Finite Group Databasebeta

← C3×#2041 #72756

Subgroups — C3×#2041 #72756

Order 384.

402 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 108 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×#127C211yesmaximal, characteristic
2C2×C3×#102C212yesmaximal, complemented
3C2×C3×#102C212yesmaximal, complemented
4C3×#389C211yesmaximal, characteristic, complemented
5C3×#336C211yesmaximal, characteristic, complemented
6#2041C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×D4—21no—
8C22×C3×D4—21no—
9C22×C3×D4C2212yes—
10C22×C3×C8C2212yes—
11C3×#127—24no—
12C2×C3×C4⋊C4—21no—
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C42C2211yescharacteristic
16C3×#102—24no—
17C3×#102—24no—
18C3×#95—21no—
19C2×#127C2×C311yescharacteristic
20C2×#102C2×C312yescomplemented
21C2×#102C2×C312yescomplemented
22#389C2×C311yescharacteristic, complemented
23#336C2×C311yescharacteristic, complemented
24C24×C3—22no—
25C2×C3×D4—24no—
26C2×C3×D4—24no—
27C2×C3×D4—22no—
28C2×C3×D4—22no—
29C2×C3×D4—22no—
30C2×C3×D4—22no—
31C2×C3×D4—22no—
32C2×C3×D4—42no—
33C2×C3×D4—42no—
34C2×C3×D4—42no—
35C2×C3×D4—42no—
36C2×C3×D4D414yes—
37C22×C3×C4—21no—
38C22×C3×C4—21no—
39C22×C3×C4D411yescharacteristic
40C22×C3×C4D411yescharacteristic
41C22×C3×C4C2311yescharacteristic
42C2×C3×C8—22no—
43C2×C3×C8—22no—
44C2×C3×C8—22no—
45C3×C4⋊C4—21no—
46C3×C4⋊C4—22no—
47C3×C4⋊C4—42no—
48C3×C4⋊C4D412yescomplemented
49C3×C42—22no—
50C22×D4—21no—
51C22×D4—21no—
52C22×D4C22×C312yes—
53C22×C8C22×C312yes—
54#127—24no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4C22×C311yescharacteristic
57C2×C4⋊C4C22×C311yescharacteristic
58C2×C42C22×C311yescharacteristic
59#102—24no—
60#102—24no—
61#95—21no—
62C23×C3—24no—
63C23×C3—42no—
64C23×C3—42no—
65C23×C3—42no—
66C23×C3—42no—
67C23×C3—42no—
68C23×C3—42no—
69C23×C3C2×D411yescharacteristic
70C3×D4—24no—
71C3×D4—24no—
72C3×D4—48no—
73C3×D4—48no—
74C3×D4—42no—
75C3×D4—42no—
76C2×C3×C4—21no—
77C2×C3×C4—21no—
78C2×C3×C4—21no—
79C2×C3×C4—21no—
80C2×C3×C4—22no—
81C2×C3×C4—41no—
82C2×C3×C4—41no—
83C2×C3×C4—41no—
84C2×C3×C4—41no—
85C2×C3×C4—41no—
86C2×C3×C4D812yes—
87C2×C3×C4SD1612yes—
88C2×C3×C4C2×D411yescharacteristic
89C2×C3×C4C2×D412yes—
90C2×C3×C4#4112yes—
91C2×C3×C4#4111yescharacteristic
92C3×C8—42no—
93C24—22no—
94C2×D4—24no—
95C2×D4—24no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—42no—
102C2×D4—42no—
103C2×D4—42no—
104C2×D4—42no—
105C2×D4C3×D414yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4C3×D411yescharacteristic
109C22×C4C3×D411yescharacteristic
110C22×C4C23×C311yescharacteristic
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C4⋊C4—21no—
115C4⋊C4—22no—
116C4⋊C4—42no—
117C4⋊C4C3×D412yescomplemented
118C42—22no—
119C22×C3—44no—
120C22×C3—44no—
121C22×C3—44no—
122C22×C3—82no—
123C22×C3—82no—
124C22×C3—82no—
125C22×C3—82no—
126C22×C3#12011yescharacteristic
127C22×C3#12111yescharacteristic
128C22×C3#12411yescharacteristic
129C22×C3C2×D811yescharacteristic
130C22×C3C2×SD1611yescharacteristic
131C22×C3AGL(1,ℤ/8)11yescharacteristic
132C22×C3AGL(1,ℤ/8)11yescharacteristic
133C3×C4—22no—
134C3×C4—22no—
135C3×C4—42no—
136C3×C4—81no—
137C3×C4#12112yes—
138C3×C4#12312yes—
139C23—24no—
140C23—42no—
141C23—42no—
142C23—42no—
143C23—42no—
144C23—42no—
145C23—42no—
146C23C2×C3×D411yescharacteristic
147D4—24no—
148D4—24no—
149D4—48no—
150D4—48no—
151D4—42no—
152D4—42no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4C3×D812yes—
164C2×C4C3×SD1612yes—
165C2×C4C2×C3×D411yescharacteristic
166C2×C4C2×C3×D412yes—
167C2×C4C3×#4112yes—
168C2×C4C3×#4111yescharacteristic
169C8—42no—
170C2×C3—84no—
171C2×C3#39411yescharacteristic
172C2×C3#44711yescharacteristic
173C2×C3#45011yescharacteristic
174C2×C3#45911yescharacteristic
175C2×C3#46011yescharacteristic
176C2×C3#48611yescharacteristic
177C2×C3#49011yescharacteristic
178C22—44no—
179C22—44no—
180C22—44no—
181C22—82no—
182C22—82no—
183C22—82no—
184C22—82no—
185C22C3×#12011yescharacteristic
186C22C3×#12111yescharacteristic
187C22C3×#12411yescharacteristic
188C22C2×C3×D811yescharacteristic
189C22C2×C3×SD1611yescharacteristic
190C22C3×AGL(1,ℤ/8)11yescharacteristic
191C22C3×AGL(1,ℤ/8)11yescharacteristic
192C4—22no—
193C4—22no—
194C4—42no—
195C4—81no—
196C4C3×#12112yes—
197C4C3×#12312yes—
198C3#204111yescharacteristic, direct product factor, Sylow
199C2—84no—
200C2C3×#39411yescharacteristic
201C2C3×#44711yescharacteristic
202C2C3×#45011yescharacteristic
203C2C3×#45911yescharacteristic
204C2C3×#46011yescharacteristic
205C2C3×#48611yescharacteristic
206C2C3×#49011yescharacteristic
subgroupG/H
center, socleC23×C3C2×D4
commutator, Frattini, cosocleC22×C4C23×C3
prime radicalC2×C3×#127C2
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
2. C2×C3×#102C2semidirect
3. C2×C3×#102C2semidirect
4. C3×#389C2semidirect
5. C3×#336C2semidirect
6. #2041C3direct (N × H)
20. C2×#102C2×C3semidirect
21. C2×#102C2×C3semidirect
22. #389C2×C3semidirect
23. #336C2×C3semidirect
48. C3×C4⋊C4D4semidirect
117. C4⋊C4C3×D4semidirect
198. C3#2041direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1272
C2×C3×#1022
C3×#3892
C3×#3362
#20413