Subgroups — C₃×#3340 #85204

Order 384.

352 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×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#555	C₂	1	1	yes	maximal, characteristic
3	C₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#485	C₂	1	2	yes	maximal, complemented
5	C₃×#482	C₂	1	2	yes	maximal, complemented
6	C₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#465	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#427	C₂	1	2	yes	maximal, complemented
12	#3340	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₈	C₂²	1	2	yes	—
15	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃×#126	—	2	2	no	—
18	C₃×#124	—	2	1	no	—
19	C₃×#124	—	2	2	no	—
20	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₃×#121	—	2	1	no	—
22	C₃×#121	C₂²	1	2	yes	—
23	C₃×C₄×Q₈	—	2	1	no	—
24	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
25	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
26	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
27	C₃×#117	—	2	1	no	—
28	C₃×#117	C₂²	1	2	yes	—
29	C₃×#117	C₂²	1	2	yes	—
30	C₃×#107	C₂²	1	2	yes	—
31	C₃×#107	C₂²	1	2	yes	—
32	C₃×#106	C₂²	1	1	yes	characteristic
33	C₃×#106	C₂²	1	2	yes	—
34	C₃×#106	C₂²	1	2	yes	—
35	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₃×#103	C₂²	1	1	yes	characteristic
37	C₃×#103	C₂²	1	2	yes	—
38	C₃×#102	C₂²	1	1	yes	characteristic
39	C₃×#102	C₂²	1	2	yes	—
40	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
42	#556	C₂×C₃	1	1	yes	characteristic, complemented
43	#555	C₂×C₃	1	1	yes	characteristic
44	#498	C₂×C₃	1	1	yes	characteristic, complemented
45	#485	C₂×C₃	1	2	yes	complemented
46	#482	C₂×C₃	1	2	yes	complemented
47	#477	C₂×C₃	1	1	yes	characteristic, complemented
48	#474	C₂×C₃	1	1	yes	characteristic, complemented
49	#465	C₂×C₃	1	2	yes	complemented
50	C₄×SD₁₆	C₂×C₃	1	1	yes	characteristic, complemented
51	C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
52	#427	C₂×C₃	1	2	yes	complemented
53	C₂×C₃×Q₈	—	2	1	no	—
54	C₂×C₃×Q₈	—	2	1	no	—
55	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
56	C₂×C₃×D₄	—	2	2	no	—
57	C₂×C₃×D₄	D₄	1	1	yes	characteristic
58	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
59	C₂²×C₃×C₄	—	2	1	no	—
60	C₂²×C₃×C₄	C₂³	1	2	yes	—
61	C₃×SD₁₆	—	2	2	no	—
62	C₂×C₃×C₈	—	2	4	no	—
63	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂³	1	2	yes	—
65	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
66	C₃×C₄⋊C₄	—	2	1	no	—
67	C₃×C₄⋊C₄	—	2	2	no	—
68	C₃×C₄⋊C₄	—	2	1	no	—
69	C₃×C₄⋊C₄	—	2	1	no	—
70	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
71	C₃×C₄⋊C₄	C₂³	1	2	yes	—
72	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
73	C₃×C₄⋊C₄	C₂³	1	2	yes	—
74	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
75	C₃×C₂²⋊C₄	—	2	2	no	—
76	C₃×C₂²⋊C₄	—	2	2	no	—
77	C₃×C₂²⋊C₄	—	2	2	no	—
78	C₃×C₂²⋊C₄	—	2	2	no	—
79	C₃×C₂²⋊C₄	—	2	1	no	—
80	C₃×C₂²⋊C₄	D₄	1	2	yes	—
81	C₃×C₄²	—	2	2	no	—
82	C₃×C₄²	—	2	2	no	—
83	C₃×C₄²	—	2	1	no	—
84	C₃×C₄²	C₂³	1	1	yes	characteristic
85	C₂×SD₁₆	C₂²×C₃	1	1	yes	characteristic
86	C₂²×C₈	C₂²×C₃	1	2	yes	—
87	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
88	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
89	#126	—	2	2	no	—
90	#124	—	2	1	no	—
91	#124	—	2	2	no	—
92	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
93	#121	—	2	1	no	—
94	#121	C₂²×C₃	1	2	yes	—
95	C₄×Q₈	—	2	1	no	—
96	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
97	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
98	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
99	#117	—	2	1	no	—
100	#117	C₂²×C₃	1	2	yes	—
101	#117	C₂²×C₃	1	2	yes	—
102	#107	C₂²×C₃	1	2	yes	—
103	#107	C₂²×C₃	1	2	yes	—
104	#106	C₂²×C₃	1	1	yes	characteristic
105	#106	C₂²×C₃	1	2	yes	—
106	#106	C₂²×C₃	1	2	yes	—
107	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
108	#103	C₂²×C₃	1	1	yes	characteristic
109	#103	C₂²×C₃	1	2	yes	—
110	#102	C₂²×C₃	1	1	yes	characteristic
111	#102	C₂²×C₃	1	2	yes	—
