Finite Group Databasebeta

← C2×#126⋊H(3,𝔽3) #13152186

Subgroups — C2×#126⋊H(3,𝔽3) #13152186

Order 1728.

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

normal subgroups only

This group has 381 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#126⋊H(3,𝔽3)C212yesmaximal, direct product factor
2C2×C32⋊(C42⋊C3)C211yesmaximal, characteristic, complemented
3C2×C32×#126C311yesmaximal, characteristic, complemented
4C2×C3×#126⋊C3C313yesmaximal, complemented
5C22×C22⋊H(3,𝔽3)—41nomaximal
6C32⋊(C42⋊C3)C2211yescharacteristic, complemented
7C32×#126—32no—
8C32×#126C2×C312yescomplemented
9C2×C32×C4⋊C4—31no—
10C2×C32×C22⋊C4—31no—
11C2×C32×C42C2×C311yescharacteristic, complemented
12C3×#126⋊C3C2×C316yescomplemented
13C2×C3×C42⋊C3C2×C313yescomplemented
14C2×C22⋊H(3,𝔽3)—41no—
15C2×C22⋊H(3,𝔽3)—42no—
16C2×C3×#126—31no—
17C2×C3×#126C3211yescharacteristic
18C2×#126⋊C3—33no—
19C24×C32A411yescharacteristic
20C22×C3×A4—43no—
21C22×C32×C4—31no—
22C22×C32×C4—31no—
23C32×C4⋊C4—32no—
24C32×C4⋊C4—32no—
25C32×C22⋊C4—32no—
26C32×C22⋊C4—32no—
27C32×C42—31no—
28C32×C42C22×C311yescharacteristic, complemented
29C3×C42⋊C3C22×C313yescomplemented
30C22×H(3,𝔽3)—161no—
31C22⋊H(3,𝔽3)—41no—
32C3×#126—36no—
33C3×#126—32no—
34C3×#126—32no—
35C3×#126C2×C3212yes—
36C2×C3×C4⋊C4—31no—
37C2×C3×C4⋊C4—33no—
38C2×C3×C22⋊C4—33no—
39C2×C3×C22⋊C4—31no—
40C2×C3×C42—31no—
41C2×C3×C42C2×C3211yescharacteristic
42#126⋊C3—36no—
43C2×C42⋊C3—33no—
44C23×C32—61no—
45C23×C32—61no—
46C23×C32C2×A411yescharacteristic
47C23×C32C2×A412yes—
48C2×C3×A4—43no—
49C2×C3×A4—46no—
50C2×C32×C4—31no—
51C2×C32×C4—31no—
52C2×C32×C4—32no—
53C2×C32×C4—61no—
54C2×C32×C4—61no—
55C2×C32×C4—61no—
56C2×C32×C4—61no—
57C2×#126H(3,𝔽3)11yescharacteristic, complemented, Sylow
58C2×H(3,𝔽3)—162no—
59C2×H(3,𝔽3)—161no—
60C24×C3—31no—
61C24×C3C3×A411yescharacteristic
62C22×A4—123no—
63C22×C3×C4—33no—
64C22×C3×C4—33no—
65C22×C3×C4—31no—
66C22×C3×C4—31no—
67C3×C4⋊C4—32no—
68C3×C4⋊C4—36no—
69C3×C4⋊C4—36no—
70C3×C4⋊C4—32no—
71C3×C22⋊C4—36no—
72C3×C22⋊C4—36no—
73C3×C22⋊C4—32no—
74C3×C22⋊C4—32no—
75C3×C42—31no—
76C3×C42—33no—
77C3×C42—31no—
78C3×C42C22×C3211yescharacteristic
79C42⋊C3—33no—
80C22×C32—31no—
81C22×C32—31no—
82C22×C32—41no—
83C22×C32—62no—
84C22×C32—121no—
85C22×C32—163no—
86C22×C32C22×A411yescharacteristic
87C3×A4—43no—
88C32×C4—61no—
89C32×C4—61no—
90C32×C4—62no—
91#126—32no—
92#126C2×H(3,𝔽3)12yescomplemented
93C2×C4⋊C4—31no—
94C2×C22⋊C4—31no—
95C2×C42C2×H(3,𝔽3)11yescharacteristic, complemented
96H(3,𝔽3)—161noSylow
97C23×C3—31no—
98C23×C3—32no—
99C23×C3—61no—
100C23×C3—61no—
101C23×C3—63no—
102C23×C3—63no—
103C23×C3C2×C3×A411yescharacteristic
