Subgroups — C₂×#4556 #71992

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#4556	C₂	1	4	yes	maximal, direct product factor
5	C₂×#344	—	3	1	no	maximal, Sylow
6	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂×C₃×M₁₆	—	2	1	no	—
8	C₂×C₃×M₁₆	—	2	1	no	—
9	C₂×C₃×M₁₆	—	2	1	no	—
10	C₂×C₃×M₁₆	—	2	1	no	—
11	C₂×C₃×M₁₆	C₄	1	2	yes	complemented
12	C₂×C₃×M₁₆	C₂²	1	2	yes	—
13	C₂²×C₃×C₈	C₄	1	2	yes	complemented
14	C₂×#217	C₄	1	2	yes	—
15	#926	—	2	1	no	—
16	#926	—	2	1	no	—
17	#926	C₄	1	2	yes	—
18	#926	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₃	—	2	1	no	—
20	C₂×C₄×Dic₃	—	2	1	no	—
21	C₂×#208	—	2	1	no	—
22	C₂×#208	—	2	1	no	—
23	C₂×#208	—	2	1	no	—
24	C₂×#208	C₄	1	2	yes	complemented
25	C₂×#208	C₄	1	2	yes	complemented
26	C₂×#208	C₂²	1	2	yes	—
27	C₂²×C₃⋊C₈	—	2	1	no	—
28	C₂²×M₁₆	—	3	1	no	—
29	C₂²×M₁₆	D₃	1	1	yes	characteristic
30	C₂×#117	—	3	1	no	—
31	#344	—	3	4	no	—
32	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
39	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
40	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
41	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
42	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
43	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
44	C₂²×C₃×C₄	Q₈	1	1	yes	characteristic
45	C₂²×C₃×C₄	Q₈	1	1	yes	characteristic
46	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
47	C₂²×Dic₃	—	2	2	no	—
48	C₃×M₁₆	—	2	2	no	—
49	C₃×M₁₆	—	2	2	no	—
50	C₃×M₁₆	—	2	2	no	—
51	C₃×M₁₆	—	2	2	no	—
52	C₂×C₃×C₈	—	2	2	no	—
53	C₂×C₃×C₈	—	2	2	no	—
54	C₂×C₃×C₈	C₂×C₄	1	4	yes	complemented
55	#217	—	2	2	no	—
56	#217	C₂×C₄	1	4	yes	—
57	C₄⋊Dic₃	—	2	2	no	—
58	C₄⋊Dic₃	—	2	2	no	—
59	C₄×Dic₃	—	2	2	no	—
60	C₄×Dic₃	—	2	2	no	—
61	#208	—	2	2	no	—
62	#208	—	2	2	no	—
63	#208	—	2	2	no	—
64	#208	C₂×C₄	1	4	yes	complemented
65	C₂×C₃⋊C₈	—	2	2	no	—
66	C₂×C₃⋊C₈	—	2	2	no	—
67	C₂×C₃⋊C₈	—	2	2	no	—
68	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
69	C₂×M₁₆	—	2	1	no	—
70	C₂×M₁₆	—	2	1	no	—
71	C₂×M₁₆	—	2	1	no	—
72	C₂×M₁₆	—	2	1	no	—
73	C₂×M₁₆	—	3	2	no	—
74	C₂×M₁₆	—	3	2	no	—
75	C₂×M₁₆	—	3	2	no	—
76	C₂×M₁₆	—	6	1	no	—
77	C₂×M₁₆	—	6	1	no	—
78	C₂×M₁₆	—	6	1	no	—
79	C₂×M₁₆	Dic₃	1	2	yes	complemented
80	C₂×M₁₆	C₂×D₃	1	2	yes	—
81	C₂²×C₈	—	6	1	no	—
82	C₂²×C₈	Dic₃	1	2	yes	complemented
83	#117	—	3	2	no	—
84	#117	—	3	2	no	—
85	#117	—	6	1	no	—
86	#117	—	6	1	no	—
87	C₂×C₄⋊C₄	—	6	1	no	—
88	C₂×C₂²⋊C₄	—	3	2	no	—
89	C₂×C₄²	—	6	1	no	—
90	C₂³×C₃	—	4	2	no	—
91	C₂³×C₃	C₄²	1	1	yes	characteristic
92	C₂³×C₃	C₄²	1	1	yes	characteristic
93	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
99	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
100	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
126	C₂×Dic₃	—	2	4	no	—
127	C₂×Dic₃	—	4	4	no	—
128	C₃×C₈	—	2	4	no	—
129	C₃⋊C₈	—	2	4	no	—
130	C₂⁴	C₄×D₃	1	1	yes	characteristic
131	C₂²×C₄	—	6	2	no	—
132	C₂²×C₄	Dic₆	1	1	yes	characteristic
133	C₂²×C₄	Dic₆	1	1	yes	characteristic
134	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
135	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
136	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
137	C₂²×C₄	D₁₂	1	1	yes	characteristic
138	C₂²×C₄	D₁₂	1	1	yes	characteristic
139	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
141	C₂²×C₄	#67	1	1	yes	characteristic
142	C₂²×C₄	#67	1	1	yes	characteristic
143	C₂²×C₄	#67	1	1	yes	characteristic
144	C₂²×C₄	#67	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
146	M₁₆	—	2	2	no	—
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	2	no	—
149	M₁₆	—	2	2	no	—
150	M₁₆	—	3	4	no	—
151	M₁₆	—	6	2	no	—
152	M₁₆	—	6	2	no	—
153	M₁₆	—	6	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	6	2	no	—
157	C₂×C₈	—	6	2	no	—
158	C₂×C₈	—	6	2	no	—
159	C₂×C₈	C₂×Dic₃	1	4	yes	complemented
160	C₄⋊C₄	—	6	2	no	—
161	C₄⋊C₄	—	6	2	no	—
162	C₂²⋊C₄	—	3	4	no	—
163	C₂²⋊C₄	—	6	2	no	—
164	C₄²	—	6	2	no	—
165	C₄²	—	6	2	no	—
166	C₂²×C₃	—	2	2	no	—
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	—	2	2	no	—
169	C₂²×C₃	—	4	1	no	—
170	C₂²×C₃	—	4	1	no	—
171	C₂²×C₃	—	4	1	no	—
172	C₂²×C₃	—	4	1	no	—
173	C₂²×C₃	#95	1	1	yes	characteristic
174	C₂²×C₃	#95	1	1	yes	characteristic
175	C₂²×C₃	#95	1	1	yes	characteristic
176	C₂²×C₃	#95	1	1	yes	characteristic
177	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
