Subgroups — C₃×#3338 #85202

Order 384.

356 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#554	C₂	1	2	yes	maximal
2	C₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#484	C₂	1	4	yes	maximal, complemented
4	C₃×#477	C₂	1	2	yes	maximal, complemented
5	C₃×#467	C₂	1	2	yes	maximal, complemented
6	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃×#107	C₂	1	2	yes	maximal, complemented
9	#3338	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₈	C₂²	1	2	yes	—
12	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₃×#125	—	2	2	no	—
14	C₃×C₂²⋊Q₈	—	2	4	no	—
15	C₃×C₂²⋊Q₈	C₂²	1	4	yes	—
16	C₃×C₄×Q₈	C₂²	1	2	yes	—
17	C₃×C₄×D₄	—	2	2	no	—
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×#103	C₂²	1	2	yes	—
26	C₃×#103	C₂²	1	4	yes	—
27	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
29	#554	C₂×C₃	1	2	yes	—
30	#500	C₂×C₃	1	1	yes	characteristic, complemented
31	#484	C₂×C₃	1	4	yes	complemented
32	#477	C₂×C₃	1	2	yes	complemented
33	#467	C₂×C₃	1	2	yes	complemented
34	C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
35	C₈×D₄	C₂×C₃	1	1	yes	characteristic
36	C₂×#107	C₂×C₃	1	2	yes	complemented
37	C₂×C₃×Q₈	—	2	2	no	—
38	C₂×C₃×Q₈	C₂³	1	2	yes	—
39	C₂×C₃×D₄	D₄	1	1	yes	characteristic
40	C₂²×C₃×C₄	—	2	4	no	—
41	C₂²×C₃×C₄	C₂³	1	2	yes	—
42	C₃×Q₁₆	—	2	2	no	—
43	C₂×C₃×C₈	—	2	4	no	—
44	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
45	C₂×C₃×C₈	C₂³	1	2	yes	—
46	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
47	C₃×C₄⋊C₄	—	2	4	no	—
48	C₃×C₄⋊C₄	—	2	2	no	—
49	C₃×C₄⋊C₄	—	2	2	no	—
50	C₃×C₄⋊C₄	—	2	4	no	—
51	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
52	C₃×C₄⋊C₄	C₂³	1	2	yes	—
53	C₃×C₄⋊C₄	C₂³	1	4	yes	—
54	C₃×C₂²⋊C₄	—	2	4	no	—
55	C₃×C₂²⋊C₄	D₄	1	2	yes	—
56	C₃×C₄²	—	2	2	no	—
57	C₃×C₄²	C₂³	1	1	yes	characteristic
58	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
59	C₂²×C₈	C₂²×C₃	1	2	yes	—
60	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
61	#125	—	2	2	no	—
62	C₂²⋊Q₈	—	2	4	no	—
63	C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
64	C₄×Q₈	C₂²×C₃	1	2	yes	—
65	C₄×D₄	—	2	2	no	—
66	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
68	#107	C₂²×C₃	1	1	yes	characteristic
69	#107	C₂²×C₃	1	2	yes	—
70	#107	C₂²×C₃	1	4	yes	—
71	#107	C₂²×C₃	1	2	yes	—
72	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
73	#103	C₂²×C₃	1	2	yes	—
74	#103	C₂²×C₃	1	4	yes	—
75	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
76	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
77	C₂³×C₃	C₂×D₄	1	2	yes	—
78	C₃×Q₈	—	2	2	no	—
79	C₃×Q₈	—	2	2	no	—
80	C₃×Q₈	—	8	2	no	—
81	C₃×D₄	Q₁₆	1	4	yes	complemented
82	C₂×C₃×C₄	—	2	2	no	—
83	C₂×C₃×C₄	—	2	2	no	—
84	C₂×C₃×C₄	—	2	2	no	—
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	2	4	no	—
87	C₂×C₃×C₄	—	8	4	no	—
88	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
90	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
92	C₃×C₈	—	2	2	no	—
