Subgroups — #19765

Order 256.

334 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3385	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#3295	C₂	1	1	yes	maximal, characteristic
4	D₄×SD₁₆	C₂	1	1	yes	maximal, characteristic
5	#1714	C₂	1	1	yes	maximal, characteristic, complemented
6	#1686	C₂	1	1	yes	maximal, characteristic, complemented
7	#1678	C₂	1	1	yes	maximal, characteristic, complemented
8	#1675	C₂	1	1	yes	maximal, characteristic, complemented
9	#1671	C₂	1	1	yes	maximal, characteristic, complemented
10	#1643	C₂	1	1	yes	maximal, characteristic, complemented
11	#1637	C₂	1	1	yes	maximal, characteristic, complemented
12	#1629	C₂	1	1	yes	maximal, characteristic
13	#1623	C₂	1	1	yes	maximal, characteristic, complemented
14	#1582	C₂	1	1	yes	maximal, characteristic, complemented
15	SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×SD₁₆	—	2	1	no	—
17	Q₈²	C₂²	1	1	yes	characteristic, complemented
18	#554	C₂²	1	1	yes	characteristic
19	D₄×Q₈	C₂²	1	1	yes	characteristic
20	D₄²	C₂²	1	1	yes	characteristic
21	#501	—	2	1	no	—
22	#500	C₂²	1	1	yes	characteristic, complemented
23	#498	—	2	1	no	—
24	#498	C₂²	1	1	yes	characteristic, complemented
25	#492	C₂²	1	1	yes	characteristic
26	#492	C₂²	1	1	yes	characteristic
27	#487	C₂²	1	1	yes	characteristic, complemented
28	#486	C₂²	1	1	yes	characteristic
29	#481	—	2	1	no	—
30	#479	—	2	1	no	—
31	#477	C₂²	1	1	yes	characteristic, complemented
32	#476	—	2	1	no	—
33	#476	C₂²	1	1	yes	characteristic
34	#476	C₂²	1	1	yes	characteristic
35	#476	C₂²	1	1	yes	characteristic
36	#475	—	2	1	no	—
37	#475	C₂²	1	1	yes	characteristic, complemented
38	#474	—	2	1	no	—
39	#465	—	2	1	no	—
40	#465	—	2	1	no	—
41	#461	C₂²	1	1	yes	characteristic
42	#460	C₂²	1	1	yes	characteristic
43	#460	C₂²	1	1	yes	characteristic
44	#450	—	2	1	no	—
45	#450	—	2	1	no	—
46	#448	—	2	1	no	—
47	#446	C₂²	1	1	yes	characteristic, complemented
48	C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
49	#440	—	2	1	no	—
50	C₄×SD₁₆	—	2	1	no	—
51	C₄×SD₁₆	C₂²	1	1	yes	characteristic
52	C₄×SD₁₆	C₂²	1	1	yes	characteristic
53	C₈×D₄	C₂²	1	1	yes	characteristic
54	C₈×D₄	C₂²	1	1	yes	characteristic
55	C₂×#102	—	2	1	no	—
56	#335	C₂²	1	1	yes	characteristic, complemented
57	#334	C₂²	1	1	yes	characteristic, complemented
58	#333	C₂²	1	1	yes	characteristic, complemented
59	#333	C₂²	1	1	yes	characteristic, complemented
60	#332	C₂²	1	1	yes	characteristic, complemented
61	#331	C₂²	1	1	yes	characteristic
62	Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
63	Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
64	D₄⋊C₈	C₂²	1	1	yes	characteristic
65	D₄⋊C₈	C₂²	1	1	yes	characteristic
66	#322	C₂²	1	1	yes	characteristic, complemented
67	C₂²×Q₈	—	2	1	no	—
68	C₂²×D₄	—	2	1	no	—
69	C₂²×D₄	—	2	1	no	—
70	C₂×SD₁₆	—	2	1	no	—
