Subgroups — #10869

Order 256.

387 conjugacy classes of subgroups.

The subgroup lattice has 199 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	C₂²×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×#405	C₂	1	1	yes	maximal, characteristic
4	C₂×#343	C₂	1	2	yes	maximal
5	#1736	C₂	1	2	yes	maximal
6	C₂²×M₁₆	C₂²	1	2	yes	—
7	C₂²×M₁₆	C₂²	1	1	yes	characteristic
8	C₂³×C₈	C₂²	1	1	yes	characteristic
9	C₂×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×#108	—	2	2	no	—
12	C₂×#108	—	2	2	no	—
13	C₂×#108	C₄	1	4	yes	complemented
14	C₂×#106	—	2	1	no	—
15	C₂×#106	—	2	1	no	—
16	C₂×#106	—	2	2	no	—
17	C₂×#106	C₄	1	4	yes	complemented
18	#405	—	2	2	no	—
19	#405	C₄	1	4	yes	complemented
20	C₂×#97	—	2	1	no	—
21	C₂×C₄×C₈	—	2	1	no	—
22	#343	—	2	4	no	—
23	#340	—	2	4	no	—
24	C₂³×C₄	—	2	2	no	—
25	C₂³×C₄	C₂³	1	1	yes	characteristic
26	C₂×M₁₆	—	2	1	no	—
27	C₂×M₁₆	—	2	1	no	—
28	C₂×M₁₆	—	2	1	no	—
29	C₂×M₁₆	—	2	2	no	—
30	C₂×M₁₆	—	2	2	no	—
31	C₂×M₁₆	—	2	2	no	—
32	C₂×M₁₆	—	2	2	no	—
33	C₂×M₁₆	—	2	2	no	—
34	C₂×M₁₆	C₂×C₄	1	4	yes	—
35	C₂×M₁₆	C₂×C₄	1	2	yes	—
36	C₂×M₁₆	D₄	1	2	yes	—
37	C₂×M₁₆	Q₈	1	2	yes	—
38	C₂²×C₈	—	2	1	no	—
39	C₂²×C₈	—	2	2	no	—
40	C₂²×C₈	C₂×C₄	1	2	yes	—
41	C₂²×C₈	C₂×C₄	1	2	yes	—
42	C₂²×C₈	C₂×C₄	1	2	yes	—
43	C₂²×C₈	D₄	1	1	yes	characteristic
44	C₂²×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₈	D₄	1	2	yes	—
46	C₂²×C₈	D₄	1	2	yes	—
47	C₂²×C₈	Q₈	1	2	yes	—
48	#117	—	2	1	no	—
49	#117	—	2	1	no	—
50	#117	C₂×C₄	1	2	yes	—
51	#117	Q₈	1	2	yes	—
52	C₂×C₄⋊C₄	—	2	2	no	—
53	C₂×C₄⋊C₄	—	2	2	no	—
54	C₂×C₄⋊C₄	—	2	2	no	—
55	C₂×C₄⋊C₄	—	2	2	no	—
56	C₂×C₄⋊C₄	—	2	2	no	—
57	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
58	C₂×C₄⋊C₄	D₄	1	2	yes	—
59	C₂×C₂²⋊C₄	—	2	1	no	—
60	C₂×C₄²	—	2	1	no	—
61	#108	—	2	8	no	—
62	#106	—	2	4	no	—
63	#106	—	2	4	no	—
64	#97	—	2	4	no	—
65	C₄×C₈	—	2	4	no	—
66	C₂⁴	#41	1	1	yes	characteristic
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	2	4	no	—
69	C₂²×C₄	—	4	2	no	—
70	C₂²×C₄	—	4	2	no	—
71	C₂²×C₄	—	8	4	no	—
72	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄	C₂×D₄	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×D₄	1	1	yes	characteristic
79	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄	C₂×Q₈	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	M₁₆	—	2	2	no	—
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	4	no	—
90	M₁₆	—	2	4	no	—
91	M₁₆	—	4	2	no	—
92	M₁₆	—	4	2	no	—
93	M₁₆	C₄⋊C₄	1	4	yes	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	4	2	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	C₄²	1	4	yes	—
101	C₂×C₈	C₂²⋊C₄	1	2	yes	—
102	C₂×C₈	C₂²⋊C₄	1	2	yes	—
103	C₂×C₈	C₂²⋊C₄	1	2	yes	—
104	C₂×C₈	C₂²⋊C₄	1	2	yes	—
105	C₂×C₈	C₂²⋊C₄	1	4	yes	—
106	C₂×C₈	C₄⋊C₄	1	4	yes	—
107	C₂×C₈	C₄⋊C₄	1	4	yes	—
108	C₂×C₈	C₄⋊C₄	1	2	yes	—
109	C₂×C₈	C₄⋊C₄	1	2	yes	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	4	no	—
112	C₄⋊C₄	—	2	4	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	C₄⋊C₄	1	4	yes	—
116	C₂²⋊C₄	—	2	2	no	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₄²	—	2	2	no	—
119	C₄²	—	2	2	no	—
120	C₂³	—	2	4	no	—
121	C₂³	#117	1	1	yes	characteristic
122	C₂³	C₄×D₄	1	1	yes	characteristic
123	C₂³	C₄×Q₈	1	1	yes	characteristic
124	C₂³	#121	1	1	yes	characteristic
125	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂³	#124	1	1	yes	characteristic
127	C₂³	#125	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	4	4	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	—	8	4	no	—
132	C₂×C₄	—	8	4	no	—
133	C₂×C₄	—	8	4	no	—
134	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄	#117	1	1	yes	characteristic
144	C₂×C₄	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
148	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
149	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
150	C₂×C₄	#121	1	1	yes	characteristic
151	C₂×C₄	#121	1	1	yes	characteristic
152	C₂×C₄	#121	1	1	yes	characteristic
153	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
154	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
156	C₂×C₄	#124	1	1	yes	characteristic
157	C₂×C₄	#125	1	1	yes	characteristic
158	C₂×C₄	#127	1	1	yes	characteristic
159	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
160	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
161	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
162	C₈	—	2	4	no	—
163	C₈	—	4	4	no	—
164	C₈	#95	1	4	yes	—
165	C₈	#95	1	4	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
173	C₂²	#380	1	1	yes	characteristic
174	C₂²	#382	1	1	yes	characteristic
175	C₂²	#389	1	1	yes	characteristic
176	C₂²	#389	1	1	yes	characteristic
177	C₂²	#390	1	1	yes	characteristic
178	C₂²	#391	1	1	yes	characteristic
179	C₂²	#471	1	4	yes	—
180	C₄	—	4	4	no	—
181	C₄	—	8	4	no	—
182	C₄	C₂×#95	1	1	yes	characteristic
183	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
184	C₄	#380	1	1	yes	characteristic
185	C₄	#384	1	1	yes	characteristic
186	C₄	#389	1	1	yes	characteristic
187	C₄	#389	1	1	yes	characteristic
188	C₄	#390	1	1	yes	characteristic
189	C₄	#391	1	1	yes	characteristic
190	C₂	—	2	1	no	—
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	C₄⋊#95	1	1	yes	characteristic
195	C₂	#1815	1	2	yes	—
196	C₂	#1951	1	2	yes	—
197	C₂	#1977	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	#121
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₄	C₂³
prime radical	C₂⁴	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
13. C₂×#108	C₄	semidirect
17. C₂×#106	C₄	semidirect
19. #405	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#108	2
C₂²×#106	2
C₂×#405	2
C₂×#343	2
#1736	2