Finite Group Databasebeta

← #64147⋊C3 #11721640

Subgroups — #64147⋊C3 #11721640

Order 864.

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

normal subgroups only

This group has 383 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×D4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#638⋊C3C212yesmaximal, complemented
3C2×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4#3757⋊C3C212yesmaximal, complemented
5C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
6#64147C311yesmaximal, characteristic, complemented
7C3×#938—31nomaximal
8C23×H(3,𝔽3)C2212yes—
9C22×(C3⋊D3)⋊C3C2211yescharacteristic
10D4×H(3,𝔽3)—22no—
11C2×C4×H(3,𝔽3)C2211yescharacteristic
12#638⋊C3—24no—
13C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×(C3⋊Dic3)⋊C3C2212yes—
15C4×(C3⋊D3)⋊C3—22no—
16C2×C32×D4—21no—
17C2×C32×D4D311yescharacteristic, complemented
18C2×C32×D4C2×C311yescharacteristic, complemented
19C2×#638C2×C312yescomplemented
20C2×C4×C3⋊D3C2×C311yescharacteristic, complemented
21C2×C3×#67—32no—
22C2×C3×C4×D3—31no—
23#3757C2×C312yescomplemented
24C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
25C3×#217—32no—
26C3×C4⋊Dic3—31no—
27C22×H(3,𝔽3)—22no—
28C22×H(3,𝔽3)—22no—
29C22×H(3,𝔽3)—22no—
30C22×H(3,𝔽3)C2311yescharacteristic
31C2×(C3⋊D3)⋊C3—21no—
32C2×(C3⋊D3)⋊C3—21no—
33C2×(C3⋊D3)⋊C3D412yescomplemented
34C4×H(3,𝔽3)D412yescomplemented
35(C3⋊Dic3)⋊C3—21no—
36(C3⋊Dic3)⋊C3—22no—
37C3×#121—91no—
38#938—31no—
39#938—91no—
40C23×C32—22no—
41C23×C32C2×D312yes—
42C23×C32C22×C312yes—
43C22×C3⋊D3C22×C311yescharacteristic
44C22×C3×D3—31no—
45C32×D4—24no—
46C32×D4—22no—
47C32×D4—22no—
48C2×C32×C4—21no—
49C2×C32×C4C2×D311yescharacteristic
50C2×C32×C4C22×C311yescharacteristic
51#638—24no—
52C2×C3⋊Dic3C22×C312yes—
53C2×C3⋊Dic3C22×C311yescharacteristic
54C4×C3⋊D3—22no—
55C3×#67—64no—
56C2×C3×Dic3—32no—
57C2×C3×Dic3—31no—
58C3×C4×D3—62no—
59C2×H(3,𝔽3)—42no—
60C2×H(3,𝔽3)C2×D411yescharacteristic
61C2×H(3,𝔽3)C2×D411yescharacteristic
62C2×H(3,𝔽3)#4111yescharacteristic
63(C3⋊D3)⋊C3—22no—
64C2×C3×D4—31no—
65C2×C3×D4—31no—
66C2×C3×D4—61no—
67C2×C3×D4—92no—
68C2×C3×D4C3×D311yescharacteristic
69C22×C3×C4—91no—
70C2×#67—32no—
71C2×#67—92no—
72C2×C4×D3—31no—
73C2×C4×D3—91no—
74C3×C4⋊C4—91no—
75C3×C22⋊C4—92no—
76#217—32no—
77#217—92no—
78C4⋊Dic3—31no—
79C4⋊Dic3—91no—
80C22×C32—24no—
81C22×C32—22no—
82C22×C32—22no—
83C22×C32—22no—
84C22×C32—22no—
85C22×C32—22no—
86C22×C32—22no—
87C22×C32—21no—
88C22×C32—42no—
89C22×C32—42no—
90C22×C32C22×D311yescharacteristic
91C22×C32C23×C311yescharacteristic
92C2×C3⋊D3—21no—
93C2×C3⋊D3—21no—
94C2×C3⋊D3C3×D412yescomplemented
95C2×C3×D3—32no—
96C2×C3×D3—61no—
97C2×C3×D3—61no—
98C32×C4—22no—
99C32×C4#6712yescomplemented
100C32×C4C3×D412yescomplemented
101C3⋊Dic3—21no—
102C3⋊Dic3—22no—
103C3×Dic3—62no—
104C3×Dic3—61no—
105#121—91noSylow
106H(3,𝔽3)#12111yescharacteristic, complemented, Sylow
107C23×C3—32no—
108C23×C3—32no—
109C23×C3—62no—
110C23×C3—91no—
111C23×C3C2×C3×D312yes—
112C22×D3—31no—
113C22×D3—91no—
114C3×D4—22no—
115C3×D4—62no—
116C3×D4—62no—
117C3×D4—64no—
118C3×D4—184no—
119C2×C3×C4—31no—
120C2×C3×C4—31no—
121C2×C3×C4—61no—
122C2×C3×C4—91no—
123C2×C3×C4—92no—
124C2×C3×C4—182no—
