Finite Group Databasebeta

← H(3,𝔽3)×#484 #13145211

Subgroups — H(3,𝔽3)×#484 #13145211

Order 1728.

572 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 189 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
1H(3,𝔽3)×C22⋊Q8C211yesmaximal, characteristic
2C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
3H(3,𝔽3)×#107C212yesmaximal, complemented
4H(3,𝔽3)×#103C212yesmaximal, complemented
5H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic
6C32×#484C314yesmaximal, complemented
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8C22×C4×H(3,𝔽3)—21no—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×C8×H(3,𝔽3)C2212yes—
11C4⋊C4×H(3,𝔽3)—21no—
12C4⋊C4×H(3,𝔽3)—21no—
13C4⋊C4×H(3,𝔽3)C2212yes—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C22⋊C4×H(3,𝔽3)—21no—
16C32×C22⋊Q8C2×C314yes—
17C2×C32×C4⋊C4C2×C314yes—
18C32×#107C2×C318yescomplemented
19C32×#103C2×C318yescomplemented
20C32×C22⋊C8C2×C314yes—
21C23×H(3,𝔽3)D411yescharacteristic
22Q8×H(3,𝔽3)—22no—
23C2×C4×H(3,𝔽3)—22no—
24C2×C4×H(3,𝔽3)—22no—
25C2×C4×H(3,𝔽3)—21no—
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)—42no—
28C2×C4×H(3,𝔽3)D411yescharacteristic
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C8×H(3,𝔽3)—22no—
31C3×#484—34no—
32C3×#484C3211yescharacteristic
33C2×C32×Q8C22×C314yes—
34C22×C32×C4—24no—
35C22×C32×C4C22×C314yes—
36C2×C32×C8C22×C318yes—
37C32×C4⋊C4—24no—
38C32×C4⋊C4—24no—
39C32×C4⋊C4C22×C318yes—
40C32×C4⋊C4C22×C314yes—
41C32×C22⋊C4—24no—
42C22×H(3,𝔽3)—21no—
43C22×H(3,𝔽3)—21no—
44C22×H(3,𝔽3)Q1612yes—
45C22×H(3,𝔽3)C2×D411yescharacteristic
46C4×H(3,𝔽3)—21no—
47C4×H(3,𝔽3)—41no—
48C4×H(3,𝔽3)—41no—
49C4×H(3,𝔽3)—42no—
50C4×H(3,𝔽3)#4112yes—
51C3×C22⋊Q8—34no—
52C3×C22⋊Q8C2×C3211yescharacteristic
53C2×C3×C4⋊C4—34no—
54C2×C3×C4⋊C4C2×C3211yescharacteristic
55C3×#107—38no—
56C3×#107C2×C3212yes—
57C3×#103—38no—
58C3×#103C2×C3212yes—
59C3×C22⋊C8—34no—
60C3×C22⋊C8C2×C3211yescharacteristic
61C23×C32C3×D414yes—
62C32×Q8—28no—
63C2×C32×C4—28no—
64C2×C32×C4—24no—
65C2×C32×C4—24no—
66C2×C32×C4—28no—
67C2×C32×C4—48no—
68C2×C32×C4C3×D414yes—
69C2×C32×C4C23×C314yes—
70C32×C8—28no—
71#484H(3,𝔽3)11yescharacteristic, direct product factor, Sylow
72C2×H(3,𝔽3)—22no—
73C2×H(3,𝔽3)#12311yescharacteristic
74C2×H(3,𝔽3)C2×Q1611yescharacteristic
75C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
76C2×C3×Q8—34no—
77C2×C3×Q8C22×C3211yescharacteristic
78C22×C3×C4—21no—
79C22×C3×C4—34no—
80C22×C3×C4—64no—
81C22×C3×C4C22×C3211yescharacteristic
82C2×C3×C8—38no—
83C2×C3×C8C22×C3212yes—
84C3×C4⋊C4—21no—
85C3×C4⋊C4—21no—
86C3×C4⋊C4—38no—
87C3×C4⋊C4—34no—
88C3×C4⋊C4—64no—
89C3×C4⋊C4—64no—
90C3×C4⋊C4C22×C3212yes—
91C3×C4⋊C4C22×C3211yescharacteristic
92C3×C22⋊C4—21no—
93C3×C22⋊C4—64no—
94C22×C32—24no—
95C22×C32—24no—
96C22×C32C3×Q1618yes—
97C22×C32C2×C3×D414yes—
98C32×C4—24no—
99C32×C4—44no—
