Finite Group Databasebeta

← #13164890

Subgroups — #13164890

Order 1728.

972 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 972 conjugacy classes of subgroups — too many to draw. Only its 53 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
1C32⋊#997C211yesmaximal, characteristic, complemented
2C32×#585C311yesmaximal, characteristic, complemented
3C3×#5950C313yesmaximal, complemented
4#41⋊H(3,𝔽3)—42nomaximal
5C4×C22⋊H(3,𝔽3)—43nomaximal
6C32×2−1+4—32no—
7C32×H(5,𝔽2)—33no—
8C32×H(5,𝔽2)C2×C311yescharacteristic
9C2×C32×#41—32no—
10C2×C32×#41—33no—
11C3×#997C2×C313yescomplemented
12C2×C22⋊H(3,𝔽3)—43no—
13Q8⋊H(3,𝔽3)—42no—
14C3×#585—31no—
15C3×#585C3211yescharacteristic
16#5950—33no—
17C32×#41—36no—
18C32×#41—36no—
19C32×#41—312no—
20C32×#41—32no—
21C32×#41A412yescomplemented
22C2×C32×Q8—33no—
23C2×C32×Q8—32no—
24C2×C32×D4—36no—
25C2×C32×D4—33no—
26C2×C32×D4—33no—
27C2×C32×D4—33no—
28C22×C32×C4—33no—
29C22×C32×C4—31no—
30C22×C32×C4A413yescomplemented
31C3×#41⋊C3—46no—
32C3×C4×A4—49no—
33C22⋊H(3,𝔽3)—43no—
34C4×H(3,𝔽3)—161no—
35C3×2−1+4—36no—
36C3×2−1+4—32no—
37C3×H(5,𝔽2)—39no—
38C3×H(5,𝔽2)—33no—
39C3×H(5,𝔽2)—31no—
40C3×H(5,𝔽2)C2×C3211yescharacteristic
41C2×C3×#41—36no—
42C2×C3×#41—39no—
43C2×C3×#41—32no—
44C2×C3×#41—33no—
45#997—33no—
46C23×C32—31no—
47C23×C32—33no—
48C23×C32C2×A413yes—
49C2×C3×A4—49no—
50C32×Q8—36no—
51C32×Q8C2×A412yes—
52C32×D4—32no—
53C32×D4—36no—
54C32×D4—312no—
55C2×C32×C4—33no—
56C2×C32×C4—36no—
57C2×C32×C4—33no—
58C2×C32×C4—32no—
59C2×C32×C4—33no—
60C2×C32×C4—33no—
61C3×SL(2,𝔽3)—46no—
62#585H(3,𝔽3)11yescharacteristic, complemented, Sylow
63C2×H(3,𝔽3)—161no—
64C3×#41—318no—
65C3×#41—36no—
66C3×#41—36no—
67C3×#41—318no—
68C3×#41—336no—
69C3×#41—312no—
70C3×#41—36no—
71C3×#41—32no—
72C3×#41—32no—
73C3×#41C3×A412yes—
74C2×C3×Q8—39no—
75C2×C3×Q8—33no—
76C2×C3×Q8—36no—
77C2×C3×Q8—32no—
78C2×C3×D4—36no—
79C2×C3×D4—318no—
80C2×C3×D4—39no—
81C2×C3×D4—33no—
82C2×C3×D4—39no—
83C2×C3×D4—39no—
84C2×C3×D4—33no—
85C2×C3×D4—33no—
86C22×C3×C4—39no—
87C22×C3×C4—33no—
88C22×C3×C4—33no—
89C22×C3×C4—31no—
90C22×C3×C4—33no—
91C22×C3×C4C3×A413yes—
92#41⋊C3—126no—
93C4×A4—129no—
94C22×C32—33no—
95C22×C32—32no—
96C22×C32—43no—
97C22×C32—121no—
98C22×C32—123no—
99C3×A4—49no—
100C32×C4—32no—
101C32×C4—33no—
102C32×C4—163no—
103C32×C4C22⋊A411yescharacteristic
1042−1+4—32no—
105H(5,𝔽2)—33no—
106H(5,𝔽2)C2×H(3,𝔽3)11yescharacteristic
107C2×#41—32no—
108C2×#41—33no—
109H(3,𝔽3)—161noSylow
110C23×C3—33no—
111C23×C3—39no—
112C23×C3—31no—
