Subgroups — #19720

Order 256.

328 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	#3231	C₂	1	1	yes	maximal, characteristic
2	#3231	C₂	1	1	yes	maximal, characteristic
3	#3229	C₂	1	1	yes	maximal, characteristic
4	#3209	C₂	1	1	yes	maximal, characteristic
5	#3185	C₂	1	1	yes	maximal, characteristic
6	#3165	C₂	1	1	yes	maximal, characteristic
7	#1723	C₂	1	1	yes	maximal, characteristic, complemented
8	#1723	C₂	1	1	yes	maximal, characteristic, complemented
9	#1718	C₂	1	1	yes	maximal, characteristic, complemented
10	#1718	C₂	1	1	yes	maximal, characteristic, complemented
11	#1716	C₂	1	1	yes	maximal, characteristic, complemented
12	#1716	C₂	1	1	yes	maximal, characteristic, complemented
13	#1707	C₂	1	1	yes	maximal, characteristic, complemented
14	#1707	C₂	1	1	yes	maximal, characteristic, complemented
15	#1453	C₂	1	1	yes	maximal, characteristic
16	#533	C₂²	1	1	yes	characteristic
17	#533	C₂²	1	1	yes	characteristic
18	#533	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	#501	—	2	2	no	—
21	#500	C₂²	1	1	yes	characteristic
22	#500	C₂²	1	1	yes	characteristic
23	#500	C₂²	1	1	yes	characteristic
24	#500	C₂²	1	1	yes	characteristic
25	#498	C₂²	1	1	yes	characteristic
26	#498	C₂²	1	1	yes	characteristic
27	#498	C₂²	1	1	yes	characteristic
28	#498	C₂²	1	1	yes	characteristic
29	#498	C₂²	1	1	yes	characteristic
30	#498	C₂²	1	1	yes	characteristic
31	#498	C₂²	1	1	yes	characteristic
32	#494	C₂²	1	1	yes	characteristic
33	#494	C₂²	1	1	yes	characteristic
34	#494	C₂²	1	1	yes	characteristic
35	#487	C₂²	1	1	yes	characteristic
36	#487	C₂²	1	1	yes	characteristic
37	#487	C₂²	1	1	yes	characteristic
38	#487	C₂²	1	1	yes	characteristic
39	#487	C₂²	1	1	yes	characteristic
40	#487	C₂²	1	1	yes	characteristic
41	#487	C₂²	1	1	yes	characteristic
42	#487	C₂²	1	1	yes	characteristic
43	#487	C₂²	1	1	yes	characteristic
44	#487	C₂²	1	1	yes	characteristic
45	#485	—	2	2	no	—
46	#485	—	2	2	no	—
47	#483	—	2	2	no	—
48	#479	—	2	2	no	—
49	#479	—	2	2	no	—
50	#475	—	2	2	no	—
51	#470	—	2	2	no	—
52	#462	—	2	2	no	—
53	#451	—	2	2	no	—
54	#441	—	2	2	no	—
55	#431	C₂²	1	1	yes	characteristic
56	#431	C₂²	1	1	yes	characteristic
57	#428	—	2	2	no	—
58	C₄×M₁₆	C₂²	1	1	yes	characteristic
59	#322	C₂²	1	1	yes	characteristic
60	#322	C₂²	1	1	yes	characteristic
61	#322	C₂²	1	1	yes	characteristic
62	C₈²	C₂²	1	1	yes	characteristic
63	C₂²×Q₈	—	2	1	no	—
64	C₂×Q₁₆	—	2	2	no	—
65	C₂×Q₁₆	—	2	2	no	—
66	C₂×Q₁₆	—	2	2	no	—
67	C₂×M₁₆	D₄	1	2	yes	—
68	C₄⋊Q₈	—	2	1	no	—
69	C₄⋊Q₈	—	2	1	no	—
70	C₄⋊Q₈	—	2	1	no	—
71	C₄⋊Q₈	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈	C₂³	1	1	yes	characteristic
73	C₄⋊Q₈	C₂³	1	1	yes	characteristic
74	C₄⋊Q₈	C₂³	1	1	yes	characteristic
75	C₄⋊Q₈	C₂³	1	1	yes	characteristic
76	C₄⋊Q₈	C₂³	1	1	yes	characteristic
77	C₄⋊Q₈	C₂³	1	1	yes	characteristic
78	C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	#125	—	2	1	no	—
80	#125	—	2	1	no	—
81	#125	—	2	1	no	—
82	C₂²⋊Q₈	—	2	1	no	—
83	C₂²⋊Q₈	—	2	1	no	—
84	C₂²⋊Q₈	—	2	1	no	—
85	C₂²⋊Q₈	—	2	1	no	—
86	C₂²⋊Q₈	—	2	1	no	—
87	C₂²⋊Q₈	—	2	1	no	—
88	C₄×Q₈	—	2	1	no	—
89	C₄×Q₈	—	2	1	no	—
90	C₄×Q₈	—	2	1	no	—
91	C₄×Q₈	—	2	1	no	—
92	C₄×Q₈	—	2	1	no	—
93	C₄×Q₈	—	2	1	no	—
94	#117	—	2	1	no	—
