Subgroups — #71994

Order 384.

372 conjugacy classes of subgroups.

The subgroup lattice has 194 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	1	yes	maximal, characteristic
3	C₂²×#208	C₂	1	1	yes	maximal, characteristic
4	#4558	C₂	1	4	yes	maximal, complemented
5	#1738	—	3	1	no	maximal, Sylow
6	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂×C₃×M₁₆	—	2	2	no	—
8	C₂×C₃×M₁₆	—	2	1	no	—
9	C₂×C₃×M₁₆	C₄	1	4	yes	—
10	C₂×C₃×M₁₆	C₂²	1	2	yes	—
11	C₂²×C₃×C₈	—	2	1	no	—
12	C₂×#208	—	2	2	no	—
13	C₂×#208	—	2	2	no	—
14	C₂×#208	—	2	1	no	—
15	C₂×#208	—	2	2	no	—
16	C₂×#208	C₄	1	4	yes	—
17	C₂×#208	C₄	1	2	yes	—
18	C₂×#208	C₄	1	2	yes	—
19	C₂×#208	C₂²	1	2	yes	—
20	C₂²×C₃⋊C₈	—	2	1	no	—
21	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
22	C₂²×M₁₆	—	3	1	no	—
23	C₂²×M₁₆	—	3	1	no	—
24	C₂²×M₁₆	D₃	1	1	yes	characteristic
25	#341	—	3	4	no	—
26	C₂⁴×C₃	Q₈	1	1	yes	characteristic
27	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
31	C₂²×C₃×C₄	D₄	1	2	yes	—
32	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
33	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
34	C₂²×C₃×C₄	D₄	1	2	yes	—
35	C₂²×C₃×C₄	Q₈	1	1	yes	characteristic
36	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
37	C₃×M₁₆	—	2	4	no	—
38	C₃×M₁₆	—	2	2	no	—
39	C₃×M₁₆	C₂×C₄	1	4	yes	—
40	C₂×C₃×C₈	—	2	4	no	—
41	C₂×C₃×C₈	—	2	2	no	—
42	#208	—	2	4	no	—
43	#208	—	2	4	no	—
44	#208	—	2	4	no	—
45	#208	—	2	2	no	—
46	#208	C₂×C₄	1	4	yes	—
47	C₂×C₃⋊C₈	—	2	4	no	—
48	C₂×C₃⋊C₈	—	2	2	no	—
49	C₂×C₃⋊C₈	—	2	2	no	—
50	C₂×C₃⋊C₈	—	2	2	no	—
51	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
52	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
53	C₂×M₁₆	—	2	2	no	—
54	C₂×M₁₆	—	2	1	no	—
55	C₂×M₁₆	—	3	2	no	—
56	C₂×M₁₆	—	3	2	no	—
57	C₂×M₁₆	—	3	4	no	—
58	C₂×M₁₆	—	3	2	no	—
59	C₂×M₁₆	—	6	2	no	—
60	C₂×M₁₆	—	6	2	no	—
61	C₂×M₁₆	—	6	2	no	—
62	C₂×M₁₆	—	6	1	no	—
63	C₂×M₁₆	Dic₃	1	4	yes	—
64	C₂×M₁₆	C₂×D₃	1	2	yes	—
65	C₂²×C₈	—	2	1	no	—
66	C₂²×C₈	—	3	2	no	—
67	C₂²×C₈	—	6	1	no	—
68	C₂³×C₃	—	4	2	no	—
69	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
70	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂³×C₃	C₄⋊C₄	1	2	yes	—
72	C₂³×C₃	C₄⋊C₄	1	2	yes	—
73	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₃×C₄	C₄²	1	2	yes	—
75	C₂×C₃×C₄	C₄²	1	2	yes	—
76	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
77	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
78	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
79	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
80	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
81	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
82	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
83	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
86	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
92	C₃×C₈	—	2	4	no	—
93	C₃⋊C₈	—	2	4	no	—
94	C₃⋊C₈	—	2	4	no	—
95	C₂⁴	Dic₆	1	1	yes	characteristic
96	C₂²×C₄	Dic₆	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₃	1	2	yes	—
98	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
99	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
100	C₂²×C₄	D₁₂	1	2	yes	—
101	C₂²×C₄	C₂×Dic₃	1	2	yes	—
102	C₂²×C₄	#67	1	2	yes	—
103	C₂²×C₄	#67	1	1	yes	characteristic
