Finite Group Databasebeta

← C3×#64242 #11724297

Subgroups — C3×#64242 #11724297

Order 864.

328 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 54 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
1C22×C3×#157C211yesmaximal, characteristic, complemented
2C2×C3×GO+(4,𝔽2)C211yesmaximal, characteristic, complemented
3C3×#3811C212yesmaximal, complemented
4C3×#3773C211yesmaximal, characteristic, complemented
5C3×D32⋊C4C212yesmaximal, complemented
6#64242C311yesmaximal, characteristic, direct product factor
7C22×C3×C3⋊D3C2211yescharacteristic
8C2×C3×D32C2212yes—
9C2×C3×#157—22no—
10C2×C3×#157C2211yescharacteristic
11C2×C3×#157C2211yescharacteristic
12C3×GO+(4,𝔽2)—22no—
13C32×#67—22no—
14C3×#626—22no—
15C3×#624C2212yes—
16C22×#157C2×C311yescharacteristic, complemented
17C2×GO+(4,𝔽2)C2×C311yescharacteristic, complemented
18C3×D3×D4—62no—
19#3811C2×C312yescomplemented
20#3773C2×C311yescharacteristic, complemented
21D32⋊C4C2×C312yescomplemented
22C22×C33D411yescharacteristic, complemented
23C2×C3×C3⋊D3—21no—
24C2×C3×C3⋊D3—21no—
25C2×C3×C3⋊D3D411yescharacteristic, complemented
26C2×C3×C3⋊D3C2311yescharacteristic
27C2×C32×D3—22no—
28C3×D32—22no—
29C3×D32—22no—
30C3×#157—21no—
31C3×#157D412yescomplemented
32C32×Dic3—22no—
33C3×#121—91nomaximal
34C22×C3⋊D3C22×C311yescharacteristic
35C22×C3×D3—62no—
36C22×C3×D3—62no—
37C2×D32C22×C312yes—
38C2×#157—22no—
39C2×#157C22×C311yescharacteristic
40C2×#157C22×C311yescharacteristic
41GO+(4,𝔽2)—22no—
42C32×D4—62no—
43C3×#67—22no—
44C3×#67—22no—
45C3×#67—42no—
46C3×#67—62no—
47C3×D12—62no—
48C3×C4×D3—62no—
49#626—22no—
50#624C22×C312yes—
51C2×C33—21no—
52C2×C33C2×D411yescharacteristic
53C3×C3⋊D3—21no—
54C3×C3⋊D3C2×D411yescharacteristic
55C3×C3⋊D3#4111yescharacteristic
56C32×D3—42no—
57C2×C3×D4—92no—
58C2×C3×D4—91no—
59C22×C3×C4—91no—
60D3×D4—62no—
61C3×C4⋊C4—91no—
62C3×C22⋊C4—92no—
63C22×C32—22no—
64C22×C32—42no—
65C22×C32—62no—
66C22×C32C3×D411yescharacteristic, complemented
67C2×C3⋊D3—21no—
68C2×C3⋊D3—21no—
69C2×C3⋊D3C3×D411yescharacteristic, complemented
70C2×C3⋊D3C23×C311yescharacteristic
71C2×C3×D3—22no—
72C2×C3×D3—22no—
73C2×C3×D3—42no—
74C2×C3×D3—62no—
75C2×C3×D3—62no—
76C2×C3×D3—122no—
77C2×C3×D3—122no—
78C2×C3×D3—122no—
79C2×C3×D3—122no—
80D32—22no—
81D32—22no—
82#157—21no—
83#157C3×D412yescomplemented
84C32×C4—62no—
85C3×Dic3—22no—
86C3×Dic3—22no—
87C3×Dic3—42no—
88#121—91noSylow
89C33#12111yescharacteristic, complemented, Sylow
90C23×C3—92no—
91C23×C3—91no—
92C22×D3—62no—
93C22×D3—62no—
94C3×D4—62no—
95C3×D4—62no—
96C3×D4—122no—
97C3×D4—182no—
98C3×D4—182no—
99C2×C3×C4—92no—
100C2×C3×C4—91no—
101C2×C3×C4—91no—
