Finite Group Databasebeta

← #85846⋊C3 #11923194

Subgroups — #85846⋊C3 #11923194

Order 1152.

809 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 809 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×C3⋊#994C212yesmaximal, complemented
2C3×C4×#41⋊C3C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊(#41⋊C3))C214yesmaximal, complemented
4#85846C311yesmaximal, characteristic, complemented
5C4⋊#994—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7C3⋊#994—24no—
8C2×C3⋊(#41⋊C3)C2214yescomplemented
9C3×#869—41nomaximal
10C3×C4×SL(2,𝔽3)C2212yescomplemented
11C2×#1009C2×C312yes—
12C2×#994—32no—
13C3×C4×#41C2×C311yescharacteristic
14#5577C2×C314yes—
15D4×D12—34no—
16#5525—32no—
17#5477—31no—
18C2×#874—31no—
19C4⋊(#41⋊C3)—34no—
20C4×#41⋊C3—21no—
21C4×#41⋊C3D311yescharacteristic, complemented
22C2×C3×Dic6—42no—
23C3×#41⋊C3D412yescomplemented
24C2×C3×SL(2,𝔽3)C2311yescharacteristic
25C3⋊(#41⋊C3)—24no—
26C32×C42—41no—
27C3×C4⋊Dic3—44no—
28#3498—31noSylow
29C2×C3×#41C22×C311yescharacteristic
30#1009—24no—
31#1009—64no—
32C2×#239C22×C314yes—
33C2×D3×D4—34no—
34C2×#235—32no—
35C22×D12—34no—
36C22×D12—32no—
37#994—64no—
38C2×#41⋊C3—21no—
39C2×#41⋊C3—34no—
40C2×#41⋊C3C2×D311yescharacteristic
41C3×C4⋊Q8—121no—
42C3×C4×Q8C22×C312yes—
43C3×C4×D4—32no—
44C3×#117—31no—
45C2×C3×C42—31no—
46#930—34no—
47#895—38no—
48#882—38no—
49#875—32no—
50#874—34no—
51#874—32no—
52#874—31no—
53C4×D12—34no—
54#869A411yescharacteristic
55C4×SL(2,𝔽3)—22no—
56C4×SL(2,𝔽3)C2×D312yescomplemented
57C2×C32×C4—41no—
58C2×C32×C4—42no—
59C2×C3×Dic3—44no—
60C3×Dic6—84no—
61C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
62C2×H(5,𝔽2)—32no—
63#550—34no—
64D42—94no—
65#535—92no—
66#532—91no—
67C2×#127—91no—
68C4×#41C3×D311yescharacteristic
69C23×D3—34no—
70C3×#41—32no—
71C3×#41C3×D412yes—
72C2×C3×Q8—122no—
73C2×C3×Q8C23×C311yescharacteristic
74C2×C3×D4—31no—
75C22×C3×C4—31no—
76C22×C3×C4—32no—
77C2×#67—34no—
78#239—24no—
79#239—64no—
80D3×D4—68no—
81D3×D4—68no—
82#235—64no—
83#235—64no—
84C2×D12—34no—
85C2×D12—34no—
86C2×D12—32no—
87C2×D12—38no—
88C2×D12—64no—
89C2×D12—64no—
90C2×D12—128no—
91C2×C4×D3—34no—
92C2×Dic6C2×A412yes—
93#41⋊C3—22no—
94#41⋊C3—64no—
95#41⋊C3D1212yescomplemented
96C2×SL(2,𝔽3)—21no—
97C2×SL(2,𝔽3)C22×D311yescharacteristic
98C3×C4⋊C4—32no—
99C3×C4⋊C4—124no—
100C3×C22⋊C4—32no—
101C3×C42—32no—
102C3×C42—31no—
103C3×C42—41no—
104C3×C42—81no—
105C3×C42C2×A411yescharacteristic
106(C2×D3)⋊C4—38no—
107C4⋊Dic3C2×A414yes—
108C22×C32—41no—
109C32×C4—42no—
110C32×C4—44no—
111C3×Dic3—84no—
112H(5,𝔽2)—64no—
113H(5,𝔽2)—184no—
114C2×#41—34no—
115C2×#41—92no—
116C2×#41C2×C3×D311yescharacteristic
117C22×D4—94no—
118C22×D4—92no—
119C22×D4—94no—
120C4⋊Q8—31no—
121#127—94no—
122#127—92no—
123#127—91no—
124#124—92no—
125#121—94no—
126#121—98no—
127#120—98no—
128C4×Q8C2×C3×D312yes—
129C4×D4—32no—
130C4×D4—94no—
131#117—31no—
132C2×C42—31no—
133C23×C3—31no—
134C22×D3—34no—
135C22×D3—68no—
136C22×D3—128no—
137C3×Q8—31no—
138C3×Q8—244no—
