Subgroups — #19748

Order 256.

396 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3184	C₂	1	1	yes	maximal, characteristic
2	#3182	C₂	1	1	yes	maximal, characteristic, complemented
3	#3182	C₂	1	1	yes	maximal, characteristic, complemented
4	#3165	C₂	1	1	yes	maximal, characteristic, complemented
5	#3152	C₂	1	1	yes	maximal, characteristic, complemented
6	#3141	C₂	1	1	yes	maximal, characteristic, complemented
7	#1723	C₂	1	1	yes	maximal, characteristic, complemented
8	#1722	C₂	1	1	yes	maximal, characteristic, complemented
9	#1720	C₂	1	1	yes	maximal, characteristic, complemented
10	#1719	C₂	1	1	yes	maximal, characteristic, complemented
11	#1715	C₂	1	1	yes	maximal, characteristic, complemented
12	#1714	C₂	1	1	yes	maximal, characteristic, complemented
13	#1707	C₂	1	1	yes	maximal, characteristic, complemented
14	#1706	C₂	1	1	yes	maximal, characteristic, complemented
15	#1455	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#137	—	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	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	#501	—	2	2	no	—
22	#500	C₂²	1	1	yes	characteristic
23	#500	C₂²	1	1	yes	characteristic
24	#500	C₂²	1	1	yes	characteristic
25	#500	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	#497	—	2	2	no	—
30	#494	C₂²	1	1	yes	characteristic
31	#494	C₂²	1	1	yes	characteristic
32	#493	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	#492	C₂²	1	1	yes	characteristic
37	#489	—	2	2	no	—
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	#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	#486	C₂²	1	1	yes	characteristic
47	#486	C₂²	1	1	yes	characteristic
48	#469	—	2	2	no	—
49	#468	—	2	2	no	—
50	#468	—	2	2	no	—
51	#442	—	2	2	no	—
52	#440	—	2	2	no	—
53	#440	—	2	2	no	—
54	#428	—	2	2	no	—
55	#417	—	2	2	no	—
56	C₄×M₁₆	C₂²	1	1	yes	characteristic
57	C₄×M₁₆	C₂²	1	1	yes	characteristic
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	C₂×#41	—	2	1	no	—
64	C₂×#41	—	2	1	no	—
65	C₂²×Q₈	—	2	1	no	—
66	#137	—	4	8	no	—
67	C₂×Q₁₆	—	2	2	no	—
68	C₂×Q₁₆	—	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×M₁₆	D₄	1	2	yes	—
75	C₂×M₁₆	D₄	1	2	yes	—
76	C₂×M₁₆	D₄	1	2	yes	—
77	C₄⋊Q₈	—	2	1	no	—
78	C₄⋊Q₈	—	2	1	no	—
79	C₄⋊Q₈	—	2	1	no	—
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	C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	#127	C₂³	1	1	yes	characteristic
87	#127	C₂³	1	1	yes	characteristic
88	#125	—	2	1	no	—
89	#124	—	2	1	no	—
90	#124	—	2	1	no	—
91	C₂²⋊Q₈	—	2	1	no	—
92	C₂²⋊Q₈	—	2	1	no	—
93	#121	—	2	1	no	—
94	#121	—	2	1	no	—
95	#121	—	2	1	no	—
96	#121	—	2	1	no	—
97	C₄×Q₈	—	2	1	no	—
98	C₄×Q₈	—	2	1	no	—
99	C₄×D₄	—	2	1	no	—
100	C₄×D₄	—	2	1	no	—
101	C₄×D₄	—	2	1	no	—
102	C₄×D₄	—	2	1	no	—
103	#117	—	2	1	no	—
104	C₂×C₄⋊C₄	—	2	1	no	—
105	C₂×C₄⋊C₄	—	2	1	no	—
106	C₂×C₄²	C₂³	1	1	yes	characteristic
107	#107	—	2	2	no	—
108	#107	—	2	2	no	—
109	#107	—	2	2	no	—
110	#107	—	2	2	no	—
111	#106	—	2	2	no	—
112	#106	—	2	2	no	—
113	#106	—	2	2	no	—
114	#103	—	2	2	no	—
115	#103	—	2	2	no	—
116	#103	—	2	2	no	—
