Finite Group Databasebeta

← #13147073

Subgroups — #13147073

Order 1728.

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

normal subgroups only

This group has 402 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#11720738C212yesmaximal, complemented
3C3×#5426—33nomaximal
4C3×#5232—31nomaximal
5#41⋊H(3,𝔽3)C412yescomplemented
6C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
7C4⋊#1084—41nomaximal
8C2×C32×#41D311yescharacteristic
9C2×C3×#41⋊C3D313yes—
10C2×C3×C4⋊Dic3—31no—
11C2×C3×#208—31no—
12C3×#859—36no—
13C3×#835—32no—
14C3×#832—32no—
15C2×C4×H(3,𝔽3)—41no—
16C2×#1084—42no—
17Q8⋊H(3,𝔽3)C2×C411yescharacteristic, complemented
18#5426—33no—
19C3×#417—91no—
20#5232—31no—
21C32×#41—32no—
22C32×#41Dic312yescomplemented
23C2×C32×Q8C2×D311yescharacteristic
24C2×C32×D4—31no—
25C22×C32×C4—31no—
26C22×C3×Dic3—61no—
27C3×#41⋊C3Dic316yescomplemented
28C2×C3×SL(2,𝔽3)C2×D313yes—
29C3×C4⋊Dic3—32no—
30C3×C4⋊Dic3—61no—
31C3×C4⋊Dic3—123no—
32C3×#208—62no—
33C2×C3×C3⋊C8—32no—
34C22×H(3,𝔽3)—41no—
35C4×H(3,𝔽3)—42no—
36#1084—82no—
37C2×C3×#41—31no—
38C2×C3×#41C3⋊D311yescharacteristic
39C2×#41⋊C3—33no—
40C2×C3×M16—91no—
41C2×C3×C4⋊C4—91no—
42C2×C4⋊Dic3—31no—
43C2×#208—31no—
44#859—36no—
45C3×#103—92no—
46C3×#102—92no—
47#835—32no—
48#832—32no—
49C23×C32—31no—
50C32×Q8—31no—
51C32×Q8C2×Dic311yescharacteristic
52C32×D4—32no—
53C32×D4—61no—
54C2×C32×C4—31no—
55C2×C32×C4—32no—
56C2×C32×C4—32no—
57C2×C32×C4—43no—
58C2×C32×C4S411yescharacteristic
59C2×C3×Dic3—62no—
60C2×C3×Dic3—126no—
61C2×C3×Dic3—122no—
62C3×SL(2,𝔽3)C2×Dic313yescomplemented
63C3×C3⋊C8—62no—
64#417—91noSylow
65C2×H(3,𝔽3)—41no—
66C2×H(3,𝔽3)—41no—
67C2×H(3,𝔽3)—41no—
68C3×#41—32no—
69C3×#41—32no—
70C3×#41—32no—
71C3×#41—62no—
72C3×#41C3⋊Dic312yes—
73C2×C3×Q8—31no—
74C2×C3×Q8C2×C3⋊D311yescharacteristic
75C2×C3×D4—31no—
76C2×C3×D4—31no—
77C2×C3×D4—61no—
78C22×C3×C4—31no—
79C22×C3×C4—31no—
80C22×C3×C4—61no—
81C22×C3×C4—181no—
82C22×Dic3—61no—
83#41⋊C3—36no—
84C2×SL(2,𝔽3)—33no—
85C3×M16—182no—
86C2×C3×C8—92no—
87C3×C4⋊C4—92no—
88C3×C4⋊C4—181no—
89C4⋊Dic3—32no—
90C4⋊Dic3—61no—
91C4⋊Dic3—123no—
92#208—62no—
93C2×C3⋊C8—32no—
94C22×C32—32no—
95C22×C32—43no—
96C22×C32—61no—
97C22×C32—61no—
98C22×C32C2×S411yescharacteristic
99C32×C4—31no—
100C32×C4—31no—
101C32×C4—46no—
102C32×C4SL(2,ℤ/4)12yes—
103C3×Dic3—122no—
104C3×Dic3—246no—
105C2×#41#27211yescharacteristic
106C2×M16—91no—
107C2×C4⋊C4—91no—
108#103—92no—
109#102—92no—
110H(3,𝔽3)—41noSylow
111C23×C3—31no—
112C23×C3—31no—
113C23×C3—61no—
114C3×Q8—31no—
115C3×Q8—31no—
116C3×Q8—31no—
117C3×Q8—61no—
118C3×Q8C2×C3⋊Dic311yescharacteristic
119C3×D4—32no—