100C32×C4—44no—
101C32×C4—48no—
102C32×C4C3×#4118yes—
103C22⋊Q8C2×H(3,𝔽3)11yescharacteristic
104C2×C4⋊C4C2×H(3,𝔽3)11yescharacteristic
105#107C2×H(3,𝔽3)12yescomplemented
106#103C2×H(3,𝔽3)12yescomplemented
107C22⋊C8C2×H(3,𝔽3)11yescharacteristic
108H(3,𝔽3)#48411yescharacteristic, direct product factor, Sylow
109C23×C3—34no—
110C23×C3C32×D411yescharacteristic
111C3×Q8—22no—
112C3×Q8—68no—
113C2×C3×C4—22no—
114C2×C3×C4—21no—
115C2×C3×C4—21no—
116C2×C3×C4—22no—
117C2×C3×C4—34no—
118C2×C3×C4—34no—
119C2×C3×C4—42no—
120C2×C3×C4—68no—
121C2×C3×C4—64no—
122C2×C3×C4—64no—
123C2×C3×C4—68no—
124C2×C3×C4—128no—
125C2×C3×C4C32×D411yescharacteristic
126C2×C3×C4C23×C3211yescharacteristic
127C3×C8—22no—
128C3×C8—68no—
129C2×C32—28no—
130C2×C32C3×#12314yes—
131C2×C32C2×C3×Q1614yes—
132C2×C32C3×AGL(1,ℤ/8)14yes—
133C2×Q8C22×H(3,𝔽3)11yescharacteristic
134C22×C4—21no—
135C22×C4C22×H(3,𝔽3)11yescharacteristic
136C2×C8C22×H(3,𝔽3)12yes—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4C22×H(3,𝔽3)12yes—
140C4⋊C4C22×H(3,𝔽3)11yescharacteristic
141C22⋊C4—21no—
142C22×C3—21no—
143C22×C3—21no—
144C22×C3—34no—
145C22×C3—38no—
146C22×C3—64no—
147C22×C3—64no—
148C22×C3C32×Q1612yes—
149C22×C3C2×C32×D411yescharacteristic
150C3×C4—21no—
151C3×C4—38no—
152C3×C4—41no—
153C3×C4—42no—
154C3×C4—41no—
155C3×C4—64no—
156C3×C4—128no—
157C3×C4—124no—
158C3×C4—124no—
159C3×C4C32×#4112yes—
160C32C3×#48414yescomplemented
161C23D4×H(3,𝔽3)11yescharacteristic
162Q8—22no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4D4×H(3,𝔽3)11yescharacteristic
169C2×C4C23×H(3,𝔽3)11yescharacteristic
170C8—22no—
171C2×C3—22no—
172C2×C3—34no—
173C2×C3—34no—
174C2×C3—34no—
175C2×C3—68no—
176C2×C3C32×#12311yescharacteristic
177C2×C3C2×C32×Q1611yescharacteristic
178C2×C3C32×AGL(1,ℤ/8)11yescharacteristic
179C22—21no—
180C22—21no—
181C22Q16×H(3,𝔽3)12yes—
182C22C2×D4×H(3,𝔽3)11yescharacteristic
183C4—21no—
184C4—41no—
185C4—42no—
186C4—41no—
187C4#41×H(3,𝔽3)12yes—
188C3—34no—
189C3C32×#48411yescharacteristic
190C2—22no—
191C2H(3,𝔽3)×#12311yescharacteristic
192C2C2×Q16×H(3,𝔽3)11yescharacteristic
193C2H(3,𝔽3)×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
center, socleC22×C3C2×C32×D4
commutator, Frattini, cosocleC2×C3×C4C23×C32
prime radicalC23×H(3,𝔽3)D4
LCSG ▷ C2×C3×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C4×H(3,𝔽3) ◁ G

Complements

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

NH ≅ G/Ntype
3. H(3,𝔽3)×#107C2semidirect
4. H(3,𝔽3)×#103C2semidirect
6. C32×#484C3semidirect
18. C32×#107C2×C3semidirect
19. C32×#103C2×C3semidirect
71. #484H(3,𝔽3)direct (N × H)
105. #107C2×H(3,𝔽3)semidirect
106. #103C2×H(3,𝔽3)semidirect
108. H(3,𝔽3)#484direct (N × H)
160. C32C3×#484semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C22⋊Q82
C2×C4⋊C4×H(3,𝔽3)2
H(3,𝔽3)×#1072
H(3,𝔽3)×#1032
H(3,𝔽3)×C22⋊C82
C32×#4843