63no—
105C23×C3—69no—
106C23×C3—63no—
107C23×C3—61no—
108C23×C3C2×C3×A411yescharacteristic
109C2×A4—123no—
110C3×Q8—312no—
111C3×Q8—34no—
112C3×Q8—34no—
113C3×Q8C2×C3×A414yes—
114C3×D4—34no—
115C3×D4—312no—
116C3×D4—336no—
117C3×D4—312no—
118C2×C3×C4—33no—
119C2×C3×C4—33no—
120C2×C3×C4—318no—
121C2×C3×C4—36no—
122C2×C3×C4—336no—
123C2×C3×C4—312no—
124C2×C3×C4—312no—
125C2×C3×C4—31no—
126C2×C3×C4—34no—
127C2×C3×C4—31no—
128SL(2,𝔽3)—1212no—
129C2×C32—31no—
130C2×C32—31no—
131C2×C32—64no—
132C2×C32—163no—
133C2×C32C2×C22⋊A411yescharacteristic
134C24—31no—
135#41—312no—
136#41—34no—
137#41—34no—
138#41C22⋊H(3,𝔽3)14yescomplemented
139C2×Q8—34no—
140C2×D4—312no—
141C22×C4—36no—
142C22×C4—36no—
143C22×C4—31no—
144C22×C4C22⋊H(3,𝔽3)11yescharacteristic
145C22×C3—31no—
146C22×C3—33no—
147C22×C3—33no—
148C22×C3—31no—
149C22×C3—312no—
150C22×C3—34no—
151C22×C3—31no—
152C22×C3—618no—
153C22×C3—618no—
154C22×C3—66no—
155C22×C3—66no—
156C22×C3C3×#41⋊C311yescharacteristic
157A4—123no—
158C3×C4—33no—
159C3×C4—312no—
160C3×C4—31no—
161C3×C4—34no—
162C3×C4—31no—
163C3×C4—483no—
164C3×C4C3×C22⋊A411yescharacteristic
165C32—163no—
166C32#594811yescharacteristic, complemented
167C23—36no—
168C23—63no—
169C23—61no—
170C23C2×C22⋊H(3,𝔽3)11yescharacteristic
171Q8—34no—
172Q8C2×C22⋊H(3,𝔽3)14yes—
173D4—34no—
174D4—312no—
175C2×C4—36no—
176C2×C4—312no—
177C2×C4—31no—
178C2×C4—34no—
179C2×C4—31no—
180C2×C3—33no—
181C2×C3—33no—
182C2×C3—31no—
183C2×C3—31no—
184C2×C3—31no—
185C2×C3—612no—
186C2×C3—64no—
187C2×C3—483no—
188C2×C3C2×C3×C22⋊A411yescharacteristic
189C22—31no—
190C22—31no—
191C22—34no—
192C22—66no—
193C22—66no—
194C22#41⋊H(3,𝔽3)11yescharacteristic, complemented
195C4—31no—
196C4—34no—
197C4C32⋊(C22⋊A4)11yescharacteristic
198C3—31no—
199C3—483no—
200C3C3×#594811yescharacteristic
201C2—31no—
202C2—31no—
203C2—64no—
204C2C2×C32⋊(C22⋊A4)11yescharacteristic
subgroupG/H
centerC3×C4C3×C22⋊A4
commutator, cosocleC22×C3×Q8C2×C32
FrattiniC2×C3C2×C3×C22⋊A4
FittingC22×C32×#41C3
socleC23×C3C2×C3×A4
hypercenterC32×C4C22⋊A4
hypocenterC22×Q8C2×H(3,𝔽3)
LCSG ▷ C22×C3×Q8 ▷ C22×Q8
UCSC1 ◁ C3×C4 ◁ C32×C4

Complements

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

NH ≅ G/Ntype
1. #11722587C2semidirect
2. C22×C32×#41C3semidirect
3. C3×#5948C3semidirect
11. C3×C22⋊SL(2,𝔽3)C2×C3semidirect
21. C32×#41A4semidirect
58. C22×#41H(3,𝔽3)semidirect
89. C22×C32#41⋊C3semidirect
138. #41C22⋊H(3,𝔽3)semidirect
166. C32#5948semidirect
194. C22#41⋊H(3,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
#117225872
C22×C32×#413
C3×#59483
#41⋊H(3,𝔽3)4
C4×C22⋊H(3,𝔽3)4