Finite Group Databasebeta

← #64104⋊C3 #11721469

Subgroups — #64104⋊C3 #11721469

Order 864.

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

normal subgroups only

This group has 355 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)⋊C3)C212yesmaximal, complemented
2C2×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3C2×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
5#3752⋊C3C212yesmaximal, complemented
6#64104C311yesmaximal, characteristic, complemented
7C3×#895—31nomaximal
8C22×(C3⋊D3)⋊C3C2212yes—
9C22×(C3⋊D3)⋊C3C2211yescharacteristic
10C2×C4×H(3,𝔽3)C2212yes—
11C2×C4×H(3,𝔽3)C2211yescharacteristic
12C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C4⋊((C3⋊D3)⋊C3)—24no—
14C4⋊((C3⋊D3)⋊C3)—22no—
15C4×(C3⋊D3)⋊C3—22no—
16C2×C4⋊(C3⋊D3)C2×C312yescomplemented
17C2×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
18C2×C4×C3⋊D3C2×C311yescharacteristic, complemented
19C2×C3×D12—32no—
20C2×C3×D12—31no—
21C2×C3×C4×D3—31no—
22C32×C4⋊C4—21no—
23C32×C4⋊C4D311yescharacteristic, complemented
24C32×C4⋊C4C2×C311yescharacteristic, complemented
25#3752C2×C312yescomplemented
26C3×(C2×D3)⋊C4—32no—
27C22×H(3,𝔽3)C2311yescharacteristic
28C2×(C3⋊D3)⋊C3—21no—
29C2×(C3⋊D3)⋊C3—21no—
30C2×(C3⋊D3)⋊C3—22no—
31C2×(C3⋊D3)⋊C3—22no—
32C2×(C3⋊D3)⋊C3—22no—
33C2×(C3⋊D3)⋊C3D412yescomplemented
34C4×H(3,𝔽3)—22no—
35C4×H(3,𝔽3)D412yescomplemented
36(C3⋊Dic3)⋊C3—21no—
37C3×#121—91no—
38#895—31no—
39#895—91no—
40C22×C3⋊D3C22×C312yes—
41C22×C3⋊D3C22×C311yescharacteristic
42C22×C3×D3—31no—
43C22×C3×D3—32no—
44C2×C32×C4—21no—
45C2×C32×C4—22no—
46C2×C32×C4C2×D312yes—
47C2×C32×C4C2×D311yescharacteristic
48C2×C32×C4C22×C312yes—
49C2×C32×C4C22×C311yescharacteristic
50C2×C3⋊Dic3C22×C311yescharacteristic
51C4⋊(C3⋊D3)—24no—
52C4⋊(C3⋊D3)—22no—
53C4×C3⋊D3—22no—
54C2×C3×Dic3—31no—
55C3×D12—64no—
56C3×D12—62no—
57C3×C4×D3—62no—
58C2×H(3,𝔽3)C2×D411yescharacteristic
59C2×H(3,𝔽3)C2×D411yescharacteristic
60C2×H(3,𝔽3)#4111yescharacteristic
61(C3⋊D3)⋊C3—22no—
62(C3⋊D3)⋊C3—42no—
63C2×C3×D4—91no—
64C2×C3×D4—92no—
65C22×C3×C4—91no—
66C2×D12—32no—
67C2×D12—31no—
68C2×D12—91no—
69C2×D12—92no—
70C2×C4×D3—31no—
71C2×C4×D3—91no—
72C3×C4⋊C4—31no—
73C3×C4⋊C4—61no—
74C3×C4⋊C4—61no—
75C3×C4⋊C4C3×D311yescharacteristic
76C3×C22⋊C4—92no—
77(C2×D3)⋊C4—32no—
78(C2×D3)⋊C4—92no—
79C22×C32—21no—
80C22×C32C22×D311yescharacteristic
81C22×C32C23×C311yescharacteristic
82C2×C3⋊D3—21no—
83C2×C3⋊D3—21no—
84C2×C3⋊D3—22no—
85C2×C3⋊D3—22no—
86C2×C3⋊D3—22no—
87C2×C3⋊D3C3×D412yescomplemented
88C2×C3×D3—32no—
89C2×C3×D3—61no—
90C2×C3×D3—61no—
91C2×C3×D3—62no—
92C2×C3×D3—62no—
93C2×C3×D3—62no—
94C32×C4—22no—
95C32×C4—22no—
96C32×C4—22no—
97C32×C4—42no—
98C32×C4D1212yescomplemented
99C32×C4C3×D412yescomplemented
100C3⋊Dic3—21no—
101C3×Dic3—61no—
102#121—91noSylow
103H(3,𝔽3)#12111yescharacteristic, complemented, Sylow
104C23×C3—91no—
105C23×C3—92no—
106C22×D3—31no—
107C22×D3—32no—
108C22×D3—91no—
109C22×D3—92no—
110C3×D4—182no—
111C3×D4—184no—
112C2×C3×C4—31no—
113C2×C3×C4—32no—
114C2×C3×C4—61no—
115C2×C3×C4—61no—
116C2×C3×C4—62no—
117C2×C3×C4—62no—
118C2×C3×C4—91no—
119C2×C3×C4—182no—
120C2×C3×C4C2×C3×D312yes—
121C2×C3×C4C2×C3×D311yescharacteristic
