Subgroups — #22131

Order 256.

318 conjugacy classes of subgroups.

The subgroup lattice has 245 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3076	C₂	1	1	yes	maximal, characteristic
2	#2960	C₂	1	1	yes	maximal, characteristic, complemented
3	#2922	C₂	1	1	yes	maximal, characteristic
4	#1986	C₂	1	1	yes	maximal, characteristic, complemented
5	#1950	C₂	1	1	yes	maximal, characteristic, complemented
6	#1946	C₂	1	2	yes	maximal, complemented
7	#1938	C₂	1	2	yes	maximal
8	#1929	C₂	1	2	yes	maximal
9	#1922	C₂	1	2	yes	maximal, complemented
10	#1806	C₂	1	1	yes	maximal, characteristic
11	#1770	C₂	1	1	yes	maximal, characteristic, complemented
12	#533	C₂²	1	1	yes	characteristic
13	C₄×#41	C₂²	1	1	yes	characteristic
14	#479	C₄	1	2	yes	complemented
15	#478	C₄	1	2	yes	complemented
16	#477	C₄	1	1	yes	characteristic, complemented
17	#476	C₄	1	1	yes	characteristic, complemented
18	#475	C₄	1	1	yes	characteristic, complemented
19	#474	C₄	1	1	yes	characteristic, complemented
20	#446	—	2	1	no	—
21	#446	—	2	1	no	—
22	C₈×Q₈	—	2	1	no	—
23	C₈×Q₈	—	2	1	no	—
24	#433	—	2	1	no	—
25	#433	—	2	1	no	—
26	#431	—	2	1	no	—
27	#431	—	2	1	no	—
28	C₂×#108	C₂²	1	2	yes	—
29	#427	C₂²	1	2	yes	—
30	#424	C₂²	1	1	yes	characteristic
31	#424	C₂²	1	1	yes	characteristic
32	#424	C₂²	1	2	yes	—
33	#423	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×#104	C₂²	1	2	yes	—
36	#416	C₂²	1	2	yes	—
37	#408	C₂²	1	1	yes	characteristic
38	#408	C₂²	1	1	yes	characteristic
39	#405	C₂²	1	2	yes	—
40	C₄×M₁₆	C₂²	1	1	yes	characteristic
41	C₂×#97	C₂²	1	2	yes	—
42	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
44	#341	C₂²	1	2	yes	—
45	#339	C₂²	1	2	yes	—
46	#339	C₂²	1	2	yes	—
47	#339	C₂²	1	2	yes	—
48	#336	C₂²	1	2	yes	—
49	C₂×#41	C₂³	1	1	yes	characteristic
50	C₂×M₁₆	C₂³	1	1	yes	characteristic
51	C₂×M₁₆	C₂³	1	1	yes	characteristic
52	C₂×M₁₆	C₂³	1	2	yes	—
53	C₂²×C₈	C₂³	1	2	yes	—
54	C₂²×C₈	C₂³	1	1	yes	characteristic
55	C₂²×C₈	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
58	#125	C₂×C₄	1	2	yes	—
59	C₂²⋊Q₈	—	2	1	no	—
60	C₂²⋊Q₈	—	2	1	no	—
61	C₄×Q₈	—	2	1	no	—
62	C₄×Q₈	—	2	1	no	—
63	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
64	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
65	C₄×D₄	—	2	1	no	—
66	C₄×D₄	—	2	1	no	—
67	C₄×D₄	C₂×C₄	1	1	yes	characteristic
68	C₄×D₄	C₂×C₄	1	1	yes	characteristic
69	#117	—	2	1	no	—
70	#117	C₂³	1	1	yes	characteristic
71	#117	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄	—	2	1	no	—
73	C₂×C₄²	—	2	1	no	—
74	C₂×C₄²	C₂³	1	2	yes	—
75	C₂×C₄²	C₂³	1	1	yes	characteristic
76	#108	—	4	2	no	—
77	#107	C₂×C₄	1	2	yes	—
78	#107	C₂×C₄	1	2	yes	—
79	#106	C₂×C₄	1	2	yes	—
80	#106	C₂×C₄	1	2	yes	—
81	C₄⋊C₈	—	2	1	no	—
82	C₄⋊C₈	—	2	1	no	—
83	C₄⋊C₈	—	2	1	no	—
84	C₄⋊C₈	—	2	1	no	—
85	C₄⋊C₈	—	2	2	no	—
86	C₄⋊C₈	—	2	2	no	—
87	C₄⋊C₈	—	2	1	no	—
88	C₄⋊C₈	—	2	1	no	—
89	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
90	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
91	C₄⋊C₈	C₂×C₄	1	2	yes	—
92	#104	—	4	2	no	—
93	#103	C₂×C₄	1	2	yes	—
94	#103	C₂×C₄	1	2	yes	—
95	#102	C₂×C₄	1	2	yes	—
96	#102	C₂×C₄	1	2	yes	—
97	C₂²⋊C₈	—	2	1	no	—
98	C₂²⋊C₈	—	2	1	no	—
99	C₂²⋊C₈	—	2	1	no	—
100	C₂²⋊C₈	—	2	1	no	—
101	#97	—	2	2	no	—
102	#97	—	2	1	no	—
103	#97	—	4	2	no	—
104	C₄×C₈	—	2	2	no	—
105	C₄×C₈	—	2	1	no	—
106	C₄×C₈	D₄	1	1	yes	characteristic
107	C₄×C₈	D₄	1	1	yes	characteristic