120C3×D4—32no—
121C3×D4—62no—
122C3×D4—61no—
123C3×D4—61no—
124C3×D4—62no—
125C2×C3×C4—31no—
126C2×C3×C4—32no—
127C2×C3×C4—32no—
128C2×C3×C4—31no—
129C2×C3×C4—32no—
130C2×C3×C4—32no—
131C2×C3×C4—31no—
132C2×C3×C4—62no—
133C2×C3×C4—62no—
134C2×C3×C4—61no—
135C2×C3×C4—123no—
136C2×C3×C4—182no—
137C2×C3×C4—362no—
138C2×C3×C4C3⋊S411yescharacteristic
139C2×Dic3—62no—
140C2×Dic3—126no—
141C2×Dic3—122no—
142SL(2,𝔽3)—33no—
143C3×C8—182no—
144C3⋊C8—62no—
145C2×C32—43no—
146C2×C32—43no—
147C2×C32—43no—
148C2×C32—62no—
149C2×C32#98411yescharacteristic
150C2×C32#98611yescharacteristic
151C2×C32C2×SL(2,ℤ/4)11yescharacteristic
152#41—32no—
153#41#108412yescomplemented
154C2×Q8C2×#27211yescharacteristic
155C2×D4—31no—
156C22×C4—31no—
157C22×C4—181no—
158M16—182no—
159C2×C8—92no—
160C4⋊C4—92no—
161C4⋊C4—181no—
162C22×C3—32no—
163C22×C3—32no—
164C22×C3—31no—
165C22×C3—62no—
166C22×C3—61no—
167C22×C3—61no—
168C22×C3—61no—
169C22×C3—61no—
170C22×C3—121no—
171C22×C3—121no—
172C22×C3—123no—
173C22×C3C2×C3⋊S411yescharacteristic
174C3×C4—31no—
175C3×C4—31no—
176C3×C4—31no—
177C3×C4—31no—
178C3×C4—32no—
179C3×C4—61no—
180C3×C4—61no—
181C3×C4—126no—
182C3×C4—362no—
183C3×C4C3⋊SL(2,ℤ/4)12yes—
184Dic3—122no—
185Dic3—246no—
186C32—43no—
187C32#542611yescharacteristic, complemented
188C23—31no—
189Q8—31no—
190Q8C2×#108411yescharacteristic
191D4—32no—
192D4—61no—
193C2×C4—31no—
194C2×C4—32no—
195C2×C4—32no—
196C2×C4—182no—
197C2×C4—362no—
198C2×C4#625811yescharacteristic
199C8—182no—
200C2×C3—31no—
201C2×C3—31no—
202C2×C3—31no—
203C2×C3—62no—
204C2×C3—62no—
205C2×C3—122no—
206C2×C3—123no—
207C2×C3—123no—
208C2×C3—123no—
209C2×C3#6426511yescharacteristic
210C2×C3#6426711yescharacteristic
211C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
212C22—32no—
213C22—61no—
214C22—61no—
215C22C2×#625811yescharacteristic
216C4—31no—
217C4—31no—
218C4—362no—
219C4#8861312yes—
220C3—31no—
221C3—123no—
222C3#1059366011yescharacteristic
223C2—62no—
224C2#1172239211yescharacteristic
225C2#1172239411yescharacteristic
226C2C2×#8861311yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C2×C3⋊S4
commutator, hypocenterQ8⋊H(3,𝔽3)C2×C4
FittingC2×C32×#41D3
prime radicalC2×#41⋊H(3,𝔽3)C2
cosocleC2×Q8⋊H(3,𝔽3)C22
hypercenterC2×C3×C4C3⋊S4
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. #11720738C2semidirect
5. #41⋊H(3,𝔽3)C4semidirect
17. Q8⋊H(3,𝔽3)C2×C4semidirect
22. C32×#41Dic3semidirect
27. C3×#41⋊C3Dic3semidirect
62. C3×SL(2,𝔽3)C2×Dic3semidirect
153. #41#1084semidirect
187. C32#5426semidirect

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
#117207382
C3×#54263
C3×#52323
C4⋊#10844