Subgroups — C₃×#3213 #85077

Order 384.

444 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#496	C₂	1	2	yes	maximal, complemented
4	C₃×#489	C₂	1	2	yes	maximal, complemented
5	C₃×#463	C₂	1	4	yes	maximal, complemented
6	C₃×#448	C₂	1	2	yes	maximal, complemented
7	C₃×#447	C₂	1	2	yes	maximal, complemented
8	C₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	#3213	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#41	—	2	1	no	—
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×SD₁₆	C₂²	1	4	yes	complemented
14	C₂×C₃×D₈	C₂²	1	4	yes	—
15	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
16	C₃×#127	C₂²	1	1	yes	characteristic, complemented
17	C₃×#124	—	2	1	no	—
18	C₃×#124	—	2	1	no	—
19	C₃×#124	—	2	1	no	—
20	C₃×#124	C₂²	1	2	yes	—
21	C₃×#124	C₂²	1	2	yes	—
22	C₃×#121	—	2	1	no	—
23	C₃×#121	C₂²	1	2	yes	—
24	C₃×C₄×D₄	—	2	1	no	—
25	C₃×C₄×D₄	C₂²	1	2	yes	complemented
26	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
27	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
28	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
29	C₃×C₄⋊C₈	C₂²	1	2	yes	complemented
30	C₃×#103	C₂²	1	4	yes	complemented
31	C₃×#102	C₂²	1	4	yes	—
32	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₃×#97	C₂²	1	2	yes	complemented
34	#532	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂×#124	C₂×C₃	1	1	yes	characteristic, complemented
36	#496	C₂×C₃	1	2	yes	complemented
37	#489	C₂×C₃	1	2	yes	complemented
38	#463	C₂×C₃	1	4	yes	complemented
39	#448	C₂×C₃	1	2	yes	complemented
40	#447	C₂×C₃	1	2	yes	complemented
41	#432	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂⁴×C₃	—	2	1	no	—
43	C₃×#41	—	4	4	no	—
44	C₂×C₃×Q₈	—	2	1	no	—
45	C₂×C₃×Q₈	—	2	1	no	—
46	C₂×C₃×Q₈	—	4	2	no	—
47	C₂×C₃×Q₈	C₂³	1	2	yes	—
48	C₂×C₃×D₄	—	2	2	no	—
49	C₂×C₃×D₄	—	2	1	no	—
50	C₂×C₃×D₄	—	2	1	no	—
51	C₂×C₃×D₄	—	2	1	no	—
52	C₂×C₃×D₄	—	4	2	no	—
53	C₂×C₃×D₄	C₂³	1	2	yes	—
54	C₂×C₃×D₄	C₂³	1	2	yes	—
55	C₂²×C₃×C₄	—	2	1	no	—
56	C₂²×C₃×C₄	—	2	2	no	—
57	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
58	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
60	C₃×SD₁₆	—	4	4	no	—
61	C₃×D₈	—	4	4	no	—
62	C₂×C₃×C₈	C₂³	1	4	yes	—
63	C₃×C₄⋊C₄	—	2	1	no	—
64	C₃×C₄⋊C₄	C₂³	1	2	yes	—
65	C₃×C₂²⋊C₄	—	2	2	no	—
66	C₃×C₂²⋊C₄	—	2	2	no	—
67	C₃×C₂²⋊C₄	—	2	2	no	—
68	C₃×C₂²⋊C₄	—	2	2	no	—
69	C₃×C₂²⋊C₄	—	2	2	no	—
70	C₃×C₂²⋊C₄	—	2	2	no	—
71	C₃×C₄²	D₄	1	1	yes	characteristic
72	C₃×C₄²	D₄	1	1	yes	characteristic
73	C₃×C₄²	C₂³	1	1	yes	characteristic
74	C₃×C₄²	C₂³	1	1	yes	characteristic
75	C₂×#41	—	2	1	no	—
76	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
77	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
78	C₂×SD₁₆	C₂²×C₃	1	4	yes	complemented
79	C₂×D₈	C₂²×C₃	1	4	yes	—
80	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
81	#127	C₂²×C₃	1	1	yes	characteristic, complemented
82	#124	—	2	1	no	—
83	#124	—	2	1	no	—
84	#124	—	2	1	no	—
85	#124	C₂²×C₃	1	2	yes	—
86	#124	C₂²×C₃	1	2	yes	—
87	#121	—	2	1	no	—
88	#121	C₂²×C₃	1	2	yes	—
89	C₄×D₄	—	2	1	no	—
90	C₄×D₄	C₂²×C₃	1	2	yes	complemented
91	C₂×C₂²⋊C₄	—	2	1	no	—
92	C₂×C₂²⋊C₄	—	2	1	no	—
93	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
94	C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
95	#103	C₂²×C₃	1	4	yes	complemented
96	#102	C₂²×C₃	1	4	yes	—
97	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
98	#97	C₂²×C₃	1	2	yes	complemented
99	C₂³×C₃	—	2	2	no	—
100	C₂³×C₃	—	2	2	no	—
101	C₂³×C₃	—	4	2	no	—
102	C₂³×C₃	—	4	2	no	—
103	C₂³×C₃	—	4	2	no	—
104	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
105	C₃×Q₈	—	2	4	no	—
106	C₃×Q₈	—	4	2	no	—
107	C₃×Q₈	—	4	2	no	—
108	C₃×D₄	—	2	4	no	—
109	C₃×D₄	—	2	4	no	—
110	C₃×D₄	—	4	2	no	—
111	C₃×D₄	—	4	2	no	—
112	C₃×D₄	—	4	2	no	—
113	C₃×D₄	—	4	4	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	4	1	no	—
119	C₂×C₃×C₄	—	4	4	no	—
120	C₂×C₃×C₄	—	4	2	no	—
121	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
127	C₃×C₈	—	2	4	no	—
128	C₂⁴	—	2	1	no	—
129	#41	—	4	4	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	—	4	2	no	—
133	C₂×Q₈	C₂³×C₃	1	2	yes	—
134	C₂×D₄	—	2	2	no	—
135	C₂×D₄	—	2	1	no	—
136	C₂×D₄	—	2	1	no	—
137	C₂×D₄	—	2	1	no	—
138	C₂×D₄	—	4	2	no	—
139	C₂×D₄	C₂³×C₃	1	2	yes	—
140	C₂×D₄	C₂³×C₃	1	2	yes	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
144	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
145	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
146	SD₁₆	—	4	4	no	—
147	D₈	—	4	4	no	—
148	C₂×C₈	C₂³×C₃	1	4	yes	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	C₂³×C₃	1	2	yes	—
151	C₂²⋊C₄	—	2	2	no	—
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₂²⋊C₄	—	2	2	no	—
155	C₂²⋊C₄	—	2	2	no	—
156	C₂²⋊C₄	—	2	2	no	—
157	C₄²	C₃×D₄	1	1	yes	characteristic
158	C₄²	C₃×D₄	1	1	yes	characteristic
159	C₄²	C₂³×C₃	1	1	yes	characteristic
160	C₄²	C₂³×C₃	1	1	yes	characteristic
161	C₂²×C₃	—	2	1	no	—
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	—	4	2	no	—
164	C₂²×C₃	—	4	2	no	—
165	C₂²×C₃	—	4	2	no	—
166	C₂²×C₃	—	4	2	no	—
167	C₂²×C₃	—	4	2	no	—
168	C₂²×C₃	—	4	2	no	—
169	C₂²×C₃	—	8	4	no	—
170	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
171	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	—	2	1	no	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	4	2	no	—
178	C₃×C₄	—	4	2	no	—
179	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	4	2	no	—
183	C₂³	—	4	2	no	—
184	C₂³	—	4	2	no	—
185	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
186	Q₈	—	2	4	no	—
187	Q₈	—	4	2	no	—
188	Q₈	—	4	2	no	—
189	D₄	—	2	4	no	—
190	D₄	—	2	4	no	—
191	D₄	—	4	2	no	—
192	D₄	—	4	2	no	—
193	D₄	—	4	2	no	—
194	D₄	—	4	4	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	—	4	1	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
203	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
204	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
205	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
206	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
207	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
208	C₈	—	2	4	no	—
209	C₂×C₃	—	2	2	no	—
210	C₂×C₃	—	8	2	no	—
211	C₂×C₃	—	8	2	no	—
212	C₂×C₃	#535	1	1	yes	characteristic
213	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
214	C₂×C₃	#575	1	1	yes	characteristic
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	8	4	no	—
224	C₂²	C₃×AGL(1,ℤ/8)	1	2	yes	—
225	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	C₃×H(5,𝔽₂)	1	2	yes	—
234	C₃	#3213	1	1	yes	characteristic, direct product factor, Sylow
235	C₂	—	2	2	no	—
236	C₂	—	8	2	no	—
237	C₂	—	8	2	no	—
238	C₂	C₃×#535	1	1	yes	characteristic
239	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
240	C₂	C₃×#575	1	1	yes	characteristic

