Finite Group Databasebeta

← C3×#3019 #84883

Subgroups — C3×#3019 #84883

Order 384.

762 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 212 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×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C3×#575C212yesmaximal, complemented
3C3×#535C211yesmaximal, characteristic, complemented
4C3×#455C214yesmaximal, complemented
5C3×#453C214yesmaximal, complemented
6C3×#421C212yesmaximal, complemented
7C2×C3×#100C211yesmaximal, characteristic, complemented
8#3019C311yesmaximal, characteristic, direct product factor, Sylow
9C3×H(5,𝔽2)—24no—
10C3×H(5,𝔽2)—22no—
11C3×H(5,𝔽2)C2214yes—
12C2×C3×#41—22no—
13C2×C3×#41C2212yes—
14C2×C3×#41C2211yescharacteristic, complemented
15C22×C3×D4—22no—
16C22×C3×D4—22no—
17C22×C3×D4C2211yescharacteristic
18C3×#137—22no—
19C3×#137C2214yescomplemented
20C3×AGL(1,ℤ/8)—22no—
21C3×AGL(1,ℤ/8)C2214yescomplemented
22C3×#135—24no—
23C2×C3×M16C2212yescomplemented
24C3×#127C2212yescomplemented
25C3×#124C2212yescomplemented
26C3×#121—22no—
27C3×#120—22no—
28C3×#117C2211yescharacteristic, complemented
29C3×#104C2218yescomplemented
30C3×#100C2214yescomplemented
31C2×H(5,𝔽2)C2×C311yescharacteristic, complemented
32#575C2×C312yescomplemented
33#535C2×C311yescharacteristic, complemented
34#455C2×C314yescomplemented
35#453C2×C314yescomplemented
36#421C2×C312yescomplemented
37C2×#100C2×C311yescharacteristic, complemented
38C24×C3—22no—
39C24×C3D412yes—
40C3×#41—28no—
41C3×#41—28no—
42C3×#41—22no—
43C3×#41—22no—
44C3×#41—22no—
45C3×#41—22no—
46C3×#41D414yescomplemented
47C3×#41C2314yes—
48C2×C3×Q8D412yescomplemented
49C2×C3×Q8C2311yescharacteristic, complemented
50C2×C3×D4—24no—
51C2×C3×D4—28no—
52C2×C3×D4—28no—
53C2×C3×D4—22no—
54C2×C3×D4—28no—
55C2×C3×D4—28no—
56C2×C3×D4—21no—
57C2×C3×D4—24no—
58C2×C3×D4—22no—
59C2×C3×D4—22no—
60C2×C3×D4D412yes—
61C2×C3×D4D412yescomplemented
62C2×C3×D4C2312yes—
63C2×C3×D4C2311yescharacteristic, complemented
64C22×C3×C4—22no—
65C22×C3×C4—22no—
66C22×C3×C4—21no—
67C22×C3×C4C2311yescharacteristic
68C3×Q16—24no—
69C3×SD16—24no—
70C3×SD16—24no—
71C3×D8—24no—
72C3×M16—22no—
73C3×M16C2314yescomplemented
74C2×C3×C8—22no—
75C3×C4⋊C4—21no—
76C3×C22⋊C4—21no—
77C3×C22⋊C4—24no—
78C3×C42C2312yescomplemented
79H(5,𝔽2)—24no—
80H(5,𝔽2)—22no—
81H(5,𝔽2)C22×C314yes—
82C2×#41—22no—
83C2×#41C22×C312yes—
84C2×#41C22×C311yescharacteristic, complemented
85C22×D4—22no—
86C22×D4—22no—
87C22×D4C22×C311yescharacteristic
88#137—22no—
89#137C22×C314yescomplemented
90AGL(1,ℤ/8)—22no—
91AGL(1,ℤ/8)C22×C314yescomplemented
92#135—24no—
93C2×M16C22×C312yescomplemented
94#127C22×C312yescomplemented
95#124C22×C312yescomplemented
96#121—22no—
97#120—22no—
98#117C22×C311yescharacteristic, complemented
99#104C22×C318yescomplemented
100#100C22×C314yescomplemented
101C23×C3—28no—
102C23×C3—22no—
103C23×C3—22no—
104C23×C3—22no—
105C23×C3—21no—
106C23×C3—82no—
107C23×C3—84no—
108C23×C3C2×D414yes—
109C23×C3C2×D411yescharacteristic
110C3×Q8—22no—
111C3×Q8—22no—
112C3×Q8C2×D414yes—
113C3×D4—216no—
114C3×D4—24no—
115C3×D4—24no—
116C3×D4—24no—
117C3×D4—22no—
118C3×D4—24no—
119C3×D4—22no—
120C3×D4—41no—
121C3×D4—41no—
122C3×D4—84no—
123C3×D4C2×D414yes—
124C2×C3×C4—28no—
125C2×C3×C4—22no—
126C2×C3×C4—24no—
127C2×C3×C4—24no—
128C2×C3×C4—22no—
129C2×C3×C4—21no—
130C2×C3×C4—42no—
131C2×C3×C4C2×D411yescharacteristic
132C2×C3×C4C2×D412yes—
133C2×C3×C4C2×D412yes—
134C2×C3×C4C2411yescharacteristic
135C3×C8—24no—
136C24—22no—
137C24C3×D412yes—
138#41—28no—
139#41—28no—
140#41—22no—
141#41—22no—
142#41—22no—
143#41—22no—
144#41C3×D414yescomplemented
145#41C23×C314yes—
