Finite Group Databasebeta

← #11922886

Subgroups — #11922886

Order 1152.

540 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 89 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⋊C3C211yesmaximal, characteristic
2#10593660C212yesmaximal
3#10593625C212yesmaximal, complemented
4#10593620C212yesmaximal, complemented
5#85337—33nomaximal
6#82508—31nomaximal
7C2×C3×#41⋊C3C2211yescharacteristic
8#64078—41nomaximal
9C3×C4×SL(2,𝔽3)C2211yescharacteristic
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11C3⋊#859C2214yes—
12C3×C4×#41D311yescharacteristic
13C2×#869—31no—
14C4×#41⋊C3D313yes—
15#5426—36no—
16#5391—36no—
17#5386—36no—
18#5232—32no—
19#5021—32no—
20#5020—32no—
21#5008—32no—
22#5007—32no—
23#4995—32no—
24#4924—31no—
25C2×#634—42no—
26C3×#41⋊C3Q812yescomplemented
27C2×C3×SL(2,𝔽3)C2311yescharacteristic
28C32×C42—41no—
29C4⋊(C3⋊Dic3)—42no—
30C4⋊(C3⋊Dic3)—42no—
31#3079—91noSylow
32C2×C3×#41C2×D311yescharacteristic
33C22×Dic6—61no—
34C2×#41⋊C3C2×D313yes—
35C3×C4×Q8C2×D311yescharacteristic
36C3×C4×Q8C2×D311yescharacteristic
37C3×C4×D4—31no—
38C3×C4×D4—31no—
39C3×#117—31no—
40C2×C3×C42—31no—
41C2×C4⋊Dic3—32no—
42C2×C4⋊Dic3—61no—
43C2×#208—32no—
44#869—32no—
45#869—32no—
46#869—61no—
47#869—61no—
48#869—123no—
49C4×SL(2,𝔽3)C2×D313yes—
50C4×SL(2,𝔽3)C2×D313yes—
51#859—312no—
52#835—34no—
53#832—34no—
54#808—34no—
55#807—34no—
56C4⋊(C3⋊C8)—32no—
57C4⋊(C3⋊C8)—32no—
58C2×C32×C4—41no—
59C2×C32×C4—41no—
60C2×C32×C4—41no—
61C2×C3⋊Dic3—44no—
62#634—84no—
63C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
64C2×C4⋊Q8—91no—
65C4×#41C3⋊D311yescharacteristic
66#477—92no—
67#476—92no—
68#475—92no—
69#474—92no—
70#428—92no—
71#423—91no—
72#417—92no—
73C3×#41—32no—
74C3×#41Dic612yescomplemented
75C2×C3×Q8C22×D311yescharacteristic
76C2×C3×D4—31no—
77C22×C3×C4—31no—
78C22×C3×C4—31no—
79C22×C3×C4—31no—
80C22×Dic3—62no—
81C2×Dic6—62no—
82C2×Dic6—62no—
83C2×Dic6—126no—
84C2×Dic6—124no—
85#41⋊C3Dic616yescomplemented
86C2×SL(2,𝔽3)C22×D313yes—
87C3×C4⋊C4—31no—
88C3×C4⋊C4—31no—
89C3×C22⋊C4—61no—
90C3×C42—31no—
91C3×C42—31no—
92C3×C42—31no—
93C3×C42—43no—
94C3×C42S411yescharacteristic
95C4⋊Dic3—34no—
96C4⋊Dic3—62no—
97C4⋊Dic3—62no—
98C4⋊Dic3—62no—
99C4⋊Dic3—126no—
100C4⋊Dic3—126no—
101#208—64no—
102C2×C3⋊C8—34no—
103C22×C32—41no—
104C32×C4—42no—
105C32×C4—42no—
106C32×C4—42no—
107C3⋊Dic3—84no—
108C2×#41C2×C3⋊D311yescharacteristic
109C22×Q8—181no—
110C2×M16—92no—
111C4⋊Q8—92no—
112C4⋊Q8—92no—
113C4⋊Q8—181no—
114C4⋊Q8—181no—
115C4×Q8C2×C3⋊D311yescharacteristic
116C4×Q8C2×C3⋊D311yescharacteristic
117C4×D4—31no—
118C4×D4—31no—
119#117—31no—
120C2×C4⋊C4—92no—
121C2×C4⋊C4—181no—
122C2×C42—31no—
123#107—94no—
124#106—94no—
125C4⋊C8—92no—
126C4⋊C8—92no—
127#103—94no—
128#102—94no—
129C23×C3—31no—
130C3×Q8—31no—
