Subgroups — #71987

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 206 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×#208	C₂	1	1	yes	maximal, characteristic
4	#4554	C₂	1	4	yes	maximal, complemented
5	#1742	—	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	complemented
10	C₂×C₃×M₁₆	C₂²	1	2	yes	—
11	C₂²×C₃×C₈	—	2	1	no	—
12	C₂×#217	C₄	1	2	yes	—
13	#926	—	2	1	no	—
14	#926	—	2	1	no	—
15	#926	—	2	1	no	—
16	#926	—	2	1	no	—
17	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
18	C₂×C₄×Dic₃	C₄	1	2	yes	—
19	C₂×#208	—	2	1	no	—
20	C₂×#208	—	2	2	no	—
21	C₂×#208	C₄	1	4	yes	complemented
22	C₂×#208	C₂²	1	2	yes	—
23	C₂²×C₃⋊C₈	—	2	1	no	—
24	C₂²×M₁₆	—	3	1	no	—
25	C₂²×M₁₆	D₃	1	1	yes	characteristic
26	C₂×#117	—	3	1	no	—
27	#343	—	3	4	no	—
28	C₂⁴×C₃	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	C₂×C₄	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	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
37	C₂²×C₃×C₄	Q₈	1	2	yes	—
38	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₂²×Dic₃	C₂×C₄	1	4	yes	—
40	C₃×M₁₆	—	2	4	no	—
41	C₃×M₁₆	—	2	2	no	—
42	C₃×M₁₆	C₂×C₄	1	4	yes	—
43	C₂×C₃×C₈	—	2	4	no	—
44	C₂×C₃×C₈	—	2	2	no	—
45	#217	—	2	2	no	—
46	#217	—	2	2	no	—
47	C₄⋊Dic₃	—	2	2	no	—
48	C₄⋊Dic₃	—	2	2	no	—
49	C₄×Dic₃	—	2	2	no	—
50	C₄×Dic₃	—	2	2	no	—
51	#208	—	2	4	no	—
52	#208	—	2	2	no	—
53	#208	C₂×C₄	1	4	yes	—
54	C₂×C₃⋊C₈	—	2	4	no	—
55	C₂×C₃⋊C₈	—	2	2	no	—
56	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
57	C₂×M₁₆	—	2	2	no	—
58	C₂×M₁₆	—	2	1	no	—
59	C₂×M₁₆	—	3	4	no	—
60	C₂×M₁₆	—	3	2	no	—
61	C₂×M₁₆	—	6	1	no	—
62	C₂×M₁₆	—	6	2	no	—
63	C₂×M₁₆	Dic₃	1	4	yes	complemented
64	C₂×M₁₆	C₂×D₃	1	2	yes	—
65	C₂²×C₈	—	2	1	no	—
66	C₂²×C₈	—	6	1	no	—
67	#117	—	6	1	no	—
68	#117	—	6	1	no	—
69	#117	—	6	1	no	—
70	#117	—	6	1	no	—
71	C₂×C₄⋊C₄	—	3	2	no	—
72	C₂×C₂²⋊C₄	—	3	2	no	—
73	C₂×C₄²	—	3	2	no	—
74	C₂³×C₃	—	4	2	no	—
75	C₂³×C₃	C₄²	1	2	yes	—
76	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂³×C₃	C₄⋊C₄	1	2	yes	—
79	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄	C₄²	1	2	yes	—
81	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
83	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
84	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
86	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
87	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
88	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
89	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
90	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
91	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
92	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
98	C₂×Dic₃	—	2	4	no	—
99	C₂×Dic₃	—	4	4	no	—
100	C₃×C₈	—	2	4	no	—
101	C₃⋊C₈	—	2	4	no	—
102	C₂⁴	C₄×D₃	1	1	yes	characteristic
103	C₂²×C₄	—	3	4	no	—
104	C₂²×C₄	Dic₆	1	2	yes	—
105	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
106	C₂²×C₄	C₄×D₃	1	2	yes	—
107	C₂²×C₄	D₁₂	1	1	yes	characteristic
108	C₂²×C₄	D₁₂	1	1	yes	characteristic
109	C₂²×C₄	C₂×Dic₃	1	2	yes	—
110	C₂²×C₄	#67	1	1	yes	characteristic
111	C₂²×C₄	#67	1	1	yes	characteristic
112	C₂²×C₄	#67	1	2	yes	—
113	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
114	M₁₆	—	2	4	no	—
115	M₁₆	—	2	2	no	—
116	M₁₆	—	3	4	no	—
