Subgroups — #33832

Order 256.

343 conjugacy classes of subgroups.

The subgroup lattice has 291 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#3280	C₂	1	1	yes	maximal, characteristic
5	#2251	C₂	1	1	yes	maximal, characteristic, complemented
6	#2220	C₂	1	1	yes	maximal, characteristic, complemented
7	#2216	C₂	1	1	yes	maximal, characteristic, complemented
8	#2211	C₂	1	1	yes	maximal, characteristic, complemented
9	#2208	C₂	1	1	yes	maximal, characteristic, complemented
10	#2189	C₂	1	1	yes	maximal, characteristic, complemented
11	#2188	C₂	1	1	yes	maximal, characteristic, complemented
12	#2187	C₂	1	1	yes	maximal, characteristic, complemented
13	#2185	C₂	1	1	yes	maximal, characteristic, complemented
14	#2175	C₂	1	1	yes	maximal, characteristic, complemented
15	#2168	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#135	C₂²	1	1	yes	characteristic
17	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
18	C₂²×Q₁₆	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₈	C₂²	1	1	yes	characteristic
20	D₄×Q₈	—	2	1	no	—
21	#546	—	2	1	no	—
22	#510	—	2	4	no	—
23	C₂×Q₃₂	C₂²	1	1	yes	characteristic, complemented
24	C₂×Q₃₂	C₂²	1	1	yes	characteristic, complemented
25	C₂×Q₃₂	C₂²	1	1	yes	characteristic, complemented
26	C₂×SD₃₂	—	2	4	no	—
27	C₂×SD₃₂	C₂²	1	1	yes	characteristic
28	C₂×SD₃₂	C₂²	1	1	yes	characteristic
29	C₂×SD₃₂	C₂²	1	1	yes	characteristic
30	C₂×SD₃₂	C₂²	1	1	yes	characteristic
31	C₂×SD₃₂	C₂²	1	1	yes	characteristic
32	C₂×SD₃₂	C₂²	1	1	yes	characteristic
33	C₂×M₃₂	C₂²	1	1	yes	characteristic, complemented
34	C₂²×C₁₆	C₂²	1	1	yes	characteristic
35	#495	C₂²	1	1	yes	characteristic
36	#494	C₂²	1	1	yes	characteristic, complemented
37	#467	C₂²	1	1	yes	characteristic
38	#467	C₂²	1	1	yes	characteristic
39	#467	C₂²	1	1	yes	characteristic, complemented
40	#466	C₂²	1	1	yes	characteristic
41	#463	—	2	1	no	—
42	#462	—	2	1	no	—
43	#451	—	2	1	no	—
44	#451	—	2	1	no	—
45	#450	—	2	1	no	—
46	#449	—	2	1	no	—
47	C₄×Q₁₆	C₂²	1	1	yes	characteristic, complemented
48	C₄×D₈	C₂²	1	1	yes	characteristic
49	C₈×D₄	C₂²	1	1	yes	characteristic
50	#366	C₂²	1	1	yes	characteristic, complemented
51	C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
52	#358	C₂²	1	1	yes	characteristic, complemented
53	#358	C₂²	1	1	yes	characteristic, complemented
54	#358	C₂²	1	1	yes	characteristic, complemented
55	#358	C₂²	1	1	yes	characteristic, complemented
56	#357	C₂²	1	1	yes	characteristic
57	#357	C₂²	1	1	yes	characteristic
58	C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
59	C₂²⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
60	#346	C₂²	1	1	yes	characteristic, complemented
61	C₂×#41	—	2	1	no	—
62	C₂²×Q₈	—	2	1	no	—
63	C₂²×Q₈	—	2	1	no	—
64	C₂²×D₄	—	2	1	no	—
65	#135	—	2	2	no	—
66	#135	—	4	1	no	—
67	C₂×Q₁₆	—	2	2	no	—
68	C₂×Q₁₆	—	2	2	no	—
69	C₂×Q₁₆	—	2	1	no	—
70	C₂×Q₁₆	—	2	1	no	—
71	C₂×Q₁₆	—	2	1	no	—
