Subgroups — C₂×#1378 #12552

Order 256.

293 conjugacy classes of subgroups.

The subgroup lattice has 213 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×#405	C₂	1	1	yes	maximal, characteristic
4	#1378	C₂	1	4	yes	maximal, direct product factor
5	C₂²×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₈	C₂²	1	1	yes	characteristic
7	C₂×#117	C₄	1	1	yes	characteristic
8	C₂×M₃₂	—	2	1	no	—
9	C₂×M₃₂	—	2	1	no	—
10	C₂×M₃₂	—	2	1	no	—
11	C₂×M₃₂	—	2	1	no	—
12	C₂×M₃₂	—	2	1	no	—
13	C₂×M₃₂	—	2	1	no	—
14	C₂×M₃₂	—	2	1	no	—
15	C₂×M₃₂	C₄	1	2	yes	complemented
16	C₂×M₃₂	C₄	1	2	yes	complemented
17	C₂×M₃₂	C₄	1	2	yes	complemented
18	C₂×M₃₂	C₂²	1	2	yes	—
19	C₂×M₃₂	C₂²	1	2	yes	—
20	C₂²×C₁₆	—	2	1	no	—
21	C₂²×C₁₆	C₄	1	2	yes	complemented
22	#405	—	2	1	no	—
23	#405	—	2	1	no	—
24	#405	C₄	1	2	yes	—
25	#405	C₂²	1	2	yes	—
26	C₂×#97	—	2	1	no	—
27	C₂×C₄×C₈	—	2	1	no	—
28	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂×M₁₆	—	2	1	no	—
30	C₂×M₁₆	—	2	1	no	—
31	C₂×M₁₆	—	2	1	no	—
32	C₂×M₁₆	C₈	1	2	yes	—
33	C₂×M₁₆	C₈	1	2	yes	—
34	C₂×M₁₆	C₂×C₄	1	2	yes	—
35	C₂²×C₈	—	2	1	no	—
36	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈	D₄	1	1	yes	characteristic
42	C₂²×C₈	D₄	1	1	yes	characteristic
43	C₂²×C₈	D₄	1	1	yes	characteristic
44	C₂²×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₈	D₄	1	1	yes	characteristic
46	C₂²×C₈	D₄	1	1	yes	characteristic
47	C₂²×C₈	Q₈	1	1	yes	characteristic
48	C₂²×C₈	Q₈	1	1	yes	characteristic
49	C₂²×C₈	C₂³	1	1	yes	characteristic
50	#117	—	2	1	no	—
51	#117	—	2	1	no	—
52	#117	C₈	1	2	yes	—
53	#117	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄	—	2	1	no	—
55	C₂×C₂²⋊C₄	C₈	1	2	yes	—
56	C₂×C₄²	—	2	1	no	—
57	M₃₂	—	2	2	no	—
58	M₃₂	—	2	2	no	—
59	M₃₂	—	2	2	no	—
60	M₃₂	—	2	2	no	—
61	M₃₂	—	2	2	no	—
62	M₃₂	—	2	2	no	—
63	M₃₂	—	2	2	no	—
64	M₃₂	C₂×C₄	1	4	yes	complemented
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	—	2	2	no	—
67	C₂×C₁₆	—	2	2	no	—
68	C₂×C₁₆	—	2	2	no	—
69	C₂×C₁₆	—	2	2	no	—
70	C₂×C₁₆	C₂×C₄	1	4	yes	complemented
71	#97	—	2	2	no	—
72	#97	—	2	2	no	—
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	—	2	2	no	—
75	C₂⁴	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄	—	2	2	no	—
77	C₂²×C₄	C₄²	1	1	yes	characteristic
78	C₂²×C₄	C₄²	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
86	C₂²×C₄	M₁₆	1	1	yes	characteristic
87	C₂²×C₄	M₁₆	1	1	yes	characteristic
88	C₂²×C₄	M₁₆	1	1	yes	characteristic
89	C₂²×C₄	M₁₆	1	1	yes	characteristic
90	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
91	M₁₆	—	2	2	no	—
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	C₂×C₈	1	4	yes	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	C₄²	1	1	yes	characteristic
99	C₂×C₈	C₄²	1	1	yes	characteristic
100	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	2	2	no	—
128	C₂²⋊C₄	—	2	2	no	—
129	C₂²⋊C₄	C₂×C₈	1	4	yes	—
130	C₄²	—	2	2	no	—
131	C₄²	—	2	2	no	—
132	C₁₆	—	2	4	no	—
133	C₁₆	—	2	4	no	—
134	C₂³	—	4	2	no	—
135	C₂³	C₄×C₈	1	1	yes	characteristic
136	C₂³	C₄×C₈	1	1	yes	characteristic
137	C₂³	C₂²⋊C₈	1	1	yes	characteristic
138	C₂³	C₂²⋊C₈	1	1	yes	characteristic
139	C₂³	C₄⋊C₈	1	1	yes	characteristic
140	C₂³	C₄⋊C₈	1	1	yes	characteristic
141	C₂³	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	#95	1	1	yes	characteristic
145	C₂×C₄	#95	1	1	yes	characteristic
146	C₂×C₄	#95	1	1	yes	characteristic
147	C₂×C₄	#95	1	1	yes	characteristic
148	C₂×C₄	C₄×C₈	1	1	yes	characteristic
149	C₂×C₄	C₄×C₈	1	1	yes	characteristic
150	C₂×C₄	#97	1	1	yes	characteristic
151	C₂×C₄	#97	1	1	yes	characteristic
152	C₂×C₄	#97	1	1	yes	characteristic
153	C₂×C₄	#97	1	1	yes	characteristic
154	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
155	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
156	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
157	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
158	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
159	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
160	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
161	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
162	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
163	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
164	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
165	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
167	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
170	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
171	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
172	C₈	—	2	4	no	—
173	C₈	#95	1	1	yes	characteristic
174	C₈	#95	1	1	yes	characteristic
175	C₈	#95	1	1	yes	characteristic
176	C₈	#95	1	1	yes	characteristic
177	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
178	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
179	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	1	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	1	no	—
188	C₂²	#336	1	1	yes	characteristic
189	C₂²	#336	1	1	yes	characteristic
190	C₂²	#336	1	1	yes	characteristic
191	C₂²	#336	1	1	yes	characteristic
192	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
193	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
194	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
195	C₄	—	4	4	no	—
196	C₄	#336	1	1	yes	characteristic
197	C₄	#336	1	1	yes	characteristic
198	C₄	#336	1	1	yes	characteristic
199	C₄	#336	1	1	yes	characteristic
200	C₄	C₂×#95	1	1	yes	characteristic
201	C₄	C₂×#97	1	1	yes	characteristic
202	C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
203	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
204	C₂	—	2	1	no	—
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	#1378	1	2	yes	direct product factor
211	C₂	C₂×#336	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊C₄
commutator	C₂²	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	#336
prime radical	C₂⁴	C₂×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1378	C₂	direct (N × H)
15. C₂×M₃₂	C₄	semidirect
16. C₂×M₃₂	C₄	semidirect
17. C₂×M₃₂	C₄	semidirect
21. C₂²×C₁₆	C₄	semidirect
64. M₃₂	C₂×C₄	semidirect
70. C₂×C₁₆	C₂×C₄	semidirect
210. C₂	#1378	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×M₃₂	2
C₂×#405	2
#1378	2

Subgroups — C2×#1378 #12552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1378 #12552