102C2×C3×C4—182no—
103#67—22no—
104#67—62no—
105D12—62no—
106C4×D3—62no—
107C2×C32—22no—
108C2×C32—21no—
109C2×C32—42no—
110C2×C32—42no—
111C2×C32—82no—
112C2×C32—122no—
113C2×C32C2×C3×D411yescharacteristic
114C3⋊D3—21no—
115C3⋊D3C2×C3×D411yescharacteristic
116C3⋊D3C3×#4111yescharacteristic
117C3×D3—42no—
118C3×D3—42no—
119C3×D3—62no—
120C3×D3—62no—
121C3×D3—82no—
122C3×D3—122no—
123C2×D4—92no—
124C2×D4—91no—
125C22×C4—91no—
126C4⋊C4—91no—
127C22⋊C4—92no—
128C22×C3—22no—
129C22×C3—42no—
130C22×C3—62no—
131C22×C3—62no—
132C22×C3—91no—
133C22×C3—91no—
134C22×C3—122no—
135C22×C3—181no—
136C22×C3—181no—
137C22×C3—182no—
138C22×C3—182no—
139C22×C3GO+(4,𝔽2)11yescharacteristic, complemented
140C2×D3—22no—
141C2×D3—62no—
142C2×D3—62no—
143C2×D3—122no—
144C2×D3—122no—
145C2×D3—122no—
146C2×D3—122no—
147C3×C4—62no—
148C3×C4—62no—
149C3×C4—92no—
150C3×C4—122no—
151C3×C4—181no—
152Dic3—22no—
153C32—22no—
154C32—42no—
155C32C3×#12111yescharacteristic, complemented
156C23—92no—
157C23—91no—
158D4—62no—
159D4—182no—
160D4—182no—
161C2×C4—92no—
162C2×C4—91no—
163C2×C4—91no—
164C2×C4—182no—
165C2×C3—21no—
166C2×C3—22no—
167C2×C3—42no—
168C2×C3—42no—
169C2×C3—82no—
170C2×C3—91no—
171C2×C3—91no—
172C2×C3—122no—
173C2×C3—122no—
174C2×C3—181no—
175C2×C3—242no—
176C2×C3C2×GO+(4,𝔽2)11yescharacteristic
177D3—42no—
178D3—62no—
179D3—62no—
180D3—122no—
181C22—62no—
182C22—91no—
183C22—91no—
184C22—181no—
185C22—181no—
186C22—182no—
187C22—182no—
188C22C3×GO+(4,𝔽2)11yescharacteristic, complemented
189C4—62no—
190C4—92no—
191C4—181no—
192C3—22no—
193C3—42no—
194C3#6424211yescharacteristic, direct product factor
195C2—21no—
196C2—91no—
197C2—91no—
198C2—122no—
199C2—181no—
200C2C2×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C23×C3
FrattiniC2C2×C3×GO+(4,𝔽2)
FittingC22×C33D4
socleC2×C33C2×D4
hypercenterC22×C3GO+(4,𝔽2)
hypocenterC32C3×#121
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×#157C2semidirect
2. C2×C3×GO+(4,𝔽2)C2semidirect
3. C3×#3811C2semidirect
4. C3×#3773C2semidirect
5. C3×D32⋊C4C2semidirect
6. #64242C3direct (N × H)
16. C22×#157C2×C3semidirect
17. C2×GO+(4,𝔽2)C2×C3semidirect
19. #3811C2×C3semidirect
20. #3773C2×C3semidirect
21. D32⋊C4C2×C3semidirect
22. C22×C33D4semidirect
25. C2×C3×C3⋊D3D4semidirect
31. C3×#157D4semidirect
66. C22×C32C3×D4semidirect
69. C2×C3⋊D3C3×D4semidirect
83. #157C3×D4semidirect
89. C33#121semidirect
139. C22×C3GO+(4,𝔽2)semidirect
155. C32C3×#121semidirect
188. C22C3×GO+(4,𝔽2)semidirect
194. C3#64242direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1572
C2×C3×GO+(4,𝔽2)2
C3×#38112
C3×#37732
C3×D32⋊C42
#642423
C3×#1219