Finite Group Databasebeta

← C3×#73099 #11920877

Subgroups — C3×#73099 #11920877

Order 1152.

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

normal subgroups only

This group has 249 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×Q8×SL(2,𝔽3)C211yesmaximal, characteristic
2C3×#5386C211yesmaximal, characteristic
3C3×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic
4#73099C311yesmaximal, characteristic, direct product factor
5C3×#1645—31nomaximal
6C2×C3×#216—41nomaximal
7C3×C4×SL(2,𝔽3)—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C3×#859—21no—
10C3×Q82D311yescharacteristic
11Q8×SL(2,𝔽3)—21no—
12Q8×SL(2,𝔽3)C2×C311yescharacteristic
13#5386C2×C311yescharacteristic
14C3×#487—31no—
15C3×#475—31no—
16C3×#462—31no—
17SL(2,𝔽3)⋊C8C2×C311yescharacteristic
18C3×#333—31no—
19C3×Q8⋊C8—31no—
20C3×Q8⋊C8—31no—
21C2×C32×Q8—41no—
22C2×C3×Dic6—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C3×#216—44no—
25C2×C3×C3⋊C8—41no—
26#1645—31noSylow
27C2×C3×Q16—61no—
28C3×C4⋊Q8—31no—
29C3×C4⋊Q8—31no—
30C3×C4⋊Q8—61no—
31C3×C4×Q8—21no—
32C3×C4×Q8—31no—
33C3×C4×Q8C2×D311yescharacteristic
34C2×#216—41no—
35C4×SL(2,𝔽3)—22no—
36C4×SL(2,𝔽3)—21no—
37C4×SL(2,𝔽3)—21no—
38C4×SL(2,𝔽3)C22×C311yescharacteristic
39#859—21no—
40C3×#106—61no—
41C3×C4⋊C8—31no—
42C3×C4⋊C8—31no—
43C3×#103—61no—
44C3×#103—62no—
45C3×#103—61no—
46C3×C4×C8—31no—
47C32×Q8—42no—
48C32×Q8—81no—
49C2×C32×C4—41no—
50C2×C32×C4—81no—
51C2×C3×Dic3—81no—
52C3×Dic6—42no—
53C3×Dic6—81no—
54C3×SL(2,𝔽3)Q1611yescharacteristic, complemented
55C3×C3⋊C8—42no—
56Q82C3×D311yescharacteristic
57#487—31no—
58#475—31no—
59#462—31no—
60#333—31no—
61Q8⋊C8—31no—
62Q8⋊C8—31no—
63C2×C3×Q8—32no—
64C2×C3×Q8—41no—
65C2×C3×Q8—61no—
66C2×C3×Q8—81no—
67C2×C3×Q8#6711yescharacteristic
68C2×C3×Q8S411yescharacteristic
69C2×Dic6—41no—
70C2×SL(2,𝔽3)—21no—
71C2×SL(2,𝔽3)C3×D411yescharacteristic
72C3×Q16—122no—
73C2×C3×C8—61no—
74C2×C3×C8—61no—
75C2×C3×C8—61no—
76C3×C4⋊C4—31no—
77C3×C4⋊C4—31no—
78C3×C4⋊C4—61no—
79C3×C4⋊C4—61no—
80C3×C4⋊C4—61no—
81C3×C4⋊C4—61no—
82C3×C42—31no—
83C3×C42—61no—
84#216—44no—
85C2×C3⋊C8—41no—
86C22×C32—41no—
87C32×C4—41no—
88C32×C4—41no—
89C32×C4—82no—
90C3×Dic3—82no—
91C2×Q16—61no—
92C4⋊Q8—31no—
93C4⋊Q8—31no—
94C4⋊Q8—61no—
95C4×Q8—21no—
96C4×Q8—31no—
97C4×Q8C2×C3×D311yescharacteristic
98#106—61no—
99C4⋊C8—31no—
100C4⋊C8—31no—
101#103—61no—
102#103—62no—
103#103—61no—
104C4×C8—31no—
105C3×Q8—21no—
106C3×Q8—31no—
107C3×Q8—42no—
108C3×Q8—81no—
109C3×Q8—82no—
110C3×Q8—82no—
111C3×Q8—122no—
112C3×Q8—121no—
113C3×Q8—121no—
