Subgroups — #19804

Order 256.

360 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#3362	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#3274	C₂	1	1	yes	maximal, characteristic
5	#1715	C₂	1	1	yes	maximal, characteristic, complemented
6	#1691	C₂	1	1	yes	maximal, characteristic, complemented
7	#1677	C₂	1	1	yes	maximal, characteristic, complemented
8	#1668	C₂	1	1	yes	maximal, characteristic, complemented
9	#1666	C₂	1	1	yes	maximal, characteristic, complemented
10	#1642	C₂	1	1	yes	maximal, characteristic, complemented
11	#1629	C₂	1	1	yes	maximal, characteristic, complemented
12	#1627	C₂	1	1	yes	maximal, characteristic, complemented
13	#1623	C₂	1	1	yes	maximal, characteristic, complemented
14	#1583	C₂	1	1	yes	maximal, characteristic, complemented
15	SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×AGL(1,ℤ/8)	—	2	1	no	—
17	C₂²×SD₁₆	—	2	1	no	—
18	Q₈²	C₂²	1	1	yes	characteristic, complemented
19	#550	C₂²	1	1	yes	characteristic
20	D₄×Q₈	C₂²	1	1	yes	characteristic
21	D₄²	C₂²	1	1	yes	characteristic
22	#498	—	2	1	no	—
23	#498	C₂²	1	1	yes	characteristic, complemented
24	#493	C₂²	1	1	yes	characteristic
25	#492	C₂²	1	1	yes	characteristic
26	#492	C₂²	1	1	yes	characteristic
27	#486	—	2	1	no	—
28	#486	C₂²	1	1	yes	characteristic, complemented
29	#486	C₂²	1	1	yes	characteristic, complemented
30	#476	—	2	1	no	—
31	#476	—	2	1	no	—
32	#476	C₂²	1	1	yes	characteristic
33	#476	C₂²	1	1	yes	characteristic
34	#475	—	2	1	no	—
35	#475	C₂²	1	1	yes	characteristic, complemented
36	#469	—	2	1	no	—
37	#465	—	2	1	no	—
38	#461	C₂²	1	1	yes	characteristic
39	#460	—	2	1	no	—
40	#460	C₂²	1	1	yes	characteristic
41	#460	C₂²	1	1	yes	characteristic
42	#460	C₂²	1	1	yes	characteristic, complemented
43	#459	C₂²	1	1	yes	characteristic
44	#450	—	2	1	no	—
45	#449	—	2	1	no	—
46	#448	—	2	1	no	—
47	#447	—	2	1	no	—
48	C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
49	C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
50	C₄×SD₁₆	—	2	1	no	—
51	C₄×SD₁₆	—	2	1	no	—
52	C₄×SD₁₆	C₂²	1	1	yes	characteristic
53	C₄×SD₁₆	C₂²	1	1	yes	characteristic
54	#436	C₂²	1	1	yes	characteristic
55	C₈×D₄	C₂²	1	1	yes	characteristic
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	#332	C₂²	1	1	yes	characteristic, complemented
60	#331	C₂²	1	1	yes	characteristic
61	#331	C₂²	1	1	yes	characteristic, complemented
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	C₂×#41	—	2	1	no	—
68	C₂²×Q₈	—	2	1	no	—
69	C₂²×D₄	—	2	1	no	—
70	C₂²×D₄	—	2	1	no	—
71	AGL(1,ℤ/8)	—	4	4	no	—
72	C₂×SD₁₆	—	2	1	no	—
73	C₂×SD₁₆	—	2	1	no	—
74	C₂×SD₁₆	—	2	1	no	—
75	C₂×SD₁₆	—	2	1	no	—
76	C₂×SD₁₆	—	2	1	no	—
77	C₂×SD₁₆	—	2	1	no	—
78	C₂×SD₁₆	—	4	1	no	—
79	C₂×SD₁₆	—	4	4	no	—
80	C₂×SD₁₆	D₄	1	1	yes	characteristic
81	C₂×SD₁₆	D₄	1	1	yes	characteristic
82	C₂×D₈	—	2	1	no	—
83	C₂×D₈	—	2	1	no	—
84	C₂×D₈	—	2	1	no	—
85	C₂×M₁₆	—	2	1	no	—
86	C₂²×C₈	—	2	1	no	—
87	C₄⋊Q₈	—	2	1	no	—
88	C₄⋊Q₈	—	2	1	no	—