178	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
179	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₃×C₄	#95	1	1	yes	characteristic
181	C₃×C₄	#95	1	1	yes	characteristic
182	C₃×C₄	#95	1	1	yes	characteristic
183	C₃×C₄	#95	1	1	yes	characteristic
184	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
186	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
188	Dic₃	—	4	4	no	—
189	C₂³	—	4	2	no	—
190	C₂³	C₄×Dic₃	1	1	yes	characteristic
191	C₂³	C₄×Dic₃	1	1	yes	characteristic
192	C₂³	Dic₃⋊C₄	1	1	yes	characteristic
193	C₂³	Dic₃⋊C₄	1	1	yes	characteristic
194	C₂³	(C₂×D₃)⋊C₄	1	1	yes	characteristic
195	C₂³	(C₂×D₃)⋊C₄	1	1	yes	characteristic
196	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
197	C₂×C₄	—	6	4	no	—
198	C₂×C₄	—	12	4	no	—
199	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
200	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
201	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
202	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
203	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
204	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
205	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
206	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
207	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
208	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
209	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
210	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
211	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
212	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
213	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
214	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
215	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
216	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
217	C₂×C₄	#217	1	1	yes	characteristic
218	C₂×C₄	#217	1	1	yes	characteristic
219	C₂×C₄	#217	1	1	yes	characteristic
220	C₂×C₄	#217	1	1	yes	characteristic
221	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
222	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
223	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
224	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
225	C₂×C₄	C₂×#67	1	1	yes	characteristic
226	C₂×C₄	C₂×#67	1	1	yes	characteristic
227	C₈	—	2	4	no	—
228	C₈	—	6	4	no	—
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	2	1	no	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	#344	1	2	yes	complemented
236	C₂×C₃	C₂×#95	1	1	yes	characteristic
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	C₃⋊#95	1	1	yes	characteristic
245	C₂²	C₃⋊#95	1	1	yes	characteristic
246	C₂²	C₃⋊#95	1	1	yes	characteristic
247	C₂²	C₃⋊#95	1	1	yes	characteristic
248	C₂²	C₂×C₄×Dic₃	1	1	yes	characteristic
249	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
250	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
251	C₄	—	12	4	no	—
252	C₄	C₃⋊#95	1	1	yes	characteristic
253	C₄	C₃⋊#95	1	1	yes	characteristic
254	C₄	C₃⋊#95	1	1	yes	characteristic
255	C₄	C₃⋊#95	1	1	yes	characteristic
256	C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
257	C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
258	C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
259	C₄	C₂×#217	1	1	yes	characteristic
260	C₃	C₂×#344	1	1	yes	characteristic, complemented, Sylow
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	#4556	1	2	yes	direct product factor
268	C₂	C₂×C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#217
commutator	C₂²×C₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂²×C₃×M₁₆	C₂
socle	C₂²×C₃	#95
prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	C₂×#344
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. #4556	C₂	direct (N × H)
11. C₂×C₃×M₁₆	C₄	semidirect
13. C₂²×C₃×C₈	C₄	semidirect
24. C₂×#208	C₄	semidirect
25. C₂×#208	C₄	semidirect
54. C₂×C₃×C₈	C₂×C₄	semidirect
64. #208	C₂×C₄	semidirect
79. C₂×M₁₆	Dic₃	semidirect
82. C₂²×C₈	Dic₃	semidirect
159. C₂×C₈	C₂×Dic₃	semidirect
235. C₂×C₃	#344	semidirect
260. C₃	C₂×#344	semidirect
267. C₂	#4556	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
C₂×#926	2
C₂²×#208	2
#4556	2
C₂×#344	3

Subgroups — C2×#4556 #71992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4556 #71992