Finite Group Databasebeta

← #64139⋊C3 #11721601

Subgroups — #64139⋊C3 #11721601

Order 864.

383 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 87 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×C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#638⋊C3C212yesmaximal, complemented
3C2×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
4#3752⋊C3C212yesmaximal, complemented
5C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
6#64139C311yesmaximal, characteristic, complemented
7C3×#930—31nomaximal
8C23×H(3,𝔽3)C2211yescharacteristic
9C22×(C3⋊D3)⋊C3C2212yes—
10C2×C4×H(3,𝔽3)—22no—
11C2×C4×H(3,𝔽3)C2211yescharacteristic
12C2×C4×H(3,𝔽3)C2211yescharacteristic
13#638⋊C3—24no—
14C2×(C3⋊Dic3)⋊C3C2212yes—
15C4⋊((C3⋊D3)⋊C3)—22no—
16C22×C32×C4—21no—
17C22×C32×C4D311yescharacteristic, complemented
18C22×C32×C4C2×C311yescharacteristic, complemented
19C2×#638C2×C312yescomplemented
20C2×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
21C2×C3×#67—32no—
22C2×C3×D12—31no—
23#3752C2×C312yescomplemented
24C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
25C3×(C2×D3)⋊C4—32no—
26C3×C4⋊Dic3—31no—
27C22×H(3,𝔽3)—21no—
28C22×H(3,𝔽3)—21no—
29C22×H(3,𝔽3)D412yescomplemented
30C22×H(3,𝔽3)C2311yescharacteristic
31C2×(C3⋊D3)⋊C3—22no—
32C2×(C3⋊D3)⋊C3—22no—
33C2×(C3⋊D3)⋊C3—22no—
34C4×H(3,𝔽3)—21no—
35C4×H(3,𝔽3)D412yescomplemented
36(C3⋊Dic3)⋊C3—22no—
37C3×#121—91no—
38#930—31no—
39#930—91no—
40C23×C32—21no—
41C23×C32C2×D311yescharacteristic
42C23×C32C22×C311yescharacteristic
43C22×C3⋊D3C22×C312yes—
44C22×C3×D3—32no—
45C2×C32×C4—24no—
46C2×C32×C4—22no—
47C2×C32×C4—22no—
48C2×C32×C4—21no—
49C2×C32×C4—21no—
50C2×C32×C4C2×D311yescharacteristic
51C2×C32×C4C2×D311yescharacteristic
52C2×C32×C4C22×C311yescharacteristic
53C2×C32×C4C22×C311yescharacteristic
54#638—24no—
55C2×C3⋊Dic3C22×C312yes—
56C4⋊(C3⋊D3)—22no—
57C3×#67—64no—
58C2×C3×Dic3—32no—
59C3×D12—62no—
60C2×H(3,𝔽3)—22no—
61C2×H(3,𝔽3)C2×D411yescharacteristic
62C2×H(3,𝔽3)C2×D411yescharacteristic
63C2×H(3,𝔽3)#4111yescharacteristic
64(C3⋊D3)⋊C3—42no—
65C2×C3×D4—91no—
66C2×C3×D4—92no—
67C22×C3×C4—31no—
68C22×C3×C4—31no—
69C22×C3×C4—61no—
70C22×C3×C4C3×D311yescharacteristic
71C2×#67—32no—
72C2×#67—92no—
73C2×D12—31no—
74C2×D12—91no—
75C3×C4⋊C4—91no—
76C3×C22⋊C4—92no—
77(C2×D3)⋊C4—32no—
78(C2×D3)⋊C4—92no—
79C4⋊Dic3—31no—
80C4⋊Dic3—91no—
81C22×C32—22no—
82C22×C32—22no—
83C22×C32—21no—
84C22×C32—21no—
85C22×C32—21no—
86C22×C32—21no—
87C22×C32—22no—
88C22×C32—21no—
89C22×C32D1212yescomplemented
90C22×C32C3×D412yescomplemented
91C22×C32C22×D311yescharacteristic
92C22×C32C23×C311yescharacteristic
93C2×C3⋊D3—22no—
94C2×C3⋊D3—22no—
95C2×C3⋊D3—22no—
96C2×C3×D3—62no—
97C2×C3×D3—62no—
98C2×C3×D3—62no—
99C32×C4—22no—
100C32×C4—21no—
101C32×C4—21no—
102C32×C4—22no—
103C32×C4#6712yescomplemented
104C32×C4C3×D412yescomplemented
105C3⋊Dic3—22no—
106C3×Dic3—62no—
107#121—91noSylow
108H(3,𝔽3)#12111yescharacteristic, complemented, Sylow
109C23×C3—31no—
110C23×C3—31no—
111C23×C3—61no—
112C23×C3—92no—
113C23×C3C2×C3×D311yescharacteristic
114C22×D3—32no—
115C22×D3—92no—
116C3×D4—182no—
117C3×D4—184no—
118C2×C3×C4—22no—
119C2×C3×C4—31no—
120C2×C3×C4—31no—
121C2×C3×C4—31no—
122C2×C3×C4—31no—
123C2×C3×C4—62no—
124C2×C3×C4—62no—
125C2×C3×C4—64no—
126C2×C3×C4—61no—
127C2×C3×C4—61no—
128C2×C3×C4—92no—
129C2×C3×C4C2×C3×D311yescharacteristic
