Finite Group Databasebeta

← C3×#73086 #11920864

Subgroups — C3×#73086 #11920864

Order 1152.

263 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 44 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×D4×SL(2,𝔽3)C211yesmaximal, characteristic
2C3×#5391C211yesmaximal, characteristic, complemented
3C3×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic, complemented
4#73086C311yesmaximal, characteristic, direct product factor
5C3×#1637—31nomaximal
6C22×C3×SL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C3×#859—21no—
9C3×#832—41nomaximal
10C3×D4×Q8D311yescharacteristic
11D4×SL(2,𝔽3)—21no—
12D4×SL(2,𝔽3)C2×C311yescharacteristic
13#5391C2×C311yescharacteristic, complemented
14C3×#487—31no—
15C3×#477—31no—
16C3×#476—31no—
17SL(2,𝔽3)⋊C8C2×C311yescharacteristic, complemented
18C3×#332—31no—
19C3×Q8⋊C8—31no—
20C3×D4⋊C8—31no—
21C2×C32×D4—41no—
22C2×C3×SL(2,𝔽3)—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C3×C4⋊Dic3—41no—
25C2×C3×C3⋊C8—41no—
26#1637—31noSylow
27C22×C3×Q8—21no—
28C22×SL(2,𝔽3)—22no—
29C22×SL(2,𝔽3)—21no—
30C3×C4⋊Q8—31no—
31C3×C4⋊Q8—31no—
32C3×C22⋊Q8—61no—
33C3×C4×Q8C2×D311yescharacteristic
34C3×C4×D4—31no—
35C4×SL(2,𝔽3)—21no—
36C4×SL(2,𝔽3)C22×C311yescharacteristic
37#859—21no—
38C3×#107—61no—
39C3×#106—61no—
40C3×C4⋊C8—31no—
41C3×C4⋊C8—31no—
42C3×#103—62no—
43C3×#103—61no—
44C3×#102—61no—
45C3×C4×C8—31no—
46#832—41no—
47C23×C32—81no—
48C32×D4—42no—
49C32×D4—81no—
50C2×C32×C4—41no—
51C2×C3×Dic3—81no—
52C3×SL(2,𝔽3)SD1611yescharacteristic
53C3×C3⋊C8—81no—
54D4×Q8C3×D311yescharacteristic
55#487—31no—
56#477—31no—
57#476—31no—
58#332—31no—
59Q8⋊C8—31no—
60D4⋊C8—31no—
61C2×C3×Q8—41no—
62C2×C3×Q8—61no—
63C2×C3×Q8—61no—
64C2×C3×Q8—121no—
65C2×C3×Q8#6711yescharacteristic
66C2×C3×D4—41no—
67C2×C3×D4—81no—
68C2×C3×D4S411yescharacteristic
69C22×C3×C4—61no—
70C2×SL(2,𝔽3)—21no—
71C2×SL(2,𝔽3)—42no—
72C2×SL(2,𝔽3)—41no—
73C2×SL(2,𝔽3)C3×D411yescharacteristic
74C2×C3×C8—61no—
75C2×C3×C8—61no—
76C2×C3×C8—61no—
77C3×C4⋊C4—31no—
78C3×C4⋊C4—32no—
79C3×C4⋊C4—31no—
80C3×C4⋊C4—61no—
81C3×C4⋊C4—61no—
82C3×C22⋊C4—61no—
83C3×C42—31no—
84C4⋊Dic3—41no—
85C2×C3⋊C8—41no—
86C22×C32—41no—
87C22×C32—82no—
88C22×C32—161no—
89C22×C32—161no—
90C32×C4—41no—
91C32×C4—41no—
92C3×Dic3—161no—
93C22×Q8—21no—
94C4⋊Q8—31no—
95C4⋊Q8—31no—
96C22⋊Q8—61no—
97C4×Q8C2×C3×D311yescharacteristic
98C4×D4—31no—
99#107—61no—
100#106—61no—
101C4⋊C8—31no—
102C4⋊C8—31no—
103#103—62no—
104#103—61no—
105#102—61no—
106C4×C8—31no—
107C23×C3—21no—
108C23×C3—81no—
109C23×C3—82no—
110C3×Q8—31no—
111C3×Q8—121no—
112C3×Q8—121no—
113C3×Q8—122no—
114C3×Q8#21411yescharacteristic
115C3×D4—21no—
116C3×D4—42no—
117C3×D4—81no—
