Subgroups — #21707

Order 256.

499 conjugacy classes of subgroups.

The subgroup lattice has 179 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×#444	C₂	1	2	yes	maximal, complemented
3	C₂×#443	C₂	1	2	yes	maximal, complemented
4	#2932	C₂	1	2	yes	maximal, complemented
5	#1975	C₂	1	2	yes	maximal, complemented
6	#1974	C₂	1	2	yes	maximal, complemented
7	#1963	C₂	1	2	yes	maximal, complemented
8	#1848	C₂	1	1	yes	maximal, characteristic, complemented
9	#1726	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135	C₂²	1	2	yes	—
11	C₂×#135	C₂²	1	2	yes	—
12	C₂×#131	C₂²	1	4	yes	—
13	#532	C₂²	1	2	yes	—
14	#495	C₄	1	2	yes	complemented
15	#495	C₄	1	2	yes	complemented
16	#494	C₄	1	1	yes	characteristic, complemented
17	#493	C₄	1	1	yes	characteristic, complemented
18	#492	C₄	1	2	yes	complemented
19	#444	—	2	8	no	—
20	#443	—	2	8	no	—
21	#435	—	2	8	no	—
22	C₂×#108	C₂²	1	2	yes	—
23	C₂×#108	C₂²	1	2	yes	—
24	#423	C₂²	1	2	yes	—
25	C₂×#104	C₂²	1	4	yes	—
26	C₂×#104	C₂²	1	4	yes	—
27	#408	C₂²	1	4	yes	—
28	C₂×#97	C₂²	1	1	yes	characteristic
29	C₂×#97	C₂²	1	1	yes	characteristic
30	C₂×#97	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
34	#376	C₂²	1	1	yes	characteristic
35	C₂×#41	C₂³	1	4	yes	—
36	#135	—	2	4	no	—
37	#135	—	2	4	no	—
38	#135	—	2	4	no	—
39	#135	—	2	4	no	—
40	C₂×Q₁₆	C₂×C₄	1	2	yes	—
41	C₂×Q₁₆	C₂×C₄	1	2	yes	—
42	C₂×SD₁₆	C₂×C₄	1	4	yes	—
43	C₂×SD₁₆	C₂×C₄	1	4	yes	—
44	C₂×D₈	C₂×C₄	1	2	yes	—
45	C₂×D₈	C₂×C₄	1	2	yes	—
46	#131	—	2	8	no	—
47	#131	—	2	8	no	—
48	C₂×M₁₆	—	2	4	no	—
49	C₂×M₁₆	C₂³	1	4	yes	—
50	C₂²×C₈	—	2	4	no	—
51	C₂²×C₈	C₂³	1	1	yes	characteristic
52	C₂²×C₈	C₂³	1	1	yes	characteristic
53	C₂²×C₈	C₂³	1	1	yes	characteristic
54	C₂²×C₈	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈	C₂×C₄	1	2	yes	—
56	#127	C₂×C₄	1	2	yes	—
57	#124	C₂×C₄	1	4	yes	—
58	#121	—	2	4	no	—
59	C₄×D₄	—	2	4	no	—
60	C₂×C₄²	C₂³	1	1	yes	characteristic
61	C₂×C₄²	C₂³	1	1	yes	characteristic
62	C₂×C₄²	C₂³	1	1	yes	characteristic
63	#108	—	2	4	no	—
64	#108	—	2	4	no	—
65	C₄⋊C₈	—	2	4	no	—
66	#104	—	2	8	no	—
67	#104	—	2	8	no	—
68	C₂²⋊C₈	—	2	8	no	—
69	#97	—	2	2	no	—
70	#97	D₄	1	4	yes	—
71	#97	D₄	1	4	yes	complemented
72	C₄×C₈	—	2	2	no	—
73	C₄×C₈	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈	C₂×C₄	1	1	yes	characteristic
75	C₄×C₈	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈	D₄	1	4	yes	complemented
78	#95	—	2	2	no	—
79	#41	—	2	4	no	—
80	#41	—	2	4	no	—
81	#41	—	2	4	no	—
82	#41	—	2	4	no	—
83	C₂×Q₈	C₂²×C₄	1	4	yes	—
84	C₂×D₄	—	2	4	no	—
85	C₂×D₄	—	2	4	no	—
