Finite Group Databasebeta

← C3×#73129 #11920907

Subgroups — C3×#73129 #11920907

Order 1152.

262 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 48 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
1C3×#568⋊C3C211yesmaximal, characteristic
2C3×#5423C211yesmaximal, characteristic
3C2×C3×GU(2,𝔽3)C211yesmaximal, characteristic
4#73129C311yesmaximal, characteristic, direct product factor
5C3×#1861—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7C3×#131⋊C3—21no—
8C3×GU(2,𝔽3)—21no—
9C3×GU(2,𝔽3)C411yescharacteristic, complemented
10C3×GU(2,𝔽3)C411yescharacteristic, complemented
11C3×#822—41nomaximal
12C3×#568D311yescharacteristic
13#568⋊C3—21no—
14#568⋊C3C2×C311yescharacteristic
15#5423C2×C311yescharacteristic
16C2×GU(2,𝔽3)C2×C311yescharacteristic
17C3×#430—31no—
18C3×#423—31no—
19C3×#421—31no—
20C2×C3×#104—31no—
21C3×#344—31no—
22C3×#339—31no—
23C3×#41⋊C3C2×C411yescharacteristic
24C3×#41⋊C3D411yescharacteristic
25C2×C3×SL(2,𝔽3)Q811yescharacteristic
26C32×M16—41no—
27C3×#208—41no—
28C2×C3×C3⋊C8—41no—
29#1861—31noSylow
30C2×C3×#41C2×D311yescharacteristic
31C2×#41⋊C3—21no—
32C2×#41⋊C3C22×C311yescharacteristic
33C3×#131—21no—
34C3×#131—61no—
35C2×C3×M16—31no—
36C2×C3×M16—31no—
37C2×C3×M16—31no—
38C2×C3×M16—31no—
39C3×#117—31no—
40C2×C3×C42—31no—
41#131⋊C3—22no—
42#131⋊C3—21no—
43GU(2,𝔽3)—21no—
44GU(2,𝔽3)C3×C411yescharacteristic, complemented
45GU(2,𝔽3)C3×C411yescharacteristic, complemented
46C3×#108—61no—
47C3×#108—61no—
48C3×C4⋊C8—62no—
49C3×#104—31no—
50C3×#104—31no—
51C3×#104—31no—
52C3×#104—31no—
53C3×#104—61no—
54C3×#104—61no—
55#822—41no—
56C2×C32×C4—41no—
57C3×SL(2,𝔽3)C4⋊C411yescharacteristic
58C32×C8—81no—
59C3×C3⋊C8—41no—
60C3×C3⋊C8—41no—
61C3×C3⋊C8—81no—
62#568C3×D311yescharacteristic
63#430—31no—
64#423—31no—
65#421—31no—
66C2×#104—31no—
67#344—31no—
68#339—31no—
69C3×#41—31no—
70C3×#41—31no—
71C3×#41C4×D311yescharacteristic
72C3×#41#6711yescharacteristic
73C2×C3×Q8Dic611yescharacteristic
74C2×C3×D4—31no—
75C22×C3×C4—31no—
76C22×C3×C4—61no—
77#41⋊C3—21no—
78#41⋊C3—21no—
79#41⋊C3C2×C3×C411yescharacteristic
80#41⋊C3C3×D411yescharacteristic
81C2×SL(2,𝔽3)—21no—
82C2×SL(2,𝔽3)C3×Q811yescharacteristic
83C3×M16—31no—
84C3×M16—31no—
85C3×M16—31no—
86C3×M16—41no—
87C3×M16—61no—
88C3×M16—61no—
89C3×M16—61no—
90C3×M16—61no—
91C3×M16—61no—
92C3×M16—81no—
93C3×M16S411yescharacteristic
94C2×C3×C8—32no—
95C2×C3×C8—61no—
96C2×C3×C8—61no—
97C2×C3×C8—61no—
98C3×C4⋊C4—61no—
99C3×C22⋊C4—61no—
100C3×C42—31no—
101C3×C42—31no—
102C3×C42—61no—
103C3×C42—61no—
104#208—41no—
105C2×C3⋊C8—41no—
106C22×C32—41no—
107C32×C4—41no—
108C32×C4—41no—
109C2×#41C2×C3×D311yescharacteristic
110#131—21no—
111#131—61no—
112C2×M16—31no—
113C2×M16—31no—
114C2×M16—31no—
115C2×M16—31no—
116#117—31no—
117C2×C42—31no—
118#108—61no—
119#108—61no—
120C4⋊C8—62no—
121#104—31no—
122#104—31no—
