Subgroups — #71985

Order 384.

448 conjugacy classes of subgroups.

The subgroup lattice has 188 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	C₂×#926	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	#4551	C₂	1	4	yes	maximal, complemented
5	#1736	—	3	1	no	maximal, Sylow
6	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂³×Dic₃	—	2	1	no	—
8	C₂×C₃×M₁₆	—	2	2	no	—
9	C₂×C₃×M₁₆	—	2	2	no	—
10	C₂×C₃×M₁₆	C₄	1	4	yes	complemented
11	C₂²×C₃×C₈	C₂²	1	2	yes	—
12	C₂×#217	—	2	1	no	—
13	#926	—	2	2	no	—
14	#926	C₄	1	4	yes	complemented
15	C₂×C₄⋊Dic₃	—	2	1	no	—
16	C₂×C₄⋊Dic₃	—	2	2	no	—
17	C₂×C₄⋊Dic₃	C₄	1	4	yes	complemented
18	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
20	C₂×C₄×Dic₃	—	2	1	no	—
21	C₂²×M₁₆	D₃	1	1	yes	characteristic
22	C₂×#117	—	3	1	no	—
23	C₂²×C₄⋊C₄	—	3	1	no	—
24	#340	—	3	4	no	—
25	C₂⁴×C₃	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
29	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄	D₄	1	2	yes	—
31	C₂²×C₃×C₄	D₄	1	2	yes	—
32	C₂²×C₃×C₄	Q₈	1	2	yes	—
33	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
34	C₂²×Dic₃	—	2	4	no	—
35	C₂²×Dic₃	—	2	2	no	—
36	C₂²×Dic₃	—	2	2	no	—
37	C₂²×Dic₃	—	4	4	no	—
38	C₃×M₁₆	—	2	8	no	—
39	C₂×C₃×C₈	—	2	2	no	—
40	C₂×C₃×C₈	—	2	2	no	—
41	C₂×C₃×C₈	C₂×C₄	1	4	yes	complemented
42	#217	—	2	4	no	—
43	C₄⋊Dic₃	—	2	4	no	—
44	C₄⋊Dic₃	—	2	2	no	—
45	C₄⋊Dic₃	—	2	2	no	—
46	C₄⋊Dic₃	C₂×C₄	1	4	yes	—
47	C₄⋊Dic₃	C₂×C₄	1	4	yes	complemented
48	C₄×Dic₃	—	2	4	no	—
49	C₂³×C₄	—	6	1	no	—
50	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
51	C₂×M₁₆	—	2	2	no	—
52	C₂×M₁₆	—	2	2	no	—
53	C₂×M₁₆	Dic₃	1	4	yes	complemented
54	C₂²×C₈	C₂×D₃	1	2	yes	—
55	#117	—	3	4	no	—
56	#117	—	6	2	no	—
57	C₂×C₄⋊C₄	—	3	4	no	—
58	C₂×C₄⋊C₄	—	3	2	no	—
59	C₂×C₄⋊C₄	—	3	2	no	—
60	C₂×C₄⋊C₄	—	6	1	no	—
61	C₂×C₄⋊C₄	—	6	2	no	—
62	C₂×C₂²⋊C₄	—	6	1	no	—
63	C₂×C₄²	—	6	1	no	—
64	C₂³×C₃	—	2	2	no	—
65	C₂³×C₃	—	2	2	no	—
66	C₂³×C₃	C₂²⋊C₄	1	2	yes	—
67	C₂³×C₃	C₂²⋊C₄	1	2	yes	—
68	C₂³×C₃	C₄⋊C₄	1	2	yes	—
69	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄	C₄²	1	4	yes	—
71	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
72	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
73	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
74	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
75	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
76	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
77	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
83	C₂×Dic₃	—	2	4	no	—
84	C₂×Dic₃	—	2	4	no	—
85	C₂×Dic₃	—	4	8	no	—
86	C₂×Dic₃	—	4	2	no	—
87	C₂×Dic₃	—	4	4	no	—
88	C₂×Dic₃	—	4	2	no	—
89	C₃×C₈	—	2	4	no	—
90	C₂⁴	D₁₂	1	1	yes	characteristic
91	C₂²×C₄	—	6	4	no	—
92	C₂²×C₄	—	6	2	no	—
93	C₂²×C₄	—	6	2	no	—
94	C₂²×C₄	—	12	4	no	—
95	C₂²×C₄	Dic₆	1	2	yes	—
96	C₂²×C₄	C₄×D₃	1	2	yes	—
97	C₂²×C₄	C₄×D₃	1	2	yes	—
98	C₂²×C₄	D₁₂	1	1	yes	characteristic
99	C₂²×C₄	C₂×Dic₃	1	2	yes	—
100	C₂²×C₄	#67	1	2	yes	—
101	C₂²×C₄	#67	1	2	yes	—
102	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
103	M₁₆	—	2	8	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	C₂×Dic₃	1	4	yes	complemented
