Finite Group Databasebeta

← #64106⋊C3 #11721471

Subgroups — #64106⋊C3 #11721471

Order 864.

285 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 81 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×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×#634⋊C3C211yesmaximal, characteristic, complemented
3C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#3752⋊C3C212yesmaximal
5C4⋊((C3⋊Dic3)⋊C3)C212yesmaximal, complemented
6#64106C311yesmaximal, characteristic, complemented
7C3×#897—31nomaximal
8C22×(C3⋊D3)⋊C3C2211yescharacteristic
9C2×C4×H(3,𝔽3)C2212yes—
10C2×C4×H(3,𝔽3)C2211yescharacteristic
11C2×(C3⋊Dic3)⋊C3C2212yes—
12C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C4×(C3⋊D3)⋊C3—22no—
14#634⋊C3—22no—
15C2×C4×C3⋊D3C2×C311yescharacteristic
16C2×#634C2×C311yescharacteristic, complemented
17C2×C3×C4×D3—31no—
18C2×C3×Dic6—31no—
19C32×C4⋊C4—21no—
20C32×C4⋊C4D311yescharacteristic, complemented
21C32×C4⋊C4C2×C311yescharacteristic, complemented
22#3752C2×C312yes—
23C4⋊(C3⋊Dic3)C2×C312yescomplemented
24C3×(C2×D3)⋊C4—32no—
25C3×C4⋊Dic3—32no—
26C22×H(3,𝔽3)C2311yescharacteristic
27C2×(C3⋊D3)⋊C3—21no—
28C2×(C3⋊D3)⋊C3—21no—
29C2×(C3⋊D3)⋊C3Q812yescomplemented
30C4×H(3,𝔽3)—22no—
31C4×H(3,𝔽3)D412yes—
32(C3⋊Dic3)⋊C3—22no—
33(C3⋊Dic3)⋊C3—21no—
34C3×C22⋊Q8—91no—
35#897—31no—
36#897—91no—
37C22×C3⋊D3C22×C311yescharacteristic
38C22×C3×D3—31no—
39C2×C32×C4—21no—
40C2×C32×C4—22no—
41C2×C32×C4C2×D312yes—
42C2×C32×C4C2×D311yescharacteristic
43C2×C32×C4C22×C312yes—
44C2×C32×C4C22×C311yescharacteristic
45C2×C3⋊Dic3C22×C311yescharacteristic
46C2×C3⋊Dic3C22×C312yes—
47C4×C3⋊D3—22no—
48#634—22no—
49C2×C3×Dic3—31no—
50C2×C3×Dic3—32no—
51C3×C4×D3—62no—
52C3×Dic6—62no—
53C2×H(3,𝔽3)C2×D411yescharacteristic
54C2×H(3,𝔽3)C2×Q811yescharacteristic
55C2×H(3,𝔽3)#4111yescharacteristic
56(C3⋊D3)⋊C3—22no—
57C2×C3×Q8—91no—
58C22×C3×C4—91no—
59C2×C4×D3—31no—
60C2×C4×D3—91no—
61C2×Dic6—31no—
62C2×Dic6—91no—
63C3×C4⋊C4—31no—
64C3×C4⋊C4—61no—
65C3×C4⋊C4—61no—
66C3×C4⋊C4—92no—
67C3×C4⋊C4C3×D311yescharacteristic
68C3×C22⋊C4—92no—
69(C2×D3)⋊C4—32no—
70(C2×D3)⋊C4—92no—
71C4⋊Dic3—32no—
72C4⋊Dic3—92no—
73C22×C32—21no—
74C22×C32C22×D311yescharacteristic
75C22×C32C23×C311yescharacteristic
76C2×C3⋊D3—21no—
77C2×C3⋊D3—21no—
78C2×C3⋊D3C3×Q812yescomplemented
79C2×C3×D3—32no—
80C2×C3×D3—61no—
81C2×C3×D3—61no—
82C32×C4—22no—
83C32×C4—22no—
84C32×C4—22no—
85C32×C4—42no—
86C32×C4D1212yes—
87C32×C4C3×D412yes—
88C3⋊Dic3—22no—
89C3⋊Dic3—21no—
90C3×Dic3—62no—
91C3×Dic3—61no—
92C22⋊Q8—91noSylow
93H(3,𝔽3)C22⋊Q811yescharacteristic, complemented, Sylow
94C23×C3—91no—
95C22×D3—31no—
96C22×D3—91no—
97C3×Q8—182no—
98C2×C3×C4—31no—
99C2×C3×C4—32no—
100C2×C3×C4—61no—
101C2×C3×C4—61no—
102C2×C3×C4—62no—
103C2×C3×C4—62no—
104C2×C3×C4—92no—