130C2×C3×C4C2×C3×D311yescharacteristic
131#67—64no—
132#67—184no—
133C2×Dic3—32no—
134C2×Dic3—92no—
135D12—62no—
136D12—182no—
137C2×C32—24no—
138C2×C32—22no—
139C2×C32—22no—
140C2×C32—21no—
141C2×C32—21no—
142C2×C32—21no—
143C2×C32C2×D1211yescharacteristic
144C2×C32#23511yescharacteristic
145C2×C32C2×#6711yescharacteristic
146C2×C32C2×C3×D411yescharacteristic
147C2×C32C2×C3×D411yescharacteristic
148C2×C32C3×#4111yescharacteristic
149C3⋊D3—42no—
150C3×D3—122no—
151C2×D4—92no—
152C2×D4—91no—
153C22×C4(C3⋊D3)⋊C311yescharacteristic, complemented
154C4⋊C4—91no—
155C22⋊C4—92no—
156C22×C3—21no—
157C22×C3—21no—
158C22×C3—32no—
159C22×C3—32no—
160C22×C3—31no—
161C22×C3—31no—
162C22×C3—61no—
163C22×C3—61no—
164C22×C3—61no—
165C22×C3—61no—
166C22×C3—62no—
167C22×C3—62no—
168C22×C3—62no—
169C22×C3—61no—
170C22×C3—182no—
171C22×C3—182no—
172C22×C3—182no—
173C22×C3C3×D1212yes—
174C22×C3C22×C3×D311yescharacteristic
175C2×D3—62no—
176C2×D3—62no—
177C2×D3—62no—
178C2×D3—182no—
179C2×D3—182no—
180C2×D3—182no—
181C3×C4—21no—
182C3×C4—32no—
183C3×C4—32no—
184C3×C4—61no—
185C3×C4—61no—
186C3×C4—62no—
187C3×C4—62no—
188C3×C4—182no—
189C3×C4C3×#6712yes—
190Dic3—62no—
191Dic3—182no—
192C32—21no—
193C32#93011yescharacteristic, complemented
194C32C3×#12111yescharacteristic, complemented
195C23—92no—
196C23C2×(C3⋊D3)⋊C311yescharacteristic
197D4—184no—
198D4—182no—
199C2×C4—22no—
200C2×C4—92no—
201C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
202C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
203C2×C3—22no—
204C2×C3—31no—
205C2×C3—31no—
206C2×C3—31no—
207C2×C3—31no—
208C2×C3—31no—
209C2×C3—31no—
210C2×C3—62no—
211C2×C3—62no—
212C2×C3—64no—
213C2×C3—61no—
214C2×C3—61no—
215C2×C3—61no—
216C2×C3—362no—
217C2×C3C2×C3×D1211yescharacteristic
218C2×C3C3×#23511yescharacteristic
219C2×C3C2×C3×#6711yescharacteristic
220D3—122no—
221D3—362no—
222C22—21no—
223C22—21no—
224C22—182no—
225C22—182no—
226C22—182no—
227C22C4⋊((C3⋊D3)⋊C3)12yescomplemented
228C22C22×(C3⋊D3)⋊C311yescharacteristic
229C4—21no—
230C4—182no—
231C4#638⋊C312yescomplemented
232C3—31no—
233C3—31no—
234C3—61no—
235C3C3×#93011yescharacteristic
236C2—22no—
237C2—362no—
238C2C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
239C2#3828⋊C311yescharacteristic
240C2C2×#638⋊C311yescharacteristic
subgroupG/H
centerC22C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C23×C3
Frattini, socleC22×C3C22×C3×D3
FittingC22×C4×H(3,𝔽3)C2
hypercenterC22×C4(C3⋊D3)⋊C3
hypocenterC32C3×#121
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×H(3,𝔽3)C2semidirect
2. C2×#638⋊C3C2semidirect
3. C2×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. #3752⋊C3C2semidirect
5. C4⋊((C3⋊Dic3)⋊C3)C2semidirect
6. #64139C3semidirect
17. C22×C32×C4D3semidirect
18. C22×C32×C4C2×C3semidirect
19. C2×#638C2×C3semidirect
20. C2×C4⋊(C3⋊D3)C2×C3semidirect
23. #3752C2×C3semidirect
24. C4⋊(C3⋊Dic3)C2×C3semidirect
29. C22×H(3,𝔽3)D4semidirect
35. C4×H(3,𝔽3)D4semidirect
89. C22×C32D12semidirect
90. C22×C32C3×D4semidirect
103. C32×C4#67semidirect
104. C32×C4C3×D4semidirect
108. H(3,𝔽3)#121semidirect
153. C22×C4(C3⋊D3)⋊C3semidirect
193. C32#930semidirect
194. C32C3×#121semidirect
227. C22C4⋊((C3⋊D3)⋊C3)semidirect
231. C4#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C4×H(3,𝔽3)2
C2×#638⋊C32
C2×C4⋊((C3⋊D3)⋊C3)2
#3752⋊C32
C4⋊((C3⋊Dic3)⋊C3)2
#641393
C3×#9303