Subgroups — #10871

Order 256.

373 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#427	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	#1740	C₂	1	2	yes	maximal
5	#1733	C₂	1	2	yes	maximal
6	C₂²×M₁₆	C₂²	1	1	yes	characteristic
7	C₂³×C₈	C₂²	1	1	yes	characteristic
8	C₂×#117	C₂²	1	2	yes	—
9	C₂×#117	C₂²	1	2	yes	—
10	C₂×#117	C₂²	1	1	yes	characteristic
11	#427	—	2	2	no	—
12	#427	—	2	1	no	—
13	#427	—	2	1	no	—
14	#427	—	2	1	no	—
15	#427	—	2	1	no	—
16	#427	C₄	1	4	yes	complemented
17	C₂×#107	—	2	1	no	—
18	C₂×#107	C₄	1	2	yes	complemented
19	C₂×#106	—	2	1	no	—
20	C₂×#106	C₄	1	2	yes	complemented
21	#405	—	2	2	no	—
22	#405	C₄	1	4	yes	complemented
23	C₂×#97	—	2	1	no	—
24	C₂×C₄×C₈	—	2	1	no	—
25	#342	—	2	4	no	—
26	#339	—	2	4	no	—
27	C₂³×C₄	C₂³	1	1	yes	characteristic
28	C₂×M₁₆	—	2	1	no	—
29	C₂×M₁₆	—	2	1	no	—
30	C₂×M₁₆	—	2	1	no	—
31	C₂×M₁₆	C₂×C₄	1	2	yes	—
32	C₂×M₁₆	D₄	1	2	yes	—
33	C₂×M₁₆	Q₈	1	2	yes	—
34	C₂²×C₈	—	2	1	no	—
35	C₂²×C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂×C₄	1	2	yes	—
39	C₂²×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₈	D₄	1	2	yes	—
42	C₂²×C₈	D₄	1	1	yes	characteristic
43	C₂²×C₈	D₄	1	1	yes	characteristic
44	C₂²×C₈	Q₈	1	2	yes	—
45	#117	—	2	1	no	—
46	#117	—	2	1	no	—
47	#117	—	2	2	no	—
48	#117	—	2	2	no	—
49	#117	—	2	2	no	—
50	#117	—	2	2	no	—
51	#117	—	2	2	no	—
52	#117	—	2	2	no	—
53	#117	—	2	2	no	—
54	#117	C₂×C₄	1	4	yes	—
55	#117	C₂×C₄	1	2	yes	—
56	#117	Q₈	1	2	yes	—
57	C₂×C₄⋊C₄	—	2	1	no	—
58	C₂×C₄⋊C₄	—	2	2	no	—
59	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
60	C₂×C₂²⋊C₄	—	2	2	no	—
61	C₂×C₂²⋊C₄	—	2	2	no	—
62	C₂×C₂²⋊C₄	D₄	1	2	yes	—
63	C₂×C₄²	—	2	1	no	—
64	C₂×C₄²	—	2	2	no	—
65	C₂×C₄²	—	2	2	no	—
66	#107	—	2	4	no	—
67	#107	—	2	2	no	—
68	#107	—	2	2	no	—
69	#106	—	2	4	no	—
70	#106	—	2	2	no	—
71	#106	—	2	2	no	—
72	#97	—	2	4	no	—
73	C₄×C₈	—	2	4	no	—
74	C₂⁴	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	—	2	4	no	—
77	C₂²×C₄	—	4	2	no	—
78	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
88	C₂²×C₄	#41	1	1	yes	characteristic
89	C₂²×C₄	#41	1	1	yes	characteristic
90	C₂²×C₄	#41	1	1	yes	characteristic
91	C₂²×C₄	#41	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	C₄⋊C₄	1	4	yes	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	C₄²	1	2	yes	—
100	C₂×C₈	C₄²	1	2	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	1	yes	characteristic
105	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₈	C₂²⋊C₄	1	2	yes	—
107	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₈	C₄⋊C₄	1	2	yes	—
110	C₂×C₈	C₄⋊C₄	1	2	yes	—
111	C₂×C₈	C₄⋊C₄	1	2	yes	—
112	C₂×C₈	C₄⋊C₄	1	2	yes	—
113	C₂×C₈	C₄⋊C₄	1	2	yes	—
114	C₂×C₈	C₄⋊C₄	1	2	yes	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₂²⋊C₄	—	2	2	no	—
122	C₂²⋊C₄	—	2	4	no	—
123	C₂²⋊C₄	—	4	2	no	—
124	C₂²⋊C₄	—	4	2	no	—
125	C₂²⋊C₄	—	4	2	no	—
126	C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
127	C₄²	—	2	2	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	2	4	no	—
130	C₄²	—	4	2	no	—
131	C₄²	—	4	2	no	—
132	C₄²	—	4	2	no	—
133	C₂³	—	4	2	no	—
134	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂³	C₄×D₄	1	1	yes	characteristic
136	C₂³	C₄×D₄	1	1	yes	characteristic
137	C₂³	#121	1	1	yes	characteristic
138	C₂³	#121	1	1	yes	characteristic
139	C₂³	#127	1	1	yes	characteristic
140	C₂³	C₄⋊Q₈	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	4	4	no	—
145	C₂×C₄	—	8	4	no	—
146	C₂×C₄	—	8	4	no	—
147	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄	#117	1	1	yes	characteristic
156	C₂×C₄	#117	1	1	yes	characteristic
157	C₂×C₄	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄	C₄×D₄	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	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
163	C₂×C₄	#121	1	1	yes	characteristic
164	C₂×C₄	#121	1	1	yes	characteristic
165	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
166	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
167	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
168	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
169	C₂×C₄	#124	1	1	yes	characteristic
170	C₂×C₄	#124	1	1	yes	characteristic
171	C₂×C₄	#125	1	1	yes	characteristic
172	C₂×C₄	#125	1	1	yes	characteristic
173	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
174	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
175	C₈	—	2	4	no	—
176	C₈	#95	1	2	yes	—
177	C₈	#95	1	2	yes	—
178	C₈	#95	1	2	yes	—
179	C₈	#95	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	4	1	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	1	no	—
186	C₂²	—	4	1	no	—
187	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
188	C₂²	#380	1	1	yes	characteristic
189	C₂²	#384	1	1	yes	characteristic
190	C₂²	#389	1	1	yes	characteristic
191	C₂²	#389	1	1	yes	characteristic
192	C₂²	#390	1	1	yes	characteristic
193	C₂²	#390	1	1	yes	characteristic
194	C₄	—	4	4	no	—
195	C₄	—	8	4	no	—
196	C₄	—	8	4	no	—
197	C₄	C₂×#95	1	1	yes	characteristic
198	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
199	C₄	#380	1	1	yes	characteristic
200	C₄	#382	1	1	yes	characteristic
201	C₄	#389	1	1	yes	characteristic
202	C₄	#389	1	1	yes	characteristic
203	C₄	#391	1	1	yes	characteristic
204	C₄	#391	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	—	2	1	no	—
211	C₂	C₄⋊#95	1	1	yes	characteristic
212	C₂	#1976	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₄	C₂³
socle	C₂²	#390
prime radical	C₂⁴	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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
16. #427	C₄	semidirect
18. C₂×#107	C₄	semidirect
20. C₂×#106	C₄	semidirect
22. #405	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#427	2
C₂×#405	2
#1740	2
#1733	2