Subgroups — C₂×C₃⋊#338 #71997

Order 384.

372 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#208	C₂	1	2	yes	maximal
3	C₃⋊#338	C₂	1	4	yes	maximal, direct product factor
4	C₂×#338	—	3	1	no	maximal, Sylow
5	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
6	C₂×C₃×M₁₆	—	2	2	no	—
7	C₂×C₃×M₁₆	—	2	2	no	—
8	C₂×C₃×M₁₆	C₄	1	2	yes	—
9	C₂×C₃×M₁₆	C₂²	1	2	yes	—
10	C₂²×C₃×C₈	C₄	1	2	yes	—
11	C₂×#208	—	2	2	no	—
12	C₂×#208	—	2	2	no	—
13	C₂×#208	—	2	2	no	—
14	C₂×#208	—	2	2	no	—
15	C₂×#208	C₄	1	4	yes	—
16	C₂×#208	C₂²	1	4	yes	—
17	C₂²×C₃⋊C₈	C₄	1	4	yes	—
18	C₂²×M₁₆	—	3	2	no	—
19	C₂²×M₁₆	D₃	1	1	yes	characteristic
20	#338	—	3	4	no	—
21	C₂⁴×C₃	Q₈	1	1	yes	characteristic
22	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄	D₄	1	2	yes	—
29	C₂²×C₃×C₄	D₄	1	2	yes	—
30	C₂²×C₃×C₄	Q₈	1	1	yes	characteristic
31	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₃×M₁₆	—	2	4	no	—
33	C₃×M₁₆	—	2	4	no	—
34	C₂×C₃×C₈	—	2	2	no	—
35	C₂×C₃×C₈	—	2	2	no	—
36	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
37	#208	—	2	4	no	—
38	#208	—	2	4	no	—
39	#208	—	2	4	no	—
40	#208	—	2	4	no	—
41	C₂×C₃⋊C₈	—	2	4	no	—
42	C₂×C₃⋊C₈	—	2	4	no	—
43	C₂×C₃⋊C₈	C₂×C₄	1	8	yes	—
44	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
45	C₂×M₁₆	—	2	2	no	—
46	C₂×M₁₆	—	2	2	no	—
47	C₂×M₁₆	—	3	4	no	—
48	C₂×M₁₆	—	3	4	no	—
49	C₂×M₁₆	—	6	2	no	—
50	C₂×M₁₆	—	6	2	no	—
51	C₂×M₁₆	—	6	2	no	—
52	C₂×M₁₆	—	6	2	no	—
53	C₂×M₁₆	Dic₃	1	2	yes	—
54	C₂×M₁₆	C₂×D₃	1	2	yes	—
55	C₂²×C₈	—	3	4	no	—
56	C₂²×C₈	Dic₃	1	2	yes	—
57	C₂³×C₃	—	4	2	no	—
58	C₂³×C₃	C₄⋊C₄	1	2	yes	—
59	C₂³×C₃	C₄⋊C₄	1	2	yes	—
60	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₃×C₄	C₄²	1	2	yes	—
64	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
65	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
66	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
67	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
68	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
69	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
70	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
73	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
74	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
75	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
78	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
81	C₃×C₈	—	2	4	no	—
82	C₃⋊C₈	—	2	8	no	—
83	C₂⁴	Dic₆	1	1	yes	characteristic
84	C₂²×C₄	Dic₆	1	1	yes	characteristic
85	C₂²×C₄	C₄×D₃	1	2	yes	—
86	C₂²×C₄	C₄×D₃	1	2	yes	—
87	C₂²×C₄	D₁₂	1	1	yes	characteristic
88	C₂²×C₄	D₁₂	1	1	yes	characteristic
89	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
90	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
91	C₂²×C₄	#67	1	2	yes	—
92	C₂²×C₄	#67	1	2	yes	—
93	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
94	M₁₆	—	2	4	no	—
95	M₁₆	—	2	4	no	—
96	M₁₆	—	6	4	no	—
97	M₁₆	—	6	4	no	—
98	M₁₆	—	6	4	no	—
99	M₁₆	—	6	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	3	8	no	—
103	C₂×C₈	—	6	4	no	—
104	C₂×C₈	—	6	4	no	—
105	C₂×C₈	C₂×Dic₃	1	4	yes	—
106	C₂²×C₃	—	2	4	no	—
107	C₂²×C₃	—	2	2	no	—
108	C₂²×C₃	—	4	2	no	—
109	C₂²×C₃	—	4	1	no	—
