Finite Group Databasebeta

← C2×C3×#395⋊C3 #11919436

Subgroups — C2×C3×#395⋊C3 #11919436

Order 1152.

436 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 56 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×C3×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic
2C3×#395⋊C3C212yesmaximal, direct product factor
3C2×C3×#395C311yesmaximal, characteristic, complemented
4C2×#395⋊C3C313yesmaximal, direct product factor
5C3×C22⋊SL(2,𝔽3)C2211yescharacteristic
6C2×C3×C4×A4—41nomaximal
7C23×C3×Q8C2×C311yescharacteristic
8C2×C22⋊SL(2,𝔽3)C2×C313yes—
9C22×C3×C4⋊C4—31no—
10C3×#395—32no—
11C3×#395C2×C312yescomplemented
12C2×C3×#95—31no—
13#395⋊C3C2×C316yesdirect product factor
14C22×C3×A4—41no—
15C2×C3×SL(2,𝔽3)—82no—
16C3×C4×A4—42no—
17C2×#395C3211yescharacteristic, complemented, Sylow
18C22×C3×Q8—31no—
19C22×C3×Q8—62no—
20C22×C3×Q8—62no—
21C22×C3×Q8C22×C311yescharacteristic
22C23×C3×C4—31no—
23C23×C3×C4—31no—
24C23×C3×C4A411yescharacteristic
25C22⋊SL(2,𝔽3)C22×C313yes—
26C2×C4×A4—43no—
27C2×C3×C4⋊C4—32no—
28C2×C3×C4⋊C4—32no—
29C2×C3×C4⋊C4—62no—
30C2×C3×C4⋊C4—62no—
31C3×#95—32no—
32C3×#95—32no—
33C2×C3×A4—42no—
34C2×C3×A4—41no—
35C2×C32×C4—161no—
36C3×SL(2,𝔽3)—82no—
37C23×Q8C2×C3211yescharacteristic
38C22×C4⋊C4—31no—
39#395—32no—
40#395C2×C3212yescomplemented
41C2×#95—31no—
42C24×C3C2×A411yescharacteristic
43C22×A4—43no—
44C2×C3×Q8—22no—
45C2×C3×Q8—62no—
46C2×C3×Q8—62no—
47C2×C3×Q8—64no—
48C2×C3×Q8—62no—
49C2×C3×Q8—62no—
50C2×C3×Q8—64no—
51C2×C3×Q8—62no—
52C22×C3×C4—32no—
53C22×C3×C4—32no—
54C22×C3×C4—34no—
55C22×C3×C4—32no—
56C22×C3×C4—31no—
57C22×C3×C4—31no—
58C22×C3×C4—61no—
59C22×C3×C4—61no—
60C22×C3×C4—61no—
61C22×C3×C4—62no—
62C22×C3×C4—62no—
63C22×C3×C4—62no—
64C22×C3×C4—61no—
65C22×C3×C4C2×A412yes—
66C2×SL(2,𝔽3)—86no—
67C4×A4—46no—
68C3×C4⋊C4—64no—
69C3×C4⋊C4—64no—
70C22×C32—161no—
71C3×A4—41no—
72C32×C4—162no—
73C22×Q8—31no—
74C22×Q8—62no—
75C22×Q8—62no—
76C22×Q8C22×C3211yescharacteristic
77C23×C4—31no—
78C23×C4—31no—
79C23×C4C3×A411yescharacteristic
80C2×C4⋊C4—32no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4—62no—
83C2×C4⋊C4—62no—
84#95—32no—
85#95—32no—
86C23×C3—31no—
87C23×C3—32no—
88C23×C3—31no—
89C23×C3#41⋊C312yes—
90C23×C3C22×A411yescharacteristic
91C2×A4—43no—
92C2×A4—46no—
93C3×Q8—22no—
94C3×Q8—62no—
95C3×Q8—62no—
96C3×Q8—62no—
97C3×Q8—64no—
98C2×C3×C4—32no—
99C2×C3×C4—32no—
100C2×C3×C4—34no—
101C2×C3×C4—41no—
102C2×C3×C4—62no—
103C2×C3×C4—62no—
104C2×C3×C4—64no—
105C2×C3×C4—64no—
106C2×C3×C4—62no—
107C2×C3×C4—64no—