93	C₃×C₈	—	2	1	no	—
94	C₃×C₈	#41	1	2	yes	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	C₂³×C₃	1	2	yes	—
97	C₂×D₄	C₃×D₄	1	1	yes	characteristic
98	C₂²×C₄	—	2	4	no	—
99	C₂²×C₄	C₂³×C₃	1	2	yes	—
100	Q₁₆	—	2	2	no	—
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
103	C₂×C₈	C₂³×C₃	1	2	yes	—
104	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
105	C₄⋊C₄	—	2	4	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	4	no	—
109	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
110	C₄⋊C₄	C₂³×C₃	1	2	yes	—
111	C₄⋊C₄	C₂³×C₃	1	4	yes	—
112	C₂²⋊C₄	—	2	4	no	—
113	C₂²⋊C₄	C₃×D₄	1	2	yes	—
114	C₄²	—	2	2	no	—
115	C₄²	C₂³×C₃	1	1	yes	characteristic
116	C₂²×C₃	—	2	2	no	—
117	C₂²×C₃	—	2	2	no	—
118	C₂²×C₃	C₂×Q₁₆	1	4	yes	—
119	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
120	C₃×C₄	—	2	2	no	—
121	C₃×C₄	—	2	1	no	—
122	C₃×C₄	—	4	2	no	—
123	C₃×C₄	—	4	2	no	—
124	C₃×C₄	—	4	4	no	—
125	C₃×C₄	C₂×Q₁₆	1	2	yes	—
126	C₃×C₄	C₂×#41	1	1	yes	characteristic
127	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂³	C₂×C₃×D₄	1	2	yes	—
129	Q₈	—	2	2	no	—
130	Q₈	—	2	2	no	—
131	Q₈	—	8	2	no	—
132	D₄	C₃×Q₁₆	1	4	yes	complemented
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	8	4	no	—
139	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
140	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
141	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
142	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	—	2	1	no	—
145	C₈	C₃×#41	1	2	yes	—
146	C₂×C₃	—	2	4	no	—
147	C₂×C₃	#547	1	1	yes	characteristic
148	C₂×C₃	C₂²×Q₁₆	1	1	yes	characteristic
149	C₂×C₃	#576	1	1	yes	characteristic
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	C₂×C₃×Q₁₆	1	4	yes	—
153	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	2	no	—
156	C₄	—	4	2	no	—
157	C₄	—	4	4	no	—
158	C₄	—	4	2	no	—
159	C₄	C₂×C₃×Q₁₆	1	2	yes	—
160	C₄	C₂×C₃×#41	1	1	yes	characteristic
161	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
162	C₃	#3338	1	1	yes	characteristic, direct product factor, Sylow
163	C₂	—	2	4	no	—
164	C₂	C₃×#547	1	1	yes	characteristic
165	C₂	C₂²×C₃×Q₁₆	1	1	yes	characteristic
166	C₂	C₃×#576	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	D₄
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
2. C₃×#500	C₂	semidirect
3. C₃×#484	C₂	semidirect
4. C₃×#477	C₂	semidirect
5. C₃×#467	C₂	semidirect
6. C₃×C₄×Q₁₆	C₂	semidirect
8. C₂×C₃×#107	C₂	semidirect
9. #3338	C₃	direct (N × H)
30. #500	C₂×C₃	semidirect
31. #484	C₂×C₃	semidirect
32. #477	C₂×C₃	semidirect
33. #467	C₂×C₃	semidirect
34. C₄×Q₁₆	C₂×C₃	semidirect
36. C₂×#107	C₂×C₃	semidirect
81. C₃×D₄	Q₁₆	semidirect
132. D₄	C₃×Q₁₆	semidirect
162. C₃	#3338	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×#554	2
C₃×#500	2
C₃×#484	2
C₃×#477	2
C₃×#467	2
C₃×C₄×Q₁₆	2
C₃×C₈×D₄	2
C₂×C₃×#107	2
#3338	3

Subgroups — C3×#3338 #85202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3338 #85202