110	C₂²×C₃	—	4	1	no	—
111	C₂²×C₃	#95	1	2	yes	—
112	C₂²×C₃	#95	1	1	yes	characteristic
113	C₂²×C₃	#95	1	1	yes	characteristic
114	C₂²×C₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₃×C₄	#95	1	2	yes	—
117	C₃×C₄	#95	1	1	yes	characteristic
118	C₃×C₄	#95	1	1	yes	characteristic
119	C₃×C₄	C₂×C₄²	1	1	yes	characteristic
120	C₃×C₄	C₂×C₂²⋊C₄	1	2	yes	—
121	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂³	—	4	2	no	—
123	C₂³	Dic₃⋊C₄	1	2	yes	—
124	C₂³	Dic₃⋊C₄	1	2	yes	—
125	C₂³	C₄⋊Dic₃	1	1	yes	characteristic
126	C₂³	C₄⋊Dic₃	1	1	yes	characteristic
127	C₂³	C₂×Dic₆	1	1	yes	characteristic
128	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
129	C₂×C₄	C₄×Dic₃	1	2	yes	—
130	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
131	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
132	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
133	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
134	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
135	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
136	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
137	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
138	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
139	C₂×C₄	#217	1	1	yes	characteristic
140	C₂×C₄	#217	1	1	yes	characteristic
141	C₂×C₄	#217	1	2	yes	—
142	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
143	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
144	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
145	C₂×C₄	C₂×#67	1	2	yes	—
146	C₈	—	2	4	no	—
147	C₈	—	6	8	no	—
148	C₂×C₃	—	2	2	no	—
149	C₂×C₃	—	2	1	no	—
150	C₂×C₃	—	2	2	no	—
151	C₂×C₃	—	2	1	no	—
152	C₂×C₃	#338	1	2	yes	complemented
153	C₂×C₃	C₂×#95	1	1	yes	characteristic
154	C₂²	—	2	4	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	4	2	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	4	1	no	—
159	C₂²	C₃⋊#95	1	2	yes	—
160	C₂²	C₃⋊#95	1	1	yes	characteristic
161	C₂²	C₃⋊#95	1	1	yes	characteristic
162	C₂²	C₂×Dic₃⋊C₄	1	2	yes	—
163	C₂²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
164	C₄	C₃⋊#95	1	2	yes	—
165	C₄	C₃⋊#95	1	1	yes	characteristic
166	C₄	C₃⋊#95	1	1	yes	characteristic
167	C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
168	C₄	C₂×(C₂×D₃)⋊C₄	1	2	yes	—
169	C₄	C₂×#217	1	1	yes	characteristic
170	C₃	C₂×#338	1	1	yes	characteristic, complemented, Sylow
171	C₂	—	2	2	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	2	no	—
174	C₂	—	2	1	no	—
175	C₂	C₃⋊#338	1	2	yes	direct product factor
176	C₂	C₂×C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₃
commutator	C₃×C₄	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂²×C₃×M₁₆	C₂
socle	C₂²×C₃	#95
prime radical	C₂⁴×C₃	Q₈
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	C₂×#338
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₃⋊#338	C₂	direct (N × H)
152. C₂×C₃	#338	semidirect
170. C₃	C₂×#338	semidirect
175. C₂	C₃⋊#338	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×M₁₆	2
C₂²×#208	2
C₃⋊#338	2
C₂×#338	3

Subgroups — C2×C3⋊#338 #71997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃⋊#338 #71997