Subgroups — C₃×#3341 #85205

Order 384.

364 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#555	C₂	1	2	yes	maximal, complemented
2	C₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#482	C₂	1	4	yes	maximal, complemented
4	C₃×#476	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	C₃×#427	C₂	1	2	yes	maximal, complemented
9	#3341	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₈	C₂²	1	2	yes	—
12	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₃×#124	—	2	4	no	—
14	C₃×#121	—	2	2	no	—
15	C₃×#121	C₂²	1	4	yes	—
16	C₃×C₄×D₄	C₂²	1	2	yes	—
17	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
18	C₃×#117	—	2	2	no	—
19	C₃×#117	C₂²	1	4	yes	—
20	C₃×#107	C₂²	1	1	yes	characteristic
21	C₃×#107	C₂²	1	2	yes	—
22	C₃×#107	C₂²	1	2	yes	—
23	C₃×#106	C₂²	1	4	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	#555	C₂×C₃	1	2	yes	complemented
30	#500	C₂×C₃	1	1	yes	characteristic, complemented
31	#482	C₂×C₃	1	4	yes	complemented
32	#476	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	#427	C₂×C₃	1	2	yes	complemented
37	C₂×C₃×Q₈	—	2	2	no	—
38	C₂×C₃×D₄	—	2	4	no	—
39	C₂×C₃×D₄	D₄	1	1	yes	characteristic
40	C₂×C₃×D₄	C₂³	1	2	yes	—
41	C₂²×C₃×C₄	—	2	2	no	—
42	C₂²×C₃×C₄	C₂³	1	2	yes	—
43	C₃×D₈	—	2	2	no	—
44	C₂×C₃×C₈	—	2	4	no	—
45	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
46	C₂×C₃×C₈	C₂³	1	2	yes	—
47	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
48	C₃×C₄⋊C₄	—	2	2	no	—
49	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
50	C₃×C₄⋊C₄	C₂³	1	2	yes	—
51	C₃×C₄⋊C₄	C₂³	1	4	yes	—
52	C₃×C₂²⋊C₄	—	2	4	no	—
53	C₃×C₂²⋊C₄	—	2	4	no	—
54	C₃×C₂²⋊C₄	—	2	2	no	—
55	C₃×C₂²⋊C₄	D₄	1	2	yes	—
56	C₃×C₄²	—	2	4	no	—
57	C₃×C₄²	C₂³	1	1	yes	characteristic
58	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
59	C₂²×C₈	C₂²×C₃	1	2	yes	—
60	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
61	#124	—	2	4	no	—
62	#121	—	2	2	no	—
63	#121	C₂²×C₃	1	4	yes	—
64	C₄×D₄	C₂²×C₃	1	2	yes	—
65	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
66	#117	—	2	2	no	—
67	#117	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	2	yes	—
71	#106	C₂²×C₃	1	4	yes	—
72	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
73	#102	C₂²×C₃	1	2	yes	—
74	#102	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	—	2	2	no	—
78	C₂³×C₃	C₂×D₄	1	2	yes	—
79	C₃×Q₈	—	8	2	no	—
80	C₃×D₄	—	2	2	no	—
81	C₃×D₄	—	2	2	no	—
82	C₃×D₄	—	2	2	no	—
83	C₃×D₄	—	8	4	no	—
84	C₂×C₃×C₄	—	2	2	no	—
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	2	2	no	—
87	C₂×C₃×C₄	—	2	4	no	—
88	C₂×C₃×C₄	—	4	4	no	—
89	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
90	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
93	C₃×C₈	—	2	2	no	—
94	C₃×C₈	—	2	1	no	—
95	C₃×C₈	#41	1	2	yes	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×D₄	—	2	4	no	—
98	C₂×D₄	C₃×D₄	1	1	yes	characteristic
99	C₂×D₄	C₂³×C₃	1	2	yes	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	C₂³×C₃	1	2	yes	—
102	D₈	—	2	2	no	—
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
105	C₂×C₈	C₂³×C₃	1	2	yes	—
106	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
109	C₄⋊C₄	C₂³×C₃	1	2	yes	—
110	C₄⋊C₄	C₂³×C₃	1	4	yes	—
111	C₂²⋊C₄	—	2	4	no	—
112	C₂²⋊C₄	—	2	4	no	—
113	C₂²⋊C₄	—	2	2	no	—
114	C₂²⋊C₄	C₃×D₄	1	2	yes	—
115	C₄²	—	2	4	no	—
116	C₄²	C₂³×C₃	1	1	yes	characteristic
117	C₂²×C₃	—	2	2	no	—
118	C₂²×C₃	—	2	2	no	—
119	C₂²×C₃	—	2	2	no	—
120	C₂²×C₃	—	4	2	no	—
121	C₂²×C₃	—	4	2	no	—
122	C₂²×C₃	—	4	2	no	—
123	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
124	C₃×C₄	—	2	2	no	—
125	C₃×C₄	—	2	1	no	—
126	C₃×C₄	—	4	2	no	—
127	C₃×C₄	—	4	4	no	—
128	C₃×C₄	#135	1	2	yes	—
129	C₃×C₄	C₂×#41	1	1	yes	characteristic
130	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
131	C₂³	—	2	2	no	—
132	C₂³	C₂×C₃×D₄	1	2	yes	—
133	Q₈	—	8	2	no	—
134	D₄	—	2	2	no	—
135	D₄	—	2	2	no	—
136	D₄	—	2	2	no	—
137	D₄	—	8	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
144	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
145	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
146	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	2	1	no	—
149	C₈	C₃×#41	1	2	yes	—
150	C₂×C₃	—	4	2	no	—
151	C₂×C₃	—	8	2	no	—
152	C₂×C₃	#547	1	1	yes	characteristic
153	C₂×C₃	C₂×#135	1	1	yes	characteristic
154	C₂×C₃	#576	1	1	yes	characteristic
155	C₂²	—	2	2	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	4	2	no	—
159	C₂²	—	4	2	no	—
160	C₂²	—	4	2	no	—
161	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	2	no	—
165	C₄	—	4	4	no	—
166	C₄	C₃×#135	1	2	yes	—
167	C₄	C₂×C₃×#41	1	1	yes	characteristic
168	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
169	C₃	#3341	1	1	yes	characteristic, direct product factor, Sylow
170	C₂	—	4	2	no	—
171	C₂	—	8	2	no	—
172	C₂	C₃×#547	1	1	yes	characteristic
173	C₂	C₂×C₃×#135	1	1	yes	characteristic
174	C₂	C₃×#576	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×#555	C₂	semidirect
2. C₃×#500	C₂	semidirect
3. C₃×#482	C₂	semidirect
4. C₃×#476	C₂	semidirect
5. C₃×#466	C₂	semidirect
6. C₃×C₄×D₈	C₂	semidirect
7. C₃×C₈×D₄	C₂	semidirect
8. C₃×#427	C₂	semidirect
9. #3341	C₃	direct (N × H)
29. #555	C₂×C₃	semidirect
30. #500	C₂×C₃	semidirect
31. #482	C₂×C₃	semidirect
32. #476	C₂×C₃	semidirect
33. #466	C₂×C₃	semidirect
34. C₄×D₈	C₂×C₃	semidirect
35. C₈×D₄	C₂×C₃	semidirect
36. #427	C₂×C₃	semidirect
169. C₃	#3341	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×#555	2
C₃×#500	2
C₃×#482	2
C₃×#476	2
C₃×#466	2
C₃×C₄×D₈	2
C₃×C₈×D₄	2
C₃×#427	2
#3341	3

Subgroups — C3×#3341 #85205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3341 #85205