Finite Group Databasebeta

← C3×#73152 #11920930

Subgroups — C3×#73152 #11920930

Order 1152.

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

normal subgroups only

This group has 419 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×#994C211yesmaximal, characteristic
2C3×#5423C212yesmaximal, complemented
3#73152C311yesmaximal, characteristic, direct product factor
4C3×#1795—31nomaximal
5C3×#994—22no—
6C2×C3×#41⋊C3C412yescomplemented
7C2×C3×#41⋊C3C2211yescharacteristic
8C3×GU(2,𝔽3)—22no—
9C3×#836—41nomaximal
10C2×C3×H(5,𝔽2)D311yescharacteristic
11C2×#994—21no—
12C2×#994C2×C311yescharacteristic
13#5423C2×C312yescomplemented
14C3×#421—32no—
15C3×#418—32no—
16C3×#413—31no—
17C3×#410—31no—
18C2×C32×Q8—41no—
19C3×#41⋊C3—22no—
20C3×#41⋊C3D412yes—
21C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
22C3×#208—42no—
23#1795—31noSylow
24C3×H(5,𝔽2)—22no—
25C3×H(5,𝔽2)—62no—
26C2×C3×#41—31no—
27C2×C3×#41Dic312yes—
28C2×C3×#41C2×D311yescharacteristic
29C22×C3×D4—31no—
30C22×C3×D4—32no—
31#994—24no—
32#994—22no—
33C2×#41⋊C3—22no—
34C2×#41⋊C3—21no—
35C2×#41⋊C3C3×C412yescomplemented
36C2×#41⋊C3C22×C311yescharacteristic
37C2×C3×M16—32no—
38C3×#117—32no—
39GU(2,𝔽3)—22no—
40C3×#104—62no—
41C3×#104—62no—
42C3×#102—64no—
43C3×#101—61no—
44C3×#100—61no—
45C3×#99—62no—
46#836—41no—
47C32×Q8—82no—
48C2×C32×C4—42no—
49C2×C32×C4—41no—
50C3×SL(2,𝔽3)C22⋊C411yescharacteristic
51C3×C3⋊C8—82no—
52C2×H(5,𝔽2)C3×D311yescharacteristic
53#421—32no—
54#418—32no—
55#413—31no—
56#410—31no—
57C24×C3—61no—
58C3×#41—22no—
59C3×#41—32no—
60C3×#41—62no—
61C3×#41—62no—
62C3×#41—62no—
63C3×#41#6712yes—
64C2×C3×Q8—41no—
65C2×C3×Q8—81no—
66C2×C3×Q8C2×Dic311yescharacteristic
67C2×C3×Q8S411yescharacteristic
68C2×C3×D4—32no—
69C2×C3×D4—31no—
70C2×C3×D4—34no—
71C2×C3×D4—32no—
72C2×C3×D4—31no—
73C2×C3×D4—62no—
74C2×C3×D4—64no—
75C2×C3×D4—64no—
76C22×C3×C4—31no—
77C22×C3×C4—32no—
78#41⋊C3—24no—
79#41⋊C3—22no—
80#41⋊C3—22no—
81#41⋊C3C3×D412yes—
82C2×SL(2,𝔽3)—21no—
83C2×SL(2,𝔽3)C2×C3×C411yescharacteristic
84C3×M16—62no—
85C3×M16—62no—
86C2×C3×C8—62no—
87C3×C4⋊C4—62no—
88C3×C22⋊C4—62no—
89C3×C42—62no—
90#208—42no—
91C22×C32—41no—
92C32×C4—42no—
93C32×C4—82no—
94H(5,𝔽2)—22no—
95H(5,𝔽2)—62no—
96C2×#41—31no—
97C2×#41C3×Dic312yes—
98C2×#41C2×C3×D311yescharacteristic
99C22×D4—31no—
100C22×D4—32no—
101C2×M16—32no—
102#117—32no—
103#104—62no—
104#104—62no—
105#102—64no—
106#101—61no—
107#100—61no—
108#99—62no—
109C23×C3—31no—
110C23×C3—32no—
111C23×C3—64no—
112C23×C3—242no—
113C3×Q8—22no—
114C3×Q8—31no—
115C3×Q8—82no—
116C3×Q8—84no—
