Subgroups — #23248

Order 256.

358 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#1985	C₂	1	2	yes	maximal
3	#1984	C₂	1	2	yes	maximal
4	#1940	C₂	1	2	yes	maximal
5	#1939	C₂	1	1	yes	maximal, characteristic
6	#1939	C₂	1	1	yes	maximal, characteristic
7	#1771	C₂	1	2	yes	maximal
8	#1770	C₂	1	1	yes	maximal, characteristic
9	C₈×C₄⋊C₄	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	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×#97	C₂²	1	2	yes	—
21	C₂×#97	C₂²	1	2	yes	—
22	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂²	1	2	yes	—
25	#391	C₄	1	2	yes	—
26	#389	C₄	1	1	yes	characteristic
27	#389	C₄	1	1	yes	characteristic
28	#389	C₄	1	2	yes	—
29	#384	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	2	yes	—
33	C₄³	C₂²	1	1	yes	characteristic
34	#336	C₂²	1	2	yes	—
35	#336	C₂²	1	2	yes	—
36	#336	C₂²	1	2	yes	—
37	#336	C₂²	1	2	yes	—
38	#336	C₂²	1	1	yes	characteristic
39	#336	C₂²	1	2	yes	—
40	#336	C₂²	1	1	yes	characteristic
41	C₂²×Q₈	C₂×C₄	1	2	yes	—
42	C₂²×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₈	C₂³	1	2	yes	—
45	C₂²×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₈	C₂³	1	2	yes	—
48	C₄⋊Q₈	—	2	4	no	—
49	C₄×Q₈	—	4	2	no	—
50	C₄×Q₈	—	4	2	no	—
51	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
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	2	yes	—
56	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²	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	C₂³	1	1	yes	characteristic
65	C₂×C₄²	C₂³	1	2	yes	—
66	C₂×C₄²	C₂³	1	2	yes	—
67	C₂×C₄²	C₂³	1	1	yes	characteristic
68	C₂×C₄²	C₂³	1	1	yes	characteristic
69	C₄⋊C₈	—	2	2	no	—
70	C₄⋊C₈	—	2	2	no	—
71	C₄⋊C₈	—	4	2	no	—
72	C₄⋊C₈	—	4	2	no	—
73	C₄⋊C₈	Q₈	1	4	yes	complemented
74	C₄⋊C₈	Q₈	1	4	yes	complemented
75	#97	—	2	4	no	—
76	#97	—	4	2	no	—
77	C₄×C₈	—	2	2	no	—
78	C₄×C₈	—	2	2	no	—
79	C₄×C₈	—	4	2	no	—
80	#95	C₂×C₄	1	2	yes	—
81	#95	C₂×C₄	1	2	yes	—
82	#95	C₂×C₄	1	2	yes	—
83	#95	C₂×C₄	1	2	yes	—
84	C₂×Q₈	—	2	4	no	—
85	C₂×Q₈	—	2	4	no	—
86	C₂×Q₈	—	4	2	no	—
87	C₂²×C₄	C₂²×C₄	1	2	yes	—
88	C₂²×C₄	C₂²×C₄	1	2	yes	—
89	C₂²×C₄	C₂²×C₄	1	2	yes	—
90	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄	C₂²×C₄	1	2	yes	—
95	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄	C₂⁴	1	1	yes	characteristic
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	C₂×Q₈	1	2	yes	—
114	C₂×C₈	C₂×Q₈	1	2	yes	—
115	C₂×C₈	C₂×Q₈	1	4	yes	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	M₁₆	1	4	yes	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	2	2	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	2	2	no	—
130	C₄²	—	2	2	no	—
131	C₄²	—	2	2	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	2	1	no	—
134	C₄²	—	4	1	no	—
135	C₄²	C₂×Q₈	1	2	yes	—
136	C₄²	C₂×Q₈	1	2	yes	—
137	C₄²	#41	1	2	yes	—
138	C₄²	#41	1	2	yes	—
139	C₂³	C₂³×C₄	1	1	yes	characteristic
140	Q₈	—	4	8	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	C₄×Q₈	1	2	yes	—
164	C₂×C₄	C₄×Q₈	1	2	yes	—
165	C₂×C₄	C₄×Q₈	1	2	yes	—
166	C₂×C₄	C₄×Q₈	1	2	yes	—
167	C₂×C₄	C₂×M₁₆	1	4	yes	—
168	C₂×C₄	C₂×M₁₆	1	2	yes	—
169	C₂×C₄	#131	1	2	yes	—
170	C₂×C₄	#131	1	2	yes	—
171	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
172	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
173	C₂×C₄	C₂×#41	1	1	yes	characteristic
174	C₂×C₄	C₂×#41	1	1	yes	characteristic
175	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
176	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
177	C₈	—	2	2	no	—
178	C₈	—	2	2	no	—
179	C₈	—	2	4	no	—
180	C₈	—	4	1	no	—
181	C₈	—	4	1	no	—
182	C₈	—	4	2	no	—
183	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
184	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
185	C₂²	#518	1	1	yes	characteristic
186	C₂²	C₂²×M₁₆	1	1	yes	characteristic
187	C₂²	C₂×#131	1	1	yes	characteristic
188	C₂²	C₂×#131	1	1	yes	characteristic
189	C₂²	#568	1	1	yes	characteristic
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	2	4	no	—
194	C₄	—	2	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	4	1	no	—
197	C₄	—	4	2	no	—
198	C₄	—	4	2	no	—
199	C₄	#446	1	4	yes	—
200	C₄	#554	1	1	yes	characteristic
201	C₄	#554	1	1	yes	characteristic
202	C₄	#555	1	1	yes	characteristic
203	C₄	Q₈²	1	1	yes	characteristic
204	C₂	#2369	1	1	yes	characteristic
205	C₂	C₂×#446	1	1	yes	characteristic
206	C₂	#2960	1	1	yes	characteristic
207	C₂	#2962	1	1	yes	characteristic
208	C₂	Q₈×M₁₆	1	1	yes	characteristic
209	C₂	#2974	1	1	yes	characteristic
210	C₂	#2981	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
73. C₄⋊C₈	Q₈	semidirect
74. 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
C₄×C₄⋊Q₈	2
#1985	2
#1984	2
#1940	2
#1939	2
#1771	2
#1770	2
C₈×C₄⋊C₄	2
#1767	2
C₄×C₄⋊C₈	2