Subgroups — #71982

Order 384.

404 conjugacy classes of subgroups.

The subgroup lattice has 210 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	C₂×#926	C₂	1	1	yes	maximal, characteristic
4	#4555	C₂	1	4	yes	maximal, complemented
5	#1733	—	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	—	2	1	no	—
13	C₂×#217	—	2	1	no	—
14	#926	—	2	1	no	—
15	#926	—	2	1	no	—
16	#926	—	2	2	no	—
17	#926	C₄	1	2	yes	complemented
18	#926	C₄	1	4	yes	complemented
19	#926	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₃	—	2	1	no	—
21	C₂×C₄⋊Dic₃	C₄	1	2	yes	complemented
22	C₂×C₄×Dic₃	—	2	1	no	—
23	C₂×C₄×Dic₃	C₂²	1	2	yes	—
24	C₂²×M₁₆	D₃	1	1	yes	characteristic
25	C₂×#117	—	3	1	no	—
26	C₂×#117	—	3	1	no	—
27	#339	—	3	4	no	—
28	C₂⁴×C₃	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
32	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
33	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
34	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
35	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₄	D₄	1	2	yes	—
37	C₂²×C₃×C₄	Q₈	1	2	yes	—
38	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₂²×Dic₃	—	2	2	no	—
40	C₂²×Dic₃	—	2	2	no	—
41	C₃×M₁₆	—	2	4	no	—
42	C₃×M₁₆	—	2	2	no	—
43	C₃×M₁₆	C₂×C₄	1	4	yes	complemented
44	C₂×C₃×C₈	—	2	4	no	—
45	C₂×C₃×C₈	—	2	2	no	—
46	#217	—	2	2	no	—
47	#217	—	2	4	no	—
48	#217	—	2	2	no	—
49	C₄⋊Dic₃	—	2	4	no	—
50	C₄⋊Dic₃	—	2	2	no	—
51	C₄⋊Dic₃	C₂×C₄	1	4	yes	—
52	C₄×Dic₃	—	2	2	no	—
53	C₄×Dic₃	—	2	2	no	—
54	C₄×Dic₃	—	2	2	no	—
55	C₄×Dic₃	C₂×C₄	1	4	yes	complemented
56	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
57	C₂×M₁₆	—	2	2	no	—
58	C₂×M₁₆	—	2	1	no	—
59	C₂×M₁₆	Dic₃	1	4	yes	complemented
60	C₂×M₁₆	C₂×D₃	1	2	yes	—
61	C₂²×C₈	—	2	1	no	—
62	#117	—	3	4	no	—
63	#117	—	3	2	no	—
64	#117	—	3	2	no	—
65	#117	—	6	1	no	—
66	#117	—	6	2	no	—
67	#117	—	6	1	no	—
68	C₂×C₄⋊C₄	—	3	2	no	—
69	C₂×C₄⋊C₄	—	6	1	no	—
70	C₂×C₂²⋊C₄	—	6	1	no	—
71	C₂×C₂²⋊C₄	—	6	1	no	—
72	C₂×C₄²	—	3	2	no	—
73	C₂×C₄²	—	6	1	no	—
74	C₂³×C₃	—	4	2	no	—
75	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂³×C₃	C₂²⋊C₄	1	2	yes	—
78	C₂³×C₃	C₄⋊C₄	1	2	yes	—
79	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄	C₄²	1	2	yes	—
81	C₂×C₃×C₄	C₄²	1	2	yes	—
82	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
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	C₂²×C₄	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	—	2	4	no	—
100	C₂×Dic₃	—	4	4	no	—
101	C₂×Dic₃	—	4	4	no	—
102	C₃×C₈	—	2	4	no	—
103	C₂⁴	D₁₂	1	1	yes	characteristic
104	C₂²×C₄	—	6	2	no	—
105	C₂²×C₄	—	6	2	no	—
106	C₂²×C₄	Dic₆	1	2	yes	—
107	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
108	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
109	C₂²×C₄	C₄×D₃	1	2	yes	—
110	C₂²×C₄	D₁₂	1	1	yes	characteristic
111	C₂²×C₄	C₂×Dic₃	1	2	yes	—
112	C₂²×C₄	#67	1	2	yes	—
113	C₂²×C₄	#67	1	1	yes	characteristic
114	C₂²×C₄	#67	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
116	M₁₆	—	2	4	no	—
117	M₁₆	—	2	2	no	—
118	M₁₆	C₂×Dic₃	1	4	yes	complemented
119	C₂×C₈	—	2	4	no	—
120	C₂×C₈	—	2	2	no	—
121	C₄⋊C₄	—	3	4	no	—
