Subgroups — #10911

Order 256.

367 conjugacy classes of subgroups.

The subgroup lattice has 239 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	C₂×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#341	C₂	1	1	yes	maximal, characteristic
4	C₂×#341	C₂	1	1	yes	maximal, characteristic
5	C₂×#341	C₂	1	1	yes	maximal, characteristic
6	#1728	C₂	1	1	yes	maximal, characteristic
7	C₂×#336	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	C₂²×M₁₆	C₂²	1	1	yes	characteristic
12	C₂³×C₈	C₂²	1	1	yes	characteristic
13	C₂³×C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₄²	C₂²	1	1	yes	characteristic
15	C₂×#108	—	2	2	no	—
16	C₂×#108	—	2	2	no	—
17	C₂×#108	C₄	1	4	yes	complemented
18	C₄×M₁₆	—	2	2	no	—
19	C₄×M₁₆	—	2	2	no	—
20	C₂×#97	—	2	1	no	—
21	C₂×C₄×C₈	—	2	1	no	—
22	#341	—	2	2	no	—
23	#341	—	2	2	no	—
24	#341	C₄	1	4	yes	complemented
25	#336	—	2	1	no	—
26	#336	—	2	1	no	—
27	#336	C₄	1	4	yes	—
28	C₂³×C₄	Q₈	1	2	yes	—
29	C₂³×C₄	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
41	C₂×M₁₆	—	2	1	no	—
42	C₂×M₁₆	—	2	1	no	—
43	C₂×M₁₆	—	2	1	no	—
44	C₂×M₁₆	—	2	1	no	—
45	C₂×M₁₆	—	2	1	no	—
46	C₂×M₁₆	C₂×C₄	1	2	yes	—
47	C₂×M₁₆	C₂×C₄	1	2	yes	—
48	C₂×M₁₆	C₂×C₄	1	2	yes	—
49	C₂×M₁₆	C₂×C₄	1	2	yes	—
50	C₂×M₁₆	D₄	1	2	yes	—
51	C₂×M₁₆	D₄	1	2	yes	—
52	C₂×M₁₆	D₄	1	2	yes	—
53	C₂×M₁₆	Q₈	1	2	yes	—
54	C₂²×C₈	—	2	1	no	—
55	C₂²×C₈	—	2	1	no	—
56	C₂²×C₈	—	2	1	no	—
57	C₂²×C₈	—	2	1	no	—
58	C₂²×C₈	—	2	1	no	—
59	C₂²×C₈	—	2	1	no	—
60	C₂²×C₈	—	2	1	no	—
61	C₂²×C₈	—	2	1	no	—
62	C₂²×C₈	—	2	1	no	—
63	C₂²×C₈	—	2	1	no	—
64	C₂²×C₈	—	2	1	no	—
65	C₂²×C₈	—	2	1	no	—
66	C₂²×C₈	—	2	1	no	—
67	C₂²×C₈	—	2	1	no	—
68	C₂²×C₈	C₂×C₄	1	2	yes	—
69	C₂²×C₈	C₂×C₄	1	2	yes	—
70	C₂²×C₈	C₂×C₄	1	2	yes	—
71	C₂²×C₈	C₂×C₄	1	2	yes	—
72	C₂×C₄²	—	2	1	no	—
73	C₂×C₄²	—	2	1	no	—
74	C₂×C₄²	—	2	1	no	—
75	C₂×C₄²	—	2	1	no	—
76	C₂×C₄²	C₂×C₄	1	2	yes	—
77	C₂×C₄²	D₄	1	1	yes	characteristic
78	C₂×C₄²	D₄	1	1	yes	characteristic
79	#108	—	2	8	no	—
80	#97	—	2	4	no	—
81	C₄×C₈	—	2	4	no	—
82	C₂⁴	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	C₄⋊C₄	1	2	yes	—
88	C₂²×C₄	C₄⋊C₄	1	2	yes	—
89	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
97	C₂²×C₄	#41	1	1	yes	characteristic
98	C₂²×C₄	#41	1	1	yes	characteristic
99	C₂²×C₄	#41	1	1	yes	characteristic
100	C₂²×C₄	#41	1	1	yes	characteristic
101	C₂²×C₄	#41	1	1	yes	characteristic
102	C₂²×C₄	#41	1	1	yes	characteristic
103	M₁₆	—	2	2	no	—
104	M₁₆	—	2	2	no	—
105	M₁₆	—	2	2	no	—
106	M₁₆	—	2	2	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	2	2	no	—
113	M₁₆	—	4	1	no	—
114	M₁₆	—	4	1	no	—
115	M₁₆	—	4	1	no	—
116	M₁₆	—	4	1	no	—
117	M₁₆	C₂²⋊C₄	1	4	yes	—
118	M₁₆	C₄⋊C₄	1	4	yes	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	—	4	1	no	—
143	C₂×C₈	—	4	1	no	—
144	C₂×C₈	C₄²	1	4	yes	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₂³	—	2	2	no	—
152	C₂³	—	2	2	no	—