89	C₄⋊Q₈	—	2	1	no	—
90	C₄⋊Q₈	—	4	1	no	—
91	C₄⋊Q₈	C₂³	1	1	yes	characteristic
92	C₄⋊Q₈	C₂³	1	1	yes	characteristic
93	#127	—	2	1	no	—
94	#127	C₂³	1	1	yes	characteristic
95	#127	C₂³	1	1	yes	characteristic
96	C₂²⋊Q₈	—	2	1	no	—
97	C₂²⋊Q₈	—	4	1	no	—
98	#121	—	2	1	no	—
99	#121	—	2	1	no	—
100	#121	—	2	1	no	—
101	#121	—	4	1	no	—
102	#120	—	4	1	no	—
103	C₄×Q₈	—	2	1	no	—
104	C₄×Q₈	—	2	1	no	—
105	C₄×Q₈	C₂³	1	1	yes	characteristic
106	C₄×Q₈	C₂³	1	1	yes	characteristic
107	C₄×D₄	—	2	1	no	—
108	C₄×D₄	—	2	1	no	—
109	C₄×D₄	C₂³	1	1	yes	characteristic
110	C₄×D₄	C₂³	1	1	yes	characteristic
111	#106	—	2	1	no	—
112	#106	—	2	1	no	—
113	#106	—	2	1	no	—
114	#106	—	2	1	no	—
115	#106	—	2	1	no	—
116	#106	D₄	1	1	yes	characteristic
117	#106	D₄	1	1	yes	characteristic
118	C₄⋊C₈	—	2	1	no	—
119	C₄⋊C₈	—	2	1	no	—
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	C₄⋊C₈	C₂³	1	1	yes	characteristic
124	#103	—	2	1	no	—
125	#103	—	2	1	no	—
126	#103	D₄	1	1	yes	characteristic
127	#103	D₄	1	1	yes	characteristic
128	#102	—	2	1	no	—
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	—	4	1	no	—
137	#102	D₄	1	1	yes	characteristic
138	#102	D₄	1	1	yes	characteristic
139	C₂²⋊C₈	—	2	1	no	—
140	C₂²⋊C₈	—	2	1	no	—
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	C₂³	1	1	yes	characteristic
144	C₄×C₈	C₂³	1	1	yes	characteristic
145	C₄×C₈	C₂³	1	1	yes	characteristic
146	C₂⁴	—	4	1	no	—
147	#41	—	4	2	no	—
148	#41	—	8	1	no	—
149	C₂×Q₈	—	2	1	no	—
150	C₂×Q₈	—	2	1	no	—
151	C₂×Q₈	—	4	1	no	—
152	C₂×Q₈	—	4	1	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	8	1	no	—
155	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
156	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
157	C₂×D₄	—	2	1	no	—
158	C₂×D₄	—	2	1	no	—
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	—	2	1	no	—
161	C₂×D₄	—	2	1	no	—
162	C₂×D₄	—	2	1	no	—
163	C₂×D₄	—	4	1	no	—
164	C₂×D₄	—	4	1	no	—
165	C₂×D₄	—	4	1	no	—
166	C₂×D₄	—	4	1	no	—
167	C₂×D₄	—	4	2	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂×D₄	—	8	1	no	—
170	C₂×D₄	—	8	1	no	—
171	C₂×D₄	C₂×D₄	1	1	yes	characteristic
172	C₂×D₄	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	4	1	no	—
176	C₂²×C₄	—	4	1	no	—
177	SD₁₆	—	2	2	no	—
178	SD₁₆	—	4	2	no	—
179	SD₁₆	—	4	2	no	—
180	SD₁₆	—	4	2	no	—
181	SD₁₆	—	4	2	no	—
182	SD₁₆	—	4	2	no	—
183	SD₁₆	—	8	1	no	—
184	SD₁₆	—	8	2	no	—
185	SD₁₆	SD₁₆	1	4	yes	complemented
186	D₈	—	4	2	no	—
187	D₈	—	4	2	no	—
188	D₈	—	8	1	no	—
189	M₁₆	—	4	2	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	C₂×D₄	1	1	yes	characteristic
197	C₂×C₈	C₂×D₄	1	1	yes	characteristic
198	C₂×C₈	C₂×D₄	1	1	yes	characteristic
199	C₂×C₈	C₂×D₄	1	1	yes	characteristic
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	4	1	no	—
207	C₄⋊C₄	—	4	1	no	—
208	C₄⋊C₄	—	4	1	no	—
209	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
210	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
211	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
