Subgroups — #23080

Order 256.

337 conjugacy classes of subgroups.

The subgroup lattice has 241 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	#2962	C₂	1	1	yes	maximal, characteristic, complemented
3	#2925	C₂	1	1	yes	maximal, characteristic
4	#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	#1950	C₂	1	1	yes	maximal, characteristic, complemented
6	#1937	C₂	1	2	yes	maximal
7	#1929	C₂	1	2	yes	maximal
8	#1922	C₂	1	2	yes	maximal, complemented
9	#1772	C₂	1	2	yes	maximal, complemented
10	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#104	C₂	1	1	yes	maximal, characteristic
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	2	no	—
21	#446	—	2	1	no	—
22	C₈×Q₈	—	2	1	no	—
23	#433	—	2	2	no	—
24	#432	—	2	1	no	—
25	#431	—	2	1	no	—
26	C₂×#108	C₂²	1	2	yes	—
27	#427	C₂²	1	2	yes	—
28	#424	C₂²	1	2	yes	—
29	#423	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×#104	C₂²	1	1	yes	characteristic
33	C₂×#104	C₂²	1	1	yes	characteristic
34	#416	C₂²	1	2	yes	—
35	#408	C₂²	1	2	yes	—
36	#405	C₂²	1	2	yes	—
37	#405	C₂²	1	2	yes	—
38	C₄×M₁₆	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂²	1	2	yes	—
40	C₄³	C₂²	1	1	yes	characteristic
41	#341	C₂²	1	2	yes	—
42	#339	C₂²	1	1	yes	characteristic
43	#339	C₂²	1	2	yes	—
44	#339	C₂²	1	2	yes	—
45	#339	C₂²	1	1	yes	characteristic
46	#336	C₂²	1	2	yes	—
47	C₂×#41	C₂³	1	1	yes	characteristic
48	C₂×M₁₆	C₂³	1	2	yes	—
49	C₂×M₁₆	C₂³	1	1	yes	characteristic
50	C₂×M₁₆	C₂³	1	1	yes	characteristic
51	C₂²×C₈	C₂³	1	2	yes	—
52	C₂²×C₈	C₂³	1	2	yes	—
53	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
54	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
55	#125	C₂×C₄	1	2	yes	—
56	C₂²⋊Q₈	—	2	1	no	—
57	C₂²⋊Q₈	—	2	1	no	—
58	C₄×Q₈	—	2	1	no	—
59	C₄×Q₈	—	2	1	no	—
60	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
61	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
62	C₄×D₄	—	2	1	no	—
63	C₄×D₄	—	2	1	no	—
64	C₄×D₄	C₂×C₄	1	1	yes	characteristic
65	C₄×D₄	C₂×C₄	1	1	yes	characteristic
66	#117	—	2	1	no	—
67	#117	C₂³	1	1	yes	characteristic
68	#117	C₂³	1	2	yes	—
69	C₂×C₄²	—	2	1	no	—
70	C₂×C₄²	—	2	1	no	—
71	C₂×C₄²	—	2	1	no	—
72	C₂×C₄²	C₂³	1	1	yes	characteristic
73	C₂×C₄²	C₂³	1	1	yes	characteristic
74	C₂×C₄²	C₂³	1	1	yes	characteristic
75	#108	—	2	4	no	—
76	#107	C₂×C₄	1	2	yes	—
77	#107	C₂×C₄	1	2	yes	—
78	#106	C₂×C₄	1	2	yes	—
79	#106	C₂×C₄	1	2	yes	—
80	C₄⋊C₈	—	2	2	no	—
81	C₄⋊C₈	—	2	2	no	—
82	C₄⋊C₈	—	2	1	no	—
83	C₄⋊C₈	—	2	1	no	—
84	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
86	C₄⋊C₈	C₂×C₄	1	2	yes	—
87	C₄⋊C₈	D₄	1	1	yes	characteristic
88	C₄⋊C₈	D₄	1	1	yes	characteristic
89	C₄⋊C₈	D₄	1	2	yes	—
90	#104	—	2	2	no	—
91	#104	—	2	2	no	—
92	#103	C₂×C₄	1	2	yes	—
93	#103	C₂×C₄	1	2	yes	—
94	#102	C₂×C₄	1	2	yes	—
95	#102	C₂×C₄	1	2	yes	—
96	C₂²⋊C₈	—	2	2	no	—
97	C₂²⋊C₈	—	2	2	no	—
98	#97	—	2	2	no	—
99	#97	—	2	2	no	—
100	#97	—	2	1	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	—	2	2	no	—
103	C₄×C₈	—	2	1	no	—
104	C₄×C₈	—	2	4	no	—
105	#41	—	2	1	no	—
106	#41	—	2	1	no	—
107	#41	—	2	1	no	—