117C3×Q8#21711yescharacteristic
118C3×D4—32no—
119C3×D4—34no—
120C3×D4—64no—
121C3×D4—61no—
122C3×D4—62no—
123C3×D4—62no—
124C2×C3×C4—31no—
125C2×C3×C4—32no—
126C2×C3×C4—32no—
127C2×C3×C4—41no—
128C2×C3×C4—42no—
129C2×C3×C4—62no—
130C2×C3×C4—62no—
131C2×C3×C4—81no—
132C2×C3×C4—82no—
133C2×C3×C4—122no—
134C2×C3×C4SL(2,ℤ/4)12yes—
135C2×C3×C4C2×S411yescharacteristic
136SL(2,𝔽3)—21no—
137SL(2,𝔽3)C3×C22⋊C411yescharacteristic
138C3×C8—122no—
139C3⋊C8—82no—
140C2×C32—41no—
141C2×C32—81no—
142C24—61no—
143#41—22no—
144#41—32no—
145#41—62no—
146#41—62no—
147#41—62no—
148#41C3×#6712yes—
149C2×Q8C2×C3×Dic311yescharacteristic
150C2×Q8C3×S411yescharacteristic
151C2×D4—32no—
152C2×D4—31no—
153C2×D4—34no—
154C2×D4—31no—
155C2×D4—32no—
156C2×D4—62no—
157C2×D4—64no—
158C2×D4—64no—
159C22×C4—31no—
160C22×C4—32no—
161M16—62no—
162M16—62no—
163C2×C8—62no—
164C4⋊C4—62no—
165C22⋊C4—62no—
166C42—62no—
167C22×C3—32no—
168C22×C3—41no—
169C22×C3—62no—
170C22×C3—81no—
171C22×C3—121no—
172C22×C3—122no—
173C22×C3—244no—
174C22×C3C2×SL(2,ℤ/4)11yescharacteristic
175C3×C4—22no—
176C3×C4—31no—
177C3×C4—31no—
178C3×C4—42no—
179C3×C4—82no—
180C3×C4—84no—
181C3×C4—82no—
182C3×C4—242no—
183C3×C4GL(2,ℤ/4)12yes—
184C32—41noSylow
185C23—31no—
186C23—32no—
187C23—64no—
188C23—242no—
189Q8—22no—
190Q8—31no—
191Q8C3×#21711yescharacteristic
192D4—32no—
193D4—34no—
194D4—64no—
195D4—61no—
196D4—62no—
197D4—62no—
198C2×C4—31no—
199C2×C4—32no—
200C2×C4—32no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—122no—
204C2×C4C3×SL(2,ℤ/4)12yes—
205C2×C4C2×C3×S411yescharacteristic
206C8—122no—
207C2×C3—21no—
208C2×C3—41no—
209C2×C3—62no—
210C2×C3—81no—
211C2×C3—82no—
212C2×C3—81no—
213C2×C3—122no—
214C2×C3#543211yescharacteristic
215C22—32no—
216C22—62no—
217C22—121no—
218C22—122no—
219C22—244no—
220C22C2×C3×SL(2,ℤ/4)11yescharacteristic
221C4—22no—
222C4—31no—
223C4—31no—
224C4—242no—
225C4C3×GL(2,ℤ/4)12yes—
226C3—41no—
227C3—81no—
228C3#7315211yescharacteristic, direct product factor
229C2—21no—
230C2—62no—
231C2—122no—
232C2C3×#543211yescharacteristic
subgroupG/H
center, socleC2×C3#5432
commutatorC2×SL(2,𝔽3)C2×C3×C4
FrattiniC2×C4C2×C3×S4
FittingC2×C3×H(5,𝔽2)D3
prime radicalC2×C3×#994C2
cosocleC2×#41⋊C3C22×C3
hypercenterC2×C3×Q8S4
hypocenterSL(2,𝔽3)C3×C22⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5423C2semidirect
3. #73152C3direct (N × H)
6. C2×C3×#41⋊C3C4semidirect
13. #5423C2×C3semidirect
35. C2×#41⋊C3C3×C4semidirect
228. C3#73152direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9942
C3×#54232
#731523
C3×#17953
C3×#8364