Subgroups — #19641

Order 256.

403 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3228	C₂	1	1	yes	maximal, characteristic, complemented
2	#3207	C₂	1	1	yes	maximal, characteristic, complemented
3	#3184	C₂	1	1	yes	maximal, characteristic, complemented
4	#3182	C₂	1	1	yes	maximal, characteristic, complemented
5	#3167	C₂	1	1	yes	maximal, characteristic, complemented
6	#3147	C₂	1	1	yes	maximal, characteristic, complemented
7	#1723	C₂	1	1	yes	maximal, characteristic, complemented
8	#1721	C₂	1	1	yes	maximal, characteristic, complemented
9	#1720	C₂	1	1	yes	maximal, characteristic, complemented
10	#1717	C₂	1	1	yes	maximal, characteristic, complemented
11	#1708	C₂	1	1	yes	maximal, characteristic, complemented
12	#1706	C₂	1	1	yes	maximal, characteristic, complemented
13	#1706	C₂	1	1	yes	maximal, characteristic, complemented
14	#1705	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#135	—	2	2	no	—
17	#533	C₂²	1	1	yes	characteristic
18	#532	C₂²	1	1	yes	characteristic
19	#532	C₂²	1	1	yes	characteristic
20	#532	C₂²	1	1	yes	characteristic
21	#500	C₂²	1	1	yes	characteristic
22	#500	C₂²	1	1	yes	characteristic
23	#498	C₂²	1	1	yes	characteristic
24	#498	C₂²	1	1	yes	characteristic
25	#498	C₂²	1	1	yes	characteristic
26	#495	—	2	2	no	—
27	#494	C₂²	1	1	yes	characteristic
28	#493	C₂²	1	1	yes	characteristic
29	#493	C₂²	1	1	yes	characteristic
30	#493	C₂²	1	1	yes	characteristic
31	#492	C₂²	1	1	yes	characteristic
32	#492	C₂²	1	1	yes	characteristic
33	#492	C₂²	1	1	yes	characteristic
34	#492	C₂²	1	1	yes	characteristic
35	#492	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	#486	C₂²	1	1	yes	characteristic
40	#486	C₂²	1	1	yes	characteristic
41	#486	C₂²	1	1	yes	characteristic
42	#486	C₂²	1	1	yes	characteristic
43	#486	C₂²	1	1	yes	characteristic
44	#486	C₂²	1	1	yes	characteristic
45	#486	C₂²	1	1	yes	characteristic
46	#482	—	2	2	no	—
47	#478	—	2	2	no	—
48	#469	—	2	2	no	—
49	#468	—	2	2	no	—
50	#465	—	2	2	no	—
51	#463	—	2	2	no	—
52	#449	—	2	2	no	—
53	#442	—	2	2	no	—
54	#440	—	2	2	no	—
55	C₄×SD₁₆	—	2	2	no	—
56	#431	C₂²	1	1	yes	characteristic
57	C₄×M₁₆	C₂²	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
59	#322	C₂²	1	1	yes	characteristic
60	#322	C₂²	1	1	yes	characteristic
61	C₈²	C₂²	1	1	yes	characteristic
62	C₈²	C₂²	1	1	yes	characteristic
63	C₂×#41	—	2	1	no	—
64	C₂×#41	—	2	1	no	—
65	C₂×#41	—	2	1	no	—
66	#135	—	4	8	no	—
67	C₂×Q₁₆	—	2	2	no	—
68	C₂×SD₁₆	—	2	2	no	—
69	C₂×SD₁₆	—	2	2	no	—
70	C₂×SD₁₆	—	2	2	no	—
71	C₂×SD₁₆	—	2	2	no	—
72	C₂×SD₁₆	—	2	2	no	—
73	C₂×D₈	—	2	2	no	—
74	C₂×D₈	—	2	2	no	—
75	C₂×D₈	—	2	2	no	—
76	C₂×M₁₆	D₄	1	2	yes	—
77	C₂²×C₈	D₄	1	2	yes	—
78	C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	#127	C₂³	1	1	yes	characteristic
84	#127	C₂³	1	1	yes	characteristic
85	#127	C₂³	1	1	yes	characteristic
86	#125	—	2	1	no	—
87	#124	—	2	1	no	—
88	#124	—	2	1	no	—
89	#124	—	2	1	no	—
90	C₂²⋊Q₈	—	2	1	no	—
91	C₂²⋊Q₈	—	2	1	no	—
92	#121	—	2	1	no	—
93	#121	—	2	1	no	—
94	#121	—	2	1	no	—
95	#121	—	2	1	no	—
96	#121	—	2	1	no	—
97	#121	—	2	1	no	—
98	C₄×Q₈	—	2	1	no	—
99	C₄×Q₈	—	2	1	no	—
100	C₄×D₄	—	2	1	no	—
101	C₄×D₄	—	2	1	no	—
102	C₄×D₄	—	2	1	no	—
103	C₄×D₄	—	2	1	no	—
104	C₄×D₄	—	2	1	no	—
105	C₄×D₄	—	2	1	no	—
106	#117	—	2	1	no	—
107	C₂×C₄²	C₂³	1	1	yes	characteristic
108	#107	—	2	2	no	—
109	#107	—	2	2	no	—
110	#106	—	2	2	no	—
111	#106	—	2	2	no	—
112	#106	—	2	2	no	—
113	C₄⋊C₈	D₄	1	2	yes	—
114	#103	—	2	2	no	—
115	#103	—	2	2	no	—
116	#103	—	2	2	no	—
117	#102	—	2	2	no	—
118	#102	—	2	2	no	—
119	#102	—	2	2	no	—
120	#102	—	2	2	no	—