86	C₂×D₄	C₂²×C₄	1	4	yes	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	2	4	no	—
90	C₂²×C₄	#41	1	2	yes	—
91	C₂²×C₄	C₂⁴	1	1	yes	characteristic
92	Q₁₆	—	2	4	no	—
93	Q₁₆	—	2	4	no	—
94	SD₁₆	—	2	8	no	—
95	SD₁₆	—	2	8	no	—
96	D₈	—	2	4	no	—
97	D₈	—	2	4	no	—
98	M₁₆	—	2	4	no	—
99	M₁₆	—	2	4	no	—
100	M₁₆	—	4	8	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	—	4	8	no	—
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	C₂×D₄	1	2	yes	—
110	C₂×C₈	C₂×D₄	1	2	yes	—
111	C₂×C₈	C₂×D₄	1	2	yes	—
112	C₂×C₈	C₂×D₄	1	2	yes	—
113	C₂×C₈	C₂×D₄	1	2	yes	—
114	C₂×C₈	C₂×D₄	1	2	yes	—
115	C₂×C₈	#41	1	2	yes	—
116	C₂×C₈	#41	1	2	yes	—
117	C₄⋊C₄	—	2	4	no	—
118	C₂²⋊C₄	—	2	8	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	2	1	no	—
121	C₄²	C₂²×C₄	1	1	yes	characteristic
122	C₄²	C₂²×C₄	1	1	yes	characteristic
123	C₄²	C₂×D₄	1	2	yes	—
124	C₄²	C₂×D₄	1	2	yes	—
125	C₄²	C₂×D₄	1	2	yes	—
126	C₂³	—	2	4	no	—
127	C₂³	C₂×#41	1	1	yes	characteristic
128	Q₈	—	2	4	no	—
129	Q₈	—	2	4	no	—
130	D₄	—	2	4	no	—
131	D₄	—	2	4	no	—
132	D₄	—	4	4	no	—
133	D₄	—	4	4	no	—
134	D₄	—	8	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	4	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	C₄×D₄	1	2	yes	—
144	C₂×C₄	C₄×D₄	1	2	yes	—
145	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄	C₂×#41	1	1	yes	characteristic
150	C₂×C₄	C₂×#41	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	4	4	no	—
153	C₈	C₄×D₄	1	2	yes	—
154	C₈	C₄×D₄	1	2	yes	—
155	C₈	C₄×D₄	1	2	yes	—
156	C₈	C₄×D₄	1	2	yes	—
157	C₈	#127	1	4	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	—	4	4	no	—
160	C₂²	—	4	8	no	—
161	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂²	#532	1	1	yes	characteristic
163	C₂²	#532	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	2	2	no	—
166	C₄	—	4	2	no	—
167	C₄	—	4	2	no	—
168	C₄	—	4	4	no	—
169	C₄	C₂×C₄×D₄	1	1	yes	characteristic
170	C₄	C₂×C₄×D₄	1	1	yes	characteristic
171	C₄	C₂×#127	1	1	yes	characteristic
172	C₄	#532	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	—	8	4	no	—
176	C₂	C₄×#127	1	1	yes	characteristic
177	C₂	#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#444	C₂	semidirect
3. C₂×#443	C₂	semidirect
4. #2932	C₂	semidirect
5. #1975	C₂	semidirect
6. #1974	C₂	semidirect
7. #1963	C₂	semidirect
8. #1848	C₂	semidirect
9. #1726	C₂	semidirect
14. #495	C₄	semidirect
15. #495	C₄	semidirect
16. #494	C₄	semidirect
17. #493	C₄	semidirect
18. #492	C₄	semidirect
71. #97	D₄	semidirect
77. C₄×C₈	D₄	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#3147	2
C₂×#444	2
C₂×#443	2
#2932	2
#1975	2
#1974	2
#1963	2
#1848	2
#1726	2