122C2×Dic3—31no—
123C2×Dic3—91no—
124D12—64no—
125D12—62no—
126D12—182no—
127D12—184no—
128C4×D3—62no—
129C4×D3—182no—
130C2×C32—21no—
131C2×C32—21no—
132C2×C32—21no—
133C2×C32C2×D1211yescharacteristic
134C2×C32D3×D411yescharacteristic
135C2×C32#23911yescharacteristic
136C2×C32C2×C3×D411yescharacteristic
137C2×C32C2×C3×D411yescharacteristic
138C2×C32C3×#4111yescharacteristic
139C3⋊D3—22no—
140C3⋊D3—42no—
141C3×D3—62no—
142C3×D3—122no—
143C2×D4—92no—
144C2×D4—91no—
145C22×C4—91no—
146C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
147C22⋊C4—92no—
148C22×C3—31no—
149C22×C3—61no—
150C22×C3—61no—
151C22×C3—92no—
152C22×C3—181no—
153C22×C3—181no—
154C22×C3—182no—
155C22×C3—182no—
156C22×C3—182no—
157C22×C3C22×C3×D311yescharacteristic
158C2×D3—32no—
159C2×D3—61no—
160C2×D3—61no—
161C2×D3—62no—
162C2×D3—62no—
163C2×D3—62no—
164C2×D3—92no—
165C2×D3—181no—
166C2×D3—181no—
167C2×D3—182no—
168C2×D3—182no—
169C2×D3—182no—
170C3×C4—22no—
171C3×C4—32no—
172C3×C4—62no—
173C3×C4—62no—
174C3×C4—62no—
175C3×C4—122no—
176C3×C4—122no—
177C3×C4—181no—
178C3×C4C3×D1212yes—
179Dic3—61no—
180Dic3—181no—
181C32—21no—
182C32#89511yescharacteristic, complemented
183C32C3×#12111yescharacteristic, complemented
184C23—91no—
185C23—92no—
186D4—184no—
187D4—182no—
188C2×C4—91no—
189C2×C4—182no—
190C2×C4C2×(C3⋊D3)⋊C312yes—
191C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
192C2×C3—31no—
193C2×C3—31no—
194C2×C3—31no—
195C2×C3—61no—
196C2×C3—61no—
197C2×C3—61no—
198C2×C3—61no—
199C2×C3—61no—
200C2×C3—61no—
201C2×C3—182no—
202C2×C3—362no—
203C2×C3C2×C3×D1211yescharacteristic
204C2×C3C3×D3×D411yescharacteristic
205C2×C3C3×#23911yescharacteristic
206D3—62no—
207D3—122no—
208D3—182no—
209D3—362no—
210C22—92no—
211C22—181no—
212C22—181no—
213C22—182no—
214C22—182no—
215C22—182no—
216C22C22×(C3⋊D3)⋊C311yescharacteristic
217C4—22no—
218C4—181no—
219C4C4⋊((C3⋊D3)⋊C3)12yescomplemented
220C3—31no—
221C3—61no—
222C3—61no—
223C3C3×#89511yescharacteristic
224C2—182no—
225C2—362no—
226C2C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
227C2D4×(C3⋊D3)⋊C311yescharacteristic
228C2#8872211yescharacteristic
subgroupG/H
centerC22C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C23×C3
Frattini, socleC22×C3C22×C3×D3
FittingC4⋊C4×H(3,𝔽3)C2
hypercenterC4⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#121
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊((C3⋊D3)⋊C3)C2semidirect
2. C2×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. C2×C4×(C3⋊D3)⋊C3C2semidirect
4. C4⋊C4×H(3,𝔽3)C2semidirect
5. #3752⋊C3C2semidirect
6. #64104C3semidirect
16. C2×C4⋊(C3⋊D3)C2×C3semidirect
17. C2×C4⋊(C3⋊D3)C2×C3semidirect
18. C2×C4×C3⋊D3C2×C3semidirect
23. C32×C4⋊C4D3semidirect
24. C32×C4⋊C4C2×C3semidirect
25. #3752C2×C3semidirect
33. C2×(C3⋊D3)⋊C3D4semidirect
35. C4×H(3,𝔽3)D4semidirect
87. C2×C3⋊D3C3×D4semidirect
98. C32×C4D12semidirect
99. C32×C4C3×D4semidirect
103. H(3,𝔽3)#121semidirect
146. C4⋊C4(C3⋊D3)⋊C3semidirect
182. C32#895semidirect
183. C32C3×#121semidirect
219. C4C4⋊((C3⋊D3)⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊((C3⋊D3)⋊C3)2
C2×C4×(C3⋊D3)⋊C32
C4⋊C4×H(3,𝔽3)2
#3752⋊C32
#641043
C3×#8953