108	#41	—	2	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	2	1	no	—
111	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	#41	1	1	yes	characteristic
120	C₂²×C₄	#41	1	1	yes	characteristic
121	C₂²×C₄	C₂⁴	1	1	yes	characteristic
122	M₁₆	—	2	1	no	—
123	M₁₆	—	2	1	no	—
124	M₁₆	—	2	2	no	—
125	M₁₆	—	2	1	no	—
126	M₁₆	—	2	2	no	—
127	M₁₆	#41	1	2	yes	—
128	C₂×C₈	—	2	1	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	1	no	—
135	C₂×C₈	C₂²×C₄	1	2	yes	—
136	C₂×C₈	C₂²×C₄	1	2	yes	—
137	C₂×C₈	C₂×D₄	1	2	yes	—
138	C₂×C₈	C₂×D₄	1	2	yes	—
139	C₄⋊C₄	—	2	1	no	—
140	C₄⋊C₄	—	2	1	no	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	1	no	—
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	C₂²×C₄	1	2	yes	—
146	C₄⋊C₄	C₂²×C₄	1	2	yes	—
147	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
148	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
149	C₂²⋊C₄	—	2	1	no	—
150	C₂²⋊C₄	—	2	2	no	—
151	C₂²⋊C₄	—	2	1	no	—
152	C₂²⋊C₄	—	2	1	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	2	1	no	—
161	C₄²	—	4	2	no	—
162	C₄²	C₂²×C₄	1	1	yes	characteristic
163	C₄²	C₂²×C₄	1	1	yes	characteristic
164	C₄²	C₂×D₄	1	1	yes	characteristic
165	C₄²	C₂×D₄	1	1	yes	characteristic
166	C₄²	#41	1	2	yes	—
167	C₂³	—	2	1	no	—
168	C₂³	C₂×#41	1	1	yes	characteristic
169	Q₈	—	2	1	no	—
170	Q₈	—	2	1	no	—
171	Q₈	—	4	2	no	—
172	Q₈	—	8	1	no	—
173	D₄	—	2	1	no	—
174	D₄	—	2	1	no	—
175	D₄	—	4	1	no	—
176	D₄	—	4	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	8	1	no	—
194	C₂×C₄	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄	C₄×D₄	1	1	yes	characteristic
196	C₂×C₄	C₄×D₄	1	1	yes	characteristic
197	C₂×C₄	C₄×D₄	1	1	yes	characteristic
198	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
199	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄	C₂×#41	1	1	yes	characteristic
201	C₂×C₄	C₂×#41	1	1	yes	characteristic
202	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
203	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	4	1	no	—
210	C₈	—	4	2	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	4	1	no	—
214	C₂²	—	4	1	no	—
215	C₂²	—	4	1	no	—
216	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
217	C₂²	#552	1	1	yes	characteristic
218	C₂²	#556	1	1	yes	characteristic
219	C₄	—	2	4	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	1	no	—
225	C₄	—	4	1	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	1	no	—
228	C₄	—	4	2	no	—
229	C₄	#443	1	2	yes	—
230	C₄	C₄×#41	1	1	yes	characteristic
231	C₄	#519	1	1	yes	characteristic
232	C₄	#547	1	1	yes	characteristic
233	C₄	#548	1	1	yes	characteristic
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	—	8	1	no	—
237	C₂	#2362	1	1	yes	characteristic
238	C₂	C₂×#443	1	1	yes	characteristic
239	C₂	#2956	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. #2962	C₂	semidirect
4. #1994	C₂	semidirect
5. #1950	C₂	semidirect
8. #1922	C₂	semidirect
9. #1772	C₂	semidirect
10. C₄×C₄⋊C₈	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
#2962	2
#2925	2
#1994	2
#1950	2
#1937	2
#1929	2
#1922	2
#1772	2
C₄×C₄⋊C₈	2
C₄×#104	2