Subgroups — C₂×#1833 #20756

Order 256.

393 conjugacy classes of subgroups.

The subgroup lattice has 209 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×#427	C₂	1	1	yes	maximal, characteristic
3	C₂×#405	C₂	1	1	yes	maximal, characteristic
4	#1833	C₂	1	8	yes	maximal, direct product factor
5	C₂×#344	C₂	1	2	yes	maximal
6	C₂×H(3,ℤ/4)	C₂	1	2	yes	maximal
7	C₂²×M₁₆	C₂²	1	2	yes	—
8	C₂²×M₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₈	C₂²	1	1	yes	characteristic
10	C₂×#117	C₂²	1	2	yes	—
11	C₂×#117	C₂²	1	1	yes	characteristic
12	C₂×#108	—	2	1	no	—
13	C₂×#108	—	2	1	no	—
14	C₂×#108	—	2	2	no	—
15	C₂×#108	C₂²	1	4	yes	—
16	#427	—	2	2	no	—
17	#427	C₂²	1	4	yes	—
18	C₂×#107	—	2	1	no	—
19	C₂×#106	—	2	1	no	—
20	#405	C₄	1	4	yes	complemented
21	#405	C₂²	1	4	yes	—
22	C₂×#97	C₄	1	2	yes	complemented
23	C₂×C₄×C₈	C₄	1	2	yes	complemented
24	#344	C₂²	1	8	yes	—
25	H(3,ℤ/4)	C₂²	1	8	yes	—
26	C₂³×C₄	C₂³	1	1	yes	characteristic
27	C₂×M₁₆	—	2	1	no	—
28	C₂×M₁₆	—	2	1	no	—
29	C₂×M₁₆	—	2	1	no	—
30	C₂×M₁₆	—	2	1	no	—
31	C₂×M₁₆	—	2	2	no	—
32	C₂×M₁₆	—	2	2	no	—
33	C₂×M₁₆	—	2	2	no	—
34	C₂×M₁₆	—	2	2	no	—
35	C₂×M₁₆	—	2	2	no	—
36	C₂×M₁₆	C₂×C₄	1	2	yes	—
37	C₂×M₁₆	C₂³	1	4	yes	—
38	C₂×M₁₆	C₂³	1	2	yes	—
39	C₂²×C₈	—	2	2	no	—
40	C₂²×C₈	C₂×C₄	1	2	yes	—
41	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈	C₂×C₄	1	2	yes	—
44	C₂²×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₈	D₄	1	1	yes	characteristic
46	C₂²×C₈	D₄	1	2	yes	—
47	C₂²×C₈	Q₈	1	1	yes	characteristic
48	C₂²×C₈	Q₈	1	1	yes	characteristic
49	C₂²×C₈	Q₈	1	2	yes	—
50	C₂²×C₈	C₂³	1	2	yes	—
51	#117	—	2	1	no	—
52	#117	—	2	1	no	—
53	#117	—	2	2	no	—
54	#117	—	2	2	no	—
55	#117	—	2	2	no	—
56	#117	C₂×C₄	1	2	yes	—
57	#117	C₂³	1	4	yes	—
58	#117	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄	—	2	1	no	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₂²⋊C₄	—	2	1	no	—
62	C₂×C₂²⋊C₄	—	2	2	no	—
63	C₂×C₄²	—	2	2	no	—
64	C₂×C₄²	C₂×C₄	1	2	yes	—
65	#108	—	2	4	no	—
66	#108	—	2	4	no	—
67	#107	—	2	4	no	—
68	#106	—	2	4	no	—
69	#97	C₂×C₄	1	8	yes	complemented
70	C₄×C₈	C₂×C₄	1	8	yes	complemented
71	C₂⁴	#41	1	1	yes	characteristic
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	4	2	no	—
74	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄	#41	1	1	yes	characteristic
81	C₂²×C₄	#41	1	1	yes	characteristic
82	C₂²×C₄	#41	1	1	yes	characteristic
83	C₂²×C₄	#41	1	1	yes	characteristic
84	C₂²×C₄	#41	1	1	yes	characteristic
85	C₂²×C₄	#41	1	1	yes	characteristic
86	C₂²×C₄	#41	1	1	yes	characteristic
87	C₂²×C₄	C₂⁴	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	2	4	no	—
93	M₁₆	—	2	4	no	—
94	M₁₆	—	4	2	no	—
95	M₁₆	—	4	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	—	4	2	no	—
101	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₈	C₄⋊C₄	1	2	yes	—
104	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₈	C₄⋊C₄	1	2	yes	—
107	C₂×C₈	C₄⋊C₄	1	2	yes	—
108	C₂×C₈	C₄⋊C₄	1	2	yes	—
109	C₂×C₈	C₄⋊C₄	1	2	yes	—
110	C₂×C₈	C₄⋊C₄	1	2	yes	—
111	C₂×C₈	C₂²×C₄	1	4	yes	—
112	C₂×C₈	C₂²×C₄	1	2	yes	—
113	C₂×C₈	C₂²×C₄	1	2	yes	—
114	C₂×C₈	C₂×D₄	1	2	yes	—
115	C₂×C₈	C₂×D₄	1	2	yes	—
116	C₂×C₈	C₂×Q₈	1	2	yes	—
