Subgroups — C₃×#3334 #85198

Order 384.

420 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#547	C₂	1	2	yes	maximal, complemented
2	C₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#480	C₂	1	4	yes	maximal, complemented
4	C₃×#474	C₂	1	2	yes	maximal, complemented
5	C₃×#466	C₂	1	2	yes	maximal, complemented
6	C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×#107	C₂	1	2	yes	maximal, complemented
9	#3334	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#41	—	2	2	no	—
11	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₈	C₂²	1	2	yes	—
13	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₃×#123	—	2	4	no	—
15	C₃×C₂²⋊Q₈	—	2	2	no	—
16	C₃×#121	C₂²	1	4	yes	—
17	C₃×C₄×D₄	C₂²	1	2	yes	—
18	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
20	C₃×#107	C₂²	1	1	yes	characteristic
21	C₃×#107	C₂²	1	2	yes	—
22	C₃×#107	C₂²	1	4	yes	—
23	C₃×#107	C₂²	1	2	yes	—
24	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₃×#102	C₂²	1	2	yes	—
26	C₃×#102	C₂²	1	4	yes	—
27	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
29	#547	C₂×C₃	1	2	yes	complemented
30	#500	C₂×C₃	1	1	yes	characteristic, complemented
31	#480	C₂×C₃	1	4	yes	complemented
32	#474	C₂×C₃	1	2	yes	complemented
33	#466	C₂×C₃	1	2	yes	complemented
34	C₄×D₈	C₂×C₃	1	1	yes	characteristic, complemented
35	C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂×#107	C₂×C₃	1	2	yes	complemented
37	C₃×#41	—	4	8	no	—
38	C₂×C₃×Q₈	—	2	2	no	—
39	C₂×C₃×D₄	—	2	4	no	—
40	C₂×C₃×D₄	D₄	1	1	yes	characteristic
41	C₂×C₃×D₄	C₂³	1	2	yes	—
42	C₂²×C₃×C₄	—	2	4	no	—
43	C₂²×C₃×C₄	—	2	2	no	—
44	C₂²×C₃×C₄	C₂³	1	2	yes	—
45	C₃×D₈	—	2	2	no	—
46	C₂×C₃×C₈	—	2	4	no	—
47	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
48	C₂×C₃×C₈	C₂³	1	2	yes	—
49	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
50	C₃×C₄⋊C₄	—	2	4	no	—
51	C₃×C₄⋊C₄	—	2	2	no	—
52	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
53	C₃×C₄⋊C₄	C₂³	1	2	yes	—
54	C₃×C₄⋊C₄	C₂³	1	4	yes	—
55	C₃×C₂²⋊C₄	—	2	2	no	—
56	C₃×C₂²⋊C₄	—	2	4	no	—
57	C₃×C₂²⋊C₄	D₄	1	2	yes	—
58	C₃×C₄²	C₂³	1	1	yes	characteristic
59	C₂×#41	—	2	2	no	—
60	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
61	C₂²×C₈	C₂²×C₃	1	2	yes	—
62	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
63	#123	—	2	4	no	—
64	C₂²⋊Q₈	—	2	2	no	—
65	#121	C₂²×C₃	1	4	yes	—
66	C₄×D₄	C₂²×C₃	1	2	yes	—
67	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
69	#107	C₂²×C₃	1	1	yes	characteristic
70	#107	C₂²×C₃	1	2	yes	—
71	#107	C₂²×C₃	1	4	yes	—
72	#107	C₂²×C₃	1	2	yes	—
73	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
74	#102	C₂²×C₃	1	2	yes	—
75	#102	C₂²×C₃	1	4	yes	—
76	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
77	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
78	C₂³×C₃	—	2	2	no	—
79	C₂³×C₃	C₂×D₄	1	2	yes	—
80	C₃×Q₈	—	4	4	no	—
81	C₃×D₄	—	2	2	no	—
82	C₃×D₄	—	2	2	no	—
83	C₃×D₄	—	4	8	no	—
84	C₃×D₄	D₈	1	4	yes	complemented
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	2	2	no	—
87	C₂×C₃×C₄	—	2	2	no	—
88	C₂×C₃×C₄	—	2	4	no	—
89	C₂×C₃×C₄	—	4	8	no	—
90	C₂×C₃×C₄	—	4	4	no	—
91	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
93	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
95	C₃×C₈	—	2	2	no	—
96	C₃×C₈	—	2	1	no	—