117	#103	—	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	#97	D₄	1	2	yes	—
125	#97	D₄	1	2	yes	—
126	#97	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	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	—	2	1	no	—
141	C₂×Q₈	—	4	4	no	—
142	C₂×Q₈	—	4	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	—	4	1	no	—
150	C₂×D₄	—	4	1	no	—
151	C₂²×C₄	—	4	1	no	—
152	C₂²×C₄	—	4	1	no	—
153	C₂²×C₄	—	4	1	no	—
154	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
157	Q₁₆	—	4	4	no	—
158	Q₁₆	—	4	4	no	—
159	SD₁₆	—	4	4	no	—
160	SD₁₆	—	4	4	no	—
161	SD₁₆	—	4	4	no	—
162	SD₁₆	—	4	4	no	—
163	D₈	—	4	4	no	—
164	M₁₆	—	2	4	no	—
165	M₁₆	—	2	4	no	—
166	M₁₆	—	4	2	no	—
167	C₂×C₈	C₂×D₄	1	2	yes	—
168	C₂×C₈	C₂×D₄	1	2	yes	—
169	C₂×C₈	C₂×D₄	1	2	yes	—
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	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	1	no	—
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	—	4	1	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊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	C₄²	—	4	1	no	—
195	C₄²	C₂×D₄	1	1	yes	characteristic
196	C₄²	C₂×D₄	1	1	yes	characteristic
197	C₄²	C₂×D₄	1	1	yes	characteristic
198	C₄²	C₂⁴	1	1	yes	characteristic
199	C₂³	—	4	1	no	—
200	C₂³	—	4	1	no	—
201	C₂³	#127	1	1	yes	characteristic
202	Q₈	—	4	2	no	—
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	Q₈	—	4	2	no	—
206	Q₈	—	4	2	no	—
207	Q₈	—	4	2	no	—
208	Q₈	—	8	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	—	8	2	no	—
216	D₄	—	8	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	#127	1	1	yes	characteristic
230	C₂×C₄	#127	1	1	yes	characteristic
231	C₂×C₄	#127	1	1	yes	characteristic
232	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	—	2	2	no	—
237	C₈	—	2	2	no	—
238	C₈	—	2	2	no	—
239	C₈	AGL(1,ℤ/8)	1	4	yes	—
240	C₈	#137	1	4	yes	complemented
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	8	1	no	—
245	C₂²	—	8	1	no	—
246	C₂²	—	8	1	no	—
247	C₂²	—	8	1	no	—
248	C₂²	—	8	1	no	—
249	C₂²	—	8	1	no	—
250	C₂²	C₂×#127	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	8	1	no	—
255	C₄	—	8	1	no	—
256	C₄	—	8	1	no	—
257	C₄	—	8	1	no	—
258	C₄	—	8	1	no	—
259	C₄	—	8	1	no	—
260	C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
261	C₄	C₂×#137	1	2	yes	—
262	C₄	#575	1	2	yes	—
263	C₂	—	4	1	no	—
264	C₂	—	16	1	no	—
265	C₂	—	16	1	no	—
266	C₂	#3151	1	1	yes	characteristic
267	C₂	#3152	1	1	yes	characteristic
268	C₂	#3153	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3184	C₂
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
2. #3182	C₂	semidirect
3. #3182	C₂	semidirect
4. #3165	C₂	semidirect
5. #3152	C₂	semidirect
6. #3141	C₂	semidirect
7. #1723	C₂	semidirect
8. #1722	C₂	semidirect
9. #1720	C₂	semidirect
10. #1719	C₂	semidirect
11. #1715	C₂	semidirect
12. #1714	C₂	semidirect
13. #1707	C₂	semidirect
14. #1706	C₂	semidirect
15. #1455	C₂	semidirect
240. C₈	#137	semidirect

Maximal subgroups

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

subgroup	index
#3184	2
#3182	2
#3165	2
#3152	2
#3141	2
#1723	2
#1722	2
#1720	2
#1719	2
#1715	2
#1714	2
#1707	2
#1706	2
#1455	2