Subgroups — #136997

Order 512.

385 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×#1676	C₂	1	1	yes	maximal, characteristic
3	#13282	C₂	1	1	yes	maximal, characteristic
4	#13276	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#7274	C₂	1	1	yes	maximal, characteristic
7	#7171	C₂	1	1	yes	maximal, characteristic
8	C₂×#500	C₂²	1	1	yes	characteristic
9	C₂×#477	—	2	1	no	—
10	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
11	Q₈×C₄⋊C₄	—	2	1	no	—
12	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1947	—	2	2	no	—
15	#1947	—	2	1	no	—
16	#1845	—	2	1	no	—
17	#1845	C₂²	1	1	yes	characteristic
18	C₄×#107	—	2	1	no	—
19	C₄×#107	C₂²	1	1	yes	characteristic
20	#1676	—	2	2	no	—
21	#1676	C₄	1	4	yes	complemented
22	C₂×#335	C₂²	1	1	yes	characteristic
23	C₂×#333	—	2	1	no	—
24	#1380	—	2	1	no	—
25	#1380	—	2	1	no	—
26	#1380	—	2	2	no	—
27	Q₈⋊C₁₆	—	2	2	no	—
28	C₂×C₄⋊Q₈	—	2	1	no	—
29	C₂×C₄×Q₈	—	4	1	no	—
30	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
31	C₂²×C₁₆	—	2	1	no	—
32	C₂²×C₁₆	—	2	1	no	—
33	#500	—	2	1	no	—
34	#500	—	2	1	no	—
35	#500	—	2	1	no	—
36	#500	C₂×C₄	1	2	yes	—
37	#477	—	2	4	no	—
38	#477	—	4	2	no	—
39	C₈×Q₈	—	2	1	no	—
40	C₈×Q₈	—	2	1	no	—
41	C₈×Q₈	C₂×C₄	1	2	yes	—
42	C₈×Q₈	D₄	1	2	yes	—
43	C₂×#107	—	2	1	no	—
44	C₂×#107	—	2	1	no	—
45	C₂×#107	—	2	1	no	—
46	C₂×#107	—	2	2	no	—
47	C₂×#107	—	4	1	no	—
48	C₂×#107	D₄	1	1	yes	characteristic
49	C₂×#107	D₄	1	1	yes	characteristic
50	C₂×#107	D₄	1	2	yes	—
51	C₂×C₄⋊C₈	—	2	1	no	—
52	C₂×C₄⋊C₈	—	2	1	no	—
53	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
54	C₂×#103	—	4	1	no	—
55	C₂×C₄×C₈	—	2	1	no	—
56	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
57	#389	—	4	2	no	—
58	#389	—	4	1	no	—
59	#384	—	2	2	no	—
60	#384	—	2	1	no	—
61	C₄×C₄⋊C₄	—	2	2	no	—
62	C₄×C₄⋊C₄	—	2	1	no	—
63	C₄⋊C₁₆	—	2	1	no	—
64	C₄⋊C₁₆	—	2	1	no	—
65	C₄×C₁₆	—	2	1	no	—
66	C₄×C₁₆	—	2	1	no	—
67	#340	—	2	2	no	—
68	#340	—	2	1	no	—
69	#335	—	2	1	no	—
70	#335	C₂×C₄	1	2	yes	—
71	#333	—	2	4	no	—
72	C₂²×Q₈	—	4	1	no	—
73	C₂²×Q₈	D₈	1	1	yes	characteristic
74	C₂²×C₈	—	2	1	no	—
75	C₂²×C₈	—	4	1	no	—
76	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
78	C₄⋊Q₈	—	2	2	no	—
79	C₄⋊Q₈	—	4	1	no	—
80	C₄⋊Q₈	—	4	1	no	—
81	C₄⋊Q₈	—	4	1	no	—
82	C₄×Q₈	—	2	1	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	8	2	no	—
85	C₄×Q₈	—	8	2	no	—
86	C₄×Q₈	C₂²⋊C₄	1	2	yes	—
87	C₄×Q₈	SD₁₆	1	2	yes	—
88	C₂×C₄⋊C₄	—	2	1	no	—
89	C₂×C₄⋊C₄	—	2	1	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₄⋊C₄	—	4	1	no	—
93	C₂×C₄⋊C₄	—	4	1	no	—
94	C₂×C₄⋊C₄	—	4	2	no	—
95	C₂×C₄⋊C₄	—	4	1	no	—
96	C₂×C₄⋊C₄	—	4	2	no	—
97	C₂×C₄⋊C₄	—	4	1	no	—
98	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
99	C₂×C₄²	—	2	1	no	—
100	C₂×C₄²	—	4	2	no	—
101	C₂×C₄²	—	4	1	no	—
102	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	4	1	no	—
105	C₂×C₁₆	—	4	1	no	—
106	C₂×C₁₆	—	4	1	no	—
107	C₂×C₁₆	—	4	1	no	—
108	C₂×C₁₆	—	4	1	no	—
109	#107	—	2	1	no	—
110	#107	—	2	2	no	—
111	#107	—	2	2	no	—
112	#107	—	2	1	no	—
113	#107	—	2	1	no	—
114	#107	—	4	1	no	—
115	#107	—	4	1	no	—
116	#107	—	4	1	no	—
117	#107	—	4	1	no	—
118	#107	—	4	4	no	—
119	#107	—	4	2	no	—
120	#107	—	8	2	no	—
