Subgroups — C₃×#3204 #85068

Order 384.

402 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#537	C₂	1	2	yes	maximal, complemented
2	C₃×#483	C₂	1	2	yes	maximal, complemented
3	C₃×#481	C₂	1	2	yes	maximal, complemented
4	C₃×#470	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#469	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#465	C₂	1	2	yes	maximal, complemented
7	C₃×#452	C₂	1	2	yes	maximal, complemented
8	C₃×#449	C₂	1	2	yes	maximal, complemented
9	C₃×#408	C₂	1	1	yes	maximal, characteristic, complemented
10	#3204	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×#41	C₂²	1	2	yes	—
12	C₂×C₃×#41	C₂²	1	1	yes	characteristic
13	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
15	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
16	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
18	C₃×C₂²⋊Q₈	—	2	2	no	—
19	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₃×#121	—	2	2	no	—
22	C₃×#121	—	2	2	no	—
23	C₃×#121	—	2	2	no	—
24	C₃×#121	C₂²	1	2	yes	—
25	C₃×#121	C₂²	1	2	yes	—
26	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
27	C₃×#106	C₂²	1	2	yes	—
28	C₃×#106	C₂²	1	1	yes	characteristic
29	C₃×#106	C₂²	1	1	yes	characteristic
30	C₃×#103	C₂²	1	2	yes	—
31	C₃×#103	C₂²	1	2	yes	—
32	C₃×#102	C₂²	1	2	yes	—
33	C₃×#102	C₂²	1	2	yes	—
34	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
36	#537	C₂×C₃	1	2	yes	complemented
37	#483	C₂×C₃	1	2	yes	complemented
38	#481	C₂×C₃	1	2	yes	complemented
39	#470	C₂×C₃	1	1	yes	characteristic, complemented
40	#469	C₂×C₃	1	1	yes	characteristic, complemented
41	#465	C₂×C₃	1	2	yes	complemented
42	#452	C₂×C₃	1	2	yes	complemented
43	#449	C₂×C₃	1	2	yes	complemented
44	#408	C₂×C₃	1	1	yes	characteristic, complemented
45	C₃×#41	—	4	1	no	—
46	C₃×#41	—	4	1	no	—
47	C₃×#41	—	4	2	no	—
48	C₃×#41	—	4	2	no	—
49	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
50	C₂×C₃×Q₈	C₂³	1	2	yes	—
51	C₂×C₃×D₄	—	2	2	no	—
52	C₂×C₃×D₄	—	2	2	no	—
53	C₂×C₃×D₄	—	2	2	no	—
54	C₂×C₃×D₄	D₄	1	1	yes	characteristic
55	C₂×C₃×D₄	D₄	1	2	yes	—
56	C₂×C₃×D₄	C₂³	1	2	yes	—
57	C₂²×C₃×C₄	—	2	2	no	—
58	C₂²×C₃×C₄	—	2	2	no	—
59	C₂²×C₃×C₄	C₂³	1	2	yes	—
60	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
61	C₃×Q₁₆	—	4	1	no	—
62	C₃×SD₁₆	—	4	2	no	—
63	C₃×D₈	—	4	1	no	—
64	C₃×M₁₆	—	4	1	no	—
65	C₂×C₃×C₈	—	4	1	no	—
66	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
67	C₂×C₃×C₈	C₂³	1	2	yes	—
68	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
69	C₃×C₄⋊C₄	—	2	2	no	—
70	C₃×C₄⋊C₄	—	2	2	no	—
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₄⋊C₄	C₂³	1	2	yes	—
73	C₃×C₄⋊C₄	C₂³	1	2	yes	—
74	C₃×C₂²⋊C₄	—	2	2	no	—
75	C₃×C₂²⋊C₄	—	2	2	no	—
76	C₃×C₂²⋊C₄	—	2	2	no	—
77	C₃×C₂²⋊C₄	—	2	2	no	—
78	C₂×#41	C₂²×C₃	1	2	yes	—
79	C₂×#41	C₂²×C₃	1	1	yes	characteristic
80	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
81	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
82	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
83	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
84	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
85	C₂²⋊Q₈	—	2	2	no	—
86	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
87	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
88	#121	—	2	2	no	—
89	#121	—	2	2	no	—
90	#121	—	2	2	no	—
91	#121	C₂²×C₃	1	2	yes	—
92	#121	C₂²×C₃	1	2	yes	—
93	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
94	#106	C₂²×C₃	1	2	yes	—
95	#106	C₂²×C₃	1	1	yes	characteristic
96	#106	C₂²×C₃	1	1	yes	characteristic
97	#103	C₂²×C₃	1	2	yes	—
98	#103	C₂²×C₃	1	2	yes	—
99	#102	C₂²×C₃	1	2	yes	—
100	#102	C₂²×C₃	1	2	yes	—
101	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