113C23×C3—33no—
114C23×C3—33no—
115C23×C3C2×C3×A413yes—
116C2×A4—129no—
117C3×Q8—318no—
118C3×Q8—36no—
119C3×Q8—32no—
120C3×Q8C2×C3×A412yes—
121C3×D4—32no—
122C3×D4—36no—
123C3×D4—318no—
124C3×D4—336no—
125C3×D4—36no—
126C3×D4—312no—
127C2×C3×C4—36no—
128C2×C3×C4—39no—
129C2×C3×C4—39no—
130C2×C3×C4—318no—
131C2×C3×C4—39no—
132C2×C3×C4—33no—
133C2×C3×C4—36no—
134C2×C3×C4—33no—
135C2×C3×C4—39no—
136C2×C3×C4—32no—
137C2×C3×C4—33no—
138C2×C3×C4—33no—
139SL(2,𝔽3)—126no—
140C2×C32—63no—
141C2×C32—62no—
142C2×C32—163no—
143C2×C32C2×C22⋊A411yescharacteristic
144#41—36no—
145#41—36no—
146#41—312no—
147#41—32no—
148#41C22⋊H(3,𝔽3)12yescomplemented
149C2×Q8—33no—
150C2×Q8—32no—
151C2×D4—36no—
152C2×D4—33no—
153C2×D4—33no—
154C2×D4—33no—
155C22×C4—33no—
156C22×C4—31no—
157C22×C4C22⋊H(3,𝔽3)13yescomplemented
158C22×C3—39no—
159C22×C3—36no—
160C22×C3—33no—
161C22×C3—32no—
162C22×C3—43no—
163C22×C3—123no—
164C22×C3—123no—
165C22×C3—129no—
166C22×C3—121no—
167C22×C3—123no—
168A4—129no—
169C3×C4—36no—
170C3×C4—39no—
171C3×C4—32no—
172C3×C4—33no—
173C3×C4—31no—
174C3×C4—483no—
175C3×C4C3×C22⋊A411yescharacteristic
176C32—163no—
177C32#595011yescharacteristic, complemented
178C23—31no—
179C23—33no—
180C23C2×C22⋊H(3,𝔽3)13yes—
181Q8—36no—
182Q8C2×C22⋊H(3,𝔽3)12yes—
183D4—32no—
184D4—36no—
185D4—312no—
186C2×C4—33no—
187C2×C4—36no—
188C2×C4—33no—
189C2×C4—32no—
190C2×C4—33no—
191C2×C4—33no—
192C2×C3—31no—
193C2×C3—69no—
194C2×C3—66no—
195C2×C3—63no—
196C2×C3—62no—
197C2×C3—483no—
198C2×C3C2×C3×C22⋊A411yescharacteristic
199C22—33no—
200C22—32no—
201C22—43no—
202C22—121no—
203C22—123no—
204C4—32no—
205C4—33no—
206C4C32⋊(C22⋊A4)11yescharacteristic
207C3—31no—
208C3—483no—
209C3C3×#595011yescharacteristic
210C2—63no—
211C2—62no—
212C2C2×C32⋊(C22⋊A4)11yescharacteristic
subgroupG/H
centerC3×C4C3×C22⋊A4
commutator, cosocleC3×H(5,𝔽2)C2×C32
Frattini, socleC2×C3C2×C3×C22⋊A4
FittingC32×#585C3
hypercenterC32×C4C22⋊A4
hypocenterH(5,𝔽2)C2×H(3,𝔽3)
LCSG ▷ C3×H(5,𝔽2) ▷ H(5,𝔽2)
UCSC1 ◁ C3×C4 ◁ C32×C4

Complements

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

NH ≅ G/Ntype
1. C32⋊#997C2semidirect
2. C32×#585C3semidirect
3. C3×#5950C3semidirect
11. C3×#997C2×C3semidirect
21. C32×#41A4semidirect
30. C22×C32×C4A4semidirect
62. #585H(3,𝔽3)semidirect
148. #41C22⋊H(3,𝔽3)semidirect
157. C22×C4C22⋊H(3,𝔽3)semidirect
177. C32#5950semidirect

Maximal subgroups

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

subgroupindex
C32⋊#9972
C32×#5853
C3×#59503
#41⋊H(3,𝔽3)4
C4×C22⋊H(3,𝔽3)4