Subgroups — #17831

Order 256.

294 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2965	C₂	1	1	yes	maximal, characteristic
2	#2925	C₂	1	2	yes	maximal
3	#1574	C₂	1	2	yes	maximal, complemented
4	#1573	C₂	1	2	yes	maximal, complemented
5	#1572	C₂	1	2	yes	maximal, complemented
6	#1569	C₂	1	2	yes	maximal, complemented
7	#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	#1565	C₂	1	2	yes	maximal, complemented
9	#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#41	C₂²	1	1	yes	characteristic
11	#446	C₄	1	1	yes	characteristic
12	#446	C₄	1	1	yes	characteristic
13	#436	C₄	1	1	yes	characteristic
14	#436	C₄	1	1	yes	characteristic
15	#435	C₄	1	2	yes	—
16	#435	C₄	1	2	yes	—
17	#433	—	2	2	no	—
18	#433	—	2	2	no	—
19	#432	C₂²	1	2	yes	—
20	#432	C₂²	1	2	yes	—
21	#431	C₂²	1	2	yes	—
22	#431	C₂²	1	2	yes	—
23	#431	C₂²	1	1	yes	characteristic
24	#431	C₂²	1	2	yes	—
25	#423	C₂²	1	2	yes	—
26	C₂×C₄⋊C₈	C₂²	1	2	yes	—
27	#408	—	2	2	no	—
28	#405	—	2	2	no	—
29	C₄×M₁₆	C₂²	1	2	yes	—
30	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
31	#335	C₂²	1	2	yes	—
32	#335	C₂²	1	2	yes	—
33	#335	C₂²	1	2	yes	—
34	#335	C₂²	1	2	yes	—
35	#334	C₂²	1	1	yes	characteristic
36	#334	C₂²	1	1	yes	characteristic
37	#334	C₂²	1	2	yes	—
38	#334	C₂²	1	1	yes	characteristic
39	#334	C₂²	1	1	yes	characteristic
40	#334	C₂²	1	2	yes	—
41	C₂×#41	D₄	1	1	yes	characteristic
42	C₂×M₁₆	—	2	2	no	—
43	C₂×M₁₆	C₂×C₄	1	2	yes	—
44	C₂×M₁₆	C₂×C₄	1	2	yes	—
45	C₂²×C₈	—	2	2	no	—
46	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
49	C₄×Q₈	Q₈	1	1	yes	characteristic
50	C₄×D₄	C₂×C₄	1	2	yes	—
51	C₄×D₄	C₂×C₄	1	1	yes	characteristic
52	C₄×D₄	Q₈	1	1	yes	characteristic
53	C₄×D₄	Q₈	1	2	yes	—
54	#117	D₄	1	2	yes	—
55	#117	D₄	1	1	yes	characteristic
56	C₂×C₄²	C₂³	1	1	yes	characteristic
57	C₂×C₄²	C₂³	1	2	yes	—
58	C₄⋊C₈	—	2	2	no	—
59	C₄⋊C₈	—	2	2	no	—
60	C₄⋊C₈	C₂×C₄	1	2	yes	—
61	C₄⋊C₈	C₂×C₄	1	2	yes	—
62	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
63	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
64	C₄⋊C₈	C₂³	1	2	yes	—
65	C₄⋊C₈	C₂³	1	2	yes	—
66	C₄⋊C₈	C₂³	1	2	yes	—
67	C₄⋊C₈	C₂³	1	2	yes	—
68	C₂²⋊C₈	—	2	2	no	—
69	C₂²⋊C₈	—	2	2	no	—
70	C₂²⋊C₈	—	2	2	no	—
71	C₂²⋊C₈	—	2	2	no	—
72	C₂²⋊C₈	C₂×C₄	1	2	yes	—
73	C₂²⋊C₈	C₂×C₄	1	2	yes	—
74	C₂²⋊C₈	C₂×C₄	1	2	yes	—
75	#97	—	2	2	no	—
76	#97	—	2	2	no	—
77	#97	C₂×C₄	1	2	yes	—
78	#97	C₂×C₄	1	2	yes	—
79	C₄×C₈	—	2	2	no	—
80	C₄×C₈	—	2	2	no	—
81	C₄×C₈	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈	C₂³	1	1	yes	characteristic
84	C₄×C₈	C₂³	1	1	yes	characteristic
85	C₄×C₈	C₂³	1	2	yes	—
86	#41	—	4	2	no	—
87	C₂×Q₈	C₄⋊C₄	1	1	yes	characteristic
88	C₂×D₄	C₄⋊C₄	1	2	yes	—
89	C₂×D₄	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄	C₂²×C₄	1	2	yes	—
92	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄	C₂×D₄	1	2	yes	—
94	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄	C₂×Q₈	1	2	yes	—
96	M₁₆	—	2	4	no	—
97	M₁₆	—	4	2	no	—
98	M₁₆	—	4	4	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	4	1	no	—
105	C₂×C₈	—	4	4	no	—
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	2	yes	—
110	C₂×C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
113	C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
114	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
115	C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
116	C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
117	C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
118	C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
119	C₄²	C₂²×C₄	1	1	yes	characteristic
120	C₄²	C₂²×C₄	1	2	yes	—
121	C₄²	C₂×D₄	1	1	yes	characteristic
122	C₄²	C₂×D₄	1	2	yes	—
123	C₄²	C₂×Q₈	1	1	yes	characteristic
124	C₄²	C₂×Q₈	1	2	yes	—
125	C₄²	C₂⁴	1	1	yes	characteristic
126	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂³	C₂×C₄⋊C₄	1	2	yes	—
128	Q₈	—	4	1	no	—
129	D₄	—	4	2	no	—
130	D₄	—	4	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
146	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	2	no	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	2	2	no	—
154	C₈	—	4	2	no	—
155	C₈	—	4	2	no	—
156	C₈	—	4	2	no	—
157	C₈	—	4	2	no	—
158	C₈	#131	1	2	yes	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	2	no	—
165	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	2	no	—
173	C₄	C₂×#131	1	1	yes	characteristic
174	C₄	#568	1	1	yes	characteristic
175	C₄	#575	1	2	yes	—
176	C₂	—	4	1	no	—
177	C₂	—	4	2	no	—
178	C₂	#2906	1	1	yes	characteristic
179	C₂	#2911	1	1	yes	characteristic
180	C₂	#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂²×C₄⋊C₄
prime radical	C₂×#41	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1574	C₂	semidirect
4. #1573	C₂	semidirect
5. #1572	C₂	semidirect
6. #1569	C₂	semidirect
7. #1568	C₂	semidirect
8. #1565	C₂	semidirect
9. #1564	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2965	2
#2925	2
#1574	2
#1573	2
#1572	2
#1569	2
#1568	2
#1565	2
#1564	2