Finite Group Databasebeta

← C3×#1598 #72313

Subgroups — C3×#1598 #72313

Order 384.

310 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C3×#537C211yesmaximal, characteristic, complemented
2C3×#447C212yesmaximal, complemented
3C3×#394C211yesmaximal, characteristic, complemented
4C3×#327C212yesmaximal, complemented
5C3×#324C211yesmaximal, characteristic, complemented
6#1598C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×#41—21no—
8C22×C3×D4C2211yescharacteristic
9C2×C3×D8—22no—
10C3×C22⋊Q8—22no—
11C3×#121—22no—
12C3×#121—21no—
13C3×#121C2212yes—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4—21no—
16C2×C3×C22⋊C4—21no—
17C3×#102—22no—
18C3×C22⋊C8C2212yes—
19C3×#95C2211yescharacteristic
20#537C2×C311yescharacteristic, complemented
21#447C2×C312yescomplemented
22#394C2×C311yescharacteristic, complemented
23#327C2×C312yescomplemented
24#324C2×C311yescharacteristic, complemented
25C24×C3—21no—
26C3×#41—42no—
27C3×#41—42no—
28C2×C3×Q8—21no—
29C2×C3×D4—21no—
30C2×C3×D4—22no—
31C2×C3×D4—21no—
32C2×C3×D4—22no—
33C2×C3×D4—42no—
34C2×C3×D4D412yes—
35C22×C3×C4—22no—
36C22×C3×C4—21no—
37C22×C3×C4D411yescharacteristic
38C22×C3×C4D411yescharacteristic
39C22×C3×C4C2311yescharacteristic
40C3×D8—44no—
41C2×C3×C8—22no—
42C3×C4⋊C4—21no—
43C3×C4⋊C4—22no—
44C3×C4⋊C4D412yes—
45C3×C22⋊C4—22no—
46C3×C22⋊C4—22no—
47C3×C22⋊C4—42no—
48C3×C22⋊C4—42no—
49C2×#41—21no—
50C22×D4C22×C311yescharacteristic
51C2×D8—22no—
52C22⋊Q8—22no—
53#121—22no—
54#121—21no—
55#121C22×C312yes—
56C2×C4⋊C4C22×C311yescharacteristic
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59#102—22no—
60C22⋊C8C22×C312yes—
61#95C22×C311yescharacteristic
62C23×C3—22no—
63C23×C3—22no—
64C23×C3—41no—
65C23×C3—41no—
66C23×C3—41no—
67C23×C3—41no—
68C23×C3—41no—
69C23×C3—41no—
70C23×C3C2×D411yescharacteristic
71C3×Q8—42no—
72C3×D4—24no—
73C3×D4—42no—
74C3×D4—42no—
75C3×D4—42no—
76C3×D4—42no—
77C3×D4—42no—
78C3×D4—81no—
79C2×C3×C4—21no—
80C2×C3×C4—21no—
81C2×C3×C4—21no—
82C2×C3×C4—22no—
83C2×C3×C4—41no—
84C2×C3×C4—44no—
85C2×C3×C4—41no—
86C2×C3×C4—41no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4D812yescomplemented
90C2×C3×C4C2×D412yes—
91C3×C8—42no—
92C24—21no—
93#41—42no—
94#41—42no—
95C2×Q8—21no—
96C2×D4—21no—
97C2×D4—22no—
98C2×D4—21no—
99C2×D4—22no—
100C2×D4—42no—
101C2×D4C3×D412yes—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4C3×D411yescharacteristic
105C22×C4C3×D411yescharacteristic
106C22×C4C23×C311yescharacteristic
107D8—44no—
108C2×C8—22no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4C3×D412yes—
112C22⋊C4—22no—
113C22⋊C4—22no—
114C22⋊C4—42no—
115C22⋊C4—42no—
116C22×C3—21no—
117C22×C3—21no—
118C22×C3—42no—
119C22×C3—42no—
120C22×C3—42no—
121C22×C3—42no—
122C22×C3—42no—
123C22×C3—42no—
124C22×C3—81no—
125C22×C3—81no—
126C22×C3—81no—
127C22×C3—81no—
128C22×C3#12011yescharacteristic
129C22×C3C2×D811yescharacteristic
130C22×C3AGL(1,ℤ/8)11yescharacteristic
131C3×C4—22no—
132C3×C4—22no—
133C3×C4—42no—
134C3×C4—41no—
135C3×C4—81no—
136C23—22no—
137C23—22no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23C2×C3×D411yescharacteristic
145Q8—42no—
146D4—24no—
147D4—42no—
148D4—42no—
149D4—42no—
150D4—42no—
151D4—42no—
152D4—81no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—44no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4C3×D812yescomplemented
164C2×C4C2×C3×D412yes—
165C8—42no—
166C2×C3—21no—
167C2×C3—21no—
168C2×C3—82no—
169C2×C3—82no—
170C2×C3#44711yescharacteristic
171C2×C3#45411yescharacteristic
172C2×C3B(4,𝔽2)11yescharacteristic
173C22—21no—
174C22—21no—
175C22—42no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22C3×#12011yescharacteristic
186C22C2×C3×D811yescharacteristic
187C22C3×AGL(1,ℤ/8)11yescharacteristic
188C4—22no—
189C4—22no—
190C4—42no—
191C4—41no—
192C4—81no—
193C3#159811yescharacteristic, direct product factor, Sylow
194C2—21no—
195C2—21no—
196C2—82no—
197C2—82no—
198C2C3×#44711yescharacteristic
199C2C3×#45411yescharacteristic
200C2C3×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C3#120
commutator, Frattini, cosocleC22×C4C23×C3
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#537C2semidirect
2. C3×#447C2semidirect
3. C3×#394C2semidirect
4. C3×#327C2semidirect
5. C3×#324C2semidirect
6. #1598C3direct (N × H)
20. #537C2×C3semidirect
21. #447C2×C3semidirect
22. #394C2×C3semidirect
23. #327C2×C3semidirect
24. #324C2×C3semidirect
89. C2×C3×C4D8semidirect
163. C2×C4C3×D8semidirect
193. C3#1598direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5372
C3×#4472
C3×#3942
C3×#3272
C3×#3242
#15983