95	#117	—	2	1	no	—
96	#117	—	2	1	no	—
97	C₂×C₄⋊C₄	—	2	1	no	—
98	C₂×C₄⋊C₄	—	2	1	no	—
99	C₂×C₄²	C₂³	1	1	yes	characteristic
100	#107	—	2	2	no	—
101	#107	—	2	2	no	—
102	#107	—	2	2	no	—
103	#107	—	2	2	no	—
104	#106	—	2	2	no	—
105	#106	—	2	2	no	—
106	#106	—	2	2	no	—
107	#106	—	2	2	no	—
108	#106	—	2	2	no	—
109	#106	—	2	2	no	—
110	#106	—	2	2	no	—
111	C₄⋊C₈	D₄	1	2	yes	—
112	C₄⋊C₈	D₄	1	2	yes	—
113	#103	—	2	2	no	—
114	#103	—	2	2	no	—
115	#103	—	2	2	no	—
116	#103	—	2	2	no	—
117	#103	—	2	2	no	—
118	#103	—	2	2	no	—
119	#103	—	2	2	no	—
120	#103	—	2	2	no	—
121	#103	—	2	2	no	—
122	#103	—	2	2	no	—
123	C₂²⋊C₈	D₄	1	2	yes	—
124	C₂²⋊C₈	D₄	1	2	yes	—
125	#97	D₄	1	2	yes	—
126	C₄×C₈	C₂³	1	1	yes	characteristic
127	C₄×C₈	C₂³	1	1	yes	characteristic
128	C₄×C₈	C₂³	1	1	yes	characteristic
129	C₄×C₈	C₂³	1	1	yes	characteristic
130	C₄×C₈	C₂³	1	1	yes	characteristic
131	C₄×C₈	C₂³	1	1	yes	characteristic
132	C₂×Q₈	—	2	1	no	—
133	C₂×Q₈	—	2	1	no	—
134	C₂×Q₈	—	2	1	no	—
135	C₂×Q₈	—	2	1	no	—
136	C₂×Q₈	—	2	1	no	—
137	C₂×Q₈	—	2	1	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	—	4	1	no	—
141	C₂×Q₈	—	8	2	no	—
142	C₂²×C₄	—	4	1	no	—
143	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
146	Q₁₆	—	4	4	no	—
147	Q₁₆	—	8	2	no	—
148	Q₁₆	—	8	2	no	—
149	M₁₆	—	2	4	no	—
150	C₂×C₈	C₂×D₄	1	2	yes	—
151	C₂×C₈	C₂×D₄	1	2	yes	—
152	C₂×C₈	C₂×D₄	1	2	yes	—
153	C₂×C₈	C₂×D₄	1	2	yes	—
154	C₂×C₈	C₂×D₄	1	2	yes	—
155	C₂×C₈	C₂×D₄	1	2	yes	—
156	C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₄	—	2	1	no	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	—	2	1	no	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	4	1	no	—
180	C₂²⋊C₄	—	4	1	no	—
181	C₂²⋊C₄	—	4	1	no	—
182	C₂²⋊C₄	—	4	1	no	—
183	C₄²	—	4	1	no	—
184	C₄²	—	4	1	no	—
185	C₄²	—	4	1	no	—
186	C₄²	C₂×D₄	1	1	yes	characteristic
187	C₄²	C₂×D₄	1	1	yes	characteristic
188	C₄²	C₂×D₄	1	1	yes	characteristic
189	C₄²	C₂⁴	1	1	yes	characteristic
190	C₂³	#127	1	1	yes	characteristic
191	Q₈	—	4	2	no	—
192	Q₈	—	4	2	no	—
193	Q₈	—	4	2	no	—
194	Q₈	—	4	2	no	—
195	Q₈	—	4	2	no	—
196	Q₈	—	4	2	no	—
197	Q₈	—	4	2	no	—
198	Q₈	—	4	2	no	—
199	Q₈	—	8	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	8	2	no	—
212	C₂×C₄	#127	1	1	yes	characteristic
213	C₂×C₄	#127	1	1	yes	characteristic
214	C₂×C₄	#127	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	#137	1	4	yes	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	C₂×#127	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	8	1	no	—
232	C₄	—	8	1	no	—
233	C₄	—	8	1	no	—
234	C₄	—	8	1	no	—
235	C₄	—	8	1	no	—
236	C₄	—	8	1	no	—
237	C₄	—	8	1	no	—
238	C₄	—	8	1	no	—
239	C₄	C₂×#137	1	2	yes	—
240	C₄	#577	1	2	yes	—
241	C₄	#578	1	2	yes	—
242	C₂	—	4	1	no	—
243	C₂	#3152	1	1	yes	characteristic
244	C₂	#3154	1	1	yes	characteristic
245	C₂	#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂³	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
7. #1723	C₂	semidirect
8. #1723	C₂	semidirect
9. #1718	C₂	semidirect
10. #1718	C₂	semidirect
11. #1716	C₂	semidirect
12. #1716	C₂	semidirect
13. #1707	C₂	semidirect
14. #1707	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3231	2
#3229	2
#3209	2
#3185	2
#3165	2
#1723	2
#1718	2
#1716	2
#1707	2
#1453	2