108	C₄×C₈	D₄	1	1	yes	characteristic
109	C₄×C₈	D₄	1	1	yes	characteristic
110	#95	—	2	1	no	—
111	#41	—	2	1	no	—
112	#41	—	2	1	no	—
113	#41	—	2	1	no	—
114	#41	—	2	1	no	—
115	C₂×Q₈	—	2	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
118	C₂×D₄	—	2	1	no	—
119	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	#41	1	1	yes	characteristic
125	C₂²×C₄	#41	1	1	yes	characteristic
126	C₂²×C₄	C₂⁴	1	1	yes	characteristic
127	M₁₆	—	2	2	no	—
128	M₁₆	—	2	1	no	—
129	M₁₆	—	2	1	no	—
130	M₁₆	—	2	1	no	—
131	M₁₆	—	2	1	no	—
132	M₁₆	—	2	2	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	C₂²×C₄	1	2	yes	—
143	C₂×C₈	C₂²×C₄	1	2	yes	—
144	C₂×C₈	C₂×D₄	1	1	yes	characteristic
145	C₂×C₈	C₂×D₄	1	1	yes	characteristic
146	C₂×C₈	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈	C₂×D₄	1	1	yes	characteristic
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	C₂²×C₄	1	2	yes	—
156	C₄⋊C₄	C₂²×C₄	1	2	yes	—
157	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
158	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
159	C₂²⋊C₄	—	2	1	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	2	1	no	—
162	C₂²⋊C₄	—	2	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₄²	—	2	2	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	2	2	no	—
169	C₄²	C₂²×C₄	1	1	yes	characteristic
170	C₄²	C₂²×C₄	1	1	yes	characteristic
171	C₄²	C₂×D₄	1	1	yes	characteristic
172	C₄²	C₂×D₄	1	1	yes	characteristic
173	C₂³	—	2	1	no	—
174	C₂³	C₂×#41	1	1	yes	characteristic
175	Q₈	—	2	1	no	—
176	Q₈	—	2	1	no	—
177	Q₈	—	4	2	no	—
178	Q₈	—	4	2	no	—
179	D₄	—	2	1	no	—
180	D₄	—	2	1	no	—
181	D₄	—	4	1	no	—
182	D₄	—	4	1	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	C₄×D₄	1	1	yes	characteristic
198	C₂×C₄	C₄×D₄	1	1	yes	characteristic
199	C₂×C₄	C₄×D₄	1	1	yes	characteristic
200	C₂×C₄	C₄×D₄	1	1	yes	characteristic
201	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
202	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
204	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
205	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
206	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	—	4	1	no	—
214	C₈	—	4	1	no	—
215	C₈	—	4	2	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	4	1	no	—
219	C₂²	—	4	1	no	—
220	C₂²	—	4	1	no	—
221	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂²	#541	1	1	yes	characteristic
223	C₂²	#542	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	4	1	no	—
228	C₄	—	4	1	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	8	2	no	—
233	C₄	#444	1	2	yes	—
234	C₄	#519	1	1	yes	characteristic
235	C₄	#520	1	1	yes	characteristic
236	C₄	#536	1	1	yes	characteristic
237	C₄	#537	1	1	yes	characteristic
238	C₂	—	2	1	no	—
239	C₂	—	2	1	no	—
240	C₂	—	8	1	no	—
241	C₂	#2323	1	1	yes	characteristic
242	C₂	C₂×#444	1	1	yes	characteristic
243	C₂	#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2960	C₂	semidirect
4. #1986	C₂	semidirect
5. #1950	C₂	semidirect
6. #1946	C₂	semidirect
9. #1922	C₂	semidirect
11. #1770	C₂	semidirect
14. #479	C₄	semidirect
15. #478	C₄	semidirect
16. #477	C₄	semidirect
17. #476	C₄	semidirect
18. #475	C₄	semidirect
19. #474	C₄	semidirect

Maximal subgroups

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

subgroup	index
#3076	2
#2960	2
#2922	2
#1986	2
#1950	2
#1946	2
#1938	2
#1929	2
#1922	2
#1806	2
#1770	2