Finite Group Databasebeta

← #11921047

Subgroups — #11921047

Order 1152.

284 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 45 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×C4⋊(#41⋊C3)C211yesmaximal, characteristic
2#10593620C211yesmaximal, characteristic, complemented
3C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic, complemented
4#73091—33nomaximal
5#71921—31nomaximal
6C2×C3×#41⋊C3—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C3⋊#859—21no—
9#63637—41nomaximal
10C3×#550D311yescharacteristic
11C4⋊(#41⋊C3)D313yes—
12#5386—33no—
13#5070—31no—
14#5020—31no—
15#5019—31no—
16SL(2,𝔽3)⋊C8—33no—
17#4544—31no—
18#4535—31no—
19D4⋊(C3⋊C8)—31no—
20C2×#634—41no—
21C3×#41⋊C3—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23C32×C4⋊C4—41no—
24C2×C3⋊(C3⋊C8)—41no—
25#1642—91noSylow
26C2×C3×#41—21no—
27C2×#41⋊C3—23no—
28C3×#127—31no—
29C3×#121—61no—
30C3×C4×Q8C2×D311yescharacteristic
31C3×C4×D4—31no—
32C2×#214—61no—
33#869—31no—
34C4×SL(2,𝔽3)C2×D313yes—
35#859—63no—
36#835—61no—
37#832—62no—
38#810—61no—
39#810—123no—
40#808—61no—
41C4⋊(C3⋊C8)—31no—
42C4⋊(C3⋊C8)—31no—
43C4×C3⋊C8—31no—
44C2×C32×C4—41no—
45C2×C32×C4—81no—
46C2×C3⋊Dic3—81no—
47#634—42no—
48#634—81no—
49C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
50C3⋊(C3⋊C8)—81no—
51#550C3⋊D311yescharacteristic
52#486—91no—
53#475—91no—
54#461—91no—
55#331—91no—
56Q8⋊C8—91no—
57D4⋊C8—91no—
58C3×#41—41no—
59C3×#41—121no—
60C2×C3×Q8#6711yescharacteristic
61C2×C3×D4—32no—
62C2×C3×D4—31no—
63C2×C3×D4—61no—
64C22×C3×C4—61no—
65C2×Dic6—61no—
66C2×Dic6—123no—
67#41⋊C3—43no—
68C2×SL(2,𝔽3)#6713yes—
69C3×C4⋊C4—31no—
70C3×C4⋊C4—43no—
71C3×C4⋊C4S411yescharacteristic
72C3×C22⋊C4—61no—
73C3×C42—31no—
74#214—122no—
75C4⋊Dic3—61no—
76C4⋊Dic3—61no—
77C2×C3⋊C8—61no—
78C2×C3⋊C8—61no—
79C2×C3⋊C8—61no—
80C2×C3⋊C8—123no—
81C22×C32—41no—
82C32×C4—41no—
83C32×C4—41no—
84C32×C4—161no—
85C3⋊Dic3—82no—
86C2×#41—21no—
87C2×SD16—181no—
88C4⋊Q8—91no—
89#127—31no—
90#121—61no—
91C4×Q8C2×C3⋊D311yescharacteristic
92C4×D4—31no—
93#106—181no—
94C4⋊C8—91no—
95C4⋊C8—91no—
96#103—181no—
97#103—181no—
98#102—182no—
99C4×C8—91no—
100C23×C3—61no—
101C3×Q8—31no—
102C3×Q8#21411yescharacteristic
103C3×D4—61no—
104C3×D4—61no—
105C3×D4—121no—
106C3×D4—121no—
107C3×D4—122no—
108C2×C3×C4—21no—
109C2×C3×C4—31no—
110C2×C3×C4—31no—
111C2×C3×C4—43no—
112C2×C3×C4—83no—
113C2×C3×C4—121no—
114C2×C3×C4—121no—
115C2×C3×C4C2×S411yescharacteristic
116C2×Dic3—121no—
117C2×Dic3—243no—
118Dic6—126no—
119Dic6—121no—
120Dic6—121no—
121Dic6—243no—
122SL(2,𝔽3)#21413yescomplemented
123C3⋊C8—121no—
124C3⋊C8—121no—
125C3⋊C8—121no—
126C3⋊C8—243no—
127C2×C32—41no—
128C2×C32—41no—
129C2×C32—41no—
130#41—41no—
131#41—121no—
132C2×Q8—181no—
133C2×Q8#63811yescharacteristic
134C2×D4—32no—
135C2×D4—31no—
136C2×D4—61no—
137C22×C4—61no—
138SD16—362no—
139C2×C8—181no—
140C2×C8—181no—
141C2×C8—181no—
142C4⋊C4—31no—
143C4⋊C4—181no—
144C4⋊C4—181no—
145C4⋊C4C3⋊S411yescharacteristic
146C22⋊C4—61no—
147C42—31no—
148C22×C3—43no—
149C22×C3—121no—
150C22×C3—121no—
151C22×C3—121no—
152C22×C3GL(2,ℤ/4)11yescharacteristic
153C3×C4—31no—
154C3×C4—31no—
155C3×C4—41no—
156C3×C4—43no—
157C3×C4—43no—
158C3×C4—61no—
159C3×C4—163no—
160C3×C4#98511yescharacteristic
161C3×C4#98611yescharacteristic
162Dic3—246no—
163Dic3—241no—
164C32—41noSylow
165C23—61no—
166Q8—31no—
167Q8—361no—
168Q8—361no—
169Q8#375411yescharacteristic
170D4—61no—
171D4—61no—
172D4—121no—
173D4—121no—
174D4—122no—
175C2×C4—21no—
176C2×C4—31no—
177C2×C4—31no—
178C2×C4—121no—
179C2×C4—121no—
180C2×C4—361no—
181C2×C4C2×C3⋊S411yescharacteristic
182C8—361no—
183C8—361no—
184C8—361no—
185C2×C3—43no—
186C2×C3—43no—
187C2×C3—43no—
188C2×C3—241no—
189C2×C3#541411yescharacteristic
190C2×C3#542711yescharacteristic
191C2×C3#542811yescharacteristic
192C22—121no—
193C22—121no—
194C22—121no—
195C22#6426911yescharacteristic
196C4—31no—
197C4—31no—
198C4—41no—
199C4—61no—
200C4—721no—
201C4#6426611yescharacteristic
202C4#6426711yescharacteristic
203C3—43no—
204C3#7309111yescharacteristic, complemented
205C2—241no—
206C2#1059364811yescharacteristic
207C2#1059366111yescharacteristic
208C2#1059366211yescharacteristic
subgroupG/H
centerC22#64269
commutator, cosocleC3×C4×SL(2,𝔽3)C22
FrattiniC2×C4C2×C3⋊S4
FittingC3×#550D3
socleC22×C3GL(2,ℤ/4)
prime radicalC3×C4⋊(#41⋊C3)C2
hypercenterC4⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)SD16
LCSG ▷ C3×C4×SL(2,𝔽3) ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #10593620C2semidirect
3. C3⋊(SL(2,𝔽3)⋊C8)C2semidirect
49. C3×SL(2,𝔽3)SD16semidirect
122. SL(2,𝔽3)#214semidirect
204. C3#73091semidirect

Maximal subgroups

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

subgroupindex
C3×C4⋊(#41⋊C3)2
#105936202
C3⋊(SL(2,𝔽3)⋊C8)2
#730913
#719213
#636374