112	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
113	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
114	C₂³×C₃	—	2	1	no	—
115	C₂³×C₃	C₂×D₄	1	2	yes	—
116	C₃×Q₈	—	2	1	no	—
117	C₃×Q₈	—	2	1	no	—
118	C₃×Q₈	—	4	2	no	—
119	C₃×Q₈	—	8	1	no	—
120	C₃×D₄	—	2	2	no	—
121	C₃×D₄	—	2	1	no	—
122	C₃×D₄	—	2	1	no	—
123	C₃×D₄	—	8	2	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	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	4	2	no	—
132	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
133	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
136	C₃×C₈	—	2	2	no	—
137	C₃×C₈	—	2	1	no	—
138	C₃×C₈	#41	1	2	yes	—
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	—	2	1	no	—
141	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	C₃×D₄	1	1	yes	characteristic
144	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	C₂³×C₃	1	2	yes	—
147	SD₁₆	—	2	2	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
150	C₂×C₈	C₂³×C₃	1	2	yes	—
151	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	1	no	—
155	C₄⋊C₄	—	2	1	no	—
156	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
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	2	no	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	1	no	—
166	C₂²⋊C₄	C₃×D₄	1	2	yes	—
167	C₄²	—	2	2	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	1	no	—
170	C₄²	C₂³×C₃	1	1	yes	characteristic
171	C₂²×C₃	—	2	2	no	—
172	C₂²×C₃	—	2	2	no	—
173	C₂²×C₃	—	2	2	no	—
174	C₂²×C₃	—	4	1	no	—
175	C₂²×C₃	—	4	1	no	—
176	C₂²×C₃	—	4	1	no	—
177	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
178	C₃×C₄	—	2	2	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	4	1	no	—
181	C₃×C₄	—	4	1	no	—
182	C₃×C₄	—	4	2	no	—
183	C₃×C₄	—	4	1	no	—
184	C₃×C₄	—	4	2	no	—
185	C₃×C₄	#135	1	2	yes	—
186	C₃×C₄	C₂×#41	1	1	yes	characteristic
187	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
188	C₂³	—	2	1	no	—
189	C₂³	C₂×C₃×D₄	1	2	yes	—
190	Q₈	—	2	1	no	—
191	Q₈	—	2	1	no	—
192	Q₈	—	4	2	no	—
193	Q₈	—	8	1	no	—
194	D₄	—	2	2	no	—
195	D₄	—	2	1	no	—
196	D₄	—	2	1	no	—
197	D₄	—	8	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
207	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
208	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
209	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	2	1	no	—
212	C₈	C₃×#41	1	2	yes	—
213	C₂×C₃	—	4	2	no	—
214	C₂×C₃	—	8	1	no	—
215	C₂×C₃	#547	1	1	yes	characteristic
216	C₂×C₃	C₂×#135	1	1	yes	characteristic
217	C₂×C₃	#577	1	1	yes	characteristic
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	4	1	no	—
224	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	2	2	no	—
227	C₄	—	4	1	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	1	no	—
232	C₄	C₃×#135	1	2	yes	—
233	C₄	C₂×C₃×#41	1	1	yes	characteristic
234	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
235	C₃	#3340	1	1	yes	characteristic, direct product factor, Sylow
236	C₂	—	4	2	no	—
237	C₂	—	8	1	no	—
238	C₂	C₃×#547	1	1	yes	characteristic
239	C₂	C₂×C₃×#135	1	1	yes	characteristic
240	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	C₃×#555	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#556	C₂	semidirect
3. C₃×#498	C₂	semidirect
4. C₃×#485	C₂	semidirect
5. C₃×#482	C₂	semidirect
6. C₃×#477	C₂	semidirect
7. C₃×#474	C₂	semidirect
8. C₃×#465	C₂	semidirect
9. C₃×C₄×SD₁₆	C₂	semidirect
10. C₃×C₈×D₄	C₂	semidirect
11. C₃×#427	C₂	semidirect
12. #3340	C₃	direct (N × H)
42. #556	C₂×C₃	semidirect
44. #498	C₂×C₃	semidirect
45. #485	C₂×C₃	semidirect
46. #482	C₂×C₃	semidirect
47. #477	C₂×C₃	semidirect
48. #474	C₂×C₃	semidirect
49. #465	C₂×C₃	semidirect
50. C₄×SD₁₆	C₂×C₃	semidirect
51. C₈×D₄	C₂×C₃	semidirect
52. #427	C₂×C₃	semidirect
235. C₃	#3340	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×#556	2
C₃×#555	2
C₃×#498	2
C₃×#485	2
C₃×#482	2
C₃×#477	2
C₃×#474	2
C₃×#465	2
C₃×C₄×SD₁₆	2
C₃×C₈×D₄	2
C₃×#427	2
#3340	3

Subgroups — C3×#3340 #85204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3340 #85204