Subgroups — #10920

Order 256.

349 conjugacy classes of subgroups.

The subgroup lattice has 223 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#424	C₂	1	1	yes	maximal, characteristic
2	#2873	C₂	1	1	yes	maximal, characteristic
3	C₂×#344	C₂	1	1	yes	maximal, characteristic
4	C₂×#344	C₂	1	1	yes	maximal, characteristic
5	#1740	C₂	1	1	yes	maximal, characteristic
6	#1729	C₂	1	1	yes	maximal, characteristic
7	#1729	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆	C₂²	1	1	yes	characteristic
11	C₂³×C₈	C₂²	1	1	yes	characteristic
12	C₂×#117	C₂²	1	1	yes	characteristic
13	C₂×#117	C₂²	1	1	yes	characteristic
14	C₂×#117	C₂²	1	1	yes	characteristic
15	#424	—	2	2	no	—
16	#424	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₈	—	2	2	no	—
18	#405	—	2	2	no	—
19	#405	—	2	2	no	—
20	C₄×M₁₆	—	2	2	no	—
21	#344	—	2	2	no	—
22	#344	C₄	1	4	yes	complemented
23	#342	C₄	1	4	yes	—
24	#336	—	2	2	no	—
25	#336	—	2	2	no	—
26	C₂³×C₄	C₂³	1	1	yes	characteristic
27	C₂×M₁₆	—	2	1	no	—
28	C₂×M₁₆	—	2	1	no	—
29	C₂×M₁₆	—	2	1	no	—
30	C₂×M₁₆	—	2	1	no	—
31	C₂×M₁₆	—	2	1	no	—
32	C₂×M₁₆	—	2	1	no	—
33	C₂×M₁₆	—	2	1	no	—
34	C₂×M₁₆	—	2	1	no	—
35	C₂×M₁₆	—	2	1	no	—
36	C₂×M₁₆	—	2	1	no	—
37	C₂×M₁₆	—	2	1	no	—
38	C₂×M₁₆	—	2	1	no	—
39	C₂×M₁₆	—	2	1	no	—
40	C₂×M₁₆	C₂×C₄	1	2	yes	—
41	C₂×M₁₆	C₂×C₄	1	2	yes	—
42	C₂×M₁₆	C₂×C₄	1	2	yes	—
43	C₂×M₁₆	D₄	1	2	yes	—
44	C₂×M₁₆	Q₈	1	2	yes	—
45	C₂²×C₈	—	2	1	no	—
46	C₂²×C₈	—	2	1	no	—
47	C₂²×C₈	—	2	1	no	—
48	C₂²×C₈	—	2	1	no	—
49	C₂²×C₈	—	2	1	no	—
50	C₂²×C₈	—	2	1	no	—
51	C₂²×C₈	—	2	1	no	—
52	C₂²×C₈	C₂×C₄	1	2	yes	—
53	C₂²×C₈	D₄	1	2	yes	—
54	C₂²×C₈	D₄	1	2	yes	—
55	#117	—	2	1	no	—
56	#117	—	2	1	no	—
57	#117	—	2	1	no	—
58	#117	—	2	1	no	—
59	#117	—	2	1	no	—
60	#117	—	2	1	no	—
61	#117	—	2	1	no	—
62	#117	—	2	1	no	—
63	#117	—	2	1	no	—
64	#117	—	2	1	no	—
65	#117	C₂×C₄	1	2	yes	—
66	#117	C₂×C₄	1	2	yes	—
67	C₂×C₄⋊C₄	—	2	1	no	—
68	C₂×C₄⋊C₄	—	2	1	no	—
69	C₂×C₄⋊C₄	Q₈	1	2	yes	—
70	C₂×C₂²⋊C₄	C₂×C₄	1	2	yes	—
71	C₂×C₂²⋊C₄	C₂×C₄	1	2	yes	—
72	C₂×C₂²⋊C₄	C₂×C₄	1	2	yes	—
73	C₂×C₄²	—	2	1	no	—
74	C₂×C₄²	—	2	1	no	—
75	C₂×C₄²	D₄	1	2	yes	—
76	C₄⋊C₈	—	2	8	no	—
77	#97	—	2	4	no	—
78	C₄×C₈	—	2	4	no	—
79	C₂⁴	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	C₄⋊C₄	1	4	yes	—
83	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
90	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
91	C₂²×C₄	#41	1	1	yes	characteristic
92	C₂²×C₄	#41	1	1	yes	characteristic
93	C₂²×C₄	#41	1	1	yes	characteristic
94	C₂²×C₄	#41	1	1	yes	characteristic
95	C₂²×C₄	#41	1	1	yes	characteristic
96	C₂²×C₄	#41	1	1	yes	characteristic
97	M₁₆	—	2	2	no	—
98	M₁₆	—	2	2	no	—
99	M₁₆	—	2	2	no	—
100	M₁₆	—	2	2	no	—
101	M₁₆	—	2	2	no	—
102	M₁₆	—	2	2	no	—
103	M₁₆	—	2	2	no	—
104	M₁₆	—	2	2	no	—
105	M₁₆	—	2	2	no	—
106	M₁₆	—	2	2	no	—
107	M₁₆	—	4	1	no	—
108	M₁₆	—	4	1	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	C₂²⋊C₄	1	4	yes	—
124	C₂×C₈	C₄⋊C₄	1	4	yes	—
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	4	1	no	—
130	C₄⋊C₄	—	4	1	no	—
131	C₂²⋊C₄	—	2	2	no	—
