Subgroups — C₃×#3387 #85251

Order 384.

326 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈²	C₂	1	1	yes	maximal, characteristic
2	C₃×#557	C₂	1	1	yes	maximal, characteristic
3	C₃×#501	C₂	1	2	yes	maximal
4	C₃×#498	C₂	1	1	yes	maximal, characteristic
5	C₃×#479	C₂	1	2	yes	maximal
6	C₃×#477	C₂	1	1	yes	maximal, characteristic
7	C₃×#475	C₂	1	2	yes	maximal
8	C₃×#475	C₂	1	1	yes	maximal, characteristic
9	C₃×#446	C₂	1	1	yes	maximal, characteristic
10	C₃×#441	C₂	1	2	yes	maximal
11	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
12	#3387	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊Q₈	—	2	1	no	—
15	C₃×C₄⋊Q₈	—	2	1	no	—
16	C₃×C₄⋊Q₈	—	2	2	no	—
17	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₃×#125	—	2	2	no	—
21	C₃×#125	C₂²	1	2	yes	—
22	C₃×C₄×Q₈	—	2	1	no	—
23	C₃×C₄×Q₈	—	2	1	no	—
24	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
25	C₃×C₄×Q₈	C₂²	1	2	yes	—
26	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
27	C₃×C₄×Q₈	C₂²	1	2	yes	—
28	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
29	C₃×#107	C₂²	1	1	yes	characteristic
30	C₃×#107	C₂²	1	2	yes	—
31	C₃×#106	C₂²	1	2	yes	—
32	C₃×#106	C₂²	1	2	yes	—
33	C₃×#106	C₂²	1	2	yes	—
34	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₃×C₄⋊C₈	C₂²	1	2	yes	—
36	C₃×#103	C₂²	1	1	yes	characteristic
37	C₃×#103	C₂²	1	1	yes	characteristic
38	C₃×#103	C₂²	1	2	yes	—
39	C₃×#103	C₂²	1	2	yes	—
40	C₃×#97	C₂²	1	2	yes	—
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
42	Q₈²	C₂×C₃	1	1	yes	characteristic
43	#557	C₂×C₃	1	1	yes	characteristic
44	#501	C₂×C₃	1	2	yes	—
45	#498	C₂×C₃	1	1	yes	characteristic
46	#479	C₂×C₃	1	2	yes	—
47	#477	C₂×C₃	1	1	yes	characteristic
48	#475	C₂×C₃	1	2	yes	—
49	#475	C₂×C₃	1	1	yes	characteristic
50	#446	C₂×C₃	1	1	yes	characteristic
51	#441	C₂×C₃	1	2	yes	—
52	C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic
53	C₂×C₃×Q₈	—	2	1	no	—
54	C₂×C₃×Q₈	—	2	2	no	—
55	C₂×C₃×Q₈	—	2	1	no	—
56	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
57	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
58	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
59	C₃×Q₁₆	Q₈	1	4	yes	complemented
60	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
61	C₂×C₃×C₈	C₂³	1	2	yes	—
62	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
63	C₃×C₄⋊C₄	—	2	2	no	—
64	C₃×C₄⋊C₄	—	2	2	no	—
65	C₃×C₄⋊C₄	—	2	1	no	—
66	C₃×C₄⋊C₄	—	2	1	no	—
67	C₃×C₄⋊C₄	—	2	2	no	—
68	C₃×C₄⋊C₄	—	2	2	no	—
69	C₃×C₄⋊C₄	—	2	1	no	—
70	C₃×C₄⋊C₄	—	2	1	no	—
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
73	C₃×C₄⋊C₄	D₄	1	2	yes	—
74	C₃×C₄⋊C₄	C₂³	1	2	yes	—
75	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
76	C₃×C₄⋊C₄	C₂³	1	2	yes	—
77	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
78	C₃×C₄²	—	2	2	no	—
79	C₃×C₄²	—	2	1	no	—
80	C₃×C₄²	—	2	2	no	—
81	C₃×C₄²	—	2	1	no	—
82	C₃×C₄²	C₂³	1	1	yes	characteristic
83	C₃×C₄²	C₂³	1	2	yes	—
84	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
85	C₄⋊Q₈	—	2	1	no	—
86	C₄⋊Q₈	—	2	1	no	—
87	C₄⋊Q₈	—	2	2	no	—
88	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
89	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
90	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
91	#125	—	2	2	no	—
92	#125	C₂²×C₃	1	2	yes	—
93	C₄×Q₈	—	2	1	no	—
94	C₄×Q₈	—	2	1	no	—
95	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
96	C₄×Q₈	C₂²×C₃	1	2	yes	—
97	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
98	C₄×Q₈	C₂²×C₃	1	2	yes	—
99	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
100	#107	C₂²×C₃	1	1	yes	characteristic
