Subgroups — C₃×#3381 #85245

Order 384.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#554	C₂	1	1	yes	maximal, characteristic
3	C₃×#499	C₂	1	2	yes	maximal, complemented
4	C₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#479	C₂	1	2	yes	maximal, complemented
6	C₃×#478	C₂	1	2	yes	maximal
7	C₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#474	C₂	1	1	yes	maximal, characteristic
9	C₃×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic
11	C₃×C₄×SD₁₆	C₂	1	2	yes	maximal
12	#3381	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊Q₈	—	2	1	no	—
15	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃×#125	—	2	2	no	—
18	C₃×#125	C₂²	1	2	yes	—
19	C₃×#125	C₂²	1	2	yes	—
20	C₃×C₂²⋊Q₈	—	2	1	no	—
21	C₃×C₂²⋊Q₈	—	2	2	no	—
22	C₃×C₄×Q₈	—	2	1	no	—
23	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
24	C₃×C₄×Q₈	C₂²	1	2	yes	—
25	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
26	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
27	C₃×C₄×D₄	C₂²	1	2	yes	—
28	C₂×C₃×C₄⋊C₄	—	2	1	no	—
29	C₃×#107	C₂²	1	2	yes	—
30	C₃×#107	C₂²	1	2	yes	—
31	C₃×#106	C₂²	1	2	yes	—
32	C₃×#106	C₂²	1	1	yes	characteristic
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	2	yes	—
37	C₃×#103	C₂²	1	1	yes	characteristic
38	C₃×#102	C₂²	1	2	yes	—
39	C₃×#102	C₂²	1	1	yes	characteristic
40	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
41	C₃×C₄×C₈	C₂²	1	2	yes	—
42	#557	C₂×C₃	1	1	yes	characteristic, complemented
43	#554	C₂×C₃	1	1	yes	characteristic
44	#499	C₂×C₃	1	2	yes	complemented
45	#498	C₂×C₃	1	1	yes	characteristic, complemented
46	#479	C₂×C₃	1	2	yes	complemented
47	#478	C₂×C₃	1	2	yes	—
48	#477	C₂×C₃	1	1	yes	characteristic, complemented
49	#474	C₂×C₃	1	1	yes	characteristic
50	C₈×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
51	C₄×SD₁₆	C₂×C₃	1	1	yes	characteristic
52	C₄×SD₁₆	C₂×C₃	1	2	yes	—
53	C₂×C₃×Q₈	—	2	1	no	—
54	C₂×C₃×Q₈	—	2	1	no	—
55	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
56	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
57	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄	—	2	1	no	—
59	C₂²×C₃×C₄	—	2	2	no	—
60	C₃×SD₁₆	Q₈	1	4	yes	complemented
61	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₈	C₂³	1	2	yes	—
63	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
64	C₃×C₄⋊C₄	—	2	2	no	—
65	C₃×C₄⋊C₄	—	2	1	no	—
66	C₃×C₄⋊C₄	—	2	2	no	—
67	C₃×C₄⋊C₄	—	2	1	no	—
68	C₃×C₄⋊C₄	—	2	2	no	—
69	C₃×C₄⋊C₄	—	2	2	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×C₂²⋊C₄	—	2	1	no	—
79	C₃×C₂²⋊C₄	—	2	2	no	—
80	C₃×C₄²	—	2	2	no	—
81	C₃×C₄²	—	2	1	no	—
82	C₃×C₄²	C₂³	1	2	yes	—
83	C₃×C₄²	C₂³	1	1	yes	characteristic
84	C₂×SD₁₆	C₂²×C₃	1	1	yes	characteristic
85	C₄⋊Q₈	—	2	1	no	—
86	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
87	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
88	#125	—	2	2	no	—
89	#125	C₂²×C₃	1	2	yes	—
90	#125	C₂²×C₃	1	2	yes	—
91	C₂²⋊Q₈	—	2	1	no	—
92	C₂²⋊Q₈	—	2	2	no	—
93	C₄×Q₈	—	2	1	no	—
94	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
95	C₄×Q₈	C₂²×C₃	1	2	yes	—
96	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
97	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
98	C₄×D₄	C₂²×C₃	1	2	yes	—
99	C₂×C₄⋊C₄	—	2	1	no	—
100	#107	C₂²×C₃	1	2	yes	—
101	#107	C₂²×C₃	1	2	yes	—
102	#106	C₂²×C₃	1	2	yes	—
103	#106	C₂²×C₃	1	1	yes	characteristic
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	2	yes	—
108	#103	C₂²×C₃	1	1	yes	characteristic
109	#102	C₂²×C₃	1	2	yes	—
110	#102	C₂²×C₃	1	1	yes	characteristic
111	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
112	C₄×C₈	C₂²×C₃	1	2	yes	—
113	C₂³×C₃	—	2	1	no	—
114	C₃×Q₈	—	2	2	no	—
115	C₃×Q₈	—	2	1	no	—
