Subgroups — #18032

Order 256.

283 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3104	C₂	1	1	yes	maximal, characteristic
2	#2986	C₂	1	1	yes	maximal, characteristic
3	#2984	C₂	1	1	yes	maximal, characteristic
4	#1593	C₂	1	1	yes	maximal, characteristic, complemented
5	#1591	C₂	1	1	yes	maximal, characteristic, complemented
6	#1587	C₂	1	1	yes	maximal, characteristic, complemented
7	#1587	C₂	1	1	yes	maximal, characteristic, complemented
8	#1584	C₂	1	1	yes	maximal, characteristic, complemented
9	#1584	C₂	1	1	yes	maximal, characteristic, complemented
10	#1579	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#1572	C₂	1	1	yes	maximal, characteristic
13	#1497	C₂	1	1	yes	maximal, characteristic
14	#1479	C₂	1	1	yes	maximal, characteristic
15	#1454	C₂	1	1	yes	maximal, characteristic
16	#533	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
18	#484	C₄	1	1	yes	characteristic
19	#484	C₄	1	1	yes	characteristic
20	#477	C₄	1	1	yes	characteristic
21	#477	C₄	1	1	yes	characteristic
22	#462	C₄	1	1	yes	characteristic
23	#462	C₄	1	1	yes	characteristic
24	#451	C₄	1	1	yes	characteristic
25	#451	C₄	1	1	yes	characteristic
26	#446	—	2	1	no	—
27	#446	—	2	1	no	—
28	#446	—	2	1	no	—
29	#446	C₂²	1	1	yes	characteristic
30	#446	C₂²	1	1	yes	characteristic
31	#446	C₂²	1	1	yes	characteristic
32	#446	C₂²	1	1	yes	characteristic
33	C₈×Q₈	—	2	1	no	—
34	C₈×Q₈	C₂²	1	1	yes	characteristic
35	C₈×Q₈	C₂²	1	1	yes	characteristic
36	C₈×Q₈	C₂²	1	1	yes	characteristic
37	C₈×Q₈	C₂²	1	1	yes	characteristic
38	C₄×Q₁₆	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆	C₂²	1	1	yes	characteristic
41	C₄×Q₁₆	C₂²	1	1	yes	characteristic
42	#433	—	2	1	no	—
43	#433	—	2	1	no	—
44	#432	—	2	1	no	—
45	#432	C₂²	1	1	yes	characteristic
46	#431	—	2	1	no	—
47	#431	C₂²	1	1	yes	characteristic
48	#431	C₂²	1	1	yes	characteristic
49	#431	C₂²	1	1	yes	characteristic
50	#431	C₂²	1	1	yes	characteristic
51	#335	C₂²	1	1	yes	characteristic
52	#335	C₂²	1	1	yes	characteristic
53	#334	C₂²	1	1	yes	characteristic
54	#334	C₂²	1	1	yes	characteristic
55	Q₈⋊C₈	C₂²	1	1	yes	characteristic
56	Q₈⋊C₈	C₂²	1	1	yes	characteristic
57	Q₈⋊C₈	C₂²	1	1	yes	characteristic
58	Q₈⋊C₈	C₂²	1	1	yes	characteristic
59	Q₈⋊C₈	C₂²	1	1	yes	characteristic
60	Q₈⋊C₈	C₂²	1	1	yes	characteristic
61	Q₈⋊C₈	C₂²	1	1	yes	characteristic
62	Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	#322	C₂²	1	1	yes	characteristic
64	#322	C₂²	1	1	yes	characteristic
65	#322	C₂²	1	1	yes	characteristic
66	C₈²	C₂²	1	1	yes	characteristic
67	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
68	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
69	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
70	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
71	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
72	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
73	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
74	#125	—	2	1	no	—
75	C₂²⋊Q₈	—	2	1	no	—
76	C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
77	C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
78	C₄×Q₈	—	2	1	no	—
79	C₄×Q₈	—	2	1	no	—
80	C₄×Q₈	—	2	1	no	—
81	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
82	C₄×Q₈	C₂×C₄	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	C₄×Q₈	C₂³	1	1	yes	characteristic
87	#117	—	2	1	no	—
88	C₂×C₄⋊C₄	—	2	1	no	—
89	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄²	—	2	1	no	—
91	C₂×C₄²	C₂³	1	1	yes	characteristic
92	#107	C₂×C₄	1	1	yes	characteristic
93	#107	C₂×C₄	1	1	yes	characteristic
94	#107	C₂×C₄	1	1	yes	characteristic
95	#107	C₂×C₄	1	1	yes	characteristic
96	C₄⋊C₈	—	2	1	no	—
97	C₄⋊C₈	—	2	1	no	—
98	C₄⋊C₈	—	2	1	no	—
99	C₄⋊C₈	—	2	1	no	—
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	—	2	1	no	—
102	C₄⋊C₈	—	2	1	no	—
103	C₄⋊C₈	—	2	1	no	—
104	C₄⋊C₈	—	2	1	no	—
105	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
106	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
107	C₄⋊C₈	D₄	1	1	yes	characteristic
108	C₄⋊C₈	D₄	1	1	yes	characteristic