97	C₃×C₈	#41	1	2	yes	complemented
98	#41	—	4	8	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×D₄	—	2	4	no	—
101	C₂×D₄	C₃×D₄	1	1	yes	characteristic
102	C₂×D₄	C₂³×C₃	1	2	yes	—
103	C₂²×C₄	—	2	4	no	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	C₂³×C₃	1	2	yes	—
106	D₈	—	2	2	no	—
107	C₂×C₈	—	2	4	no	—
108	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
109	C₂×C₈	C₂³×C₃	1	2	yes	—
110	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
111	C₄⋊C₄	—	2	4	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
114	C₄⋊C₄	C₂³×C₃	1	2	yes	—
115	C₄⋊C₄	C₂³×C₃	1	4	yes	—
116	C₂²⋊C₄	—	2	2	no	—
117	C₂²⋊C₄	—	2	4	no	—
118	C₂²⋊C₄	C₃×D₄	1	2	yes	—
119	C₄²	C₂³×C₃	1	1	yes	characteristic
120	C₂²×C₃	—	2	2	no	—
121	C₂²×C₃	—	2	2	no	—
122	C₂²×C₃	—	4	2	no	—
123	C₂²×C₃	—	4	2	no	—
124	C₂²×C₃	—	4	2	no	—
125	C₂²×C₃	C₂×D₈	1	4	yes	—
126	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
127	C₃×C₄	—	2	2	no	—
128	C₃×C₄	—	2	1	no	—
129	C₃×C₄	—	4	2	no	—
130	C₃×C₄	—	4	4	no	—
131	C₃×C₄	C₂×D₈	1	2	yes	—
132	C₃×C₄	C₂×#41	1	1	yes	characteristic
133	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
134	C₂³	—	2	2	no	—
135	C₂³	C₂×C₃×D₄	1	2	yes	—
136	Q₈	—	4	4	no	—
137	D₄	—	2	2	no	—
138	D₄	—	2	2	no	—
139	D₄	—	4	8	no	—
140	D₄	C₃×D₈	1	4	yes	complemented
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	8	no	—
146	C₂×C₄	—	4	4	no	—
147	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
148	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
149	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	1	no	—
153	C₈	C₃×#41	1	2	yes	complemented
154	C₂×C₃	—	2	4	no	—
155	C₂×C₃	—	8	2	no	—
156	C₂×C₃	#547	1	1	yes	characteristic
157	C₂×C₃	C₂²×D₈	1	1	yes	characteristic
158	C₂×C₃	#578	1	1	yes	characteristic
159	C₂²	—	2	2	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	4	2	no	—
162	C₂²	—	4	2	no	—
163	C₂²	—	4	2	no	—
164	C₂²	C₂×C₃×D₈	1	4	yes	—
165	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	2	no	—
168	C₄	—	4	2	no	—
169	C₄	—	4	4	no	—
170	C₄	C₂×C₃×D₈	1	2	yes	—
171	C₄	C₂×C₃×#41	1	1	yes	characteristic
172	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
173	C₃	#3334	1	1	yes	characteristic, direct product factor, Sylow
174	C₂	—	2	4	no	—
175	C₂	—	8	2	no	—
176	C₂	C₃×#547	1	1	yes	characteristic
177	C₂	C₂²×C₃×D₈	1	1	yes	characteristic
178	C₂	C₃×#578	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 ◁ C₃×C₄×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#547	C₂	semidirect
2. C₃×#500	C₂	semidirect
3. C₃×#480	C₂	semidirect
4. C₃×#474	C₂	semidirect
5. C₃×#466	C₂	semidirect
6. C₃×C₄×D₈	C₂	semidirect
7. C₃×C₈×D₄	C₂	semidirect
8. C₂×C₃×#107	C₂	semidirect
9. #3334	C₃	direct (N × H)
29. #547	C₂×C₃	semidirect
30. #500	C₂×C₃	semidirect
31. #480	C₂×C₃	semidirect
32. #474	C₂×C₃	semidirect
33. #466	C₂×C₃	semidirect
34. C₄×D₈	C₂×C₃	semidirect
35. C₈×D₄	C₂×C₃	semidirect
36. C₂×#107	C₂×C₃	semidirect
84. C₃×D₄	D₈	semidirect
97. C₃×C₈	#41	semidirect
140. D₄	C₃×D₈	semidirect
153. C₈	C₃×#41	semidirect
173. C₃	#3334	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×#547	2
C₃×#500	2
C₃×#480	2
C₃×#474	2
C₃×#466	2
C₃×C₄×D₈	2
C₃×C₈×D₄	2
C₂×C₃×#107	2
#3334	3

Subgroups — C3×#3334 #85198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3334 #85198