Finite Group Databasebeta

← #64155⋊C3 #11721672

Subgroups — #64155⋊C3 #11721672

Order 864.

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

normal subgroups only

This group has 297 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×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
3#3752⋊C3C212yesmaximal
4C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
5#3750⋊C3C212yesmaximal, complemented
6#64155C311yesmaximal, characteristic, complemented
7C3×#946—31nomaximal
8C22×(C3⋊D3)⋊C3C2211yescharacteristic
9Q8×H(3,𝔽3)—22no—
10C2×C4×H(3,𝔽3)C2212yes—
11C2×C4×H(3,𝔽3)C2211yescharacteristic
12C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×(C3⋊Dic3)⋊C3C2212yes—
14C4×(C3⋊D3)⋊C3—22no—
15C2×C32×Q8—21no—
16C2×C32×Q8D311yescharacteristic, complemented
17C2×C32×Q8C2×C311yescharacteristic, complemented
18C2×C4×C3⋊D3C2×C311yescharacteristic
19C2×C3×C4×D3—31no—
20#3752C2×C312yes—
21C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
22#3750C2×C312yescomplemented
23C3×(C2×D3)⋊C4—32no—
24C3×C4⋊Dic3—31no—
25C3×Dic3⋊C4—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—21no—
33(C3⋊Dic3)⋊C3—22no—
34C3×C22⋊Q8—91no—
35#946—31no—
36#946—91no—
37C22×C3⋊D3C22×C311yescharacteristic
38C22×C3×D3—31no—
39C32×Q8—24no—
40C32×Q8—22no—
41C32×Q8—22no—
42C2×C32×C4—22no—
43C2×C32×C4—21no—
44C2×C32×C4C2×D312yes—
45C2×C32×C4C2×D311yescharacteristic
46C2×C32×C4C22×C312yes—
47C2×C32×C4C22×C311yescharacteristic
48C2×C3⋊Dic3C22×C311yescharacteristic
49C2×C3⋊Dic3C22×C312yes—
50C4×C3⋊D3—22no—
51C2×C3×Dic3—31no—
52C2×C3×Dic3—32no—
53C3×C4×D3—62no—
54C2×H(3,𝔽3)C2×D411yescharacteristic
55C2×H(3,𝔽3)C2×Q811yescharacteristic
56C2×H(3,𝔽3)#4111yescharacteristic
57(C3⋊D3)⋊C3—22no—
58C2×C3×Q8—31no—
59C2×C3×Q8—61no—
60C2×C3×Q8—61no—
61C2×C3×Q8C3×D311yescharacteristic
62C22×C3×C4—91no—
63C2×C4×D3—31no—
64C2×C4×D3—91no—
65C3×C4⋊C4—92no—
66C3×C4⋊C4—91no—
67C3×C22⋊C4—92no—
68(C2×D3)⋊C4—32no—
69(C2×D3)⋊C4—92no—
70C4⋊Dic3—31no—
71C4⋊Dic3—91no—
72Dic3⋊C4—32no—
73Dic3⋊C4—92no—
74C22×C32—21no—
75C22×C32C22×D311yescharacteristic
76C22×C32C23×C311yescharacteristic
77C2×C3⋊D3—21no—
78C2×C3⋊D3—21no—
79C2×C3⋊D3C3×Q812yescomplemented
80C2×C3×D3—32no—
81C2×C3×D3—61no—
82C2×C3×D3—61no—
83C32×C4—24no—
84C32×C4—22no—
85C32×C4—22no—
86C32×C4—22no—
87C32×C4#6712yes—
88C32×C4C3×D412yes—
89C3⋊Dic3—21no—
90C3⋊Dic3—22no—
91C3×Dic3—62no—
92C3×Dic3—61no—
93C22⋊Q8—91noSylow
94H(3,𝔽3)C22⋊Q811yescharacteristic, complemented, Sylow
95C23×C3—91no—
96C22×D3—31no—
97C22×D3—91no—
98C3×Q8—22no—
99C3×Q8—62no—
100C3×Q8—62no—
101C3×Q8—64no—
102C2×C3×C4—31no—
103C2×C3×C4—32no—
104C2×C3×C4—62no—
105C2×C3×C4—61no—
106C2×C3×C4—61no—
107C2×C3×C4—62no—
108C2×C3×C4—92no—
109C2×C3×C4—91no—
110C2×C3×C4—182no—