117	C₂×C₈	C₂×Q₈	1	2	yes	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₂²⋊C₄	—	2	2	no	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	—	2	4	no	—
125	C₂²⋊C₄	—	4	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	2	4	no	—
128	C₄²	—	4	2	no	—
129	C₄²	C₂²×C₄	1	4	yes	—
130	C₂³	—	4	2	no	—
131	C₂³	#117	1	1	yes	characteristic
132	C₂³	#117	1	1	yes	characteristic
133	C₂³	#124	1	1	yes	characteristic
134	C₂³	#124	1	1	yes	characteristic
135	C₂³	#125	1	1	yes	characteristic
136	C₂³	#125	1	1	yes	characteristic
137	C₂³	C₂×#41	1	1	yes	characteristic
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	4	4	no	—
140	C₂×C₄	—	4	4	no	—
141	C₂×C₄	—	8	4	no	—
142	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄	#117	1	1	yes	characteristic
147	C₂×C₄	#117	1	1	yes	characteristic
148	C₂×C₄	#117	1	1	yes	characteristic
149	C₂×C₄	#117	1	1	yes	characteristic
150	C₂×C₄	#117	1	1	yes	characteristic
151	C₂×C₄	#117	1	1	yes	characteristic
152	C₂×C₄	#124	1	1	yes	characteristic
153	C₂×C₄	#124	1	1	yes	characteristic
154	C₂×C₄	#124	1	1	yes	characteristic
155	C₂×C₄	#124	1	1	yes	characteristic
156	C₂×C₄	#124	1	1	yes	characteristic
157	C₂×C₄	#124	1	1	yes	characteristic
158	C₂×C₄	#125	1	1	yes	characteristic
159	C₂×C₄	#125	1	1	yes	characteristic
160	C₂×C₄	#125	1	1	yes	characteristic
161	C₂×C₄	#125	1	1	yes	characteristic
162	C₂×C₄	#125	1	1	yes	characteristic
163	C₂×C₄	#125	1	1	yes	characteristic
164	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
167	C₂×C₄	C₂×#41	1	1	yes	characteristic
168	C₂×C₄	C₂×#41	1	1	yes	characteristic
169	C₂×C₄	C₂×#41	1	1	yes	characteristic
170	C₈	—	2	4	no	—
171	C₈	—	4	4	no	—
172	C₈	C₂×C₄⋊C₄	1	2	yes	—
173	C₈	C₂×C₄⋊C₄	1	2	yes	—
174	C₈	C₂×C₄⋊C₄	1	2	yes	—
175	C₈	C₂×C₄⋊C₄	1	2	yes	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	4	1	no	—
180	C₂²	—	4	1	no	—
181	C₂²	—	4	1	no	—
182	C₂²	—	4	1	no	—
183	C₂²	#382	1	1	yes	characteristic
184	C₂²	#382	1	1	yes	characteristic
185	C₂²	#382	1	1	yes	characteristic
186	C₂²	#382	1	1	yes	characteristic
187	C₂²	C₂×#117	1	1	yes	characteristic
188	C₂²	C₂×#124	1	1	yes	characteristic
189	C₂²	C₂×#125	1	1	yes	characteristic
190	C₄	—	4	4	no	—
191	C₄	—	8	4	no	—
192	C₄	#382	1	1	yes	characteristic
193	C₄	#382	1	1	yes	characteristic
194	C₄	#382	1	1	yes	characteristic
195	C₄	#382	1	1	yes	characteristic
196	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
197	C₄	C₂×#117	1	1	yes	characteristic
198	C₄	C₂×#124	1	1	yes	characteristic
199	C₄	C₂×#125	1	1	yes	characteristic
200	C₂	—	2	1	no	—
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	—	2	1	no	—
206	C₂	#1833	1	2	yes	direct product factor
207	C₂	C₂×#382	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#117
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	#382
prime radical	C₂⁴	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1833	C₂	direct (N × H)
20. #405	C₄	semidirect
22. C₂×#97	C₄	semidirect
23. C₂×C₄×C₈	C₄	semidirect
69. #97	C₂×C₄	semidirect
70. C₄×C₈	C₂×C₄	semidirect
206. C₂	#1833	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×#108	2
C₂×#427	2
C₂×#405	2
#1833	2
C₂×#344	2
C₂×H(3,ℤ/4)	2

Subgroups — C2×#1833 #20756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1833 #20756