121	#102	—	2	2	no	—
122	#102	—	2	2	no	—
123	#102	—	2	2	no	—
124	C₂²⋊C₈	D₄	1	2	yes	—
125	#97	D₄	1	2	yes	—
126	C₄×C₈	D₄	1	2	yes	—
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	C₄×C₈	C₂³	1	1	yes	characteristic
133	#41	—	4	4	no	—
134	#41	—	4	4	no	—
135	#41	—	8	2	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	—	2	1	no	—
141	C₂×D₄	—	2	1	no	—
142	C₂×D₄	—	2	1	no	—
143	C₂×D₄	—	2	1	no	—
144	C₂×D₄	—	2	1	no	—
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	—	2	1	no	—
147	C₂×D₄	—	2	1	no	—
148	C₂×D₄	—	2	1	no	—
149	C₂×D₄	—	2	1	no	—
150	C₂×D₄	—	4	1	no	—
151	C₂×D₄	—	4	1	no	—
152	C₂×D₄	—	4	1	no	—
153	C₂²×C₄	—	4	1	no	—
154	C₂²×C₄	—	4	1	no	—
155	C₂²×C₄	—	4	1	no	—
156	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
159	Q₁₆	—	4	4	no	—
160	SD₁₆	—	4	4	no	—
161	SD₁₆	—	4	4	no	—
162	SD₁₆	—	4	4	no	—
163	SD₁₆	—	4	4	no	—
164	SD₁₆	—	8	2	no	—
165	D₈	—	4	4	no	—
166	D₈	—	4	4	no	—
167	D₈	—	8	2	no	—
168	M₁₆	—	2	4	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	C₂×D₄	1	2	yes	—
171	C₂×C₈	C₂×D₄	1	2	yes	—
172	C₂×C₈	C₂×D₄	1	2	yes	—
173	C₂×C₈	C₂×D₄	1	2	yes	—
174	C₂×C₈	C₂×D₄	1	2	yes	—
175	C₂×C₈	C₂×D₄	1	2	yes	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₂²⋊C₄	—	4	1	no	—
189	C₂²⋊C₄	—	4	1	no	—
190	C₂²⋊C₄	—	4	1	no	—
191	C₂²⋊C₄	—	4	1	no	—
192	C₂²⋊C₄	—	4	1	no	—
193	C₂²⋊C₄	—	4	1	no	—
194	C₂²⋊C₄	—	4	1	no	—
195	C₄²	—	4	1	no	—
196	C₄²	C₂×D₄	1	1	yes	characteristic
197	C₄²	C₂×D₄	1	1	yes	characteristic
198	C₄²	C₂×D₄	1	1	yes	characteristic
199	C₄²	C₂⁴	1	1	yes	characteristic
200	C₂³	—	4	1	no	—
201	C₂³	—	4	1	no	—
202	C₂³	—	4	1	no	—
203	C₂³	#127	1	1	yes	characteristic
204	Q₈	—	4	2	no	—
205	Q₈	—	4	2	no	—
206	Q₈	—	4	2	no	—
207	Q₈	—	4	2	no	—
208	Q₈	—	4	2	no	—
209	D₄	—	4	2	no	—
210	D₄	—	4	2	no	—
211	D₄	—	4	2	no	—
212	D₄	—	4	2	no	—
213	D₄	—	4	2	no	—
214	D₄	—	4	2	no	—
215	D₄	—	4	2	no	—
216	D₄	—	4	2	no	—
217	D₄	—	4	2	no	—
218	D₄	—	8	2	no	—
219	D₄	—	8	2	no	—
220	D₄	—	8	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	#127	1	1	yes	characteristic
233	C₂×C₄	#127	1	1	yes	characteristic
234	C₂×C₄	#127	1	1	yes	characteristic
235	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	#135	1	4	yes	complemented
243	C₈	AGL(1,ℤ/8)	1	4	yes	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	8	1	no	—
248	C₂²	—	8	1	no	—
249	C₂²	—	8	1	no	—
250	C₂²	—	8	1	no	—
251	C₂²	—	8	1	no	—
252	C₂²	—	8	1	no	—
253	C₂²	—	8	1	no	—
254	C₂²	—	8	1	no	—
255	C₂²	—	8	1	no	—
256	C₂²	C₂×#127	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	8	1	no	—
261	C₄	—	8	1	no	—
262	C₄	—	8	1	no	—
263	C₄	—	8	1	no	—
264	C₄	—	8	1	no	—
265	C₄	C₂×#135	1	2	yes	—
266	C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
267	C₄	#577	1	2	yes	—
268	C₂	—	4	1	no	—
269	C₂	—	16	1	no	—
270	C₂	—	16	1	no	—
271	C₂	—	16	1	no	—
272	C₂	#3147	1	1	yes	characteristic
273	C₂	#3151	1	1	yes	characteristic
274	C₂	#3154	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
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
1. #3228	C₂	semidirect
2. #3207	C₂	semidirect
3. #3184	C₂	semidirect
4. #3182	C₂	semidirect
5. #3167	C₂	semidirect
6. #3147	C₂	semidirect
7. #1723	C₂	semidirect
8. #1721	C₂	semidirect
9. #1720	C₂	semidirect
10. #1717	C₂	semidirect
11. #1708	C₂	semidirect
12. #1706	C₂	semidirect
13. #1706	C₂	semidirect
14. #1705	C₂	semidirect
15. C₈×M₁₆	C₂	semidirect
242. C₈	#135	semidirect

Maximal subgroups

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

subgroup	index
#3228	2
#3207	2
#3184	2
#3182	2
#3167	2
#3147	2
#1723	2
#1721	2
#1720	2
#1717	2
#1708	2
#1706	2
#1705	2
C₈×M₁₆	2