72	C₂×Q₁₆	—	4	1	no	—
73	C₂×Q₁₆	—	4	1	no	—
74	C₂×Q₁₆	C₂³	1	1	yes	characteristic
75	C₂×Q₁₆	C₂³	1	1	yes	characteristic
76	C₂×Q₁₆	C₂³	1	1	yes	characteristic
77	C₂×Q₁₆	C₂³	1	1	yes	characteristic
78	C₂×SD₁₆	—	2	1	no	—
79	C₂×SD₁₆	—	2	1	no	—
80	C₂×D₈	—	2	2	no	—
81	C₂×D₈	—	2	1	no	—
82	C₂×D₈	—	4	1	no	—
83	C₂×D₈	C₂³	1	1	yes	characteristic
84	C₂×D₈	C₂³	1	1	yes	characteristic
85	C₂²×C₈	C₂³	1	1	yes	characteristic
86	C₂²×C₈	C₂³	1	1	yes	characteristic
87	C₄⋊Q₈	—	2	1	no	—
88	C₄⋊Q₈	—	4	1	no	—
89	#124	—	2	1	no	—
90	#123	—	4	1	no	—
91	C₂²⋊Q₈	—	2	1	no	—
92	C₂²⋊Q₈	—	2	1	no	—
93	C₂²⋊Q₈	—	2	1	no	—
94	C₂²⋊Q₈	—	4	1	no	—
95	C₂²⋊Q₈	—	4	1	no	—
96	#121	—	2	1	no	—
97	#121	—	4	1	no	—
98	#120	—	4	1	no	—
99	C₄×Q₈	—	2	1	no	—
100	C₄×D₄	—	2	1	no	—
101	C₄×D₄	—	4	1	no	—
102	C₄×D₄	D₄	1	1	yes	characteristic
103	C₂×C₂²⋊C₄	—	4	1	no	—
104	Q₃₂	—	2	2	no	—
105	Q₃₂	—	2	2	no	—
106	Q₃₂	—	4	1	no	—
107	SD₃₂	—	2	2	no	—
108	SD₃₂	—	2	2	no	—
109	SD₃₂	—	2	2	no	—
110	SD₃₂	—	2	2	no	—
111	SD₃₂	—	4	1	no	—
112	SD₃₂	D₄	1	4	yes	complemented
113	M₃₂	—	2	2	no	—
114	C₂×C₁₆	—	2	2	no	—
115	C₂×C₁₆	C₂³	1	1	yes	characteristic
116	C₂×C₁₆	C₂³	1	1	yes	characteristic
117	C₂×C₁₆	C₂³	1	1	yes	characteristic
118	C₂×C₁₆	C₂³	1	1	yes	characteristic
119	#107	C₂³	1	1	yes	characteristic
120	#107	C₂³	1	1	yes	characteristic
121	C₄⋊C₈	D₄	1	1	yes	characteristic
122	#103	—	2	1	no	—
123	#103	—	2	1	no	—
124	#103	—	2	1	no	—
125	#103	—	2	1	no	—
126	#102	—	2	1	no	—
127	#102	—	2	1	no	—
128	C₂²⋊C₈	D₄	1	1	yes	characteristic
129	C₂²⋊C₈	D₄	1	1	yes	characteristic
130	C₄×C₈	C₂³	1	1	yes	characteristic
131	C₂⁴	—	4	1	no	—
132	#41	—	4	2	no	—
133	#41	—	8	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	—	4	2	no	—
139	C₂×Q₈	—	4	2	no	—
140	C₂×Q₈	—	4	1	no	—
141	C₂×Q₈	—	4	1	no	—
142	C₂×Q₈	—	8	1	no	—
143	C₂×Q₈	—	8	1	no	—
144	C₂×D₄	—	2	1	no	—
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	—	4	2	no	—
147	C₂×D₄	—	4	1	no	—
148	C₂×D₄	—	4	1	no	—
149	C₂×D₄	—	4	1	no	—
150	C₂×D₄	—	8	1	no	—
151	C₂×D₄	D₈	1	1	yes	characteristic
152	C₂²×C₄	—	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	Q₁₆	—	2	1	no	—
159	Q₁₆	—	2	1	no	—
160	Q₁₆	—	2	1	no	—
161	Q₁₆	—	2	1	no	—
162	Q₁₆	—	2	1	no	—
163	Q₁₆	—	2	1	no	—
164	Q₁₆	—	2	1	no	—
165	Q₁₆	—	4	2	no	—
166	Q₁₆	—	4	2	no	—
167	Q₁₆	—	8	1	no	—
168	Q₁₆	C₂×D₄	1	2	yes	—
169	SD₁₆	—	4	2	no	—
170	SD₁₆	—	8	1	no	—
171	D₈	—	2	1	no	—
172	D₈	—	2	1	no	—
173	D₈	—	2	1	no	—
174	D₈	—	4	2	no	—
175	D₈	C₂×D₄	1	2	yes	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈	C₂⁴	1	1	yes	characteristic
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	C₄⋊C₄	—	4	1	no	—
189	C₄⋊C₄	—	4	1	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	—	4	1	no	—
192	C₄⋊C₄	D₈	1	1	yes	characteristic
193	C₂²⋊C₄	—	4	1	no	—
194	C₂²⋊C₄	—	4	1	no	—
195	C₂²⋊C₄	—	4	1	no	—
196	C₂²⋊C₄	—	4	1	no	—
197	C₂²⋊C₄	—	4	1	no	—
198	C₂²⋊C₄	—	4	1	no	—
199	C₂²⋊C₄	—	4	1	no	—
200	C₂²⋊C₄	D₈	1	1	yes	characteristic