108C2×C3×C4—62no—
109C2×C3×C4—62no—
110C2×C3×C4—62no—
111C2×C3×C4—62no—
112C2×C3×C4—121no—
113C2×C3×C4—163no—
114SL(2,𝔽3)—86no—
115C2×C32—162no—
116C2×C32—161no—
117C24C2×C3×A411yescharacteristic
118C2×Q8—22no—
119C2×Q8—62no—
120C2×Q8—62no—
121C2×Q8—64no—
122C2×Q8—62no—
123C2×Q8—62no—
124C2×Q8—64no—
125C2×Q8—62no—
126C22×C4—32no—
127C22×C4—32no—
128C22×C4—34no—
129C22×C4—32no—
130C22×C4—31no—
131C22×C4—31no—
132C22×C4—61no—
133C22×C4—61no—
134C22×C4—61no—
135C22×C4—62no—
136C22×C4—62no—
137C22×C4—62no—
138C22×C4—61no—
139C22×C4C2×C3×A412yes—
140C4⋊C4—64no—
141C4⋊C4—64no—
142C22×C3—31no—
143C22×C3—32no—
144C22×C3—32no—
145C22×C3—32no—
146C22×C3—32no—
147C22×C3—31no—
148C22×C3—31no—
149C22×C3—163no—
150C22×C3#120⋊C311yescharacteristic
151C22×C3C2×#41⋊C311yescharacteristic
152A4—43no—
153C3×C4—42no—
154C3×C4—64no—
155C3×C4—62no—
156C3×C4—62no—
157C3×C4—166no—
158C32—161noSylow
159C23—31no—
160C23—32no—
161C23—31no—
162C23C3×#41⋊C312yes—
163C23C22×C3×A411yescharacteristic
164Q8—22no—
165Q8—62no—
166Q8—62no—
167Q8—62no—
168Q8—64no—
169C2×C4—32no—
170C2×C4—32no—
171C2×C4—34no—
172C2×C4—41no—
173C2×C4—62no—
174C2×C4—62no—
175C2×C4—62no—
176C2×C4—64no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—64no—
180C2×C4—64no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—121no—
184C2×C3—31no—
185C2×C3—31no—
186C2×C3—32no—
187C2×C3—163no—
188C2×C3—166no—
189C2×C3#395⋊C312yesdirect product factor
190C2×C3C2×#120⋊C311yescharacteristic
191C22—31no—
192C22—32no—
193C22—32no—
194C22—31no—
195C22—31no—
196C22—32no—
197C22—32no—
198C22C3×#120⋊C311yescharacteristic
199C22C2×C3×#41⋊C311yescharacteristic
200C4—42no—
201C4—64no—
202C4—62no—
203C4—62no—
204C3—163no—
205C3C2×#395⋊C311yescharacteristic, direct product factor
206C2—31no—
207C2—31no—
208C2—32no—
209C2C3×#395⋊C312yesdirect product factor
210C2C2×C3×#120⋊C311yescharacteristic
subgroupG/H
center, hypercenterC22×C3#120⋊C3
commutator, cosocle, hypocenterC22×Q8C22×C32
FrattiniC23C22×C3×A4
FittingC2×C3×#395C3
socleC24×C3C2×A4
prime radicalC2×C3×C22⋊SL(2,𝔽3)C2
LCSG ▷ C22×Q8
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C3×#395⋊C3C2direct (N × H)
3. C2×C3×#395C3semidirect
4. C2×#395⋊C3C3direct (N × H)
11. C3×#395C2×C3semidirect
13. #395⋊C3C2×C3direct (N × H)
17. C2×#395C32semidirect
40. #395C2×C32semidirect
189. C2×C3#395⋊C3direct (N × H)
205. C3C2×#395⋊C3direct (N × H)
209. C2C3×#395⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊SL(2,𝔽3)2
C3×#395⋊C32
C2×C3×#3953
C2×#395⋊C33
C2×C3×C4×A44