117	M₁₆	—	6	4	no	—
118	M₁₆	—	6	2	no	—
119	M₁₆	C₂×Dic₃	1	4	yes	—
120	C₂×C₈	—	2	4	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	6	4	no	—
123	C₂×C₈	—	6	2	no	—
124	C₄⋊C₄	—	6	2	no	—
125	C₄⋊C₄	—	6	2	no	—
126	C₂²⋊C₄	—	6	2	no	—
127	C₂²⋊C₄	—	6	2	no	—
128	C₄²	—	6	2	no	—
129	C₄²	—	6	2	no	—
130	C₂²×C₃	—	2	2	no	—
131	C₂²×C₃	—	2	2	no	—
132	C₂²×C₃	—	2	2	no	—
133	C₂²×C₃	—	4	2	no	—
134	C₂²×C₃	—	4	2	no	—
135	C₂²×C₃	#95	1	2	yes	—
136	C₂²×C₃	#95	1	2	yes	—
137	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
138	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₃×C₄	#95	1	2	yes	—
141	C₃×C₄	#95	1	2	yes	—
142	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
146	Dic₃	—	4	4	no	—
147	C₂³	—	4	2	no	—
148	C₂³	C₄×Dic₃	1	2	yes	—
149	C₂³	Dic₃⋊C₄	1	2	yes	—
150	C₂³	(C₂×D₃)⋊C₄	1	1	yes	characteristic
151	C₂³	(C₂×D₃)⋊C₄	1	1	yes	characteristic
152	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
153	C₂×C₄	—	6	4	no	—
154	C₂×C₄	—	12	4	no	—
155	C₂×C₄	C₄×Dic₃	1	2	yes	—
156	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
157	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
158	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
159	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
160	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
161	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
162	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
163	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
164	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
165	C₂×C₄	#217	1	2	yes	—
166	C₂×C₄	#217	1	2	yes	—
167	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
169	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
170	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
171	C₂×C₄	C₂×#67	1	1	yes	characteristic
172	C₂×C₄	C₂×#67	1	1	yes	characteristic
173	C₈	—	2	4	no	—
174	C₈	—	6	4	no	—
175	C₂×C₃	—	2	2	no	—
176	C₂×C₃	—	2	2	no	—
177	C₂×C₃	—	2	1	no	—
178	C₂×C₃	—	2	1	no	—
179	C₂×C₃	C₂×#95	1	1	yes	characteristic
180	C₂×C₃	#413	1	2	yes	—
181	C₂²	—	2	2	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₃⋊#95	1	2	yes	—
187	C₂²	C₃⋊#95	1	2	yes	—
188	C₂²	C₂×C₄×Dic₃	1	1	yes	characteristic
189	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
190	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
191	C₄	—	12	4	no	—
192	C₄	C₃⋊#95	1	2	yes	—
193	C₄	C₃⋊#95	1	2	yes	—
194	C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
195	C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
196	C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
197	C₄	C₂×#217	1	1	yes	characteristic
198	C₃	#1742	1	1	yes	characteristic, complemented, Sylow
199	C₂	—	2	2	no	—
200	C₂	—	2	2	no	—
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#5132	1	2	yes	—
204	C₂	C₂×C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₃)⋊C₄
commutator	C₂²×C₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂²×C₃×M₁₆	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	#1742
LCS	G ▷ C₂²×C₃ ▷ 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. #4554	C₂	semidirect
9. C₂×C₃×M₁₆	C₄	semidirect
21. C₂×#208	C₄	semidirect
63. C₂×M₁₆	Dic₃	semidirect
198. C₃	#1742	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×#208	2
#4554	2
#1742	3