132	C₂²⋊C₄	—	2	2	no	—
133	C₂²⋊C₄	—	2	2	no	—
134	C₂²⋊C₄	—	2	2	no	—
135	C₂²⋊C₄	—	2	2	no	—
136	C₂²⋊C₄	C₄²	1	4	yes	—
137	C₄²	—	2	2	no	—
138	C₄²	—	2	2	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	4	1	no	—
142	C₄²	—	4	1	no	—
143	C₂³	—	4	2	no	—
144	C₂³	C₂×C₄²	1	1	yes	characteristic
145	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂³	#117	1	1	yes	characteristic
147	C₂³	C₄×D₄	1	1	yes	characteristic
148	C₂³	C₄×D₄	1	1	yes	characteristic
149	C₂³	C₄×D₄	1	1	yes	characteristic
150	C₂³	C₄×Q₈	1	1	yes	characteristic
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	4	4	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄	#117	1	1	yes	characteristic
160	C₂×C₄	#117	1	1	yes	characteristic
161	C₂×C₄	C₄×D₄	1	1	yes	characteristic
162	C₂×C₄	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
167	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
168	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
169	C₂×C₄	#120	1	1	yes	characteristic
170	C₂×C₄	#121	1	1	yes	characteristic
171	C₂×C₄	#121	1	1	yes	characteristic
172	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
173	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
174	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
175	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
176	C₂×C₄	#123	1	1	yes	characteristic
177	C₂×C₄	#123	1	1	yes	characteristic
178	C₂×C₄	#123	1	1	yes	characteristic
179	C₂×C₄	#124	1	1	yes	characteristic
180	C₂×C₄	#125	1	1	yes	characteristic
181	C₂×C₄	#125	1	1	yes	characteristic
182	C₂×C₄	#126	1	1	yes	characteristic
183	C₂×C₄	#126	1	1	yes	characteristic
184	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	2	4	no	—
187	C₈	—	4	2	no	—
188	C₈	—	4	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	4	1	no	—
193	C₂²	—	4	1	no	—
194	C₂²	—	4	1	no	—
195	C₂²	—	4	1	no	—
196	C₂²	C₄×C₂²⋊C₄	1	1	yes	characteristic
197	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
198	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
199	C₂²	#385	1	1	yes	characteristic
200	C₂²	#387	1	1	yes	characteristic
201	C₂²	#388	1	1	yes	characteristic
202	C₂²	#389	1	1	yes	characteristic
203	C₄	—	4	4	no	—
204	C₄	—	4	4	no	—
205	C₄	—	8	2	no	—
206	C₄	#385	1	1	yes	characteristic
207	C₄	#386	1	1	yes	characteristic
208	C₄	#387	1	1	yes	characteristic
209	C₄	#387	1	1	yes	characteristic
210	C₄	#387	1	1	yes	characteristic
211	C₄	#388	1	1	yes	characteristic
212	C₄	#389	1	1	yes	characteristic
213	C₄	#391	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	—	2	1	no	—
218	C₂	—	2	1	no	—
219	C₂	—	2	1	no	—
220	C₂	#1436	1	1	yes	characteristic
221	C₂	#1868	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#120
commutator	C₂³	C₂×C₄²
Frattini, cosocle	C₂³×C₄	C₂³
socle	C₂²	#385
prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂³ ▷ 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. #424	C₄	semidirect
22. #344	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#424	2
#2873	2
C₂×#344	2
#1740	2
#1729	2