Finite Group Databasebeta

← C2×C3×#5386 #11922598

Subgroups — C2×C3×#5386 #11922598

Order 1152.

569 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 108 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×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3×#859C212yesmaximal
3C3×#5386C214yesmaximal, direct product factor
4C2×#5386C311yesmaximal, characteristic, direct product factor
5C2×C3×#475—31nomaximal
6C22×C3×Dic6—41nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C3×C4×SL(2,𝔽3)C2212yes—
9C3×#859C2214yes—
10C2×C3×C4⋊Q8—31no—
11C2×C3×C4×Q8D311yescharacteristic
12C2×C4×SL(2,𝔽3)—21no—
13C2×C4×SL(2,𝔽3)C2×C311yescharacteristic
14C2×#859C2×C312yes—
15#5386C2×C314yesdirect product factor
16C3×#475—34no—
17C3×#475—34no—
18C2×C3×#106—32no—
19C2×C3×C4⋊C8—31no—
20C2×C3×#103—32no—
21C22×C32×C4—41no—
22C22×C3×Dic3—42no—
23C2×C3×Dic6—44no—
24C2×C3×Dic6—48no—
25C2×C3×SL(2,𝔽3)Q812yescomplemented
26C2×C3×SL(2,𝔽3)C2311yescharacteristic
27C2×#475—31noSylow
28C22×C3×Q8—61no—
29C22×C3×Q8C2×D311yescharacteristic
30C22×Dic6—41no—
31C22×SL(2,𝔽3)—21no—
32C22×SL(2,𝔽3)C22×C311yescharacteristic
33C22×C3×C8—32no—
34C3×C4⋊Q8—34no—
35C3×C4⋊Q8—62no—
36C3×C4×Q8—32no—
37C3×C4×Q8C2×D312yes—
38C2×C3×C4⋊C4—31no—
39C2×C3×C4⋊C4—32no—
40C2×C3×C4⋊C4—61no—
41C2×C3×C42—31no—
42C4×SL(2,𝔽3)—22no—
43C4×SL(2,𝔽3)C22×C312yes—
44#859C22×C314yes—
45C3×#106—38no—
46C3×C4⋊C8—34no—
47C3×#103—34no—
48C3×#103—34no—
49C23×C32—41no—
50C2×C32×C4—44no—
51C2×C32×C4—42no—
52C2×C3×Dic3—44no—
53C2×C3×Dic3—48no—
54C3×Dic6—416no—
55C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
56C2×C4⋊Q8—31no—
57C2×C4×Q8C3×D311yescharacteristic
58#475—34no—
59#475—34no—
60C2×#106—32no—
61C2×C4⋊C8—31no—
62C2×#103—32no—
63C2×C3×Q8—31no—
64C2×C3×Q8—32no—
65C2×C3×Q8—64no—
66C2×C3×Q8—124no—
67C2×C3×Q8Dic612yescomplemented
68C2×C3×Q8C22×D311yescharacteristic
69C22×C3×C4—31no—
70C22×C3×C4—31no—
71C22×C3×C4—41no—
72C22×C3×C4—62no—
73C22×C3×C4—81no—
74C22×C3×C4S411yescharacteristic
75C22×Dic3—42no—
76C2×Dic6—44no—
77C2×Dic6—48no—
78C2×SL(2,𝔽3)—22no—
79C2×SL(2,𝔽3)—21no—
80C2×SL(2,𝔽3)C3×Q812yescomplemented
81C2×SL(2,𝔽3)C23×C311yescharacteristic
82C2×C3×C8—34no—
83C2×C3×C8—64no—
84C3×C4⋊C4—32no—
85C3×C4⋊C4—34no—
86C3×C4⋊C4—61no—
87C3×C4⋊C4—62no—
88C3×C4⋊C4—64no—
89C3×C42—32no—
90C3×C42—32no—
91C22×C32—42no—
92C22×C32—42no—
93C22×C32—42no—
94C22×C32—41no—
95C32×C4—44no—
96C3×Dic3—48no—
97C22×Q8—61no—
98C22×Q8C2×C3×D311yescharacteristic
99C22×C8—32no—
100C4⋊Q8—34no—
101C4⋊Q8—62no—
102C4×Q8—32no—
103C4×Q8C2×C3×D312yes—
104C2×C4⋊C4—31no—
105C2×C4⋊C4—32no—
106C2×C4⋊C4—61no—
107C2×C42—31no—
108#106—38no—
109C4⋊C8—34no—
110#103—34no—
111#103—34no—
112C23×C3—41no—
113C23×C3—81no—
114C23×C3C2×S411yescharacteristic
115C3×Q8—32no—
116C3×Q8—31no—
117C3×Q8—61no—
118C3×Q8—128no—
119C3×Q8C2×Dic611yescharacteristic, complemented
120C2×C3×C4—32no—
121C2×C3×C4—32no—
122C2×C3×C4—31no—
123C2×C3×C4—31no—
124C2×C3×C4—32no—