104C23×C3C2×C3×A412yes—
105C2×A4—123no—
106C2×A4—126no—
107C2×C3×C4—33no—
108C2×C3×C4—33no—
109C2×C3×C4—36no—
110C2×C3×C4—31no—
111C2×C3×C4—31no—
112C2×C3×C4—32no—
113C2×C3×C4—61no—
114C2×C3×C4—61no—
115C2×C3×C4—63no—
116C2×C3×C4—63no—
117C2×C3×C4—63no—
118C2×C3×C4—63no—
119C2×C3×C4—61no—
120C2×C3×C4—61no—
121C2×C32—31no—
122C2×C32—31no—
123C2×C32—42no—
124C2×C32—166no—
125C2×C32—163no—
126C2×C32#126⋊C311yescharacteristic, complemented
127C24C22⋊H(3,𝔽3)11yescharacteristic
128C22×C4—31no—
129C22×C4—31no—
130C4⋊C4—32no—
131C4⋊C4—32no—
132C22⋊C4—32no—
133C22⋊C4—32no—
134C42—31no—
135C42C22×H(3,𝔽3)11yescharacteristic, complemented
136C22×C3—31no—
137C22×C3—33no—
138C22×C3—33no—
139C22×C3—31no—
140C22×C3—31no—
141C22×C3—41no—
142C22×C3—62no—
143C22×C3—66no—
144C22×C3—121no—
145C22×C3—121no—
146C22×C3—123no—
147C22×C3—483no—
148C22×C3C22×C3×A411yescharacteristic
149A4—123no—
150C3×C4—63no—
151C3×C4—63no—
152C3×C4—66no—
153C3×C4—61no—
154C3×C4—61no—
155C3×C4—62no—
156C32—163no—
157C32C2×#126⋊C311yescharacteristic, complemented
158C23—61no—
159C23—61no—
160C23C2×C22⋊H(3,𝔽3)11yescharacteristic
161C23C2×C22⋊H(3,𝔽3)12yes—
162C2×C4—31no—
163C2×C4—31no—
164C2×C4—32no—
165C2×C4—61no—
166C2×C4—61no—
167C2×C4—61no—
168C2×C4—61no—
169C2×C3—33no—
170C2×C3—33no—
171C2×C3—31no—
172C2×C3—31no—
173C2×C3—31no—
174C2×C3—42no—
175C2×C3—122no—
176C2×C3—486no—
177C2×C3—483no—
178C2×C3C3×#126⋊C311yescharacteristic
179C22—31no—
180C22—31no—
181C22—41no—
182C22—62no—
183C22—121no—
184C22C22×C22⋊H(3,𝔽3)11yescharacteristic
185C4—61no—
186C4—61no—
187C4—62no—
188C3—31no—
189C3—483no—
190C3C2×C3×#126⋊C311yescharacteristic
191C2—31no—
192C2—31no—
193C2—42no—
194C2#126⋊H(3,𝔽3)11yescharacteristic, direct product factor
subgroupG/H
centerC2×C3C3×#126⋊C3
commutator, cosocleC3×C42C22×C32
FrattiniC22×C3C22×C3×A4
FittingC2×C32×#126C3
socleC23×C3C2×C3×A4
hypercenterC2×C32#126⋊C3
hypocenterC42C22×H(3,𝔽3)
LCSG ▷ C3×C42 ▷ C42
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
1. #126⋊H(3,𝔽3)C2direct (N × H)
2. C2×C32⋊(C42⋊C3)C2semidirect
3. C2×C32×#126C3semidirect
4. C2×C3×#126⋊C3C3semidirect
6. C32⋊(C42⋊C3)C22semidirect
8. C32×#126C2×C3semidirect
11. C2×C32×C42C2×C3semidirect
12. C3×#126⋊C3C2×C3semidirect
13. C2×C3×C42⋊C3C2×C3semidirect
28. C32×C42C22×C3semidirect
29. C3×C42⋊C3C22×C3semidirect
57. C2×#126H(3,𝔽3)semidirect
92. #126C2×H(3,𝔽3)semidirect
95. C2×C42C2×H(3,𝔽3)semidirect
126. C2×C32#126⋊C3semidirect
135. C42C22×H(3,𝔽3)semidirect
157. C32C2×#126⋊C3semidirect
194. C2#126⋊H(3,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
#126⋊H(3,𝔽3)2
C2×C32⋊(C42⋊C3)2
C2×C32×#1263
C2×C3×#126⋊C33
C22×C22⋊H(3,𝔽3)4