105C2×C3×C4—91no—
106C2×C3×C4—182no—
107C2×C3×C4C2×C3×D312yes—
108C2×C3×C4C2×C3×D311yescharacteristic
109C2×Dic3—31no—
110C2×Dic3—32no—
111C2×Dic3—91no—
112C2×Dic3—92no—
113C4×D3—62no—
114C4×D3—182no—
115Dic6—62no—
116Dic6—182no—
117C2×C32—21no—
118C2×C32—21no—
119C2×C32—21no—
120C2×C32C2×D1211yescharacteristic
121C2×C32#23711yescharacteristic
122C2×C32D3×Q811yescharacteristic
123C2×C32C2×C3×D411yescharacteristic
124C2×C32C2×C3×Q811yescharacteristic
125C2×C32C3×#4111yescharacteristic
126C3⋊D3—22no—
127C3×D3—62no—
128C2×Q8—91no—
129C22×C4—91no—
130C4⋊C4—92no—
131C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
132C22⋊C4—92no—
133C22×C3—31no—
134C22×C3—61no—
135C22×C3—61no—
136C22×C3—92no—
137C22×C3—181no—
138C22×C3—181no—
139C22×C3C22×C3×D311yescharacteristic
140C2×D3—32no—
141C2×D3—61no—
142C2×D3—61no—
143C2×D3—92no—
144C2×D3—181no—
145C2×D3—181no—
146C3×C4—22no—
147C3×C4—32no—
148C3×C4—62no—
149C3×C4—62no—
150C3×C4—62no—
151C3×C4—122no—
152C3×C4—122no—
153C3×C4—181no—
154C3×C4—182no—
155C3×C4C3×D1212yes—
156Dic3—62no—
157Dic3—61no—
158Dic3—182no—
159Dic3—181no—
160C32—21no—
161C32#89711yescharacteristic, complemented
162C32C3×C22⋊Q811yescharacteristic, complemented
163C23—91no—
164Q8—182no—
165C2×C4—92no—
166C2×C4—91no—
167C2×C4—182no—
168C2×C4C2×(C3⋊D3)⋊C312yes—
169C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
170C2×C3—31no—
171C2×C3—31no—
172C2×C3—31no—
173C2×C3—61no—
174C2×C3—61no—
175C2×C3—61no—
176C2×C3—61no—
177C2×C3—61no—
178C2×C3—61no—
179C2×C3—182no—
180C2×C3C2×C3×D1211yescharacteristic
181C2×C3C3×#23711yescharacteristic
182C2×C3C3×D3×Q811yescharacteristic
183D3—62no—
184D3—182no—
185C22—92no—
186C22—181no—
187C22—181no—
188C22C22×(C3⋊D3)⋊C311yescharacteristic
189C4—22no—
190C4—181no—
191C4—182no—
192C4C4⋊((C3⋊D3)⋊C3)12yes—
193C3—31no—
194C3—61no—
195C3—61no—
196C3C3×#89711yescharacteristic
197C2—182no—
198C2C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
199C2#3830⋊C311yescharacteristic
200C2Q8×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C23×C3
Frattini, socleC22×C3C22×C3×D3
FittingC4⋊C4×H(3,𝔽3)C2
prime radicalC22×(C3⋊D3)⋊C3C22
hypercenterC4⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×C22⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#634⋊C3C2semidirect
3. C4⋊C4×H(3,𝔽3)C2semidirect
5. C4⋊((C3⋊Dic3)⋊C3)C2semidirect
6. #64106C3semidirect
16. C2×#634C2×C3semidirect
20. C32×C4⋊C4D3semidirect
21. C32×C4⋊C4C2×C3semidirect
23. C4⋊(C3⋊Dic3)C2×C3semidirect
29. C2×(C3⋊D3)⋊C3Q8semidirect
78. C2×C3⋊D3C3×Q8semidirect
93. H(3,𝔽3)C22⋊Q8semidirect
131. C4⋊C4(C3⋊D3)⋊C3semidirect
161. C32#897semidirect
162. C32C3×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×(C3⋊D3)⋊C32
C2×#634⋊C32
C4⋊C4×H(3,𝔽3)2
#3752⋊C32
C4⋊((C3⋊Dic3)⋊C3)2
#641063
C3×#8973