139C3×Q8C2×C3×D411yescharacteristic
140C3×D4—32no—
141C3×D4—32no—
142C2×C3×C4—31no—
143C2×C3×C4—32no—
144C2×C3×C4—32no—
145C2×C3×C4—32no—
146C2×C3×C4—42no—
147C2×C3×C4—41no—
148C2×C3×C4—64no—
149C2×C3×C4—82no—
150C2×C3×C4—81no—
151C2×C3×C4—124no—
152C2×C3×C4C22×A411yescharacteristic
153C2×C3×C4C22×A412yes—
154#67—68no—
155C2×Dic3C22×A414yes—
156D12—64no—
157D12—64no—
158D12—64no—
159D12—68no—
160D12—128no—
161C4×D3—64no—
162C4×D3—64no—
163Dic6—24no—
164SL(2,𝔽3)—21no—
165SL(2,𝔽3)C2×D1211yescharacteristic, complemented
166C2×C32—41no—
167C2×C32—42no—
168C24—94no—
169#41—32no—
170#41—64no—
171#41—184no—
172#41—184no—
173#41—184no—
174#41C3×D1212yes—
175C2×Q8—32no—
176C2×Q8C22×C3×D311yescharacteristic
177C2×D4—31no—
178C2×D4—94no—
179C2×D4—94no—
180C2×D4—98no—
181C2×D4—94no—
182C2×D4—92no—
183C2×D4—184no—
184C2×D4—184no—
185C2×D4—188no—
186C2×D4—188no—
187C2×D4—368no—
188C22×C4—31no—
189C22×C4—32no—
190C22×C4—94no—
191C4⋊C4—32no—
192C4⋊C4—34no—
193C22⋊C4—32no—
194C22⋊C4—98no—
195C42—31no—
196C42—32no—
197C42D3×A411yescharacteristic
198C22×C3—32no—
199C22×C3—41no—
200C22×C3—62no—
201C22×C3—81no—
202C22×C3C23×A411yescharacteristic
203C2×D3—64no—
204C2×D3—68no—
205C2×D3—248no—
206C3×C4—31no—
207C3×C4—31no—
208C3×C4—42no—
209C3×C4—44no—
210C3×C4—62no—
211C3×C4—82no—
212C3×C4—84no—
213C3×C4—244no—
214C3×C4D4×A412yes—
215C3×C4#99414yescomplemented
216Dic3—24no—
217C32—41noSylow
218C23—31no—
219C23—94no—
220C23—188no—
221C23—368no—
222Q8—31no—
223Q8—64no—
224Q8C2×C3×D1211yescharacteristic
225D4—32no—
226D4—32no—
227D4—184no—
228D4—184no—
229D4—188no—
230D4—184no—
231D4—188no—
232D4—368no—
233C2×C4—31no—
234C2×C4—32no—
235C2×C4—32no—
236C2×C4—32no—
237C2×C4—34no—
238C2×C4—64no—
239C2×C4—184no—
240C2×C4—184no—
241C2×C4C2×D3×A411yescharacteristic
242C2×C4C2×D3×A412yes—
243C2×C3—41no—
244C2×C3—42no—
245C2×C3—62no—
246C2×C3—81no—
247C2×C3—82no—
248C2×C3C2×D4×A411yescharacteristic
249C2×C3C2×#99412yes—
250D3—124no—
251C22—32no—
252C22—62no—
253C22—184no—
254C22—188no—
255C22—728no—
256C22C22×D3×A411yescharacteristic
257C4—31no—
258C4—31no—
259C4—64no—
260C4—62no—
261C4A4×D1212yes—
262C4C3⋊#99414yescomplemented
263C3—41no—
264C3—81no—
265C3C4⋊#99411yescharacteristic, complemented
266C2—62no—
267C2—364no—
268C2C2×A4×D1211yescharacteristic
269C2C2×C3⋊#99412yes—
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×C4×#41C2×C3
socleC22×C3C23×A4
hypercenterC42D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊#994C2semidirect
2. C3×C4×#41⋊C3C2semidirect
3. C4⋊(C3⋊(#41⋊C3))C2semidirect
4. #85846C3semidirect
8. C2×C3⋊(#41⋊C3)C22semidirect
10. C3×C4×SL(2,𝔽3)C22semidirect
21. C4×#41⋊C3D3semidirect
23. C3×#41⋊C3D4semidirect
56. C4×SL(2,𝔽3)C2×D3semidirect
61. C3×SL(2,𝔽3)C2×D4semidirect
95. #41⋊C3D12semidirect
165. SL(2,𝔽3)C2×D12semidirect
215. C3×C4#994semidirect
262. C4C3⋊#994semidirect
265. C3C4⋊#994semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊#9942
C3×C4×#41⋊C32
C4⋊(C3⋊(#41⋊C3))2
#858463
C4⋊#9943
C3×#8694