Finite Group Databasebeta

← #64144⋊C3 #11721637

Subgroups — #64144⋊C3 #11721637

Order 864.

327 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 77 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
2C22×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3#3757⋊C3C212yesmaximal, complemented
4#3757⋊C3C211yesmaximal, characteristic, complemented
5#3750⋊C3C212yesmaximal, complemented
6#64144C311yesmaximal, characteristic, complemented
7C3×#935—31nomaximal
8C23×H(3,𝔽3)C2211yescharacteristic
9C23×H(3,𝔽3)C2211yescharacteristic
10D4×H(3,𝔽3)—22no—
11C2×C4×H(3,𝔽3)C2211yescharacteristic
12C2×(C3⋊Dic3)⋊C3—22no—
13C2×(C3⋊Dic3)⋊C3C2212yes—
14C2×(C3⋊Dic3)⋊C3C2212yes—
15C2×C32×D4—21no—
16C2×C32×D4D311yescharacteristic
17C2×C32×D4C2×C311yescharacteristic
18C22×C3⋊Dic3C2×C311yescharacteristic, complemented
19C22×C3×Dic3—31no—
20#3757C2×C311yescharacteristic, complemented
21#3757C2×C312yescomplemented
22#3750C2×C312yescomplemented
23C3×#217—31no—
24C3×#217—32no—
25C3×Dic3⋊C4—32no—
26C22×H(3,𝔽3)—22no—
27C22×H(3,𝔽3)—21no—
28C22×H(3,𝔽3)—22no—
29C22×H(3,𝔽3)D412yes—
30C22×H(3,𝔽3)C2311yescharacteristic
31C4×H(3,𝔽3)—21no—
32(C3⋊Dic3)⋊C3—22no—
33(C3⋊Dic3)⋊C3—22no—
34C3×#123—91no—
35#935—31no—
36#935—91no—
37C23×C32—21no—
38C23×C32—21no—
39C23×C32C2×D311yescharacteristic
40C23×C32C2×D311yescharacteristic
41C23×C32C22×C311yescharacteristic
42C23×C32C22×C311yescharacteristic
43C32×D4—24no—
44C32×D4—22no—
45C32×D4—22no—
46C2×C32×C4—21no—
47C2×C32×C4C2×D311yescharacteristic
48C2×C32×C4C22×C311yescharacteristic
49C2×C3⋊Dic3—22no—
50C2×C3⋊Dic3C22×C312yes—
51C2×C3⋊Dic3C22×C312yes—
52C2×C3×Dic3—32no—
53C2×C3×Dic3—32no—
54C2×C3×Dic3—62no—
55C2×H(3,𝔽3)—22no—
56C2×H(3,𝔽3)—41no—
57C2×H(3,𝔽3)C2×D411yescharacteristic
58C2×H(3,𝔽3)#4112yes—
59C2×C3×D4—31no—
60C2×C3×D4—31no—
61C2×C3×D4—61no—
62C2×C3×D4C3×D311yescharacteristic
63C22×C3×C4—91no—
64C22×Dic3—31no—
65C22×Dic3—91no—
66C3×C4⋊C4—92no—
67C3×C22⋊C4—91no—
68C3×C22⋊C4—92no—
69#217—31no—
70#217—32no—
71#217—91no—
72#217—92no—
73Dic3⋊C4—32no—
74Dic3⋊C4—92no—
75C22×C32—22no—
76C22×C32—22no—
77C22×C32—22no—
78C22×C32—22no—
79C22×C32—21no—
80C22×C32—21no—
81C22×C32—22no—
82C22×C32—22no—
83C22×C32—21no—
84C22×C32—42no—
85C22×C32—42no—
86C22×C32#6712yes—
87C22×C32C3×D412yes—
88C22×C32C22×D311yescharacteristic
89C22×C32C23×C311yescharacteristic
90C32×C4—22no—
91C32×C4—21no—
92C32×C4—21no—
93C3⋊Dic3—22no—
94C3⋊Dic3—22no—
95C3×Dic3—62no—
96C3×Dic3—62no—
97#123—91noSylow
98H(3,𝔽3)#12311yescharacteristic, complemented, Sylow
99C23×C3—31no—
100C23×C3—31no—
101C23×C3—31no—
102C23×C3—31no—
103C23×C3—61no—
104C23×C3—61no—
105C23×C3C2×C3×D311yescharacteristic
106C23×C3C2×C3×D311yescharacteristic
107C3×D4—22no—
