Finite Group Databasebeta

← #64107⋊C3 #11721472

Subgroups — #64107⋊C3 #11721472

Order 864.

236 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#3752⋊C3C211yesmaximal, characteristic
3#3752⋊C3C211yesmaximal, characteristic
4#3752⋊C3C211yesmaximal, characteristic
5C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
6#3750⋊C3C211yesmaximal, characteristic, complemented
7C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
8#64107C311yesmaximal, characteristic, complemented
9C3×#898—31nomaximal
10C22×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×C4×H(3,𝔽3)C2211yescharacteristic
12C2×C4×H(3,𝔽3)C2211yescharacteristic
13C2×C4×H(3,𝔽3)C2211yescharacteristic
14C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C32×C4⋊C4—21no—
18C32×C4⋊C4D311yescharacteristic, complemented
19C32×C4⋊C4C2×C311yescharacteristic, complemented
20#3752C2×C311yescharacteristic
21#3752C2×C311yescharacteristic
22#3752C2×C311yescharacteristic
23C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
24#3750C2×C311yescharacteristic, complemented
25C4×C3⋊Dic3C2×C311yescharacteristic, complemented
26C3×(C2×D3)⋊C4—31no—
27C3×(C2×D3)⋊C4—31no—
28C3×(C2×D3)⋊C4—31no—
29C3×C4⋊Dic3—31no—
30C3×Dic3⋊C4—31no—
31C3×C4×Dic3—31no—
32C22×H(3,𝔽3)C2311yescharacteristic
33C2×(C3⋊D3)⋊C3—21no—
34C2×(C3⋊D3)⋊C3—21no—
35C2×(C3⋊D3)⋊C3—21no—
36C4×H(3,𝔽3)—21no—
37C4×H(3,𝔽3)—21no—
38C4×H(3,𝔽3)—21no—
39(C3⋊Dic3)⋊C3—21no—
40(C3⋊Dic3)⋊C3—21no—
41(C3⋊Dic3)⋊C3—21no—
42C3×#126—91no—
43#898—31no—
44#898—91no—
45C22×C3⋊D3C22×C311yescharacteristic
46C22×C3×D3—31no—
47C2×C32×C4—21no—
48C2×C32×C4—21no—
49C2×C32×C4—21no—
50C2×C32×C4C2×D311yescharacteristic
51C2×C32×C4C2×D311yescharacteristic
52C2×C32×C4C2×D311yescharacteristic
53C2×C32×C4C22×C311yescharacteristic
54C2×C32×C4C22×C311yescharacteristic
55C2×C32×C4C22×C311yescharacteristic
56C2×C3⋊Dic3C22×C311yescharacteristic
57C2×C3⋊Dic3C22×C311yescharacteristic
58C2×C3⋊Dic3C22×C311yescharacteristic
59C2×C3×Dic3—31no—
60C2×C3×Dic3—31no—
61C2×C3×Dic3—31no—
62C2×H(3,𝔽3)#4111yescharacteristic
63C2×H(3,𝔽3)#4111yescharacteristic
64C2×H(3,𝔽3)#4111yescharacteristic
65(C3⋊D3)⋊C3—41no—
66C3×C4⋊C4—31no—
67C3×C4⋊C4—61no—
68C3×C4⋊C4—61no—
69C3×C4⋊C4—91no—
70C3×C4⋊C4—91no—
71C3×C4⋊C4C3×D311yescharacteristic
72C3×C22⋊C4—91no—
73C3×C22⋊C4—91no—
74C3×C22⋊C4—91no—
75C3×C42—91no—
76(C2×D3)⋊C4—31no—
77(C2×D3)⋊C4—31no—
78(C2×D3)⋊C4—31no—
79(C2×D3)⋊C4—91no—
80(C2×D3)⋊C4—91no—
81(C2×D3)⋊C4—91no—
82C4⋊Dic3—31no—
83C4⋊Dic3—91no—
84Dic3⋊C4—31no—
85Dic3⋊C4—91no—
86C4×Dic3—31no—
87C4×Dic3—91no—
88C22×C32—21no—
89C22×C32C22×D311yescharacteristic
90C22×C32C23×C311yescharacteristic
91C2×C3⋊D3—21no—
92C2×C3⋊D3—21no—
93C2×C3⋊D3—21no—
94C2×C3×D3—61no—
95C2×C3×D3—61no—
96C2×C3×D3—61no—
97C32×C4—22no—
98C32×C4—21no—
99C32×C4—21no—
100C32×C4—21no—
101C32×C4—21no—
102C32×C4—21no—
103C32×C4—21no—
104C32×C4—41no—
105C32×C4—41no—
106C3⋊Dic3—21no—
107C3⋊Dic3—21no—
108C3⋊Dic3—21no—
109C3×Dic3—61no—
110C3×Dic3—61no—
111C3×Dic3—61no—
112#126—91noSylow
113H(3,𝔽3)#12611yescharacteristic, complemented, Sylow
114C23×C3—91no—
