Finite Group Databasebeta

← #82078⋊C3 #11921252

Subgroups — #82078⋊C3 #11921252

Order 1152.

289 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 33 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
1C22×C3×C42⋊C3C211yesmaximal, characteristic, complemented
2C2×C3⋊(#126⋊C3)C211yesmaximal, characteristic, complemented
3C3⋊(#383⋊C3)C211yesmaximal, characteristic, complemented
4#82078C311yesmaximal, characteristic, complemented
5#2852⋊C3—31nomaximal
6A4×#67—41nomaximal
7C2×C3×C42⋊C3—21no—
8C2×C3×C42⋊C3C2211yescharacteristic
9C3⋊(#126⋊C3)—21no—
10C22×C3×C42C2×C311yescharacteristic, complemented
11C2×#876C2×C311yescharacteristic, complemented
12C2×#126⋊C3—31no—
13C22×C42⋊C3—21no—
14C22×C42⋊C3D311yescharacteristic, complemented
15#5213—31no—
16#5210—31no—
17#4940—31no—
18#4937C2×C311yescharacteristic, complemented
19#4933—31no—
20#383⋊C3—31no—
21C22×C3×A4—41no—
22C2×D3×A4—41no—
23Dic3×A4—41no—
24C3×C42⋊C3D411yescharacteristic, complemented
25#2852—31noSylow
26C23×C3×C4—31no—
27C22×#67A411yescharacteristic
28D4×A4—121no—
29C2×C3×C42—21no—
30C2×C3×C42—31no—
31C2×C3×C42—61no—
32C2×C3×C42C22×C311yescharacteristic
33C2×#217—31no—
34C2×(C2×D3)⋊C4—31no—
35C2×(C2×D3)⋊C4—31no—
36C2×Dic3⋊C4—31no—
37C2×Dic3⋊C4—31no—
38#876—21no—
39#876—61no—
40#126⋊C3—61no—
41C2×C42⋊C3—22no—
42C2×C42⋊C3—21no—
43C2×C42⋊C3—21no—
44C2×C42⋊C3C2×D311yescharacteristic
45C3⋊#95—31no—
46C3⋊#95—31no—
47C3⋊#95—31no—
48C3⋊#95—31no—
49C2×C3×A4—41no—
50C2×C3×A4—81no—
51D3×A4—81no—
52C3×#67—161no—
53C2×#126—31no—
54C22×C42C3×D311yescharacteristic, complemented
55#388—91no—
56#387—91no—
57#386—91no—
58#385—91no—
59#383—31no—
60C24×C3C2×A411yescharacteristic
61C23×D3C2×A411yescharacteristic
62C22×A4—41no—
63C22×A4—81no—
64C22×A4—121no—
65C22×C3×C4—31no—
66C22×C3×C4—31no—
67C22×C3×C4—61no—
68C22×C3×C4—61no—
69C22×C3×C4—62no—
70C22×C3×C4—61no—
71C22×C3×C4—61no—
72C2×#67—61no—
73C2×#67—61no—
74C22×Dic3—31no—
75C22×Dic3—31no—
76C22×Dic3C2×A411yescharacteristic
77C4×A4—121no—
78C3×C42—31no—
79C3×C42—61no—
80C3×C42—61no—
81C3×C42C3×D411yescharacteristic, complemented
82#217—61no—
83#217—61no—
84(C2×D3)⋊C4—61no—
85(C2×D3)⋊C4—61no—
86(C2×D3)⋊C4—62no—
87Dic3⋊C4—62no—
88Dic3⋊C4—61no—
89Dic3⋊C4—61no—
90C42⋊C3—21no—
91C42⋊C3#6711yescharacteristic, complemented
92C22×C32—161no—
93C2×C3×D3—161no—
94C3×A4—41no—
95C3×Dic3—161no—
96C22×D4—31no—
97C23×C4—31no—
98#126—61no—
99#126—181no—
100C2×C4⋊C4—91no—
101C2×C4⋊C4—91no—
102C2×C22⋊C4—91no—
103C2×C22⋊C4—91no—
104C2×C22⋊C4—91no—
105C2×C42—21no—
106C2×C42—31no—
107C2×C42—61no—
108C2×C42C2×C3×D311yescharacteristic
109#95—91no—
110#95—91no—
111#95—91no—
112#95—91no—
113C23×C3—21no—
114C23×C3—31no—
115C23×C3—31no—
116C23×C3—61no—
117C23×C3C22×A411yescharacteristic
118C22×D3—21no—
119C22×D3—61no—
120C22×D3—61no—
121C2×A4—41no—
122C2×A4—81no—
123C2×A4—82no—
124C2×A4—81no—
125C2×A4—241no—
126C3×D4—481no—
127C2×C3×C4—32no—
128C2×C3×C4—31no—
129C2×C3×C4—31no—
130C2×C3×C4—61no—
131C2×C3×C4—61no—
132C2×C3×C4—61no—
133C2×C3×C4—62no—
134C2×C3×C4—62no—
135C2×C3×C4—62no—
