Subgroups — #21558

Order 256.

438 conjugacy classes of subgroups.

The subgroup lattice has 267 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3067	C₂	1	1	yes	maximal, characteristic
2	#2955	C₂	1	2	yes	maximal, complemented
3	C₂×#443	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#131	C₂	1	1	yes	maximal, characteristic, complemented
5	#1976	C₂	1	1	yes	maximal, characteristic, complemented
6	#1953	C₂	1	1	yes	maximal, characteristic, complemented
7	#1932	C₂	1	2	yes	maximal, complemented
8	#1906	C₂	1	2	yes	maximal, complemented
9	#1814	C₂	1	1	yes	maximal, characteristic, complemented
10	#1772	C₂	1	1	yes	maximal, characteristic, complemented
11	#1757	C₂	1	2	yes	maximal, complemented
12	#575	C₂²	1	2	yes	—
13	C₂×#135	C₂²	1	1	yes	characteristic
14	C₂×#131	C₂²	1	1	yes	characteristic
15	C₂×#131	C₂²	1	2	yes	—
16	C₂×#131	C₂²	1	1	yes	characteristic
17	C₄×#41	C₂²	1	1	yes	characteristic
18	#473	C₄	1	2	yes	complemented
19	#472	C₄	1	2	yes	complemented
20	#471	C₄	1	4	yes	complemented
21	#444	—	2	4	no	—
22	#443	—	2	4	no	—
23	#443	—	2	1	no	—
24	#443	—	2	1	no	—
25	#443	—	2	1	no	—
26	#443	—	2	1	no	—
27	#430	C₂²	1	2	yes	—
28	C₂×#108	C₂²	1	1	yes	characteristic
29	C₂×#108	C₂²	1	1	yes	characteristic
30	#427	C₂²	1	1	yes	characteristic
31	#421	C₂²	1	2	yes	—
32	#421	C₂²	1	2	yes	—
33	C₂×#104	C₂²	1	2	yes	—
34	#413	C₂²	1	2	yes	—
35	#410	C₂²	1	2	yes	—
36	#405	—	2	1	no	—
37	#405	—	2	1	no	—
38	#405	C₂²	1	1	yes	characteristic
39	#405	C₂²	1	1	yes	characteristic
40	#405	C₂²	1	2	yes	—
41	C₄×M₁₆	—	2	1	no	—
42	C₄×M₁₆	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈	—	2	1	no	—
44	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
45	#344	C₂²	1	2	yes	—
46	#344	C₂²	1	2	yes	—
47	#339	C₂²	1	2	yes	—
48	C₂×#41	C₂³	1	2	yes	—
49	C₂×#41	C₂³	1	1	yes	characteristic
50	#137	—	2	2	no	—
51	#137	C₂×C₄	1	4	yes	—
52	AGL(1,ℤ/8)	—	2	2	no	—
53	AGL(1,ℤ/8)	C₂×C₄	1	4	yes	—
54	#135	—	2	4	no	—
55	#135	—	2	1	no	—
56	#135	—	2	1	no	—
57	#135	—	2	1	no	—
58	#135	—	2	1	no	—
59	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
60	C₂×SD₁₆	C₂×C₄	1	2	yes	complemented
61	C₂×D₈	C₂×C₄	1	1	yes	characteristic, complemented
62	#131	—	2	1	no	—
63	#131	—	2	1	no	—
64	#131	—	2	1	no	—
65	#131	—	2	1	no	—
66	#131	—	2	2	no	—
67	#131	—	2	2	no	—
68	#131	—	2	2	no	—
69	#131	—	2	2	no	—
70	#131	C₂×C₄	1	2	yes	—
71	#131	C₂×C₄	1	2	yes	—
72	#131	D₄	1	2	yes	—
73	#131	D₄	1	2	yes	—
74	C₂×M₁₆	—	2	1	no	—
75	C₂×M₁₆	—	2	1	no	—
