Finite Group Databasebeta

← C3×#2071 #72786

Subgroups — C3×#2071 #72786

Order 384.

274 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 100 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×C3×#102C212yesmaximal
2C3×#390C211yesmaximal, characteristic
3C3×#384C211yesmaximal, characteristic, complemented
4C3×#340C212yesmaximal, complemented
5C3×#336C211yesmaximal, characteristic, complemented
6#2071C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×D4C2211yescharacteristic
8C22×C3×C8C2212yes—
9C2×C3×C4⋊C4—21no—
10C2×C3×C4⋊C4—21no—
11C2×C3×C4⋊C4C2212yes—
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C22⋊C4—21no—
14C2×C3×C22⋊C4—21no—
15C2×C3×C42C2211yescharacteristic
16C3×#102—24no—
17C2×#102C2×C312yes—
18#390C2×C311yescharacteristic
19#384C2×C311yescharacteristic, complemented
20#340C2×C312yescomplemented
21#336C2×C311yescharacteristic, complemented
22C24×C3—21no—
23C2×C3×D4—22no—
24C2×C3×D4—22no—
25C2×C3×D4—21no—
26C2×C3×D4D412yes—
27C22×C3×C4—21no—
28C22×C3×C4—22no—
29C22×C3×C4D411yescharacteristic
30C22×C3×C4D411yescharacteristic
31C22×C3×C4C2311yescharacteristic
32C2×C3×C8—22no—
33C2×C3×C8—22no—
34C2×C3×C8—22no—
35C3×C4⋊C4—22no—
36C3×C4⋊C4—22no—
37C3×C4⋊C4—22no—
38C3×C4⋊C4—42no—
39C3×C4⋊C4—42no—
40C3×C22⋊C4—42no—
41C3×C22⋊C4—42no—
42C3×C42—22no—
43C22×D4C22×C311yescharacteristic
44C22×C8C22×C312yes—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4C22×C311yescharacteristic
48C2×C4⋊C4C22×C312yes—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—21no—
51C2×C42C22×C311yescharacteristic
52#102—24no—
53C23×C3—22no—
54C23×C3—41no—
55C23×C3—41no—
56C23×C3—41no—
57C23×C3—41no—
58C23×C3—41no—
59C23×C3—41no—
60C23×C3C2×D411yescharacteristic
61C3×D4—24no—
62C3×D4—42no—
63C2×C3×C4—21no—
64C2×C3×C4—21no—
65C2×C3×C4—21no—
66C2×C3×C4—21no—
67C2×C3×C4—42no—
68C2×C3×C4—41no—
69C2×C3×C4—41no—
70C2×C3×C4—42no—
71C2×C3×C4—41no—
72C2×C3×C4—42no—
73C2×C3×C4D812yes—
74C2×C3×C4SD1612yes—
75C2×C3×C4C2×D411yescharacteristic
76C2×C3×C4#4112yes—
77C2×C3×C4#4112yes—
78C2×C3×C4#4111yescharacteristic
79C3×C8—42no—
80C24—21no—
81C2×D4—22no—
82C2×D4—22no—
83C2×D4—21no—
84C2×D4C3×D412yes—
85C22×C4—21no—
86C22×C4—22no—
87C22×C4C3×D411yescharacteristic
88C22×C4C3×D411yescharacteristic
89C22×C4C23×C311yescharacteristic
90C2×C8—22no—
91C2×C8—22no—
92C2×C8—22no—
93C4⋊C4—22no—
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—42no—
97C4⋊C4—42no—
98C22⋊C4—42no—
99C22⋊C4—42no—
100C42—22no—
101C22×C3—42no—
102C22×C3—42no—
103C22×C3—42no—
104C22×C3—81no—
105C22×C3—81no—
106C22×C3—81no—
107C22×C3—81no—
108C22×C3#12111yescharacteristic
109C22×C3#12311yescharacteristic
110C22×C3#12411yescharacteristic
111C22×C3C2×D811yescharacteristic
112C22×C3C2×SD1611yescharacteristic
113C22×C3AGL(1,ℤ/8)11yescharacteristic
114C22×C3#13711yescharacteristic
115C3×C4—22no—
116C3×C4—22no—
117C3×C4—82no—
118C3×C4—81no—
119C3×C4#12312yes—
120C3×C4#12612yes—
121C23—22no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23C2×C3×D411yescharacteristic
129D4—24no—
130D4—42no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4C3×D812yes—
142C2×C4C3×SD1612yes—
143C2×C4C2×C3×D411yescharacteristic
144C2×C4C3×#4112yes—
145C2×C4C3×#4112yes—
146C2×C4C3×#4111yescharacteristic
147C8—42no—
148C2×C3—82no—
149C2×C3#39711yescharacteristic
150C2×C3#45911yescharacteristic
151C2×C3#46111yescharacteristic
152C2×C3#48011yescharacteristic
153C2×C3#48111yescharacteristic
154C2×C3#48611yescharacteristic
155C2×C3#48911yescharacteristic
156C22—42no—
157C22—42no—
158C22—42no—
159C22—81no—
160C22—81no—
161C22—81no—
162C22—81no—
163C22C3×#12111yescharacteristic
164C22C3×#12311yescharacteristic
165C22C3×#12411yescharacteristic
166C22C2×C3×D811yescharacteristic
167C22C2×C3×SD1611yescharacteristic
168C22C3×AGL(1,ℤ/8)11yescharacteristic
169C22C3×#13711yescharacteristic
170C4—22no—
171C4—22no—
172C4—82no—
173C4—81no—
174C4C3×#12312yes—
175C4C3×#12612yes—
176C3#207111yescharacteristic, direct product factor, Sylow
177C2—82no—
178C2C3×#39711yescharacteristic
179C2C3×#45911yescharacteristic
180C2C3×#46111yescharacteristic
181C2C3×#48011yescharacteristic
182C2C3×#48111yescharacteristic
183C2C3×#48611yescharacteristic
184C2C3×#48911yescharacteristic
subgroupG/H
center, socleC23×C3C2×D4
commutator, Frattini, cosocleC22×C4C23×C3
prime radicalC22×C3×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#384C2semidirect
4. C3×#340C2semidirect
5. C3×#336C2semidirect
6. #2071C3direct (N × H)
19. #384C2×C3semidirect
20. #340C2×C3semidirect
21. #336C2×C3semidirect
176. C3#2071direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1022
C3×#3902
C3×#3842
C3×#3402
C3×#3362
#20713