136C2×C3×C4—62no—
137C2×C3×C4—61no—
138#67—41no—
139#67—121no—
140C2×Dic3—61no—
141C2×Dic3—61no—
142C2×Dic3—61no—
143C2×Dic3—61no—
144C2×Dic3—61no—
145C2×Dic3—61no—
146C2×Dic3—61no—
147C2×C32—161no—
148C2×C32—321no—
149C3×D3—321no—
150C24—31no—
151C24D3×A411yescharacteristic
152C2×D4—181no—
153C2×D4—181no—
154C22×C4—31no—
155C22×C4—31no—
156C22×C4—31no—
157C22×C4—61no—
158C22×C4—61no—
159C22×C4—62no—
160C22×C4—61no—
161C22×C4—61no—
162C22×C4—91no—
163C22×C4—91no—
164C4⋊C4—181no—
165C4⋊C4—181no—
166C4⋊C4—182no—
167C22⋊C4—181no—
168C22⋊C4—181no—
169C22⋊C4—182no—
170C22⋊C4—181no—
171C22⋊C4—181no—
172C42—31no—
173C42—61no—
174C42—61no—
175C42C3×#6711yescharacteristic, complemented
176C22×C3—31no—
177C22×C3—31no—
178C22×C3—31no—
179C22×C3—61no—
180C22×C3—61no—
181C22×C3—61no—
182C22×C3—61no—
183C22×C3—161no—
184C22×C3—321no—
185C22×C3—481no—
186C22×C3#126⋊C311yescharacteristic, complemented
187C22×C3D4×A411yescharacteristic
188C2×D3—41no—
189C2×D3—121no—
190C2×D3—121no—
191A4—41no—
192A4—81no—
193C3×C4—62no—
194C3×C4—61no—
195C3×C4—61no—
196C3×C4—481no—
197Dic3—41no—
198Dic3—121no—
199Dic3—121no—
200C32—161noSylow
201C23—21no—
202C23—31no—
203C23—31no—
204C23—61no—
205C23—61no—
206C23—181no—
207C23—181no—
208C23C2×D3×A411yescharacteristic
209D4—121no—
210D4—361no—
211C2×C4—32no—
212C2×C4—31no—
213C2×C4—31no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—62no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—62no—
221C2×C4—61no—
222C2×C4—181no—
223C2×C4—181no—
224C2×C4—181no—
225C2×C4—181no—
226C2×C4—181no—
227C2×C4—181no—
228C2×C4—181no—
229C2×C3—21no—
230C2×C3—31no—
231C2×C3—31no—
232C2×C3—61no—
233C2×C3—161no—
234C2×C3—321no—
235C2×C3—322no—
236C2×C3—321no—
237C2×C3—961no—
238C2×C3C2×#126⋊C311yescharacteristic
239D3—81no—
240C22—31no—
241C22—31no—
242C22—31no—
243C22—61no—
244C22—61no—
245C22—61no—
246C22—61no—
247C22—121no—
248C22—361no—
249C22—361no—
250C22C3⋊(#126⋊C3)11yescharacteristic, complemented
251C22A4×#6711yescharacteristic
252C4—62no—
253C4—61no—
254C4—61no—
255C4—121no—
256C4—361no—
257C4—361no—
258C3—161no—
259C3—321no—
260C3#2852⋊C311yescharacteristic, complemented
261C2—21no—
262C2—31no—
263C2—31no—
264C2—61no—
265C2—241no—
266C2C2×C3⋊(#126⋊C3)11yescharacteristic
subgroupG/H
centerC2C2×C3⋊(#126⋊C3)
commutator, cosocleC2×C3×C42C22×C3
FrattiniC23C2×D3×A4
FittingC22×C3×C42C2×C3
socleC23×C3C22×A4
hypercenterC22C3⋊(#126⋊C3)
hypocenterC3×C42C3×D4
LCSG ▷ C2×C3×C42 ▷ C3×C42
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×C42⋊C3C2semidirect
2. C2×C3⋊(#126⋊C3)C2semidirect
3. C3⋊(#383⋊C3)C2semidirect
4. #82078C3semidirect
10. C22×C3×C42C2×C3semidirect
11. C2×#876C2×C3semidirect
14. C22×C42⋊C3D3semidirect
18. #4937C2×C3semidirect
24. C3×C42⋊C3D4semidirect
54. C22×C42C3×D3semidirect
81. C3×C42C3×D4semidirect
91. C42⋊C3#67semidirect
175. C42C3×#67semidirect
186. C22×C3#126⋊C3semidirect
250. C22C3⋊(#126⋊C3)semidirect
260. C3#2852⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C42⋊C32
C2×C3⋊(#126⋊C3)2
C3⋊(#383⋊C3)2
#820783
#2852⋊C33
A4×#674