71	C₂×SD₁₆	—	2	1	no	—
72	C₂×SD₁₆	—	2	1	no	—
73	C₂×SD₁₆	—	2	1	no	—
74	C₂×SD₁₆	—	2	1	no	—
75	C₂×SD₁₆	—	4	1	no	—
76	C₂×SD₁₆	—	4	4	no	—
77	C₂×SD₁₆	D₄	1	1	yes	characteristic
78	C₂×SD₁₆	D₄	1	1	yes	characteristic
79	C₂²×C₈	—	2	1	no	—
80	C₂²×C₈	—	2	1	no	—
81	C₄⋊Q₈	—	2	1	no	—
82	C₄⋊Q₈	—	2	1	no	—
83	C₄⋊Q₈	—	2	1	no	—
84	C₄⋊Q₈	—	4	1	no	—
85	C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	#127	C₂³	1	1	yes	characteristic
89	#125	—	2	1	no	—
90	C₂²⋊Q₈	—	2	1	no	—
91	C₂²⋊Q₈	—	2	1	no	—
92	C₂²⋊Q₈	—	4	1	no	—
93	C₂²⋊Q₈	—	4	1	no	—
94	#121	—	2	1	no	—
95	#121	—	2	1	no	—
96	#120	—	4	1	no	—
97	C₄×Q₈	—	2	1	no	—
98	C₄×Q₈	—	2	1	no	—
99	C₄×Q₈	C₂³	1	1	yes	characteristic
100	C₄×Q₈	C₂³	1	1	yes	characteristic
101	C₄×D₄	—	2	1	no	—
102	C₄×D₄	—	2	1	no	—
103	C₄×D₄	C₂³	1	1	yes	characteristic
104	C₄×D₄	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	2	1	no	—
106	#107	—	2	1	no	—
107	#107	—	2	1	no	—
108	#107	—	2	1	no	—
109	#106	—	2	1	no	—
110	#106	—	2	1	no	—
111	#106	—	2	1	no	—
112	#106	—	2	1	no	—
113	#106	—	2	1	no	—
114	#106	—	2	1	no	—
115	#106	D₄	1	1	yes	characteristic
116	#106	D₄	1	1	yes	characteristic
117	C₄⋊C₈	—	2	1	no	—
118	C₄⋊C₈	—	2	1	no	—
119	C₄⋊C₈	C₂³	1	1	yes	characteristic
120	C₄⋊C₈	C₂³	1	1	yes	characteristic
121	C₄⋊C₈	C₂³	1	1	yes	characteristic
122	C₄⋊C₈	C₂³	1	1	yes	characteristic
123	#103	—	2	1	no	—
124	#103	—	2	1	no	—
125	#103	—	2	1	no	—
126	#103	—	4	1	no	—
127	#103	D₄	1	1	yes	characteristic
128	#103	D₄	1	1	yes	characteristic
129	#102	—	2	1	no	—
130	#102	—	2	1	no	—
131	#102	—	2	1	no	—
132	#102	—	2	1	no	—
133	#102	—	2	1	no	—
134	#102	—	2	1	no	—
135	#102	—	2	1	no	—
136	#102	D₄	1	1	yes	characteristic
137	#102	D₄	1	1	yes	characteristic
138	C₂²⋊C₈	—	2	1	no	—
139	C₂²⋊C₈	—	2	1	no	—
140	#97	—	2	1	no	—
141	C₄×C₈	—	2	1	no	—
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	C₂⁴	—	4	1	no	—
146	C₂×Q₈	—	2	1	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	—	4	1	no	—
150	C₂×Q₈	—	4	1	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	8	1	no	—
153	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
154	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
155	C₂×D₄	—	2	1	no	—
156	C₂×D₄	—	2	1	no	—
157	C₂×D₄	—	2	1	no	—
158	C₂×D₄	—	4	1	no	—
159	C₂×D₄	—	4	1	no	—
160	C₂×D₄	—	4	2	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	8	1	no	—
163	C₂×D₄	—	8	1	no	—
164	C₂×D₄	C₂×D₄	1	1	yes	characteristic
165	C₂×D₄	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	4	1	no	—
169	C₂²×C₄	—	4	1	no	—
170	SD₁₆	—	2	2	no	—
171	SD₁₆	—	4	2	no	—
172	SD₁₆	—	4	2	no	—
173	SD₁₆	—	4	2	no	—
174	SD₁₆	—	8	1	no	—
175	SD₁₆	—	8	1	no	—
176	SD₁₆	—	8	2	no	—
177	SD₁₆	SD₁₆	1	4	yes	complemented
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	8	1	no	—
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	—	2	1	no	—
197	C₄⋊C₄	—	4	1	no	—