109	C₄⋊C₈	C₂³	1	1	yes	characteristic
110	C₄⋊C₈	C₂³	1	1	yes	characteristic
111	C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₄⋊C₈	C₂³	1	1	yes	characteristic
113	#103	C₂×C₄	1	1	yes	characteristic
114	#103	C₂×C₄	1	1	yes	characteristic
115	#103	C₂×C₄	1	1	yes	characteristic
116	#103	C₂×C₄	1	1	yes	characteristic
117	#103	C₂×C₄	1	1	yes	characteristic
118	#103	C₂×C₄	1	1	yes	characteristic
119	#103	C₂×C₄	1	1	yes	characteristic
120	#103	C₂×C₄	1	1	yes	characteristic
121	C₂²⋊C₈	—	2	1	no	—
122	C₂²⋊C₈	—	2	1	no	—
123	C₂²⋊C₈	—	2	1	no	—
124	C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
125	C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
126	C₂²⋊C₈	D₄	1	1	yes	characteristic
127	C₂²⋊C₈	D₄	1	1	yes	characteristic
128	#97	—	2	1	no	—
129	#97	—	2	1	no	—
130	#97	—	2	1	no	—
131	#97	—	2	1	no	—
132	#97	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₄×C₈	—	2	1	no	—
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	—	2	1	no	—
137	C₄×C₈	—	2	1	no	—
138	C₄×C₈	C₂³	1	1	yes	characteristic
139	C₄×C₈	C₂³	1	1	yes	characteristic
140	C₄×C₈	C₂³	1	1	yes	characteristic
141	C₄×C₈	C₂³	1	1	yes	characteristic
142	C₄×C₈	C₂³	1	1	yes	characteristic
143	C₄×C₈	C₂³	1	1	yes	characteristic
144	C₂×Q₈	—	2	1	no	—
145	C₂×Q₈	—	2	1	no	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×Q₈	—	4	1	no	—
148	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
149	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
150	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
151	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄	#41	1	1	yes	characteristic
157	Q₁₆	—	2	2	no	—
158	Q₁₆	—	2	2	no	—
159	Q₁₆	—	4	1	no	—
160	Q₁₆	—	4	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
168	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
169	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
171	C₂×C₈	C₂×D₄	1	1	yes	characteristic
172	C₂×C₈	C₂×D₄	1	1	yes	characteristic
173	C₂×C₈	C₂×D₄	1	1	yes	characteristic
174	C₂×C₈	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
188	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
189	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
190	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
191	C₂²⋊C₄	—	2	1	no	—
192	C₂²⋊C₄	—	2	1	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	C₂²×C₄	1	1	yes	characteristic
200	C₄²	C₂×D₄	1	1	yes	characteristic
201	C₄²	#41	1	1	yes	characteristic
202	C₄²	C₂⁴	1	1	yes	characteristic
203	C₂³	C₄×D₄	1	1	yes	characteristic
204	Q₈	—	2	1	no	—
205	Q₈	—	2	1	no	—
206	Q₈	—	2	1	no	—
207	Q₈	—	2	1	no	—
208	Q₈	—	2	1	no	—
209	Q₈	—	2	1	no	—
210	Q₈	—	2	1	no	—
211	Q₈	—	4	2	no	—
212	Q₈	—	4	2	no	—
213	Q₈	#131	1	2	yes	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	C₄×D₄	1	1	yes	characteristic
230	C₂×C₄	C₄×D₄	1	1	yes	characteristic
231	C₂×C₄	C₄×D₄	1	1	yes	characteristic
232	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
233	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄	C₂×#41	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	4	1	no	—
244	C₈	—	4	1	no	—
245	C₈	—	4	1	no	—
246	C₈	—	4	1	no	—
247	C₈	—	4	1	no	—
248	C₈	—	4	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
253	C₄	—	2	1	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	1	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	1	no	—
265	C₄	C₂×#131	1	1	yes	characteristic
266	C₄	#568	1	1	yes	characteristic
267	C₄	#578	1	1	yes	characteristic
268	C₄	#578	1	1	yes	characteristic
269	C₂	—	4	1	no	—
270	C₂	#2933	1	1	yes	characteristic
271	C₂	#2950	1	1	yes	characteristic
272	C₂	#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂³	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1593	C₂	semidirect
5. #1591	C₂	semidirect
6. #1587	C₂	semidirect
7. #1587	C₂	semidirect
8. #1584	C₂	semidirect
9. #1584	C₂	semidirect
10. #1579	C₂	semidirect
11. C₈×Q₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3104	2
#2986	2
#2984	2
#1593	2
#1591	2
#1587	2
#1584	2
#1579	2
C₈×Q₁₆	2
#1572	2
#1497	2
#1479	2
#1454	2