212	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	4	1	no	—
216	C₂²⋊C₄	—	4	1	no	—
217	C₂²⋊C₄	—	4	1	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	4	1	no	—
221	C₄²	C₂⁴	1	1	yes	characteristic
222	C₂³	—	2	1	no	—
223	C₂³	—	2	1	no	—
224	C₂³	—	4	1	no	—
225	C₂³	—	4	1	no	—
226	C₂³	—	8	2	no	—
227	C₂³	—	8	2	no	—
228	C₂³	—	16	1	no	—
229	Q₈	—	2	1	no	—
230	Q₈	—	2	1	no	—
231	Q₈	—	4	1	no	—
232	Q₈	—	4	1	no	—
233	Q₈	—	8	2	no	—
234	Q₈	—	8	1	no	—
235	Q₈	—	16	1	no	—
236	Q₈	C₂×SD₁₆	1	2	yes	complemented
237	Q₈	AGL(1,ℤ/8)	1	2	yes	complemented
238	D₄	—	2	1	no	—
239	D₄	—	2	1	no	—
240	D₄	—	2	1	no	—
241	D₄	—	4	1	no	—
242	D₄	—	4	1	no	—
243	D₄	—	4	1	no	—
244	D₄	—	4	1	no	—
245	D₄	—	4	1	no	—
246	D₄	—	4	1	no	—
247	D₄	—	4	1	no	—
248	D₄	—	4	1	no	—
249	D₄	—	8	2	no	—
250	D₄	—	8	1	no	—
251	D₄	—	8	1	no	—
252	D₄	—	8	2	no	—
253	D₄	—	8	2	no	—
254	D₄	—	16	1	no	—
255	D₄	C₂×SD₁₆	1	2	yes	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	8	2	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	4	1	no	—
277	C₈	—	4	1	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₈	—	4	1	no	—
281	C₈	C₂×SD₁₆	1	2	yes	complemented
282	C₂²	—	2	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	8	1	no	—
289	C₂²	—	8	1	no	—
290	C₂²	—	8	1	no	—
291	C₂²	—	8	1	no	—
292	C₂²	—	8	1	no	—
293	C₂²	—	8	1	no	—
294	C₂²	—	16	4	no	—
295	C₂²	D₄²	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	8	1	no	—
304	C₄	—	8	1	no	—
305	C₄	C₂²×SD₁₆	1	1	yes	characteristic
306	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
307	C₄	#577	1	1	yes	characteristic
308	C₄	#577	1	1	yes	characteristic
309	C₂	—	4	2	no	—
310	C₂	—	8	1	no	—
311	C₂	—	16	1	no	—
312	C₂	—	16	1	no	—
313	C₂	#3265	1	1	yes	characteristic
314	C₂	D₄×SD₁₆	1	1	yes	characteristic
315	C₂	#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3274	C₂
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. Q₈×SD₁₆	C₂	semidirect
2. #3362	C₂	semidirect
3. D₄×SD₁₆	C₂	semidirect
5. #1715	C₂	semidirect
6. #1691	C₂	semidirect
7. #1677	C₂	semidirect
8. #1668	C₂	semidirect
9. #1666	C₂	semidirect
10. #1642	C₂	semidirect
11. #1629	C₂	semidirect
12. #1627	C₂	semidirect
13. #1623	C₂	semidirect
14. #1583	C₂	semidirect
15. SD₁₆⋊C₈	C₂	semidirect
18. Q₈²	C₂²	semidirect
23. #498	C₂²	semidirect
28. #486	C₂²	semidirect
29. #486	C₂²	semidirect
35. #475	C₂²	semidirect
42. #460	C₂²	semidirect
48. C₈×Q₈	C₂²	semidirect
49. C₈×Q₈	C₂²	semidirect
56. #335	C₂²	semidirect
57. #334	C₂²	semidirect
58. #333	C₂²	semidirect
59. #332	C₂²	semidirect
61. #331	C₂²	semidirect
62. Q₈⋊C₈	C₂²	semidirect
63. Q₈⋊C₈	C₂²	semidirect
66. #322	C₂²	semidirect
185. SD₁₆	SD₁₆	semidirect
236. Q₈	C₂×SD₁₆	semidirect
237. Q₈	AGL(1,ℤ/8)	semidirect
281. C₈	C₂×SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×SD₁₆	2
#3362	2
D₄×SD₁₆	2
#3274	2
#1715	2
#1691	2
#1677	2
#1668	2
#1666	2
#1642	2
#1629	2
#1627	2
#1623	2
#1583	2
SD₁₆⋊C₈	2