76	C₂×M₁₆	—	2	2	no	—
77	C₂×M₁₆	C₂³	1	2	yes	—
78	C₂×M₁₆	C₂³	1	1	yes	characteristic
79	C₂×M₁₆	C₂³	1	2	yes	—
80	C₂×M₁₆	C₂³	1	1	yes	characteristic
81	C₂²×C₈	—	2	1	no	—
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	—	2	2	no	—
84	C₂²×C₈	C₂³	1	1	yes	characteristic
85	C₂²×C₈	C₂³	1	1	yes	characteristic
86	C₄×Q₈	—	2	1	no	—
87	C₄×D₄	—	2	1	no	—
88	C₄×D₄	—	2	2	no	—
89	#117	—	2	1	no	—
90	#117	C₂³	1	2	yes	—
91	#117	C₂³	1	1	yes	characteristic
92	C₂×C₄²	—	2	1	no	—
93	C₂×C₄²	C₂³	1	1	yes	characteristic
94	#108	—	2	4	no	—
95	#108	—	2	1	no	—
96	#108	—	2	1	no	—
97	#108	C₂×C₄	1	2	yes	complemented
98	#108	C₂×C₄	1	2	yes	complemented
99	#107	—	2	1	no	—
100	#106	—	2	1	no	—
101	#104	—	2	4	no	—
102	#104	—	2	4	no	—
103	#104	—	2	2	no	—
104	#104	—	2	2	no	—
105	#101	C₂×C₄	1	4	yes	complemented
106	#100	C₂×C₄	1	4	yes	complemented
107	#99	—	2	4	no	—
108	#97	—	2	1	no	—
109	#97	—	2	1	no	—
110	#97	—	2	2	no	—
111	#97	—	2	1	no	—
112	C₄×C₈	—	2	1	no	—
113	C₄×C₈	—	2	1	no	—
114	C₄×C₈	—	2	2	no	—
115	C₄×C₈	—	2	1	no	—
116	C₄×C₈	D₄	1	1	yes	characteristic
117	C₄×C₈	D₄	1	1	yes	characteristic
118	C₄×C₈	D₄	1	1	yes	characteristic
119	C₄×C₈	D₄	1	1	yes	characteristic
120	#41	—	2	1	no	—
121	#41	—	2	1	no	—
122	#41	—	2	2	no	—
123	#41	—	2	2	no	—
124	#41	—	2	2	no	—
125	#41	—	2	2	no	—
126	#41	C₂²×C₄	1	2	yes	—
127	#41	C₂×D₄	1	2	yes	—
128	C₂×Q₈	C₂²×C₄	1	2	yes	—
129	C₂×Q₈	#41	1	1	yes	characteristic
130	C₂×D₄	—	2	1	no	—
131	C₂×D₄	—	2	2	no	—
132	C₂×D₄	C₂²×C₄	1	2	yes	—
133	C₂×D₄	#41	1	1	yes	characteristic
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	C₂⁴	1	1	yes	characteristic
139	Q₁₆	—	2	4	no	—
140	Q₁₆	—	2	1	no	—
141	Q₁₆	—	2	1	no	—
142	SD₁₆	—	2	4	no	—
143	SD₁₆	—	2	4	no	—
144	SD₁₆	—	2	2	no	—
145	SD₁₆	—	2	2	no	—
146	D₈	—	2	4	no	—
147	D₈	—	2	1	no	—
148	D₈	—	2	1	no	—
149	M₁₆	—	2	1	no	—
150	M₁₆	—	2	1	no	—
151	M₁₆	—	2	2	no	—
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	2	no	—
154	M₁₆	—	2	2	no	—
155	M₁₆	—	2	2	no	—
156	M₁₆	—	2	2	no	—
157	M₁₆	—	2	2	no	—
158	M₁₆	—	4	2	no	—
159	M₁₆	—	4	2	no	—
160	M₁₆	C₂²×C₄	1	2	yes	—
161	M₁₆	C₂²×C₄	1	4	yes	—
162	M₁₆	C₂×D₄	1	2	yes	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
174	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₈	C₂×D₄	1	1	yes	characteristic
176	C₂×C₈	C₂×D₄	1	1	yes	characteristic
177	C₂×C₈	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈	#41	1	1	yes	characteristic
182	C₂×C₈	#41	1	1	yes	characteristic
183	C₂×C₈	#41	1	1	yes	characteristic