114C3×Q8#21611yescharacteristic, complemented
115C3×Q8GL(2,𝔽3)12yes—
116C2×C3×C4—21no—
117C2×C3×C4—31no—
118C2×C3×C4—31no—
119C2×C3×C4—41no—
120C2×C3×C4—61no—
121C2×C3×C4—81no—
122C2×C3×C4—82no—
123C2×C3×C4—81no—
124C2×C3×C4—121no—
125C2×C3×C4C2×S411yescharacteristic
126C2×Dic3—81no—
127Dic6—42no—
128Dic6—81no—
129SL(2,𝔽3)—21no—
130SL(2,𝔽3)C3×Q1611yescharacteristic, complemented
131C3×C8—121no—
132C3×C8—121no—
133C3×C8—121no—
134C3⋊C8—42no—
135C2×C32—41no—
136C2×C32—41no—
137C2×C32—41no—
138C2×Q8—32no—
139C2×Q8—61no—
140C2×Q8C3×#6711yescharacteristic
141C2×Q8C3×S411yescharacteristic
142Q16—122no—
143C2×C8—61no—
144C2×C8—61no—
145C2×C8—61no—
146C4⋊C4—31no—
147C4⋊C4—31no—
148C4⋊C4—61no—
149C4⋊C4—61no—
150C4⋊C4—61no—
151C4⋊C4—61no—
152C42—31no—
153C42—61no—
154C22×C3—41no—
155C22×C3—81no—
156C22×C3GL(2,ℤ/4)11yescharacteristic
157C3×C4—22no—
158C3×C4—31no—
159C3×C4—31no—
160C3×C4—41no—
161C3×C4—41no—
162C3×C4—61no—
163C3×C4—82no—
164C3×C4—84no—
165C3×C4—81no—
166C3×C4—81no—
167C3×C4—121no—
168C3×C4—241no—
169C3×C4C2×GL(2,𝔽3)11yescharacteristic
170C3×C4#98311yescharacteristic
171Dic3—82no—
172C32—41noSylow
173Q8—21no—
174Q8—31no—
175Q8—122no—
176Q8—121no—
177Q8—121no—
178Q8C3×#21611yescharacteristic, complemented
179Q8C3×GL(2,𝔽3)12yes—
180C2×C4—21no—
181C2×C4—31no—
182C2×C4—31no—
183C2×C4—61no—
184C2×C4—121no—
185C2×C4C2×C3×S411yescharacteristic
186C8—121no—
187C8—121no—
188C8—121no—
189C2×C3—41no—
190C2×C3—41no—
191C2×C3—41no—
192C2×C3—81no—
193C2×C3—81no—
194C2×C3—81no—
195C2×C3#541611yescharacteristic
196C2×C3#542111yescharacteristic
197C2×C3#542811yescharacteristic
198C22C3×GL(2,ℤ/4)11yescharacteristic
199C4—22no—
200C4—31no—
201C4—31no—
202C4—61no—
203C4—121no—
204C4—241no—
205C4C2×C3×GL(2,𝔽3)11yescharacteristic
206C4C3×#98311yescharacteristic
207C3—41no—
208C3—81no—
209C3#7309911yescharacteristic, direct product factor
210C2C3×#541611yescharacteristic
211C2C3×#542111yescharacteristic
212C2C3×#542811yescharacteristic
subgroupG/H
center, socleC22×C3GL(2,ℤ/4)
commutator, cosocleC4×SL(2,𝔽3)C22×C3
FrattiniC2×C4C2×C3×S4
FittingC3×Q82D3
prime radicalC2×C3×SL(2,𝔽3)D4
hypercenterC2×C3×Q8S4
hypocenterSL(2,𝔽3)C3×Q16
LCSG ▷ C4×SL(2,𝔽3) ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
4. #73099C3direct (N × H)
54. C3×SL(2,𝔽3)Q16semidirect
114. C3×Q8#216semidirect
130. SL(2,𝔽3)C3×Q16semidirect
178. Q8C3×#216semidirect
209. C3#73099direct (N × H)

Maximal subgroups

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

subgroupindex
C3×Q8×SL(2,𝔽3)2
C3×#53862
C3×SL(2,𝔽3)⋊C82
#730993
C3×#16453
C2×C3×#2164