116	C₃×Q₈	—	4	2	no	—
117	C₃×Q₈	—	8	1	no	—
118	C₃×Q₈	C₂×Q₈	1	2	yes	—
119	C₃×D₄	—	2	1	no	—
120	C₃×D₄	C₂×Q₈	1	2	yes	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	4	2	no	—
127	C₂×C₃×C₄	—	8	2	no	—
128	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
129	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
130	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
133	C₃×C₈	—	2	1	no	—
134	C₃×C₈	—	2	2	no	—
135	C₃×C₈	C₂×Q₈	1	2	yes	—
136	C₂×Q₈	—	2	1	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×D₄	C₂³×C₃	1	1	yes	characteristic
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	2	2	no	—
143	SD₁₆	C₃×Q₈	1	4	yes	complemented
144	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
145	C₂×C₈	C₂³×C₃	1	2	yes	—
146	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	1	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
156	C₄⋊C₄	C₃×D₄	1	2	yes	—
157	C₄⋊C₄	C₂³×C₃	1	2	yes	—
158	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
159	C₄⋊C₄	C₂³×C₃	1	2	yes	—
160	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
161	C₂²⋊C₄	—	2	1	no	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	1	no	—
165	C₄²	C₂³×C₃	1	2	yes	—
166	C₄²	C₂³×C₃	1	1	yes	characteristic
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	—	4	1	no	—
169	C₂²×C₃	—	4	1	no	—
170	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	—	2	2	no	—
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	—	2	1	no	—
175	C₃×C₄	—	4	1	no	—
176	C₃×C₄	—	4	2	no	—
177	C₃×C₄	—	4	1	no	—
178	C₃×C₄	—	4	2	no	—
179	C₃×C₄	#135	1	2	yes	—
180	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
181	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
182	C₂³	—	2	1	no	—
183	Q₈	—	2	2	no	—
184	Q₈	—	2	1	no	—
185	Q₈	—	4	2	no	—
186	Q₈	—	8	1	no	—
187	Q₈	C₂×C₃×Q₈	1	2	yes	—
188	D₄	—	2	1	no	—
189	D₄	C₂×C₃×Q₈	1	2	yes	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
198	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
199	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
200	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
201	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	2	no	—
204	C₈	C₂×C₃×Q₈	1	2	yes	—
205	C₂×C₃	—	4	2	no	—
206	C₂×C₃	D₄×Q₈	1	1	yes	characteristic
207	C₂×C₃	C₂×#135	1	1	yes	characteristic
208	C₂×C₃	#577	1	1	yes	characteristic
209	C₂²	—	2	2	no	—
210	C₂²	—	4	1	no	—
211	C₂²	—	4	1	no	—
212	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	—	4	1	no	—
218	C₄	—	4	2	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	1	no	—
221	C₄	C₃×#135	1	2	yes	—
222	C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
223	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
224	C₃	#3381	1	1	yes	characteristic, direct product factor, Sylow
225	C₂	—	4	2	no	—
226	C₂	C₃×D₄×Q₈	1	1	yes	characteristic
227	C₂	C₂×C₃×#135	1	1	yes	characteristic
228	C₂	C₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₂×C₃×D₄	C₂³
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
1. C₃×#557	C₂	semidirect
3. C₃×#499	C₂	semidirect
4. C₃×#498	C₂	semidirect
5. C₃×#479	C₂	semidirect
7. C₃×#477	C₂	semidirect
9. C₃×C₈×Q₈	C₂	semidirect
12. #3381	C₃	direct (N × H)
42. #557	C₂×C₃	semidirect
44. #499	C₂×C₃	semidirect
45. #498	C₂×C₃	semidirect
46. #479	C₂×C₃	semidirect
48. #477	C₂×C₃	semidirect
50. C₈×Q₈	C₂×C₃	semidirect
60. C₃×SD₁₆	Q₈	semidirect
143. SD₁₆	C₃×Q₈	semidirect
224. C₃	#3381	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×#557	2
C₃×#554	2
C₃×#499	2
C₃×#498	2
C₃×#479	2
C₃×#478	2
C₃×#477	2
C₃×#474	2
C₃×C₈×Q₈	2
C₃×C₄×SD₁₆	2
#3381	3

Subgroups — C3×#3381 #85245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3381 #85245