198	C₄⋊C₄	—	4	1	no	—
199	C₄⋊C₄	—	4	1	no	—
200	C₄⋊C₄	—	4	1	no	—
201	C₄⋊C₄	—	4	1	no	—
202	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
203	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
204	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
205	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	4	1	no	—
209	C₂²⋊C₄	—	4	1	no	—
210	C₂²⋊C₄	—	4	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	C₂⁴	1	1	yes	characteristic
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	4	1	no	—
218	C₂³	—	8	2	no	—
219	C₂³	—	8	2	no	—
220	C₂³	—	16	1	no	—
221	Q₈	—	2	1	no	—
222	Q₈	—	2	1	no	—
223	Q₈	—	2	1	no	—
224	Q₈	—	4	1	no	—
225	Q₈	—	4	1	no	—
226	Q₈	—	8	2	no	—
227	Q₈	—	8	1	no	—
228	Q₈	—	8	1	no	—
229	Q₈	—	16	1	no	—
230	Q₈	C₂×SD₁₆	1	2	yes	—
231	D₄	—	2	1	no	—
232	D₄	—	2	1	no	—
233	D₄	—	4	1	no	—
234	D₄	—	4	1	no	—
235	D₄	—	4	1	no	—
236	D₄	—	4	1	no	—
237	D₄	—	8	1	no	—
238	D₄	—	8	2	no	—
239	D₄	—	8	2	no	—
240	D₄	C₂×SD₁₆	1	2	yes	—
241	D₄	#137	1	2	yes	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	16	1	no	—
257	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	4	1	no	—
264	C₈	—	4	1	no	—
265	C₈	—	4	1	no	—
266	C₈	—	4	1	no	—
267	C₈	—	4	1	no	—
268	C₈	C₂×SD₁₆	1	2	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	8	1	no	—
276	C₂²	—	8	1	no	—
277	C₂²	—	8	1	no	—
278	C₂²	—	16	4	no	—
279	C₂²	D₄²	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	1	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	1	no	—
287	C₄	—	8	1	no	—
288	C₄	—	8	1	no	—
289	C₄	—	8	1	no	—
290	C₄	C₂²×SD₁₆	1	1	yes	characteristic
291	C₄	C₂×#137	1	1	yes	characteristic
292	C₄	#577	1	1	yes	characteristic
293	C₄	#577	1	1	yes	characteristic
294	C₂	—	4	2	no	—
295	C₂	—	4	2	no	—
296	C₂	—	16	1	no	—
297	C₂	#3266	1	1	yes	characteristic
298	C₂	D₄×SD₁₆	1	1	yes	characteristic
299	C₂	#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	D₄²	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3385	C₂	semidirect
2. Q₈×SD₁₆	C₂	semidirect
5. #1714	C₂	semidirect
6. #1686	C₂	semidirect
7. #1678	C₂	semidirect
8. #1675	C₂	semidirect
9. #1671	C₂	semidirect
10. #1643	C₂	semidirect
11. #1637	C₂	semidirect
13. #1623	C₂	semidirect
14. #1582	C₂	semidirect
15. SD₁₆⋊C₈	C₂	semidirect
17. Q₈²	C₂²	semidirect
22. #500	C₂²	semidirect
24. #498	C₂²	semidirect
27. #487	C₂²	semidirect
31. #477	C₂²	semidirect
37. #475	C₂²	semidirect
47. #446	C₂²	semidirect
48. C₈×Q₈	C₂²	semidirect
56. #335	C₂²	semidirect
57. #334	C₂²	semidirect
58. #333	C₂²	semidirect
59. #333	C₂²	semidirect
60. #332	C₂²	semidirect
62. Q₈⋊C₈	C₂²	semidirect
63. Q₈⋊C₈	C₂²	semidirect
66. #322	C₂²	semidirect
177. SD₁₆	SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
#3385	2
Q₈×SD₁₆	2
#3295	2
D₄×SD₁₆	2
#1714	2
#1686	2
#1678	2
#1675	2
#1671	2
#1643	2
#1637	2
#1629	2
#1623	2
#1582	2
SD₁₆⋊C₈	2