104	C₂²×C₄	#67	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
106	M₁₆	—	2	4	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	3	4	no	—
109	M₁₆	—	6	4	no	—
110	M₁₆	—	6	4	no	—
111	M₁₆	—	6	4	no	—
112	M₁₆	—	6	2	no	—
113	M₁₆	C₂×Dic₃	1	4	yes	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	3	4	no	—
117	C₂×C₈	—	6	4	no	—
118	C₂×C₈	—	6	2	no	—
119	C₂×C₈	—	6	2	no	—
120	C₂×C₈	—	6	2	no	—
121	C₂²×C₃	—	2	2	no	—
122	C₂²×C₃	—	2	2	no	—
123	C₂²×C₃	—	2	2	no	—
124	C₂²×C₃	—	4	2	no	—
125	C₂²×C₃	—	4	2	no	—
126	C₂²×C₃	#95	1	2	yes	—
127	C₂²×C₃	#95	1	2	yes	—
128	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₃×C₄	#95	1	2	yes	—
132	C₃×C₄	#95	1	2	yes	—
133	C₃×C₄	C₂×C₄²	1	1	yes	characteristic
134	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂³	—	4	2	no	—
138	C₂³	Dic₃⋊C₄	1	1	yes	characteristic
139	C₂³	Dic₃⋊C₄	1	1	yes	characteristic
140	C₂³	Dic₃⋊C₄	1	2	yes	—
141	C₂³	C₄⋊Dic₃	1	2	yes	—
142	C₂³	C₂×Dic₆	1	1	yes	characteristic
143	C₂×C₄	C₄×Dic₃	1	2	yes	—
144	C₂×C₄	C₄×Dic₃	1	2	yes	—
145	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
146	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
147	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
148	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
149	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
150	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
151	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
152	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
153	C₂×C₄	#217	1	2	yes	—
154	C₂×C₄	#217	1	2	yes	—
155	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
157	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
158	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
159	C₂×C₄	C₂×#67	1	1	yes	characteristic
160	C₂×C₄	C₂×#67	1	1	yes	characteristic
161	C₈	—	2	4	no	—
162	C₈	—	6	4	no	—
163	C₈	—	6	4	no	—
164	C₂×C₃	—	2	2	no	—
165	C₂×C₃	—	2	2	no	—
166	C₂×C₃	—	2	1	no	—
167	C₂×C₃	—	2	1	no	—
168	C₂×C₃	C₂×#95	1	1	yes	characteristic
169	C₂×C₃	#430	1	2	yes	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	4	2	no	—
174	C₂²	—	4	2	no	—
175	C₂²	C₃⋊#95	1	2	yes	—
176	C₂²	C₃⋊#95	1	2	yes	—
177	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
178	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
179	C₂²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
180	C₄	C₃⋊#95	1	2	yes	—
181	C₄	C₃⋊#95	1	2	yes	—
182	C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
183	C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
184	C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
185	C₄	C₂×#217	1	1	yes	characteristic
186	C₃	#1738	1	1	yes	characteristic, complemented, Sylow
187	C₂	—	2	2	no	—
188	C₂	—	2	2	no	—
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#5124	1	2	yes	—
192	C₂	C₂×C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#67
commutator	C₃×C₄	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂²×C₃×M₁₆	C₂
socle	C₂²×C₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃	Q₈
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	#1738
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
4. #4558	C₂	semidirect
186. C₃	#1738	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
C₂²×#208	2
#4558	2
#1738	3