	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×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#532	C₂	semidirect
2. C₂×C₃×#124	C₂	semidirect
3. C₃×#496	C₂	semidirect
4. C₃×#489	C₂	semidirect
5. C₃×#463	C₂	semidirect
6. C₃×#448	C₂	semidirect
7. C₃×#447	C₂	semidirect
8. C₃×#432	C₂	semidirect
9. #3213	C₃	direct (N × H)
13. C₂×C₃×SD₁₆	C₂²	semidirect
15. C₃×C₄⋊Q₈	C₂²	semidirect
16. C₃×#127	C₂²	semidirect
25. C₃×C₄×D₄	C₂²	semidirect
29. C₃×C₄⋊C₈	C₂²	semidirect
30. C₃×#103	C₂²	semidirect
33. C₃×#97	C₂²	semidirect
34. #532	C₂×C₃	semidirect
35. C₂×#124	C₂×C₃	semidirect
36. #496	C₂×C₃	semidirect
37. #489	C₂×C₃	semidirect
38. #463	C₂×C₃	semidirect
39. #448	C₂×C₃	semidirect
40. #447	C₂×C₃	semidirect
41. #432	C₂×C₃	semidirect
78. C₂×SD₁₆	C₂²×C₃	semidirect
80. C₄⋊Q₈	C₂²×C₃	semidirect
81. #127	C₂²×C₃	semidirect
90. C₄×D₄	C₂²×C₃	semidirect
94. C₄⋊C₈	C₂²×C₃	semidirect
95. #103	C₂²×C₃	semidirect
98. #97	C₂²×C₃	semidirect
234. C₃	#3213	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×#532	2
C₂×C₃×#124	2
C₃×#496	2
C₃×#489	2
C₃×#463	2
C₃×#448	2
C₃×#447	2
C₃×#432	2
#3213	3

Subgroups — C3×#3213 #85077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3213 #85077