131C3×Q8C2×Dic611yescharacteristic
132C3×D4—32no—
133C3×D4—61no—
134C2×C3×C4—31no—
135C2×C3×C4—32no—
136C2×C3×C4—32no—
137C2×C3×C4—31no—
138C2×C3×C4—31no—
139C2×C3×C4—43no—
140C2×C3×C4—43no—
141C2×C3×C4—43no—
142C2×C3×C4—62no—
143C2×C3×C4—62no—
144C2×C3×C4C2×S411yescharacteristic
145C2×C3×C4C2×S411yescharacteristic
146C2×C3×C4C2×S411yescharacteristic
147C2×Dic3—64no—
148C2×Dic3—1212no—
149C2×Dic3—124no—
150Dic6—124no—
151Dic6—124no—
152Dic6—2412no—
153SL(2,𝔽3)C2×Dic613yescomplemented
154C3⋊C8—64no—
155C2×C32—41no—
156C2×C32—41no—
157C2×C32—41no—
158#41—32no—
159#41#63412yescomplemented
160C2×Q8—182no—
161C2×Q8—182no—
162C2×Q8—364no—
163C2×Q8C22×C3⋊D311yescharacteristic
164C2×D4—31no—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—182no—
169M16—184no—
170C2×C8—94no—
171C4⋊C4—31no—
172C4⋊C4—31no—
173C4⋊C4—94no—
174C4⋊C4—182no—
175C4⋊C4—182no—
176C4⋊C4—182no—
177C22⋊C4—61no—
178C42—31no—
179C42—31no—
180C42—31no—
181C42C3⋊S411yescharacteristic
182C22×C3—32no—
183C22×C3—43no—
184C22×C3—61no—
185C22×C3—61no—
186C22×C3C22×S411yescharacteristic
187C3×C4—31no—
188C3×C4—31no—
189C3×C4—46no—
190C3×C4—46no—
191C3×C4—46no—
192C3×C4—61no—
193C3×C4—61no—
194C3×C4A4⋊Q812yes—
195C3×C4#98412yes—
196C3×C4#98612yes—
197Dic3—124no—
198Dic3—2412no—
199C32—41noSylow
200C23—31no—
201Q8—31no—
202Q8—364no—
203Q8—364no—
204Q8C2×#63411yescharacteristic
205D4—32no—
206D4—61no—
207C2×C4—31no—
208C2×C4—32no—
209C2×C4—32no—
210C2×C4—31no—
211C2×C4—31no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—184no—
215C2×C4—364no—
216C2×C4C2×C3⋊S411yescharacteristic
217C2×C4C2×C3⋊S411yescharacteristic
218C2×C4C2×C3⋊S411yescharacteristic
219C8—184no—
220C2×C3—43no—
221C2×C3—43no—
222C2×C3—43no—
223C2×C3—62no—
224C2×C3C2×A4⋊Q811yescharacteristic
225C2×C3C2×#98411yescharacteristic
226C2×C3C2×#98611yescharacteristic
227C22—32no—
228C22—61no—
229C22—61no—
230C22C22×C3⋊S411yescharacteristic
231C4—31no—
232C4—31no—
233C4—61no—
234C4—61no—
235C4—364no—
236C4#6425912yes—
237C4#6426512yes—
238C4#6426712yes—
239C3—43no—
240C3#8533711yescharacteristic, complemented
241C2—62no—
242C2C2×#6425911yescharacteristic
243C2C2×#6426511yescharacteristic
244C2C2×#6426711yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×C4×#41D3
socleC22×C3C22×S4
prime radicalC2×C3×#41⋊C3C22
hypercenterC42C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×Q8
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
3. #10593625C2semidirect
4. #10593620C2semidirect
26. C3×#41⋊C3Q8semidirect
63. C3×SL(2,𝔽3)C2×Q8semidirect
74. C3×#41Dic6semidirect
85. #41⋊C3Dic6semidirect
153. SL(2,𝔽3)C2×Dic6semidirect
159. #41#634semidirect
240. C3#85337semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#41⋊C32
#105936602
#105936252
#105936202
#853373
#825083
#640784