Subgroups — #33443

Order 256.

324 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3407	C₂	1	2	yes	maximal, complemented
2	C₂×#443	C₂	1	1	yes	maximal, characteristic
3	#2171	C₂	1	4	yes	maximal, complemented
4	#2170	C₂	1	4	yes	maximal, complemented
5	C₂×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	#2117	C₂	1	2	yes	maximal, complemented
7	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135	C₄	1	1	yes	characteristic
9	C₂×#131	C₂²	1	2	yes	—
10	#504	—	2	4	no	—
11	#504	C₂²	1	8	yes	complemented
12	C₂×M₃₂	—	2	4	no	—
13	C₂×M₃₂	C₂²	1	2	yes	—
14	C₂×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×M₃₂	C₂²	1	1	yes	characteristic
16	#443	C₄	1	2	yes	—
17	#443	C₄	1	2	yes	—
18	#443	C₂²	1	2	yes	—
19	#443	C₂²	1	2	yes	—
20	C₂×#108	C₄	1	1	yes	characteristic
21	C₂×#104	C₄	1	2	yes	—
22	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
23	#364	C₂²	1	2	yes	complemented
24	#364	C₂²	1	2	yes	complemented
25	#350	C₂²	1	8	yes	complemented
26	#346	C₂²	1	2	yes	complemented
27	C₄×C₁₆	C₂²	1	2	yes	complemented
28	C₂×#41	C₂×C₄	1	2	yes	—
29	#135	C₈	1	2	yes	—
30	#135	C₈	1	2	yes	—
31	#135	C₂×C₄	1	2	yes	—
32	#135	C₂×C₄	1	2	yes	—
33	C₂×Q₁₆	C₈	1	1	yes	characteristic
34	C₂×SD₁₆	C₈	1	2	yes	—
35	C₂×D₈	C₈	1	1	yes	characteristic
36	#131	—	2	2	no	—
37	#131	—	2	2	no	—
38	#131	C₂×C₄	1	4	yes	—
39	#131	C₂³	1	4	yes	—
40	C₂×M₁₆	—	2	2	no	—
41	C₂×M₁₆	C₂×C₄	1	2	yes	—
42	C₂²×C₈	—	2	2	no	—
43	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈	C₂³	1	1	yes	characteristic
45	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
46	M₃₂	—	2	2	no	—
47	M₃₂	—	2	1	no	—
48	M₃₂	—	2	1	no	—
49	M₃₂	—	2	8	no	—
50	M₃₂	D₄	1	2	yes	—
51	M₃₂	D₄	1	2	yes	—
52	M₃₂	C₂³	1	4	yes	—
53	C₂×C₁₆	—	2	2	no	—
54	C₂×C₁₆	—	2	8	no	—
55	C₂×C₁₆	C₂³	1	2	yes	—
56	C₂×C₁₆	C₂³	1	2	yes	—
57	#108	C₂×C₄	1	2	yes	—
58	#108	C₂×C₄	1	2	yes	—
59	#104	C₂×C₄	1	4	yes	—
60	#104	C₂×C₄	1	4	yes	—
61	C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈	C₂³	1	1	yes	characteristic
64	C₄×C₈	C₂³	1	1	yes	characteristic
65	#41	—	2	2	no	—
66	#41	—	2	2	no	—
67	#41	C₂×C₈	1	4	yes	—
68	#41	C₂²×C₄	1	4	yes	—
69	C₂×Q₈	C₂×C₈	1	2	yes	—
70	C₂×D₄	—	2	2	no	—
71	C₂×D₄	C₂×C₈	1	2	yes	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	2	1	no	—
74	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
75	Q₁₆	C₂×C₈	1	2	yes	—
76	Q₁₆	C₂×C₈	1	2	yes	—
77	SD₁₆	C₂×C₈	1	4	yes	—
78	SD₁₆	C₂×C₈	1	4	yes	—
79	D₈	C₂×C₈	1	2	yes	—
80	D₈	C₂×C₈	1	2	yes	—
81	M₁₆	—	2	2	no	—
82	M₁₆	—	2	4	no	—
83	M₁₆	—	2	4	no	—
84	M₁₆	C₂²×C₄	1	4	yes	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	4	no	—
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	C₂×C₈	1	1	yes	characteristic
89	C₂×C₈	C₂×C₈	1	1	yes	characteristic
90	C₂×C₈	C₂×C₈	1	1	yes	characteristic
91	C₂×C₈	C₂×C₈	1	1	yes	characteristic
92	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈	#41	1	1	yes	characteristic
98	C₂×C₈	#41	1	1	yes	characteristic
99	C₂×C₈	C₂⁴	1	1	yes	characteristic
100	C₄²	—	2	1	no	—
101	C₄²	C₂²×C₄	1	1	yes	characteristic
102	C₄²	C₂²×C₄	1	1	yes	characteristic
103	C₁₆	—	2	4	no	—
104	C₁₆	—	2	2	no	—
105	C₁₆	C₂×D₄	1	4	yes	—
106	C₂³	—	2	2	no	—
107	C₂³	#131	1	1	yes	characteristic
108	Q₈	—	2	2	no	—
109	Q₈	C₂²×C₈	1	4	yes	—
110	D₄	—	2	4	no	—
111	D₄	—	2	4	no	—
112	D₄	—	2	2	no	—
113	D₄	C₂²×C₈	1	4	yes	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄	#131	1	1	yes	characteristic
125	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
126	C₈	—	2	4	no	—
127	C₈	C₄×D₄	1	1	yes	characteristic
128	C₈	C₄×D₄	1	1	yes	characteristic
129	C₈	C₂²×C₈	1	1	yes	characteristic
130	C₈	C₂²×C₈	1	1	yes	characteristic
131	C₈	C₂²×C₈	1	1	yes	characteristic
132	C₈	C₂²×C₈	1	1	yes	characteristic
133	C₈	C₂²×D₄	1	1	yes	characteristic
134	C₈	C₂×#41	1	1	yes	characteristic
135	C₂²	—	2	4	no	—
136	C₂²	—	4	1	no	—
137	C₂²	—	8	2	no	—
138	C₂²	C₈×D₄	1	1	yes	characteristic
139	C₂²	C₈×D₄	1	1	yes	characteristic
140	C₂²	C₂×#131	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	2	4	no	—
143	C₄	—	4	1	no	—
144	C₄	C₈×D₄	1	1	yes	characteristic
145	C₄	C₈×D₄	1	1	yes	characteristic
146	C₄	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄	C₂³×C₈	1	1	yes	characteristic
148	C₂	—	2	1	no	—
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	—	4	4	no	—
152	C₂	C₂×C₈×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×D₄
commutator	C₄	C₂³×C₈
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂	C₂×C₈×D₄
prime radical	C₂×#135	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3407	C₂	semidirect
3. #2171	C₂	semidirect
4. #2170	C₂	semidirect
5. C₂×#364	C₂	semidirect
6. #2117	C₂	semidirect
7. C₄×M₃₂	C₂	semidirect
11. #504	C₂²	semidirect
23. #364	C₂²	semidirect
24. #364	C₂²	semidirect
25. #350	C₂²	semidirect
26. #346	C₂²	semidirect
27. C₄×C₁₆	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3407	2
C₂×#443	2
#2171	2
#2170	2
C₂×#364	2
#2117	2
C₄×M₃₂	2