184	C₂×C₈	#41	1	1	yes	characteristic
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	#41	1	2	yes	—
191	C₄²	—	2	2	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	2	no	—
195	C₄²	C₂×D₄	1	1	yes	characteristic
196	C₄²	C₂×D₄	1	1	yes	characteristic
197	C₂³	—	2	1	no	—
198	C₂³	—	2	2	no	—
199	C₂³	C₂×#41	1	1	yes	characteristic
200	Q₈	—	2	1	no	—
201	Q₈	—	2	2	no	—
202	Q₈	—	2	2	no	—
203	Q₈	C₄×D₄	1	2	yes	—
204	D₄	—	2	2	no	—
205	D₄	—	2	2	no	—
206	D₄	—	2	2	no	—
207	D₄	—	2	2	no	—
208	D₄	—	2	1	no	—
209	D₄	—	4	2	no	—
210	D₄	—	4	2	no	—
211	D₄	C₄×D₄	1	2	yes	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄	C₂×#41	1	1	yes	characteristic
228	C₂×C₄	C₂×#41	1	1	yes	characteristic
229	C₂×C₄	C₂×#41	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	2	4	no	—
232	C₈	—	2	2	no	—
233	C₈	—	4	2	no	—
234	C₈	C₄×D₄	1	1	yes	characteristic
235	C₈	C₄×D₄	1	1	yes	characteristic
236	C₈	C₄×D₄	1	1	yes	characteristic
237	C₈	C₄×D₄	1	1	yes	characteristic
238	C₈	#121	1	1	yes	characteristic
239	C₈	#121	1	1	yes	characteristic
240	C₈	#121	1	1	yes	characteristic
241	C₈	#121	1	1	yes	characteristic
242	C₂²	—	2	2	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	8	2	no	—
247	C₂²	C₄×#41	1	1	yes	characteristic
248	C₂²	#525	1	1	yes	characteristic
249	C₂²	#532	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	4	2	no	—
255	C₄	—	8	2	no	—
256	C₄	C₂×C₄×D₄	1	1	yes	characteristic
257	C₄	C₂×C₄×D₄	1	1	yes	characteristic
258	C₄	C₂×#121	1	1	yes	characteristic
259	C₄	#529	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	4	2	no	—
264	C₂	—	8	2	no	—
265	C₂	C₄×#121	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#121
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	C₄×#121
prime radical	#3067	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×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. #2955	C₂	semidirect
3. C₂×#443	C₂	semidirect
4. C₄×#131	C₂	semidirect
5. #1976	C₂	semidirect
6. #1953	C₂	semidirect
7. #1932	C₂	semidirect
8. #1906	C₂	semidirect
9. #1814	C₂	semidirect
10. #1772	C₂	semidirect
11. #1757	C₂	semidirect
18. #473	C₄	semidirect
19. #472	C₄	semidirect
20. #471	C₄	semidirect
59. C₂×Q₁₆	C₂×C₄	semidirect
60. C₂×SD₁₆	C₂×C₄	semidirect
61. C₂×D₈	C₂×C₄	semidirect
97. #108	C₂×C₄	semidirect
98. #108	C₂×C₄	semidirect
105. #101	C₂×C₄	semidirect
106. #100	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
#3067	2
#2955	2
C₂×#443	2
C₄×#131	2
#1976	2
#1953	2
#1932	2
#1906	2
#1814	2
#1772	2
#1757	2