122	C₄⋊C₄	—	6	2	no	—
123	C₄⋊C₄	—	6	4	no	—
124	C₂²⋊C₄	—	6	4	no	—
125	C₂²⋊C₄	—	6	2	no	—
126	C₂²⋊C₄	—	6	2	no	—
127	C₄²	—	3	4	no	—
128	C₄²	—	6	2	no	—
129	C₄²	—	6	2	no	—
130	C₄²	—	6	2	no	—
131	C₂²×C₃	—	2	2	no	—
132	C₂²×C₃	—	2	2	no	—
133	C₂²×C₃	—	2	2	no	—
134	C₂²×C₃	—	4	2	no	—
135	C₂²×C₃	—	4	2	no	—
136	C₂²×C₃	#95	1	2	yes	—
137	C₂²×C₃	#95	1	2	yes	—
138	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₃×C₄	#95	1	2	yes	—
142	C₃×C₄	#95	1	2	yes	—
143	C₃×C₄	C₂×C₄²	1	1	yes	characteristic
144	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
147	Dic₃	—	4	4	no	—
148	Dic₃	—	4	4	no	—
149	C₂³	—	4	2	no	—
150	C₂³	C₄⋊Dic₃	1	2	yes	—
151	C₂³	(C₂×D₃)⋊C₄	1	1	yes	characteristic
152	C₂³	(C₂×D₃)⋊C₄	1	1	yes	characteristic
153	C₂³	(C₂×D₃)⋊C₄	1	2	yes	—
154	C₂³	C₂×D₁₂	1	1	yes	characteristic
155	C₂×C₄	—	6	4	no	—
156	C₂×C₄	—	6	4	no	—
157	C₂×C₄	—	12	4	no	—
158	C₂×C₄	—	12	4	no	—
159	C₂×C₄	C₄×Dic₃	1	2	yes	—
160	C₂×C₄	C₄×Dic₃	1	2	yes	—
161	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
162	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
163	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
164	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
165	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
166	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
167	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
168	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
169	C₂×C₄	#217	1	2	yes	—
170	C₂×C₄	#217	1	2	yes	—
171	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
175	C₂×C₄	C₂×#67	1	1	yes	characteristic
176	C₂×C₄	C₂×#67	1	1	yes	characteristic
177	C₈	—	2	4	no	—
178	C₂×C₃	—	2	1	no	—
179	C₂×C₃	—	2	2	no	—
180	C₂×C₃	—	2	2	no	—
181	C₂×C₃	—	2	1	no	—
182	C₂×C₃	C₂×#95	1	1	yes	characteristic
183	C₂×C₃	#421	1	2	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	4	2	no	—
188	C₂²	—	4	2	no	—
189	C₂²	C₃⋊#95	1	2	yes	—
190	C₂²	C₃⋊#95	1	2	yes	—
191	C₂²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
192	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
193	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
194	C₄	—	12	4	no	—
195	C₄	—	12	4	no	—
196	C₄	C₃⋊#95	1	2	yes	—
197	C₄	C₃⋊#95	1	2	yes	—
198	C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
199	C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
200	C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
201	C₄	C₂×#217	1	1	yes	characteristic
202	C₃	#1733	1	1	yes	characteristic, complemented, Sylow
203	C₂	—	2	1	no	—
204	C₂	—	2	2	no	—
205	C₂	—	2	2	no	—
206	C₂	—	2	1	no	—
207	C₂	#5139	1	2	yes	—
208	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×C₂²⋊C₄
prime radical	C₂⁴×C₃	D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	#1733
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. #4555	C₂	semidirect
9. C₂×C₃×M₁₆	C₄	semidirect
17. #926	C₄	semidirect
18. #926	C₄	semidirect
21. C₂×C₄⋊Dic₃	C₄	semidirect
43. C₃×M₁₆	C₂×C₄	semidirect
55. C₄×Dic₃	C₂×C₄	semidirect
59. C₂×M₁₆	Dic₃	semidirect
118. M₁₆	C₂×Dic₃	semidirect
202. C₃	#1733	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
#4555	2
#1733	3