115C22×D3—31no—
116C22×D3—91no—
117C2×C3×C4—31no—
118C2×C3×C4—31no—
119C2×C3×C4—31no—
120C2×C3×C4—61no—
121C2×C3×C4—61no—
122C2×C3×C4—61no—
123C2×C3×C4—61no—
124C2×C3×C4—61no—
125C2×C3×C4—61no—
126C2×C3×C4—91no—
127C2×C3×C4—91no—
128C2×C3×C4—91no—
129C2×C3×C4C2×C3×D311yescharacteristic
130C2×C3×C4C2×C3×D311yescharacteristic
131C2×C3×C4C2×C3×D311yescharacteristic
132C2×Dic3—31no—
133C2×Dic3—31no—
134C2×Dic3—31no—
135C2×Dic3—91no—
136C2×Dic3—91no—
137C2×Dic3—91no—
138C2×C32—21no—
139C2×C32—21no—
140C2×C32—21no—
141C2×C32#23511yescharacteristic
142C2×C32#23711yescharacteristic
143C2×C32#23911yescharacteristic
144C2×C32C3×#4111yescharacteristic
145C2×C32C3×#4111yescharacteristic
146C2×C32C3×#4111yescharacteristic
147C3⋊D3—41no—
148C3×D3—121no—
149C4⋊C4—91no—
150C4⋊C4—91no—
151C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
152C22⋊C4—91no—
153C22⋊C4—91no—
154C22⋊C4—91no—
155C42—91no—
156C22×C3—31no—
157C22×C3—61no—
158C22×C3—61no—
159C22×C3—181no—
160C22×C3—181no—
161C22×C3—181no—
162C22×C3C22×C3×D311yescharacteristic
163C2×D3—61no—
164C2×D3—61no—
165C2×D3—61no—
166C2×D3—181no—
167C2×D3—181no—
168C2×D3—181no—
169C3×C4—21no—
170C3×C4—21no—
171C3×C4—21no—
172C3×C4—61no—
173C3×C4—61no—
174C3×C4—61no—
175C3×C4—62no—
176C3×C4—61no—
177C3×C4—121no—
178C3×C4—121no—
179C3×C4—121no—
180C3×C4—121no—
181C3×C4—181no—
182C3×C4—181no—
183C3×C4—181no—
184Dic3—61no—
185Dic3—61no—
186Dic3—61no—
187Dic3—181no—
188Dic3—181no—
189Dic3—181no—
190C32—21no—
191C32#89811yescharacteristic, complemented
192C32C3×#12611yescharacteristic, complemented
193C23—91no—
194C2×C4—91no—
195C2×C4—91no—
196C2×C4—91no—
197C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
198C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
199C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
200C2×C3—31no—
201C2×C3—31no—
202C2×C3—31no—
203C2×C3—61no—
204C2×C3—61no—
205C2×C3—61no—
206C2×C3—61no—
207C2×C3—61no—
208C2×C3—61no—
209C2×C3—361no—
210C2×C3C3×#23511yescharacteristic
211C2×C3C3×#23711yescharacteristic
212C2×C3C3×#23911yescharacteristic
213D3—121no—
214D3—361no—
215C22—181no—
216C22—181no—
217C22—181no—
218C22C22×(C3⋊D3)⋊C311yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—181no—
223C4—181no—
224C4—181no—
225C3—31no—
226C3—61no—
227C3—61no—
228C3C3×#89811yescharacteristic
229C2—361no—
230C2#3828⋊C311yescharacteristic
231C2#3830⋊C311yescharacteristic
232C2#8872211yescharacteristic
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×#126
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C4⋊C4×H(3,𝔽3)C2semidirect
5. C4⋊((C3⋊Dic3)⋊C3)C2semidirect
6. #3750⋊C3C2semidirect
7. C4×(C3⋊Dic3)⋊C3C2semidirect
8. #64107C3semidirect
18. C32×C4⋊C4D3semidirect
19. C32×C4⋊C4C2×C3semidirect
23. C4⋊(C3⋊Dic3)C2×C3semidirect
24. #3750C2×C3semidirect
25. C4×C3⋊Dic3C2×C3semidirect
113. H(3,𝔽3)#126semidirect
151. C4⋊C4(C3⋊D3)⋊C3semidirect
191. C32#898semidirect
192. C32C3×#126semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×H(3,𝔽3)2
#3752⋊C32
C4⋊((C3⋊Dic3)⋊C3)2
#3750⋊C32
C4×(C3⋊Dic3)⋊C32
#641073
C3×#8983