125C2×C3×C4C2×C3×D311yescharacteristic
126#67—64no—
127#67—184no—
128C2×Dic3—32no—
129C2×Dic3—31no—
130C2×Dic3—92no—
131C2×Dic3—91no—
132C4×D3—62no—
133C4×D3—182no—
134C2×C32—21no—
135C2×C32—21no—
136C2×C32—21no—
137C2×C32—44no—
138C2×C32—42no—
139C2×C32—42no—
140C2×C32D3×D411yescharacteristic
141C2×C32#23711yescharacteristic
142C2×C32C2×#6711yescharacteristic
143C2×C32C2×C3×D411yescharacteristic
144C2×C32C2×C3×D411yescharacteristic
145C2×C32C3×#4111yescharacteristic
146C3⋊D3—22no—
147C3×D3—62no—
148C2×D4—92no—
149C2×D4(C3⋊D3)⋊C311yescharacteristic, complemented
150C22×C4—91no—
151C4⋊C4—91no—
152C22⋊C4—92no—
153C22×C3—22no—
154C22×C3—22no—
155C22×C3—22no—
156C22×C3—31no—
157C22×C3—31no—
158C22×C3—62no—
159C22×C3—62no—
160C22×C3—62no—
161C22×C3—62no—
162C22×C3—62no—
163C22×C3—62no—
164C22×C3—64no—
165C22×C3—61no—
166C22×C3—92no—
167C22×C3—122no—
168C22×C3—122no—
169C22×C3—181no—
170C22×C3—181no—
171C22×C3C22×C3×D311yescharacteristic
172C2×D3—32no—
173C2×D3—61no—
174C2×D3—61no—
175C2×D3—92no—
176C2×D3—181no—
177C2×D3—181no—
178C3×C4—32no—
179C3×C4—32no—
180C3×C4—62no—
181C3×C4—181no—
182C3×C4—182no—
183C3×C4C3×#6712yes—
184Dic3—62no—
185Dic3—61no—
186Dic3—181no—
187Dic3—182no—
188C32—21no—
189C32#93811yescharacteristic, complemented
190C32C3×#12111yescharacteristic, complemented
191C23—91no—
192C23C2×(C3⋊D3)⋊C312yes—
193D4—22no—
194D4—184no—
195C2×C4—91no—
196C2×C4—92no—
197C2×C4—182no—
198C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
199C2×C3—31no—
200C2×C3—31no—
201C2×C3—31no—
202C2×C3—31no—
203C2×C3—31no—
204C2×C3—31no—
205C2×C3—42no—
206C2×C3—61no—
207C2×C3—61no—
208C2×C3—61no—
209C2×C3—122no—
210C2×C3—122no—
211C2×C3—124no—
212C2×C3—182no—
213C2×C3C3×D3×D411yescharacteristic
214C2×C3C3×#23711yescharacteristic
215C2×C3C2×C3×#6711yescharacteristic
216D3—62no—
217D3—182no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22—92no—
222C22—181no—
223C22—181no—
224C22C22×(C3⋊D3)⋊C311yescharacteristic
225C4—182no—
226C4—181no—
227C4#638⋊C312yescomplemented
228C3—31no—
229C3—31no—
230C3—61no—
231C3C3×#93811yescharacteristic
232C2—42no—
233C2—182no—
234C2D4×(C3⋊D3)⋊C311yescharacteristic
235C2#3830⋊C311yescharacteristic
236C2C2×#638⋊C311yescharacteristic
subgroupG/H
centerC22C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C23×C3
Frattini, socleC22×C3C22×C3×D3
FittingC2×D4×H(3,𝔽3)C2
hypercenterC2×D4(C3⋊D3)⋊C3
hypocenterC32C3×#121
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×H(3,𝔽3)C2semidirect
2. C2×#638⋊C3C2semidirect
3. C2×C4×(C3⋊D3)⋊C3C2semidirect
4. #3757⋊C3C2semidirect
5. C4⋊((C3⋊Dic3)⋊C3)C2semidirect
6. #64147C3semidirect
17. C2×C32×D4D3semidirect
18. C2×C32×D4C2×C3semidirect
19. C2×#638C2×C3semidirect
20. C2×C4×C3⋊D3C2×C3semidirect
23. #3757C2×C3semidirect
24. C4⋊(C3⋊Dic3)C2×C3semidirect
33. C2×(C3⋊D3)⋊C3D4semidirect
34. C4×H(3,𝔽3)D4semidirect
94. C2×C3⋊D3C3×D4semidirect
99. C32×C4#67semidirect
100. C32×C4C3×D4semidirect
106. H(3,𝔽3)#121semidirect
149. C2×D4(C3⋊D3)⋊C3semidirect
189. C32#938semidirect
190. C32C3×#121semidirect
227. C4#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D4×H(3,𝔽3)2
C2×#638⋊C32
C2×C4×(C3⋊D3)⋊C32
#3757⋊C32
C4⋊((C3⋊Dic3)⋊C3)2
#641473
C3×#9383