Finite Group Databasebeta

← C3⋊#85557 #11923127

Subgroups — C3⋊#85557 #11923127

Order 1152.

330 conjugacy classes of subgroups; only the 19 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 19 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×#5465C211yesmaximal, characteristic, complemented
2#85557—33nomaximal
3#71950—31nomaximal
4#64269—42nomaximal
5#64264—41nomaximal
6C3×H(3,𝔽4)D311yescharacteristic, complemented
7#5465D313yescomplemented
8#5061—31no—
9#5032—32no—
10#4541—31no—
11#4536—32no—
12C22×C3×A4—42no—
13C2×C3⋊S4—42no—
14C2×C3×SL(2,𝔽3)—41no—
15C2×C3×SL(2,𝔽3)—81no—
16C3⋊SL(2,ℤ/4)—42no—
17C3⋊GL(2,𝔽3)—41no—
18C3⋊#226—41no—
19#2199—91noSylow
20#1004—31no—
21GL(2,ℤ/4)—126no—
22#983—123no—
23C3×#124—31no—
24C3×#124—61no—
25C3×#120—32no—
26#953—62no—
27#942—61no—
28#834—32no—
29#833—31no—
30#806—62no—
31#805—61no—
32C2×C3×A4—42no—
33C2×C3×A4—82no—
34C3⋊S4—82no—
35#638—161no—
36C3×SL(2,𝔽3)—41no—
37C3×SL(2,𝔽3)—81no—
38H(3,𝔽4)C3⋊D311yescharacteristic, complemented
39B(4,𝔽2)—92no—
40#455—91no—
41#354—91no—
42AΣL(1,𝔽16)—92no—
43C24×C3S412yescomplemented
44C22×A4—46no—
45C2×S4—126no—
46C2×C3×Q8—21no—
47C2×C3×Q8S411yescharacteristic, complemented
48C2×C3×D4—31no—
49C2×C3×D4—61no—
50C2×#67—62no—
51C2×#67—122no—
52#237—121no—
53D3×D4—121no—
54#235—61no—
55C2×SL(2,𝔽3)—43no—
56C2×SL(2,𝔽3)—83no—
57SL(2,ℤ/4)—126no—
58GL(2,𝔽3)—123no—
59#226—123no—
60C3×C22⋊C4—32no—
61C3×C22⋊C4—62no—
62C3×C42—31no—
63#217—62no—
64#217—122no—
65#216—121no—
66#215—121no—
67#208—61no—
68C22×C32—161no—
69C2×C3⋊D3—161no—
70C3×A4—42no—
71C3⋊Dic3—161no—
72H(5,𝔽2)—91no—
73#137—181no—
74#124—31no—
75#124—61no—
76#120—32no—
77#120—182no—
78#104—181no—
79#100—91no—
80#99—92no—
81#99—182no—
82C23×C3—32no—
83C23×C3—42no—
84C23×C3—82no—
85C22×D3—62no—
86C2×A4—46no—
87C2×A4—86no—
88S4—246no—
89C3×Q8—41no—
90C3×Q8—81no—
91C3×D4—121no—
92C3×D4—241no—
93C2×C3×C4—31no—
94C2×C3×C4—61no—
95#67—122no—
96#67—121no—
97#67—242no—
98#67—242no—
99#67—483no—
100C2×Dic3—62no—
101C2×Dic3—122no—
102D12—61no—
103C4×D3—121no—
104Dic6—61no—
105SL(2,𝔽3)—43no—
106SL(2,𝔽3)—83no—
107C3⋊C8—121no—
108C2×C32—161no—
109C2×C32—321no—
110C3⋊D3—321no—
111C24C3⋊S412yescomplemented
112#41—181no—
113#41—361no—
114C2×Q8—21no—
115C2×Q8C3⋊S411yescharacteristic, complemented
116C2×D4—31no—
117C2×D4—61no—
118C2×D4—182no—
119C2×D4—361no—
120C2×D4—362no—
121Q16—361no—
122SD16—361no—
123M16—181no—
124C22⋊C4—32no—
125C22⋊C4—62no—
126C22⋊C4—182no—
127C22⋊C4—362no—
128C42—31no—
129C22×C3—42no—
130C22×C3—62no—
131C22×C3—122no—
132C22×C3—122no—
133C22×C3—163no—
134C22×C3#102011yescharacteristic
135C2×D3—121no—
136C2×D3—242no—
137C2×D3—483no—
138A4—46no—
139C3×C4—61no—
140C3×C4—121no—
141Dic3—121no—
142Dic3—242no—
143Dic3—483no—
144C32—161noSylow
145C23—32no—
146C23—42no—
147C23—82no—
148C23—182no—
149Q8—41no—
150Q8—81no—
151Q8—181no—
152D4—121no—
153D4—181no—
154D4—241no—
155D4—362no—
156D4—361no—
157D4—722no—
158D4—722no—
159C2×C4—31no—
160C2×C4—61no—
161C2×C4—182no—
162C2×C4—361no—
163C2×C4—362no—
164C8—361no—
165C2×C3—21no—
166C2×C3—122no—
167C2×C3—163no—
168C2×C3—323no—
169C2×C3#593411yescharacteristic
170D3—241no—
171D3—963no—
172C22—42no—
173C22—62no—
174C22—122no—
175C22—122no—
176C22—361no—
177C22—722no—
178C22C3⋊#102011yescharacteristic
179C4—61no—
180C4—121no—
181C4—361no—
182C4—722no—
183C3—163no—
184C3#8555711yescharacteristic, complemented
185C2—21no—
186C2—122no—
187C2—721no—
188C2C3⋊#593411yescharacteristic
subgroupG/H
centerC2C3⋊#5934
commutator, cosocle, hypocenterC3×#5465C2
Frattini, hypercenterC22C3⋊#1020
FittingC3×H(3,𝔽4)D3
socleC2×C3#5934
LCSG ▷ C3×#5465
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C3×#5465C2semidirect
6. C3×H(3,𝔽4)D3semidirect
7. #5465D3semidirect
38. H(3,𝔽4)C3⋊D3semidirect
43. C24×C3S4semidirect
47. C2×C3×Q8S4semidirect
111. C24C3⋊S4semidirect
115. C2×Q8C3⋊S4semidirect
184. C3#85557semidirect

Maximal subgroups

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

subgroupindex
C3×#54652
#855573
#719503
#642694
#642644