125C2×C3×C4—42no—
126C2×C3×C4—44no—
127C2×C3×C4—62no—
128C2×C3×C4—64no—
129C2×C3×C4—82no—
130C2×C3×C4—84no—
131C2×C3×C4—124no—
132C2×C3×C4GL(2,𝔽3)14yes—
133C2×C3×C4C2×S412yes—
134C2×Dic3—44no—
135C2×Dic3—48no—
136Dic6—416no—
137SL(2,𝔽3)—21no—
138SL(2,𝔽3)C2×C3×Q811yescharacteristic, complemented
139C3×C8—64no—
140C2×C32—44no—
141C2×C32—41no—
142C2×C32—41no—
143C2×C32—41no—
144C2×Q8—31no—
145C2×Q8—32no—
146C2×Q8—64no—
147C2×Q8—124no—
148C2×Q8C3×Dic612yescomplemented
149C2×Q8C22×C3×D311yescharacteristic
150C22×C4—31no—
151C22×C4—31no—
152C22×C4—62no—
153C22×C4C3×S411yescharacteristic
154C2×C8—34no—
155C2×C8—64no—
156C4⋊C4—32no—
157C4⋊C4—34no—
158C4⋊C4—61no—
159C4⋊C4—62no—
160C4⋊C4—64no—
161C42—32no—
162C42—32no—
163C22×C3—41no—
164C22×C3—42no—
165C22×C3—42no—
166C22×C3—42no—
167C22×C3—81no—
168C22×C3—82no—
169C22×C3—82no—
170C22×C3—82no—
171C22×C3A4⋊Q812yes—
172C22×C3C2×GL(2,𝔽3)12yes—
173C22×C3#98412yes—
174C22×C3C22×S411yescharacteristic
175C3×C4—31no—
176C3×C4—31no—
177C3×C4—32no—
178C3×C4—44no—
179C3×C4—62no—
180C3×C4—84no—
181C3×C4—124no—
182C3×C4C2×GL(2,𝔽3)14yes—
183Dic3—48no—
184C32—41noSylow
185C23C2×C3×S411yescharacteristic
186Q8—32no—
187Q8—31no—
188Q8—61no—
189Q8—128no—
190Q8C2×C3×Dic611yescharacteristic, complemented
191C2×C4—31no—
192C2×C4—31no—
193C2×C4—32no—
194C2×C4—32no—
195C2×C4—32no—
196C2×C4—62no—
197C2×C4—64no—
198C2×C4—124no—
199C2×C4C3×GL(2,𝔽3)14yes—
200C2×C4C2×C3×S412yes—
201C8—64no—
202C2×C3—44no—
203C2×C3—41no—
204C2×C3—41no—
205C2×C3—41no—
206C2×C3—84no—
207C2×C3—81no—
208C2×C3—81no—
209C2×C3—81no—
210C2×C3#538614yesdirect product factor
211C2×C3C2×A4⋊Q811yescharacteristic
212C2×C3C22×GL(2,𝔽3)11yescharacteristic
213C2×C3C2×#98411yescharacteristic
214C22C3×A4⋊Q812yes—
215C22C2×C3×GL(2,𝔽3)12yes—
216C22C3×#98412yes—
217C22C22×C3×S411yescharacteristic
218C4—31no—
219C4—31no—
220C4—32no—
221C4—62no—
222C4—124no—
223C4C2×C3×GL(2,𝔽3)14yes—
224C3—41no—
225C3—81no—
226C3C2×#538611yescharacteristic, direct product factor
227C2C3×#538614yesdirect product factor
228C2C2×C3×A4⋊Q811yescharacteristic
229C2C22×C3×GL(2,𝔽3)11yescharacteristic
230C2C2×C3×#98411yescharacteristic
subgroupG/H
center, socleC23×C3C2×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C3
FrattiniC22C22×C3×S4
FittingC2×C3×C4×Q8D3
prime radicalC22×C3×SL(2,𝔽3)C22
hypercenterC22×C3×C4S4
hypocenterSL(2,𝔽3)C2×C3×Q8
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×#5386C2direct (N × H)
4. C2×#5386C3direct (N × H)
15. #5386C2×C3direct (N × H)
25. C2×C3×SL(2,𝔽3)Q8semidirect
55. C3×SL(2,𝔽3)C2×Q8semidirect
67. C2×C3×Q8Dic6semidirect
80. C2×SL(2,𝔽3)C3×Q8semidirect
119. C3×Q8C2×Dic6semidirect
138. SL(2,𝔽3)C2×C3×Q8semidirect
148. C2×Q8C3×Dic6semidirect
190. Q8C2×C3×Dic6semidirect
210. C2×C3#5386direct (N × H)
226. C3C2×#5386direct (N × H)
227. C2C3×#5386direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
C2×C3×#8592
C3×#53862
C2×#53863
C2×C3×#4753
C22×C3×Dic64