Subgroups — C₃×#3215 #85079

Order 384.

400 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#124	C₂	1	1	yes	maximal, characteristic
3	C₃×#497	C₂	1	2	yes	maximal, complemented
4	C₃×#489	C₂	1	2	yes	maximal, complemented
5	C₃×#464	C₂	1	4	yes	maximal, complemented
6	C₃×#451	C₂	1	2	yes	maximal, complemented
7	C₃×#450	C₂	1	2	yes	maximal
8	C₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	#3215	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
12	C₂×C₃×Q₁₆	C₂²	1	4	yes	complemented
13	C₂×C₃×SD₁₆	C₂²	1	4	yes	—
14	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
16	C₃×#125	—	2	1	no	—
17	C₃×#124	—	2	1	no	—
18	C₃×#124	—	2	1	no	—
19	C₃×#124	C₂²	1	2	yes	—
20	C₃×#124	C₂²	1	2	yes	—
21	C₃×C₂²⋊Q₈	—	2	1	no	—
22	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₃×C₄×Q₈	—	2	1	no	—
24	C₃×C₄×Q₈	C₂²	1	2	yes	complemented
25	C₃×#117	—	2	1	no	—
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	#533	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂×#124	C₂×C₃	1	1	yes	characteristic
36	#497	C₂×C₃	1	2	yes	complemented
37	#489	C₂×C₃	1	2	yes	complemented
38	#464	C₂×C₃	1	4	yes	complemented
39	#451	C₂×C₃	1	2	yes	complemented
40	#450	C₂×C₃	1	2	yes	—
41	#432	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂⁴×C₃	—	2	1	no	—
43	C₂×C₃×Q₈	—	2	1	no	—
44	C₂×C₃×Q₈	—	2	1	no	—
45	C₂×C₃×Q₈	—	4	2	no	—
46	C₂×C₃×Q₈	C₂³	1	2	yes	—
47	C₂×C₃×Q₈	C₂³	1	2	yes	—
48	C₂×C₃×D₄	—	2	1	no	—
49	C₂×C₃×D₄	—	4	2	no	—
50	C₂×C₃×D₄	C₂³	1	2	yes	—
51	C₂²×C₃×C₄	—	2	1	no	—
52	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
53	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
55	C₃×Q₁₆	—	4	4	no	—
56	C₃×SD₁₆	—	4	4	no	—
57	C₂×C₃×C₈	C₂³	1	4	yes	—
58	C₃×C₄⋊C₄	—	2	1	no	—
59	C₃×C₄⋊C₄	—	2	1	no	—
60	C₃×C₄⋊C₄	—	2	1	no	—
61	C₃×C₄⋊C₄	—	2	2	no	—
62	C₃×C₄⋊C₄	—	2	2	no	—
63	C₃×C₄⋊C₄	C₂³	1	2	yes	—
64	C₃×C₂²⋊C₄	—	2	2	no	—
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×C₄²	—	2	2	no	—
70	C₃×C₄²	D₄	1	1	yes	characteristic
71	C₃×C₄²	D₄	1	1	yes	characteristic
72	C₃×C₄²	C₂³	1	1	yes	characteristic
73	C₃×C₄²	C₂³	1	1	yes	characteristic
74	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
75	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
76	C₂×Q₁₆	C₂²×C₃	1	4	yes	complemented
77	C₂×SD₁₆	C₂²×C₃	1	4	yes	—
78	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
79	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
80	#125	—	2	1	no	—
81	#124	—	2	1	no	—
82	#124	—	2	1	no	—
83	#124	C₂²×C₃	1	2	yes	—
84	#124	C₂²×C₃	1	2	yes	—
85	C₂²⋊Q₈	—	2	1	no	—
86	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
87	C₄×Q₈	—	2	1	no	—
88	C₄×Q₈	C₂²×C₃	1	2	yes	complemented
89	#117	—	2	1	no	—
90	C₂×C₂²⋊C₄	—	2	1	no	—
91	C₂×C₂²⋊C₄	—	2	1	no	—
92	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
93	C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
94	#103	C₂²×C₃	1	4	yes	complemented
95	#102	C₂²×C₃	1	4	yes	—
96	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
97	#97	C₂²×C₃	1	2	yes	complemented
98	C₂³×C₃	—	2	2	no	—
99	C₂³×C₃	—	4	2	no	—
100	C₂³×C₃	—	4	2	no	—
101	C₂³×C₃	—	4	2	no	—
102	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
103	C₃×Q₈	—	2	4	no	—
104	C₃×Q₈	—	2	4	no	—
105	C₃×Q₈	—	4	2	no	—
106	C₃×Q₈	—	4	2	no	—
107	C₃×D₄	—	2	4	no	—
108	C₃×D₄	—	4	2	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	4	1	no	—
115	C₂×C₃×C₄	—	4	2	no	—
116	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
122	C₃×C₈	—	2	4	no	—
123	C₂⁴	—	2	1	no	—
124	C₂×Q₈	—	2	1	no	—
125	C₂×Q₈	—	2	1	no	—
126	C₂×Q₈	—	4	2	no	—