111C2×C3×C4C2×C3×D312yes—
112C2×C3×C4C2×C3×D311yescharacteristic
113C2×Dic3—31no—
114C2×Dic3—32no—
115C2×Dic3—91no—
116C2×Dic3—92no—
117C4×D3—62no—
118C4×D3—182no—
119C2×C32—21no—
120C2×C32—21no—
121C2×C32—21no—
122C2×C32D3×Q811yescharacteristic
123C2×C32#23911yescharacteristic
124C2×C32C2×#6711yescharacteristic
125C2×C32C2×C3×D411yescharacteristic
126C2×C32C2×C3×Q811yescharacteristic
127C2×C32C3×#4111yescharacteristic
128C3⋊D3—22no—
129C3×D3—62no—
130C2×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
131C22×C4—91no—
132C4⋊C4—92no—
133C4⋊C4—91no—
134C22⋊C4—92no—
135C22×C3—31no—
136C22×C3—61no—
137C22×C3—61no—
138C22×C3—92no—
139C22×C3—181no—
140C22×C3—181no—
141C22×C3C22×C3×D311yescharacteristic
142C2×D3—32no—
143C2×D3—61no—
144C2×D3—61no—
145C2×D3—92no—
146C2×D3—181no—
147C2×D3—181no—
148C3×C4—22no—
149C3×C4—32no—
150C3×C4—62no—
151C3×C4—64no—
152C3×C4—62no—
153C3×C4—62no—
154C3×C4—62no—
155C3×C4—181no—
156C3×C4—182no—
157C3×C4C3×#6712yes—
158Dic3—62no—
159Dic3—61no—
160Dic3—181no—
161Dic3—182no—
162C32—21no—
163C32#94611yescharacteristic, complemented
164C32C3×C22⋊Q811yescharacteristic, complemented
165C23—91no—
166Q8—22no—
167C2×C4—92no—
168C2×C4—91no—
169C2×C4—182no—
170C2×C4C2×(C3⋊D3)⋊C312yes—
171C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
172C2×C3—31no—
173C2×C3—31no—
174C2×C3—31no—
175C2×C3—61no—
176C2×C3—61no—
177C2×C3—61no—
178C2×C3—61no—
179C2×C3—61no—
180C2×C3—61no—
181C2×C3—182no—
182C2×C3C3×D3×Q811yescharacteristic
183C2×C3C3×#23911yescharacteristic
184C2×C3C2×C3×#6711yescharacteristic
185D3—62no—
186D3—182no—
187C22—92no—
188C22—181no—
189C22—181no—
190C22C22×(C3⋊D3)⋊C311yescharacteristic
191C4—22no—
192C4—182no—
193C4—181no—
194C4#638⋊C312yes—
195C3—31no—
196C3—61no—
197C3—61no—
198C3C3×#94611yescharacteristic
199C2—182no—
200C2Q8×(C3⋊D3)⋊C311yescharacteristic
201C2#8872211yescharacteristic
202C2C2×#638⋊C311yescharacteristic
subgroupG/H
centerC22C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C23×C3
Frattini, socleC22×C3C22×C3×D3
FittingC2×Q8×H(3,𝔽3)C2
prime radicalC22×(C3⋊D3)⋊C3C22
hypercenterC2×Q8(C3⋊D3)⋊C3
hypocenterC32C3×C22⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×H(3,𝔽3)C2semidirect
4. C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. #3750⋊C3C2semidirect
6. #64155C3semidirect
16. C2×C32×Q8D3semidirect
17. C2×C32×Q8C2×C3semidirect
21. C4⋊(C3⋊Dic3)C2×C3semidirect
22. #3750C2×C3semidirect
29. C2×(C3⋊D3)⋊C3Q8semidirect
79. C2×C3⋊D3C3×Q8semidirect
94. H(3,𝔽3)C22⋊Q8semidirect
130. C2×Q8(C3⋊D3)⋊C3semidirect
163. C32#946semidirect
164. C32C3×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×H(3,𝔽3)2
C2×C4×(C3⋊D3)⋊C32
#3752⋊C32
C4⋊((C3⋊Dic3)⋊C3)2
#3750⋊C32
#641553
C3×#9463