146C2×Q8C3×D412yescomplemented
147C2×Q8C23×C311yescharacteristic, complemented
148C2×D4—24no—
149C2×D4—28no—
150C2×D4—28no—
151C2×D4—22no—
152C2×D4—28no—
153C2×D4—28no—
154C2×D4—21no—
155C2×D4—24no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4C3×D412yes—
159C2×D4C3×D412yescomplemented
160C2×D4C23×C312yes—
161C2×D4C23×C311yescharacteristic, complemented
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4C23×C311yescharacteristic
166Q16—24no—
167SD16—24no—
168SD16—24no—
169D8—24no—
170M16—22no—
171M16C23×C314yescomplemented
172C2×C8—22no—
173C4⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—24no—
176C42C23×C312yescomplemented
177C22×C3—24no—
178C22×C3—24no—
179C22×C3—41no—
180C22×C3—44no—
181C22×C3—44no—
182C22×C3—44no—
183C22×C3—42no—
184C22×C3—41no—
185C22×C3—82no—
186C22×C3—88no—
187C22×C3#12011yescharacteristic
188C22×C3#12011yescharacteristic
189C22×C3C22×D411yescharacteristic
190C3×C4—24no—
191C3×C4—42no—
192C3×C4—41no—
193C3×C4#12012yes—
194C3×C4C22×D412yes—
195C23—28no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—21no—
200C23—82no—
201C23—84no—
202C23C2×C3×D414yes—
203C23C2×C3×D411yescharacteristic
204Q8—22no—
205Q8—22no—
206Q8C2×C3×D414yes—
207D4—216no—
208D4—24no—
209D4—24no—
210D4—24no—
211D4—22no—
212D4—24no—
213D4—22no—
214D4—41no—
215D4—41no—
216D4—84no—
217D4C2×C3×D414yes—
218C2×C4—28no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4C2×C3×D411yescharacteristic
226C2×C4C2×C3×D412yes—
227C2×C4C2×C3×D412yes—
228C2×C4C24×C311yescharacteristic
229C8—24no—
230C2×C3—21no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3—44no—
234C2×C3—44no—
235C2×C3—81no—
236C2×C3C2×#12011yescharacteristic
237C22—24no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—44no—
243C22—42no—
244C22—41no—
245C22—82no—
246C22—88no—
247C22C3×#12011yescharacteristic
248C22C3×#12011yescharacteristic
249C22C22×C3×D411yescharacteristic
250C4—24no—
251C4—42no—
252C4—41no—
253C4C3×#12012yes—
254C4C22×C3×D412yes—
255C3#301911yescharacteristic, direct product factor, Sylow
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—81no—
262C2C2×C3×#12011yescharacteristic
subgroupG/H
center, socleC2×C3C2×#120
commutator, Frattini, cosocleC2×C4C24×C3
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×H(5,𝔽2)C2semidirect
2. C3×#575C2semidirect
3. C3×#535C2semidirect
4. C3×#455C2semidirect
5. C3×#453C2semidirect
6. C3×#421C2semidirect
7. C2×C3×#100C2semidirect
8. #3019C3direct (N × H)
14. C2×C3×#41C22semidirect
19. C3×#137C22semidirect
21. C3×AGL(1,ℤ/8)C22semidirect
23. C2×C3×M16C22semidirect
24. C3×#127C22semidirect
25. C3×#124C22semidirect
28. C3×#117C22semidirect
29. C3×#104C22semidirect
30. C3×#100C22semidirect
31. C2×H(5,𝔽2)C2×C3semidirect
32. #575C2×C3semidirect
33. #535C2×C3semidirect
34. #455C2×C3semidirect
35. #453C2×C3semidirect
36. #421C2×C3semidirect
37. C2×#100C2×C3semidirect
46. C3×#41D4semidirect
48. C2×C3×Q8D4semidirect
49. C2×C3×Q8C23semidirect
61. C2×C3×D4D4semidirect
63. C2×C3×D4C23semidirect
73. C3×M16C23semidirect
78. C3×C42C23semidirect
84. C2×#41C22×C3semidirect
89. #137C22×C3semidirect
91. AGL(1,ℤ/8)C22×C3semidirect
93. C2×M16C22×C3semidirect
94. #127C22×C3semidirect
95. #124C22×C3semidirect
98. #117C22×C3semidirect
99. #104C22×C3semidirect
100. #100C22×C3semidirect
144. #41C3×D4semidirect
146. C2×Q8C3×D4semidirect
147. C2×Q8C23×C3semidirect
159. C2×D4C3×D4semidirect
161. C2×D4C23×C3semidirect
171. M16C23×C3semidirect
176. C42C23×C3semidirect
255. C3#3019direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×H(5,𝔽2)2
C3×#5752
C3×#5352
C3×#4552
C3×#4532
C3×#4212
C2×C3×#1002
#30193