101	#107	C₂²×C₃	1	2	yes	—
102	#106	C₂²×C₃	1	2	yes	—
103	#106	C₂²×C₃	1	2	yes	—
104	#106	C₂²×C₃	1	2	yes	—
105	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
106	C₄⋊C₈	C₂²×C₃	1	2	yes	—
107	#103	C₂²×C₃	1	1	yes	characteristic
108	#103	C₂²×C₃	1	1	yes	characteristic
109	#103	C₂²×C₃	1	2	yes	—
110	#103	C₂²×C₃	1	2	yes	—
111	#97	C₂²×C₃	1	2	yes	—
112	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
113	C₃×Q₈	—	2	2	no	—
114	C₃×Q₈	—	2	1	no	—
115	C₃×Q₈	—	2	1	no	—
116	C₃×Q₈	—	4	2	no	—
117	C₃×Q₈	—	8	2	no	—
118	C₃×Q₈	—	8	1	no	—
119	C₃×Q₈	C₂×Q₈	1	2	yes	—
120	C₃×Q₈	C₂×Q₈	1	2	yes	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
128	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
129	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
132	C₃×C₈	—	2	1	no	—
133	C₃×C₈	—	2	2	no	—
134	C₃×C₈	C₂×Q₈	1	2	yes	—
135	C₂×Q₈	—	2	1	no	—
136	C₂×Q₈	—	2	2	no	—
137	C₂×Q₈	—	2	1	no	—
138	C₂×Q₈	C₃×D₄	1	1	yes	characteristic
139	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
140	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
141	Q₁₆	C₃×Q₈	1	4	yes	complemented
142	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
143	C₂×C₈	C₂³×C₃	1	2	yes	—
144	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	1	no	—
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
155	C₄⋊C₄	C₃×D₄	1	2	yes	—
156	C₄⋊C₄	C₂³×C₃	1	2	yes	—
157	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
158	C₄⋊C₄	C₂³×C₃	1	2	yes	—
159	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
160	C₄²	—	2	2	no	—
161	C₄²	—	2	1	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	1	no	—
164	C₄²	C₂³×C₃	1	1	yes	characteristic
165	C₄²	C₂³×C₃	1	2	yes	—
166	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
167	C₃×C₄	—	2	2	no	—
168	C₃×C₄	—	2	2	no	—
169	C₃×C₄	—	2	2	no	—
170	C₃×C₄	—	2	2	no	—
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	4	2	no	—
173	C₃×C₄	—	4	1	no	—
174	C₃×C₄	—	4	2	no	—
175	C₃×C₄	—	4	1	no	—
176	C₃×C₄	#137	1	2	yes	—
177	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
178	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
179	Q₈	—	2	2	no	—
180	Q₈	—	2	1	no	—
181	Q₈	—	2	1	no	—
182	Q₈	—	4	2	no	—
183	Q₈	—	8	2	no	—
184	Q₈	—	8	1	no	—
185	Q₈	C₂×C₃×Q₈	1	2	yes	—
186	Q₈	C₂×C₃×Q₈	1	2	yes	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
194	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
195	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
196	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
197	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	2	2	no	—
200	C₈	C₂×C₃×Q₈	1	2	yes	—
201	C₂×C₃	D₄×Q₈	1	1	yes	characteristic
202	C₂×C₃	C₂×#137	1	1	yes	characteristic
203	C₂×C₃	#577	1	1	yes	characteristic
204	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	1	no	—
213	C₄	—	4	1	no	—
214	C₄	C₃×#137	1	2	yes	—
215	C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
216	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
217	C₃	#3387	1	1	yes	characteristic, direct product factor, Sylow
218	C₂	C₃×D₄×Q₈	1	1	yes	characteristic
219	C₂	C₂×C₃×#137	1	1	yes	characteristic
220	C₂	C₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
12. #3387	C₃	direct (N × H)
59. C₃×Q₁₆	Q₈	semidirect
141. Q₁₆	C₃×Q₈	semidirect
217. C₃	#3387	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×Q₈²	2
C₃×#557	2
C₃×#501	2
C₃×#498	2
C₃×#479	2
C₃×#477	2
C₃×#475	2
C₃×#446	2
C₃×#441	2
C₃×C₄×Q₁₆	2
#3387	3

Subgroups — C3×#3387 #85251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3387 #85251