153	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂³	C₄×Q₈	1	1	yes	characteristic
155	C₂³	C₄×Q₈	1	1	yes	characteristic
156	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
157	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
158	C₂³	#125	1	1	yes	characteristic
159	C₂³	C₄⋊Q₈	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	#108	1	4	yes	complemented
168	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
169	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₄	#117	1	1	yes	characteristic
172	C₂×C₄	#117	1	1	yes	characteristic
173	C₂×C₄	#117	1	1	yes	characteristic
174	C₂×C₄	C₄×D₄	1	1	yes	characteristic
175	C₂×C₄	C₄×D₄	1	1	yes	characteristic
176	C₂×C₄	C₄×D₄	1	1	yes	characteristic
177	C₂×C₄	C₄×D₄	1	1	yes	characteristic
178	C₂×C₄	C₄×D₄	1	1	yes	characteristic
179	C₂×C₄	C₄×D₄	1	1	yes	characteristic
180	C₂×C₄	C₄×D₄	1	1	yes	characteristic
181	C₂×C₄	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
183	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
184	C₂×C₄	#120	1	1	yes	characteristic
185	C₂×C₄	#121	1	1	yes	characteristic
186	C₂×C₄	#121	1	1	yes	characteristic
187	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
188	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
189	C₂×C₄	#123	1	1	yes	characteristic
190	C₂×C₄	#123	1	1	yes	characteristic
191	C₂×C₄	#123	1	1	yes	characteristic
192	C₂×C₄	#124	1	1	yes	characteristic
193	C₂×C₄	#125	1	1	yes	characteristic
194	C₂×C₄	#126	1	1	yes	characteristic
195	C₂×C₄	#126	1	1	yes	characteristic
196	C₈	—	2	4	no	—
197	C₈	—	2	4	no	—
198	C₈	—	2	4	no	—
199	C₈	—	4	2	no	—
200	C₈	—	4	2	no	—
201	C₈	—	4	2	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
211	C₂²	#385	1	1	yes	characteristic
212	C₂²	#387	1	1	yes	characteristic
213	C₂²	#387	1	1	yes	characteristic
214	C₂²	#389	1	1	yes	characteristic
215	C₂²	#389	1	1	yes	characteristic
216	C₂²	#391	1	1	yes	characteristic
217	C₂²	C₂×#108	1	2	yes	—
218	C₂²	#430	1	2	yes	—
219	C₄	—	2	4	no	—
220	C₄	—	4	2	no	—
221	C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
222	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
223	C₄	#385	1	1	yes	characteristic
224	C₄	#386	1	1	yes	characteristic
225	C₄	#387	1	1	yes	characteristic
226	C₄	#387	1	1	yes	characteristic
227	C₄	#388	1	1	yes	characteristic
228	C₄	#388	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	—	2	1	no	—
233	C₂	#1436	1	1	yes	characteristic
234	C₂	C₄×#108	1	1	yes	characteristic
235	C₂	#1778	1	1	yes	characteristic
236	C₂	#1855	1	2	yes	—
237	C₂	#1950	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₄	C₂³
socle	C₂³	C₂×C₄⋊C₄
prime radical	C₂⁴	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
17. C₂×#108	C₄	semidirect
24. #341	C₄	semidirect
167. C₂×C₄	#108	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
C₂×C₄×M₁₆	2
C₂×#341	2
#1728	2
C₂×#336	2