Subgroups — #22823

Order 256.

368 conjugacy classes of subgroups.

The subgroup lattice has 247 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Q₈	C₂	1	1	yes	maximal, characteristic
2	#1985	C₂	1	2	yes	maximal
3	#1917	C₂	1	2	yes	maximal
4	#1916	C₂	1	2	yes	maximal
5	#1842	C₂	1	1	yes	maximal, characteristic
6	#1831	C₂	1	1	yes	maximal, characteristic
7	#1831	C₂	1	1	yes	maximal, characteristic
8	#1771	C₂	1	2	yes	maximal
9	#1767	C₂	1	1	yes	maximal, characteristic
10	#1767	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×#97	C₂²	1	2	yes	—
22	C₂×#97	C₂²	1	2	yes	—
23	C₂×C₄×C₈	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
25	C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
26	C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
28	C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
29	C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
30	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
33	C₄³	C₂²	1	1	yes	characteristic
34	C₄³	C₂²	1	1	yes	characteristic
35	C₄³	C₂²	1	1	yes	characteristic
36	#336	C₂²	1	2	yes	—
37	#336	C₂²	1	2	yes	—
38	#336	C₂²	1	2	yes	—
39	#336	C₂²	1	2	yes	—
40	#336	C₂²	1	2	yes	—
41	#336	C₂²	1	2	yes	—
42	#336	C₂²	1	2	yes	—
43	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈	C₂³	1	2	yes	—
45	C₂²×C₈	C₂³	1	2	yes	—
46	C₂²×C₈	C₂³	1	2	yes	—
47	C₂²×C₈	C₂³	1	2	yes	—
48	C₄×Q₈	—	2	4	no	—
49	C₄×Q₈	—	2	4	no	—
50	C₄×Q₈	—	2	2	no	—
51	C₄×Q₈	—	2	2	no	—
52	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²	C₂³	1	1	yes	characteristic
74	C₂×C₄²	C₂³	1	1	yes	characteristic
75	C₂×C₄²	C₂³	1	1	yes	characteristic
76	C₂×C₄²	C₂³	1	1	yes	characteristic
77	C₂×C₄²	C₂³	1	1	yes	characteristic
78	C₂×C₄²	C₂³	1	1	yes	characteristic
79	C₂×C₄²	C₂³	1	1	yes	characteristic
80	C₄⋊C₈	—	2	4	no	—
81	C₄⋊C₈	—	2	2	no	—
82	C₄⋊C₈	—	2	2	no	—
83	C₄⋊C₈	—	2	2	no	—
84	C₄⋊C₈	—	4	2	no	—
85	C₄⋊C₈	Q₈	1	4	yes	complemented
86	#97	—	2	4	no	—
87	#97	—	4	2	no	—
88	C₄×C₈	—	4	2	no	—
89	#95	C₂×C₄	1	1	yes	characteristic
90	#95	C₂×C₄	1	1	yes	characteristic
91	#95	C₂×C₄	1	1	yes	characteristic
92	#95	C₂×C₄	1	1	yes	characteristic
93	#95	C₂×C₄	1	1	yes	characteristic
94	#95	C₂×C₄	1	1	yes	characteristic
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄	C₂⁴	1	1	yes	characteristic
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	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
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	C₂×Q₈	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	—	2	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	M₁₆	1	4	yes	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	2	2	no	—
138	C₄²	—	2	2	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	1	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	—	2	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	2	1	no	—
152	C₄²	—	4	1	no	—
153	C₄²	—	4	1	no	—
154	C₄²	—	4	1	no	—
155	C₄²	M₁₆	1	4	yes	—
156	C₄²	C₂×Q₈	1	2	yes	—
157	C₄²	#41	1	2	yes	—
158	C₄²	#41	1	2	yes	—
159	C₄²	#41	1	2	yes	—
160	C₂³	C₂³×C₄	1	1	yes	characteristic
161	Q₈	—	2	8	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	#117	1	2	yes	—
194	C₂×C₄	#117	1	2	yes	—
195	C₂×C₄	C₄×Q₈	1	2	yes	—
196	C₂×C₄	C₄×Q₈	1	2	yes	—
197	C₂×C₄	C₂×M₁₆	1	2	yes	—
198	C₂×C₄	C₂×M₁₆	1	2	yes	—
199	C₂×C₄	C₂×M₁₆	1	2	yes	—
200	C₂×C₄	C₂×M₁₆	1	2	yes	—
201	C₂×C₄	C₂×M₁₆	1	2	yes	—
202	C₂×C₄	C₂×M₁₆	1	2	yes	—
203	C₂×C₄	#131	1	2	yes	—
204	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
205	C₂×C₄	C₂×#41	1	1	yes	characteristic
206	C₂×C₄	C₂×#41	1	1	yes	characteristic
207	C₂×C₄	C₂×#41	1	1	yes	characteristic
208	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
209	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
210	C₈	—	2	4	no	—
211	C₈	—	4	2	no	—
212	C₈	—	4	2	no	—
213	C₈	—	4	2	no	—
214	C₂²	C₂×#117	1	1	yes	characteristic
215	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
216	C₂²	#520	1	1	yes	characteristic
217	C₂²	C₂²×M₁₆	1	1	yes	characteristic
218	C₂²	C₂²×M₁₆	1	1	yes	characteristic
219	C₂²	C₂×#131	1	1	yes	characteristic
220	C₂²	#568	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	4	1	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	1	no	—
234	C₄	#446	1	4	yes	—
235	C₄	#554	1	1	yes	characteristic
236	C₄	#556	1	1	yes	characteristic
237	C₄	#557	1	1	yes	characteristic
238	C₄	#559	1	1	yes	characteristic
239	C₂	#2355	1	1	yes	characteristic
240	C₂	#2920	1	1	yes	characteristic
241	C₂	#2925	1	1	yes	characteristic
242	C₂	C₂×#446	1	1	yes	characteristic
243	C₂	Q₈×M₁₆	1	1	yes	characteristic
244	C₂	#2989	1	1	yes	characteristic
245	C₂	#2990	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
85. C₄⋊C₈	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₄²×Q₈	2
#1985	2
#1917	2
#1916	2
#1842	2
#1831	2
#1771	2
#1767	2
C₄×C₄⋊C₈	2