Finite Group Databasebeta

← C3×#73016 #11920794

Subgroups — C3×#73016 #11920794

Order 1152.

195 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 195 conjugacy classes of subgroups — too many to draw. Only its 40 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×#5391C211yesmaximal, characteristic
2C3×#5386C211yesmaximal, characteristic
3C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic
4#73016C311yesmaximal, characteristic, direct product factor
5C3×#1684—31nomaximal
6C3×C4×SL(2,𝔽3)C2211yescharacteristic
7C3×#859—21no—
8C3×#859—21no—
9C3×#816—41nomaximal
10#5391C2×C311yescharacteristic
11#5386C2×C311yescharacteristic
12C3×#487—31no—
13C3×#477—31no—
14C3×#475—31no—
15C3×C8×Q8D311yescharacteristic
16C8×SL(2,𝔽3)—21no—
17C8×SL(2,𝔽3)C2×C311yescharacteristic
18C3×#335—31no—
19C3×#333—31no—
20C3×#332—31no—
21C2×C3×Dic6—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23C2×C32×C8—41no—
24C3×C4⋊Dic3—41no—
25#1684—31noSylow
26C3×C4⋊Q8—31no—
27C3×C4⋊Q8—31no—
28C3×C4×Q8C2×D311yescharacteristic
29C4×SL(2,𝔽3)—21no—
30C4×SL(2,𝔽3)C22×C311yescharacteristic
31#859—21no—
32#859—21no—
33C3×#107—61no—
34C3×#106—61no—
35C3×C4⋊C8—31no—
36C3×C4⋊C8—31no—
37C3×C4⋊C8—31no—
38C3×#103—61no—
39C3×#103—61no—
40C3×#103—61no—
41C3×#103—61no—
42C3×C4×C8—31no—
43#816—41no—
44C2×C32×C4—41no—
45C2×C3×Dic3—81no—
46C2×C3×Dic3—81no—
47C3×Dic6—42no—
48C3×Dic6—81no—
49C3×SL(2,𝔽3)Q1611yescharacteristic
50C32×C8—81no—
51#487—31no—
52#477—31no—
53#475—31no—
54C8×Q8C3×D311yescharacteristic
55#335—31no—
56#333—31no—
57#332—31no—
58C2×C3×Q8—61no—
59C2×C3×Q8—61no—
60C2×C3×Q8D1211yescharacteristic
61C2×Dic6—41no—
62C2×SL(2,𝔽3)—21no—
63C2×SL(2,𝔽3)C3×D411yescharacteristic
64C2×C3×C8—31no—
65C2×C3×C8—41no—
66C2×C3×C8—61no—
67C2×C3×C8—61no—
68C2×C3×C8—81no—
69C2×C3×C8S411yescharacteristic
70C3×C4⋊C4—31no—
71C3×C4⋊C4—61no—
72C3×C4⋊C4—61no—
73C3×C4⋊C4—61no—
74C3×C4⋊C4—61no—
75C3×C42—31no—
76C4⋊Dic3—41no—
77C22×C32—41no—
78C32×C4—41no—
79C32×C4—41no—
80C3×Dic3—82no—
81C3×Dic3—161no—
82C4⋊Q8—31no—
83C4⋊Q8—31no—
84C4×Q8C2×C3×D311yescharacteristic
85#107—61no—
86#106—61no—
87C4⋊C8—31no—
88C4⋊C8—31no—
89C4⋊C8—31no—
90#103—61no—
91#103—61no—
92#103—61no—
93#103—61no—
94C4×C8—31no—
95C3×Q8—31no—
96C3×Q8—121no—
97C3×Q8—121no—
98C3×Q8—241no—
99C3×Q8Dic1211yescharacteristic
100C2×C3×C4—31no—
101C2×C3×C4—31no—
102C2×C3×C4—41no—
103C2×C3×C4—81no—
104C2×C3×C4—121no—
105C2×C3×C4—121no—
106C2×C3×C4C2×S411yescharacteristic
107C2×Dic3—81no—
108C2×Dic3—81no—
109Dic6—42no—
110Dic6—81no—
111SL(2,𝔽3)—21no—
112SL(2,𝔽3)C3×Q1611yescharacteristic
113C3×C8—21no—
114C3×C8—61no—
115C3×C8—81no—
116C3×C8—82no—
117C3×C8—121no—
118C3×C8—121no—
119C2×C32—41no—
120C2×C32—41no—
121C2×C32—41no—
122C2×Q8—61no—
123C2×Q8—61no—
124C2×Q8C3×D1211yescharacteristic
125C2×C8—31no—
126C2×C8—61no—
127C2×C8—61no—
128C2×C8C3×S411yescharacteristic
129C4⋊C4—31no—
130C4⋊C4—61no—
131C4⋊C4—61no—
132C4⋊C4—61no—
133C4⋊C4—61no—
134C42—31no—
135C22×C3—41no—
136C22×C3—81no—
137C22×C3C4⋊S411yescharacteristic
138C3×C4—31no—
139C3×C4—31no—
140C3×C4—41no—
141C3×C4—41no—
142C3×C4—61no—
143C3×C4—81no—
144C3×C4—81no—
145C3×C4—241no—
146C3×C4—241no—
147C3×C4#98311yescharacteristic
148C3×C4#98511yescharacteristic
149Dic3—82no—
150Dic3—161no—
151C32—41noSylow
152Q8—31no—
153Q8—121no—
154Q8—121no—
155Q8—241no—
156Q8C3×Dic1211yescharacteristic
157C2×C4—31no—
158C2×C4—31no—
159C2×C4—121no—
160C2×C4—121no—
161C2×C4C2×C3×S411yescharacteristic
162C8—21no—
163C8—61no—
164C8—121no—
165C8—121no—
166C2×C3—41no—
167C2×C3—41no—
168C2×C3—41no—
169C2×C3—81no—
170C2×C3—81no—
171C2×C3—81no—
172C2×C3#539511yescharacteristic
173C2×C3#539811yescharacteristic
174C2×C3#540611yescharacteristic
175C22C3×C4⋊S411yescharacteristic
176C4—31no—
177C4—31no—
178C4—61no—
179C4—241no—
180C4—241no—
181C4C3×#98311yescharacteristic
182C4C3×#98511yescharacteristic
183C3—41no—
184C3—81no—
185C3#7301611yescharacteristic, direct product factor
186C2C3×#539511yescharacteristic
187C2C3×#539811yescharacteristic
188C2C3×#540611yescharacteristic
subgroupG/H
center, socleC22×C3C4⋊S4
commutator, cosocleC4×SL(2,𝔽3)C22×C3
FrattiniC2×C4C2×C3×S4
FittingC3×C8×Q8D3
prime radicalC2×C3×SL(2,𝔽3)D4
hypercenterC2×C3×C8S4
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×C8

Complements

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

NH ≅ G/Ntype
4. #73016C3direct (N × H)
185. C3#73016direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#53912
C3×#53862
C3×C8×SL(2,𝔽3)2
#730163
C3×#16843
C3×#8164