Subgroups — #18047

Order 256.

321 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2970	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#137	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	#1585	C₂	1	1	yes	maximal, characteristic, complemented
8	#1584	C₂	1	1	yes	maximal, characteristic, complemented
9	#1582	C₂	1	1	yes	maximal, characteristic, complemented
10	SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#1569	C₂	1	1	yes	maximal, characteristic, complemented
13	#1491	C₂	1	1	yes	maximal, characteristic, complemented
14	#1489	C₂	1	1	yes	maximal, characteristic
15	#1455	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#137	C₄	1	1	yes	characteristic
17	C₄×#41	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
19	#446	—	2	1	no	—
20	#446	—	2	1	no	—
21	#446	C₂²	1	1	yes	characteristic
22	#446	C₂²	1	1	yes	characteristic
23	#446	C₂²	1	1	yes	characteristic
24	#446	C₂²	1	1	yes	characteristic
25	C₈×Q₈	C₂²	1	1	yes	characteristic
26	C₈×Q₈	C₂²	1	1	yes	characteristic
27	#441	C₄	1	1	yes	characteristic
28	#441	C₄	1	1	yes	characteristic
29	#440	C₄	1	1	yes	characteristic
30	#440	C₄	1	1	yes	characteristic
31	C₄×Q₁₆	C₂²	1	1	yes	characteristic
32	C₄×Q₁₆	C₂²	1	1	yes	characteristic
33	C₄×SD₁₆	C₂²	1	1	yes	characteristic
34	C₄×SD₁₆	C₂²	1	1	yes	characteristic
35	#435	—	2	1	no	—
36	#435	—	2	1	no	—
37	C₈×D₄	C₂²	1	1	yes	characteristic
38	C₈×D₄	C₂²	1	1	yes	characteristic
39	#432	—	2	1	no	—
40	#432	C₂²	1	1	yes	characteristic
41	#428	C₄	1	1	yes	characteristic
42	#423	—	2	1	no	—
43	#423	C₂²	1	1	yes	characteristic
44	#419	C₄	1	1	yes	characteristic
45	#417	C₄	1	1	yes	characteristic
46	C₄×M₁₆	—	2	1	no	—
47	C₄×M₁₆	C₂²	1	1	yes	characteristic
48	C₄×M₁₆	C₂²	1	1	yes	characteristic
49	C₄×M₁₆	C₂²	1	1	yes	characteristic
50	C₂×#97	—	2	1	no	—
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	D₄⋊C₈	C₂²	1	1	yes	characteristic
62	D₄⋊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	C₂×#41	C₂×C₄	1	1	yes	characteristic
68	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
69	#137	—	2	4	no	—
70	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
71	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
72	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic
73	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic
74	C₂×M₁₆	—	2	1	no	—
75	C₂×M₁₆	—	2	1	no	—
76	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
77	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
78	C₂×M₁₆	D₄	1	1	yes	characteristic
79	C₂×M₁₆	D₄	1	1	yes	characteristic
80	C₂²×C₈	—	2	1	no	—
81	C₂²×C₈	—	2	1	no	—
82	C₄×Q₈	—	2	1	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
87	C₄×Q₈	C₂³	1	1	yes	characteristic
88	C₄×Q₈	C₂³	1	1	yes	characteristic
89	C₄×Q₈	C₂³	1	1	yes	characteristic
90	C₄×D₄	—	2	1	no	—
91	C₄×D₄	—	2	1	no	—
92	C₄×D₄	C₂×C₄	1	1	yes	characteristic
93	C₄×D₄	C₂³	1	1	yes	characteristic
94	#117	—	2	1	no	—
95	#117	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄⋊C₄	—	2	1	no	—
97	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄²	—	2	1	no	—
99	C₂×C₄²	—	2	1	no	—
100	C₂×C₄²	C₂³	1	1	yes	characteristic
101	#107	C₂×C₄	1	1	yes	characteristic
102	#107	C₂×C₄	1	1	yes	characteristic
103	#106	C₂×C₄	1	1	yes	characteristic
104	#106	C₂×C₄	1	1	yes	characteristic
105	C₄⋊C₈	—	2	1	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	2	1	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	—	2	1	no	—
110	C₄⋊C₈	—	2	1	no	—
111	C₄⋊C₈	—	2	1	no	—
112	C₄⋊C₈	C₂³	1	1	yes	characteristic
113	C₄⋊C₈	C₂³	1	1	yes	characteristic
114	C₄⋊C₈	C₂³	1	1	yes	characteristic
115	C₄⋊C₈	C₂³	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	#103	C₂×C₄	1	1	yes	characteristic
122	#102	C₂×C₄	1	1	yes	characteristic