118C3×D4—82no—
119C3×D4—82no—
120C3×D4#22612yes—
121C2×C3×C4—31no—
122C2×C3×C4—31no—
123C2×C3×C4—41no—
124C2×C3×C4—61no—
125C2×C3×C4—81no—
126C2×C3×C4—121no—
127C2×C3×C4—121no—
128C2×C3×C4—121no—
129C2×C3×C4C2×S411yescharacteristic
130C2×Dic3—81no—
131SL(2,𝔽3)—21no—
132SL(2,𝔽3)C3×SD1611yescharacteristic
133C3×C8—121no—
134C3×C8—121no—
135C3×C8—121no—
136C3⋊C8—81no—
137C2×C32—41no—
138C2×C32—41no—
139C2×C32—41no—
140C2×C32—162no—
141C2×Q8—41no—
142C2×Q8—61no—
143C2×Q8—61no—
144C2×Q8—121no—
145C2×Q8C3×#6711yescharacteristic
146C2×D4C3×S411yescharacteristic
147C22×C4—61no—
148C2×C8—61no—
149C2×C8—61no—
150C2×C8—61no—
151C4⋊C4—31no—
152C4⋊C4—32no—
153C4⋊C4—31no—
154C4⋊C4—61no—
155C4⋊C4—61no—
156C22⋊C4—61no—
157C42—31no—
158C22×C3—22no—
159C22×C3—41no—
160C22×C3—41no—
161C22×C3—41no—
162C22×C3—82no—
163C22×C3—84no—
164C22×C3—81no—
165C22×C3—161no—
166C22×C3—161no—
167C22×C3—162no—
168C22×C3—162no—
169C22×C3GL(2,ℤ/4)11yescharacteristic
170C3×C4—31no—
171C3×C4—31no—
172C3×C4—41no—
173C3×C4—41no—
174C3×C4—61no—
175C3×C4—81no—
176C3×C4—81no—
177C3×C4—121no—
178C3×C4—241no—
179C3×C4C2×#22611yescharacteristic
180C3×C4#98311yescharacteristic
181Dic3—161no—
182C32—41noSylow
183C23—21no—
184Q8—31no—
185Q8—121no—
186Q8—121no—
187Q8—122no—
188Q8C3×#21411yescharacteristic
189D4—21no—
190D4C3×#22612yes—
191C2×C4—31no—
192C2×C4—31no—
193C2×C4—61no—
194C2×C4—121no—
195C2×C4—121no—
196C2×C4—121no—
197C2×C4C2×C3×S411yescharacteristic
198C8—121no—
199C8—121no—
200C8—121no—
201C2×C3—42no—
202C2×C3—41no—
203C2×C3—41no—
204C2×C3—41no—
205C2×C3—81no—
206C2×C3—81no—
207C2×C3—81no—
208C2×C3—162no—
209C2×C3—164no—
210C2×C3#541411yescharacteristic
211C2×C3#541911yescharacteristic
212C2×C3#543111yescharacteristic
213C22—22no—
214C22—41no—
215C22—41no—
216C22C3×GL(2,ℤ/4)11yescharacteristic
217C4—31no—
218C4—31no—
219C4—61no—
220C4—121no—
221C4—241no—
222C4C2×C3×#22611yescharacteristic
223C4C3×#98311yescharacteristic
224C3—41no—
225C3—81no—
226C3#7308611yescharacteristic, direct product factor
227C2—42no—
228C2C3×#541411yescharacteristic
229C2C3×#541911yescharacteristic
230C2C3×#543111yescharacteristic
subgroupG/H
center, socleC22×C3GL(2,ℤ/4)
commutator, cosocleC4×SL(2,𝔽3)C22×C3
FrattiniC2×C4C2×C3×S4
FittingC3×D4×Q8D3
prime radicalC3×D4×SL(2,𝔽3)C2
hypercenterC2×C3×D4S4
hypocenterSL(2,𝔽3)C3×SD16
LCSG ▷ C4×SL(2,𝔽3) ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×#5391C2semidirect
3. C3×SL(2,𝔽3)⋊C8C2semidirect
4. #73086C3direct (N × H)
13. #5391C2×C3semidirect
17. SL(2,𝔽3)⋊C8C2×C3semidirect
226. C3#73086direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D4×SL(2,𝔽3)2
C3×#53912
C3×SL(2,𝔽3)⋊C82
#730863
C3×#16373
C3×#8324