123#104—31no—
124#104—31no—
125#104—61no—
126#104—61no—
127C23×C3—31no—
128C3×Q8—31no—
129C3×Q8Dic3⋊C411yescharacteristic
130C3×D4—31no—
131C3×D4—31no—
132C3×D4—61no—
133C2×C3×C4—31no—
134C2×C3×C4—31no—
135C2×C3×C4—31no—
136C2×C3×C4—31no—
137C2×C3×C4—31no—
138C2×C3×C4—41no—
139C2×C3×C4—61no—
140C2×C3×C4—61no—
141C2×C3×C4—81no—
142C2×C3×C4—122no—
143C2×C3×C4—121no—
144C2×C3×C4C2×S411yescharacteristic
145SL(2,𝔽3)—21no—
146SL(2,𝔽3)C3×C4⋊C411yescharacteristic
147C3×C8—21no—
148C3×C8—61no—
149C3×C8—61no—
150C3×C8—61no—
151C3×C8—81no—
152C3×C8—82no—
153C3×C8—121no—
154C3⋊C8—41no—
155C3⋊C8—41no—
156C3⋊C8—81no—
157C2×C32—41no—
158C2×C32—81no—
159#41—31no—
160#41—31no—
161#41C3×C4×D311yescharacteristic
162#41C3×#6711yescharacteristic
163C2×Q8C3×Dic611yescharacteristic
164C2×D4—31no—
165C22×C4—31no—
166C22×C4—61no—
167M16—31no—
168M16—31no—
169M16—31no—
170M16—61no—
171M16—61no—
172M16—61no—
173M16—61no—
174M16—61no—
175M16C3×S411yescharacteristic
176C2×C8—32no—
177C2×C8—61no—
178C2×C8—61no—
179C2×C8—61no—
180C4⋊C4—61no—
181C22⋊C4—61no—
182C42—31no—
183C42—31no—
184C42—61no—
185C42—61no—
186C22×C3—31no—
187C22×C3—31no—
188C22×C3—41no—
189C22×C3—81no—
190C22×C3—121no—
191C22×C3A4⋊Q811yescharacteristic
192C3×C4—31no—
193C3×C4—31no—
194C3×C4—41no—
195C3×C4—41no—
196C3×C4—62no—
197C3×C4—81no—
198C3×C4—81no—
199C3×C4—121no—
200C3×C4—241no—
201C3×C4C4×S411yescharacteristic
202C3×C4GL(2,ℤ/4)11yescharacteristic
203C32—41noSylow
204C23—31no—
205Q8—31no—
206Q8C3×Dic3⋊C411yescharacteristic
207D4—31no—
208D4—31no—
209D4—61no—
210C2×C4—31no—
211C2×C4—31no—
212C2×C4—31no—
213C2×C4—31no—
214C2×C4—31no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—122no—
218C2×C4—121no—
219C2×C4C2×C3×S411yescharacteristic
220C8—21no—
221C8—61no—
222C8—61no—
223C8—61no—
224C8—121no—
225C2×C3—21no—
226C2×C3—41no—
227C2×C3—61no—
228C2×C3—61no—
229C2×C3—81no—
230C2×C3—81no—
231C2×C3—161no—
232C2×C3SL(2,ℤ/4)⋊C411yescharacteristic
233C22—31no—
234C22—31no—
235C22—121no—
236C22C3×A4⋊Q811yescharacteristic
237C4—31no—
238C4—31no—
239C4—62no—
240C4—121no—
241C4—241no—
242C4C3×C4×S411yescharacteristic
243C4C3×GL(2,ℤ/4)11yescharacteristic
244C3—41no—
245C3—81no—
246C3#7312911yescharacteristic, direct product factor
247C2—21no—
248C2—61no—
249C2—61no—
250C2C3×SL(2,ℤ/4)⋊C411yescharacteristic
subgroupG/H
centerC3×C4GL(2,ℤ/4)
commutator#41⋊C3C2×C3×C4
FrattiniC2×C4C2×C3×S4
FittingC3×#568D3
socleC2×C3SL(2,ℤ/4)⋊C4
prime radicalC2×C3×#41⋊C3C22
cosocleC2×#41⋊C3C22×C3
hypercenterC3×M16S4
hypocenterSL(2,𝔽3)C3×C4⋊C4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
4. #73129C3direct (N × H)
9. C3×GU(2,𝔽3)C4semidirect
10. C3×GU(2,𝔽3)C4semidirect
44. GU(2,𝔽3)C3×C4semidirect
45. GU(2,𝔽3)C3×C4semidirect
246. C3#73129direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#568⋊C32
C3×#54232
C2×C3×GU(2,𝔽3)2
#731293
C3×#18613
C3×#8224