107	C₄⋊C₄	—	3	4	no	—
108	C₄⋊C₄	—	3	4	no	—
109	C₄⋊C₄	—	6	4	no	—
110	C₄⋊C₄	—	6	2	no	—
111	C₄⋊C₄	—	6	2	no	—
112	C₂²⋊C₄	—	6	4	no	—
113	C₄²	—	6	4	no	—
114	C₂²×C₃	—	2	4	no	—
115	C₂²×C₃	—	2	2	no	—
116	C₂²×C₃	—	2	2	no	—
117	C₂²×C₃	—	2	4	no	—
118	C₂²×C₃	#95	1	4	yes	—
119	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
123	C₂²×C₃	#137	1	2	yes	—
124	C₃×C₄	#95	1	4	yes	—
125	C₃×C₄	C₂×C₄²	1	1	yes	characteristic
126	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
129	Dic₃	—	4	4	no	—
130	Dic₃	—	4	4	no	—
131	C₂³	—	2	2	no	—
132	C₂³	—	2	2	no	—
133	C₂³	C₄⋊Dic₃	1	2	yes	—
134	C₂³	(C₂×D₃)⋊C₄	1	2	yes	—
135	C₂³	(C₂×D₃)⋊C₄	1	2	yes	—
136	C₂³	C₂×D₁₂	1	1	yes	characteristic
137	C₂×C₄	—	6	4	no	—
138	C₂×C₄	—	6	4	no	—
139	C₂×C₄	—	12	8	no	—
140	C₂×C₄	—	12	4	no	—
141	C₂×C₄	—	12	2	no	—
142	C₂×C₄	—	12	2	no	—
143	C₂×C₄	C₄×Dic₃	1	4	yes	—
144	C₂×C₄	Dic₃⋊C₄	1	4	yes	—
145	C₂×C₄	Dic₃⋊C₄	1	4	yes	—
146	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
147	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
148	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
149	C₂×C₄	#217	1	4	yes	—
150	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
153	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
154	C₂×C₄	C₂×#67	1	1	yes	characteristic
155	C₂×C₄	C₂×#67	1	1	yes	characteristic
156	C₈	—	2	4	no	—
157	C₂×C₃	—	2	4	no	—
158	C₂×C₃	C₂×#95	1	1	yes	characteristic
159	C₂×C₃	#417	1	2	yes	—
160	C₂×C₃	#418	1	1	yes	characteristic
161	C₂×C₃	#419	1	1	yes	characteristic
162	C₂×C₃	#428	1	2	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	4	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	4	no	—
167	C₂²	C₃⋊#95	1	4	yes	—
168	C₂²	#908	1	2	yes	—
169	C₂²	#909	1	2	yes	—
170	C₂²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
171	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
172	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
173	C₄	—	12	4	no	—
174	C₄	—	12	4	no	—
175	C₄	C₃⋊#95	1	4	yes	—
176	C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
177	C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
178	C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
179	C₄	C₂×#217	1	1	yes	characteristic
180	C₃	#1736	1	1	yes	characteristic, complemented, Sylow
181	C₂	—	2	4	no	—
182	C₂	#5120	1	1	yes	characteristic
183	C₂	#5121	1	2	yes	—
184	C₂	#5131	1	1	yes	characteristic
185	C₂	#5133	1	2	yes	—
186	C₂	C₂×C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₂
commutator	C₃×C₄	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂²×C₃×M₁₆	C₂
socle	C₂³×C₃	C₂×D₄
prime radical	C₂⁴×C₃	D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	#1736
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
4. #4551	C₂	semidirect
10. C₂×C₃×M₁₆	C₄	semidirect
14. #926	C₄	semidirect
17. C₂×C₄⋊Dic₃	C₄	semidirect
41. C₂×C₃×C₈	C₂×C₄	semidirect
47. C₄⋊Dic₃	C₂×C₄	semidirect
53. C₂×M₁₆	Dic₃	semidirect
106. C₂×C₈	C₂×Dic₃	semidirect
180. C₃	#1736	semidirect

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
C₂×#926	2
C₂²×C₄⋊Dic₃	2
#4551	2
#1736	3