123	#102	C₂×C₄	1	1	yes	characteristic
124	C₂²⋊C₈	—	2	1	no	—
125	C₂²⋊C₈	—	2	1	no	—
126	C₂²⋊C₈	—	2	1	no	—
127	#97	—	2	1	no	—
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	#97	C₂×C₄	1	1	yes	characteristic
134	#97	C₂×C₄	1	1	yes	characteristic
135	#97	D₄	1	1	yes	characteristic
136	#97	D₄	1	1	yes	characteristic
137	C₄×C₈	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
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	C₄×C₈	C₂³	1	1	yes	characteristic
145	#41	—	2	2	no	—
146	#41	—	4	1	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	4	1	no	—
150	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
151	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
152	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
153	C₂×D₄	—	2	1	no	—
154	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄	#41	1	1	yes	characteristic
163	Q₁₆	—	2	2	no	—
164	Q₁₆	—	2	2	no	—
165	SD₁₆	—	2	2	no	—
166	SD₁₆	M₁₆	1	4	yes	complemented
167	M₁₆	—	2	2	no	—
168	M₁₆	—	2	2	no	—
169	M₁₆	—	4	1	no	—
170	M₁₆	—	4	1	no	—
171	M₁₆	—	4	2	no	—
172	M₁₆	—	4	2	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
182	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
183	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
184	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
185	C₂×C₈	C₂×D₄	1	1	yes	characteristic
186	C₂×C₈	C₂×D₄	1	1	yes	characteristic
187	C₂×C₈	C₂×D₄	1	1	yes	characteristic
188	C₂×C₈	C₂×D₄	1	1	yes	characteristic
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
197	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
198	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
199	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	C₂²×C₄	1	1	yes	characteristic
211	C₄²	C₂×D₄	1	1	yes	characteristic
212	C₄²	#41	1	1	yes	characteristic
213	C₄²	C₂⁴	1	1	yes	characteristic
214	C₂³	—	2	1	no	—
215	C₂³	C₄×D₄	1	1	yes	characteristic
216	Q₈	—	2	1	no	—
217	Q₈	—	2	1	no	—
218	Q₈	—	2	1	no	—
219	Q₈	—	2	1	no	—
220	Q₈	—	2	1	no	—
221	Q₈	—	4	2	no	—
222	Q₈	C₂×M₁₆	1	2	yes	—
223	D₄	—	2	1	no	—
224	D₄	—	4	2	no	—
225	D₄	C₂×M₁₆	1	2	yes	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	8	1	no	—
243	C₂×C₄	—	8	1	no	—
244	C₂×C₄	C₄×D₄	1	1	yes	characteristic
245	C₂×C₄	C₄×D₄	1	1	yes	characteristic
246	C₂×C₄	C₄×D₄	1	1	yes	characteristic
247	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
248	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄	C₂×#41	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	2	1	no	—
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	4	1	no	—
258	C₈	—	4	1	no	—
259	C₈	—	4	1	no	—
260	C₈	—	4	1	no	—
261	C₈	—	4	1	no	—
262	C₈	C₂×M₁₆	1	2	yes	—
263	C₈	#137	1	2	yes	complemented
264	C₂²	—	2	1	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	1	no	—
282	C₄	C₂²×M₁₆	1	1	yes	characteristic
283	C₄	#568	1	1	yes	characteristic
284	C₄	C₂×#137	1	1	yes	characteristic
285	C₄	#575	1	1	yes	characteristic
286	C₂	—	4	1	no	—
287	C₂	—	4	2	no	—
288	C₂	D₄×M₁₆	1	1	yes	characteristic
289	C₂	C₄×#137	1	1	yes	characteristic
290	C₂	#2956	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	C₂×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. Q₈×M₁₆	C₂	semidirect
4. #1593	C₂	semidirect
5. #1591	C₂	semidirect
6. #1587	C₂	semidirect
7. #1585	C₂	semidirect
8. #1584	C₂	semidirect
9. #1582	C₂	semidirect
10. SD₁₆⋊C₈	C₂	semidirect
11. C₈×SD₁₆	C₂	semidirect
12. #1569	C₂	semidirect
13. #1491	C₂	semidirect
15. #1455	C₂	semidirect
166. SD₁₆	M₁₆	semidirect
263. C₈	#137	semidirect

Maximal subgroups

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

subgroup	index
#2970	2
Q₈×M₁₆	2
C₄×#137	2
#1593	2
#1591	2
#1587	2
#1585	2
#1584	2
#1582	2
SD₁₆⋊C₈	2
C₈×SD₁₆	2
#1569	2
#1491	2
#1489	2
#1455	2