127	C₂×Q₈	C₂³×C₃	1	2	yes	—
128	C₂×Q₈	C₂³×C₃	1	2	yes	—
129	C₂×D₄	—	2	1	no	—
130	C₂×D₄	—	4	2	no	—
131	C₂×D₄	C₂³×C₃	1	2	yes	—
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
134	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
136	Q₁₆	—	4	4	no	—
137	SD₁₆	—	4	4	no	—
138	C₂×C₈	C₂³×C₃	1	4	yes	—
139	C₄⋊C₄	—	2	1	no	—
140	C₄⋊C₄	—	2	1	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	C₂³×C₃	1	2	yes	—
145	C₂²⋊C₄	—	2	2	no	—
146	C₂²⋊C₄	—	2	2	no	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₂²⋊C₄	—	2	2	no	—
149	C₂²⋊C₄	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	C₃×D₄	1	1	yes	characteristic
152	C₄²	C₃×D₄	1	1	yes	characteristic
153	C₄²	C₂³×C₃	1	1	yes	characteristic
154	C₄²	C₂³×C₃	1	1	yes	characteristic
155	C₂²×C₃	—	2	1	no	—
156	C₂²×C₃	—	2	1	no	—
157	C₂²×C₃	—	4	2	no	—
158	C₂²×C₃	—	4	2	no	—
159	C₂²×C₃	—	4	2	no	—
160	C₂²×C₃	—	8	4	no	—
161	C₂²×C₃	#137	1	2	yes	—
162	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
163	C₃×C₄	—	2	1	no	—
164	C₃×C₄	—	2	1	no	—
165	C₃×C₄	—	2	1	no	—
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	—	2	1	no	—
168	C₃×C₄	—	4	2	no	—
169	C₃×C₄	—	4	2	no	—
170	C₃×C₄	—	4	2	no	—
171	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
172	C₂³	—	2	2	no	—
173	C₂³	—	4	2	no	—
174	C₂³	—	4	2	no	—
175	C₂³	—	4	2	no	—
176	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
177	Q₈	—	2	4	no	—
178	Q₈	—	2	4	no	—
179	Q₈	—	4	2	no	—
180	Q₈	—	4	2	no	—
181	D₄	—	2	4	no	—
182	D₄	—	4	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
191	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
192	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
193	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
194	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
195	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
196	C₈	—	2	4	no	—
197	C₂×C₃	—	2	2	no	—
198	C₂×C₃	—	8	2	no	—
199	C₂×C₃	#535	1	1	yes	characteristic
200	C₂×C₃	C₂×#137	1	1	yes	characteristic
201	C₂×C₃	#575	1	1	yes	characteristic
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	8	4	no	—
208	C₂²	C₃×#137	1	2	yes	—
209	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	2	no	—
217	C₄	—	4	2	no	—
218	C₄	C₃×H(5,𝔽₂)	1	2	yes	—
219	C₃	#3215	1	1	yes	characteristic, direct product factor, Sylow
220	C₂	—	2	2	no	—
221	C₂	—	8	2	no	—
222	C₂	C₃×#535	1	1	yes	characteristic
223	C₂	C₂×C₃×#137	1	1	yes	characteristic
224	C₂	C₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	C₂²
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×#533	C₂	semidirect
3. C₃×#497	C₂	semidirect
4. C₃×#489	C₂	semidirect
5. C₃×#464	C₂	semidirect
6. C₃×#451	C₂	semidirect
8. C₃×#432	C₂	semidirect
9. #3215	C₃	direct (N × H)
12. C₂×C₃×Q₁₆	C₂²	semidirect
14. C₃×C₄⋊Q₈	C₂²	semidirect
15. C₃×C₄⋊Q₈	C₂²	semidirect
24. C₃×C₄×Q₈	C₂²	semidirect
29. C₃×C₄⋊C₈	C₂²	semidirect
30. C₃×#103	C₂²	semidirect
33. C₃×#97	C₂²	semidirect
34. #533	C₂×C₃	semidirect
36. #497	C₂×C₃	semidirect
37. #489	C₂×C₃	semidirect
38. #464	C₂×C₃	semidirect
39. #451	C₂×C₃	semidirect
41. #432	C₂×C₃	semidirect
76. C₂×Q₁₆	C₂²×C₃	semidirect
78. C₄⋊Q₈	C₂²×C₃	semidirect
79. C₄⋊Q₈	C₂²×C₃	semidirect
88. C₄×Q₈	C₂²×C₃	semidirect
93. C₄⋊C₈	C₂²×C₃	semidirect
94. #103	C₂²×C₃	semidirect
97. #97	C₂²×C₃	semidirect
219. C₃	#3215	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×#533	2
C₂×C₃×#124	2
C₃×#497	2
C₃×#489	2
C₃×#464	2
C₃×#451	2
C₃×#450	2
C₃×#432	2
#3215	3

Subgroups — C3×#3215 #85079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3215 #85079