108C3×D4—62no—
109C3×D4—62no—
110C3×D4—64no—
111C2×C3×C4—31no—
112C2×C3×C4—31no—
113C2×C3×C4—61no—
114C2×C3×C4—92no—
115C2×C3×C4—92no—
116C2×C3×C4—182no—
117C2×C3×C4C2×C3×D311yescharacteristic
118C2×Dic3—32no—
119C2×Dic3—32no—
120C2×Dic3—62no—
121C2×Dic3—92no—
122C2×Dic3—92no—
123C2×Dic3—182no—
124C2×C32—22no—
125C2×C32—22no—
126C2×C32—22no—
127C2×C32—21no—
128C2×C32—42no—
129C2×C32—42no—
130C2×C32—41no—
131C2×C32—41no—
132C2×C32#23712yes—
133C2×C32C2×#6711yescharacteristic
134C2×C32C2×C3×D411yescharacteristic
135C2×C32C3×#4112yes—
136C2×D4(C3⋊D3)⋊C311yescharacteristic
137C22×C4—91no—
138C4⋊C4—92no—
139C22⋊C4—91no—
140C22⋊C4—92no—
141C22×C3—22no—
142C22×C3—22no—
143C22×C3—21no—
144C22×C3—32no—
145C22×C3—32no—
146C22×C3—31no—
147C22×C3—31no—
148C22×C3—62no—
149C22×C3—62no—
150C22×C3—61no—
151C22×C3—61no—
152C22×C3—62no—
153C22×C3—62no—
154C22×C3—62no—
155C22×C3—62no—
156C22×C3—61no—
157C22×C3—122no—
158C22×C3—122no—
159C22×C3C3×#6712yes—
160C22×C3C22×C3×D311yescharacteristic
161C3×C4—21no—
162C3×C4—61no—
163C3×C4—61no—
164C3×C4—62no—
165C3×C4—182no—
166C3×C4—182no—
167Dic3—62no—
168Dic3—62no—
169Dic3—182no—
170Dic3—182no—
171C32—21no—
172C32#93511yescharacteristic, complemented
173C32C3×#12311yescharacteristic, complemented
174C23C2×(C3⋊D3)⋊C311yescharacteristic
175C23C2×(C3⋊D3)⋊C311yescharacteristic
176D4—22no—
177C2×C4—92no—
178C2×C4—92no—
179C2×C4—182no—
180C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
181C2×C3—22no—
182C2×C3—32no—
183C2×C3—32no—
184C2×C3—31no—
185C2×C3—31no—
186C2×C3—41no—
187C2×C3—62no—
188C2×C3—62no—
189C2×C3—62no—
190C2×C3—61no—
191C2×C3—122no—
192C2×C3—121no—
193C2×C3—121no—
194C2×C3—122no—
195C2×C3C3×#23712yes—
196C2×C3C2×C3×#6711yescharacteristic
197C22—22no—
198C22—22no—
199C22—21no—
200C22#638⋊C312yes—
201C22C22×(C3⋊D3)⋊C311yescharacteristic
202C4—21no—
203C4—182no—
204C4—182no—
205C3—31no—
206C3—31no—
207C3—61no—
208C3C3×#93511yescharacteristic
209C2—22no—
210C2—41no—
211C2#3830⋊C312yes—
212C2C2×#638⋊C311yescharacteristic
subgroupG/H
centerC22C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C23×C3
Frattini, socleC22×C3C22×C3×D3
Fitting, prime radicalC2×D4×H(3,𝔽3)C2
hypercenterC2×D4(C3⋊D3)⋊C3
hypocenterC32C3×#123
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C22×(C3⋊Dic3)⋊C3C2semidirect
3. #3757⋊C3C2semidirect
4. #3757⋊C3C2semidirect
5. #3750⋊C3C2semidirect
6. #64144C3semidirect
18. C22×C3⋊Dic3C2×C3semidirect
20. #3757C2×C3semidirect
21. #3757C2×C3semidirect
22. #3750C2×C3semidirect
98. H(3,𝔽3)#123semidirect
172. C32#935semidirect
173. C32C3×#123semidirect

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
C22×(C3⋊Dic3)⋊C32
#3757⋊C32
#3750⋊C32
#641443
C3×#9353