102	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
103	C₂³×C₃	—	2	2	no	—
104	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
105	C₂³×C₃	C₂×D₄	1	2	yes	—
106	C₃×Q₈	—	2	2	no	—
107	C₃×Q₈	—	2	2	no	—
108	C₃×Q₈	—	4	1	no	—
109	C₃×D₄	—	2	2	no	—
110	C₃×D₄	—	2	2	no	—
111	C₃×D₄	—	4	2	no	—
112	C₃×D₄	—	4	1	no	—
113	C₃×D₄	—	4	4	no	—
114	C₃×D₄	—	8	2	no	—
115	C₃×D₄	—	8	2	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	1	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	4	4	no	—
124	C₂×C₃×C₄	—	8	2	no	—
125	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
127	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
128	C₃×C₈	—	2	2	no	—
129	C₃×C₈	—	2	1	no	—
130	C₃×C₈	—	2	1	no	—
131	#41	—	4	1	no	—
132	#41	—	4	1	no	—
133	#41	—	4	2	no	—
134	#41	—	4	2	no	—
135	C₂×Q₈	C₃×D₄	1	1	yes	characteristic
136	C₂×Q₈	C₂³×C₃	1	2	yes	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	C₃×D₄	1	1	yes	characteristic
141	C₂×D₄	C₃×D₄	1	2	yes	—
142	C₂×D₄	C₂³×C₃	1	2	yes	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	C₂³×C₃	1	2	yes	—
146	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
147	Q₁₆	—	4	1	no	—
148	SD₁₆	—	4	2	no	—
149	D₈	—	4	1	no	—
150	M₁₆	—	4	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
153	C₂×C₈	C₂³×C₃	1	2	yes	—
154	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	C₂³×C₃	1	2	yes	—
159	C₄⋊C₄	C₂³×C₃	1	2	yes	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₂²×C₃	—	2	2	no	—
165	C₂²×C₃	—	2	1	no	—
166	C₂²×C₃	—	2	2	no	—
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	—	2	2	no	—
169	C₂²×C₃	—	4	2	no	—
170	C₂²×C₃	—	4	2	no	—
171	C₂²×C₃	—	4	2	no	—
172	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	—	2	1	no	—
175	C₃×C₄	—	4	2	no	—
176	C₃×C₄	—	4	2	no	—
177	C₃×C₄	—	4	2	no	—
178	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
179	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
180	C₂³	—	2	2	no	—
181	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
182	C₂³	C₂×C₃×D₄	1	2	yes	—
183	Q₈	—	2	2	no	—
184	Q₈	—	2	2	no	—
185	Q₈	—	4	1	no	—
186	D₄	—	2	2	no	—
187	D₄	—	2	2	no	—
188	D₄	—	4	2	no	—
189	D₄	—	4	1	no	—
190	D₄	—	4	4	no	—
191	D₄	—	8	2	no	—
192	D₄	—	8	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
203	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
204	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₂×C₃	—	4	2	no	—
209	C₂×C₃	—	4	1	no	—
210	C₂×C₃	—	8	2	no	—
211	C₂×C₃	#534	1	1	yes	characteristic
212	C₂×C₃	#577	1	2	yes	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	1	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
228	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
229	C₃	#3204	1	1	yes	characteristic, direct product factor, Sylow
230	C₂	—	4	2	no	—
231	C₂	—	4	1	no	—
232	C₂	—	8	2	no	—
233	C₂	C₃×#534	1	1	yes	characteristic
234	C₂	C₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#537	C₂	semidirect
2. C₃×#483	C₂	semidirect
3. C₃×#481	C₂	semidirect
4. C₃×#470	C₂	semidirect
5. C₃×#469	C₂	semidirect
6. C₃×#465	C₂	semidirect
7. C₃×#452	C₂	semidirect
8. C₃×#449	C₂	semidirect
9. C₃×#408	C₂	semidirect
10. #3204	C₃	direct (N × H)
36. #537	C₂×C₃	semidirect
37. #483	C₂×C₃	semidirect
38. #481	C₂×C₃	semidirect
39. #470	C₂×C₃	semidirect
40. #469	C₂×C₃	semidirect
41. #465	C₂×C₃	semidirect
42. #452	C₂×C₃	semidirect
43. #449	C₂×C₃	semidirect
44. #408	C₂×C₃	semidirect
229. C₃	#3204	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#537	2
C₃×#483	2
C₃×#481	2
C₃×#470	2
C₃×#469	2
C₃×#465	2
C₃×#452	2
C₃×#449	2
C₃×#408	2
#3204	3

Subgroups — C3×#3204 #85068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3204 #85068