201	C₂²⋊C₄	D₈	1	1	yes	characteristic
202	C₄²	—	4	1	no	—
203	C₄²	C₂×D₄	1	1	yes	characteristic
204	C₁₆	—	2	1	no	—
205	C₁₆	—	2	1	no	—
206	C₁₆	—	2	1	no	—
207	C₁₆	C₂×D₄	1	2	yes	—
208	C₂³	—	4	1	no	—
209	C₂³	—	4	1	no	—
210	C₂³	—	8	2	no	—
211	C₂³	—	8	2	no	—
212	C₂³	—	16	1	no	—
213	C₂³	C₂×D₈	1	1	yes	characteristic
214	C₂³	C₂×D₈	1	1	yes	characteristic
215	Q₈	—	2	2	no	—
216	Q₈	—	4	1	no	—
217	Q₈	—	4	1	no	—
218	Q₈	—	4	1	no	—
219	Q₈	—	4	1	no	—
220	Q₈	—	4	1	no	—
221	Q₈	—	4	1	no	—
222	Q₈	—	4	1	no	—
223	Q₈	—	8	2	no	—
224	Q₈	—	8	2	no	—
225	D₄	—	2	2	no	—
226	D₄	—	2	2	no	—
227	D₄	—	4	1	no	—
228	D₄	—	4	1	no	—
229	D₄	—	4	1	no	—
230	D₄	—	8	2	no	—
231	D₄	—	8	2	no	—
232	D₄	—	16	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	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	8	2	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	—	16	1	no	—
247	C₂×C₄	C₂×D₈	1	1	yes	characteristic
248	C₂×C₄	C₂×D₈	1	1	yes	characteristic
249	C₂×C₄	C₂×D₈	1	1	yes	characteristic
250	C₂×C₄	C₂×D₈	1	1	yes	characteristic
251	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	C₂²×D₄	1	1	yes	characteristic
256	C₈	H(5,𝔽₂)	1	1	yes	characteristic
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	8	1	no	—
264	C₂²	—	8	1	no	—
265	C₂²	—	8	1	no	—
266	C₂²	—	8	1	no	—
267	C₂²	—	8	1	no	—
268	C₂²	—	16	4	no	—
269	C₂²	#510	1	2	yes	complemented
270	C₂²	C₂²×D₈	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	4	2	no	—
276	C₄	—	8	1	no	—
277	C₄	—	8	1	no	—
278	C₄	—	8	1	no	—
279	C₄	—	8	1	no	—
280	C₄	—	8	1	no	—
281	C₄	D₄²	1	1	yes	characteristic
282	C₄	#576	1	1	yes	characteristic
283	C₂	—	2	2	no	—
284	C₂	—	4	1	no	—
285	C₂	—	8	2	no	—
286	C₂	—	16	1	no	—
287	C₂	D₄×D₈	1	1	yes	characteristic
288	C₂	C₂×#510	1	1	yes	characteristic
289	C₂	#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#3280	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#510	C₂	semidirect
2. C₂²×SD₃₂	C₂	semidirect
3. D₄×Q₁₆	C₂	semidirect
5. #2251	C₂	semidirect
6. #2220	C₂	semidirect
7. #2216	C₂	semidirect
8. #2211	C₂	semidirect
9. #2208	C₂	semidirect
10. #2189	C₂	semidirect
11. #2188	C₂	semidirect
12. #2187	C₂	semidirect
13. #2185	C₂	semidirect
14. #2175	C₂	semidirect
15. #2168	C₂	semidirect
18. C₂²×Q₁₆	C₂²	semidirect
23. C₂×Q₃₂	C₂²	semidirect
24. C₂×Q₃₂	C₂²	semidirect
25. C₂×Q₃₂	C₂²	semidirect
33. C₂×M₃₂	C₂²	semidirect
36. #494	C₂²	semidirect
39. #467	C₂²	semidirect
47. C₄×Q₁₆	C₂²	semidirect
50. #366	C₂²	semidirect
51. C₄⋊C₁₆	C₂²	semidirect
52. #358	C₂²	semidirect
53. #358	C₂²	semidirect
54. #358	C₂²	semidirect
55. #358	C₂²	semidirect
59. C₂²⋊C₁₆	C₂²	semidirect
60. #346	C₂²	semidirect
112. SD₃₂	D₄	semidirect
269. C₂²	#510	semidirect

Maximal subgroups

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

subgroup	index
C₂×#510	2
C₂²×SD₃₂	2
D₄×Q₁₆	2
#3280	2
#2251	2
#2220	2
#2216	2
#2211	2
#2208	2
#2189	2
#2188	2
#2187	2
#2185	2
#2175	2
#2168	2