121	#107	Q₁₆	1	2	yes	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	1	no	—
126	C₄⋊C₈	—	4	1	no	—
127	C₄⋊C₈	C₂²⋊C₄	1	2	yes	—
128	#103	—	4	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
132	C₄×C₈	C₂×D₄	1	1	yes	characteristic
133	C₄×C₈	#41	1	1	yes	characteristic
134	C₄×C₈	#41	1	1	yes	characteristic
135	#95	—	4	2	no	—
136	#95	—	4	1	no	—
137	C₂×Q₈	—	2	1	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	—	4	2	no	—
141	C₂×Q₈	—	8	1	no	—
142	C₂×Q₈	—	16	1	no	—
143	C₂×Q₈	—	16	1	no	—
144	C₂×Q₈	#102	1	2	yes	—
145	C₂×Q₈	SD₃₂	1	2	yes	—
146	C₂×Q₈	Q₃₂	1	2	yes	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	4	1	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	—	4	1	no	—
152	C₂²×C₄	—	8	1	no	—
153	C₂²×C₄	#120	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
155	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	8	1	no	—
163	C₂×C₈	—	8	1	no	—
164	C₂×C₈	C₄×D₄	1	1	yes	characteristic
165	C₂×C₈	C₄×D₄	1	1	yes	characteristic
166	C₂×C₈	#121	1	1	yes	characteristic
167	C₂×C₈	#121	1	1	yes	characteristic
168	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
169	C₂×C₈	#135	1	1	yes	characteristic
170	C₂×C₈	#137	1	1	yes	characteristic
171	C₂×C₈	#137	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	1	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄⋊C₄	—	8	4	no	—
184	C₄⋊C₄	—	8	1	no	—
185	C₄⋊C₄	—	8	1	no	—
186	C₄⋊C₄	—	8	1	no	—
187	C₄⋊C₄	—	8	1	no	—
188	C₄⋊C₄	—	16	1	no	—
189	C₄⋊C₄	—	16	2	no	—
190	C₄⋊C₄	#102	1	2	yes	—
191	C₄²	—	2	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	8	4	no	—
194	C₄²	—	8	1	no	—
195	C₄²	—	8	1	no	—
196	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	C₄²	#123	1	1	yes	characteristic
198	C₄²	C₂×SD₁₆	1	1	yes	characteristic
199	C₄²	#137	1	1	yes	characteristic
200	C₁₆	—	4	2	no	—
201	C₁₆	—	8	1	no	—
202	C₂³	#447	1	1	yes	characteristic
203	Q₈	—	2	2	no	—
204	Q₈	—	2	2	no	—
205	Q₈	—	2	2	no	—
206	Q₈	—	16	2	no	—
207	Q₈	—	16	2	no	—
208	Q₈	#358	1	4	yes	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	8	1	no	—
219	C₂×C₄	—	8	1	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	—	8	1	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	8	1	no	—
227	C₂×C₄	—	16	1	no	—
228	C₂×C₄	—	16	1	no	—
229	C₂×C₄	#386	1	1	yes	characteristic
230	C₂×C₄	C₂×#102	1	1	yes	characteristic
231	C₂×C₄	#417	1	1	yes	characteristic
232	C₂×C₄	#448	1	1	yes	characteristic
233	C₂×C₄	#451	1	1	yes	characteristic
234	C₂×C₄	#452	1	1	yes	characteristic
235	C₂×C₄	#455	1	1	yes	characteristic
236	C₂×C₄	#456	1	1	yes	characteristic
237	C₂×C₄	#480	1	1	yes	characteristic
238	C₂×C₄	#481	1	1	yes	characteristic
239	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
240	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
241	C₂×C₄	#509	1	1	yes	characteristic
242	C₂×C₄	#510	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	4	2	no	—
246	C₈	—	8	2	no	—
247	C₈	C₄×Q₁₆	1	1	yes	characteristic
248	C₈	#441	1	1	yes	characteristic
249	C₈	#462	1	1	yes	characteristic
250	C₈	#464	1	1	yes	characteristic
251	C₂²	#1641	1	1	yes	characteristic
252	C₂²	#1649	1	1	yes	characteristic
253	C₂²	#1874	1	1	yes	characteristic
254	C₂²	#2187	1	1	yes	characteristic
255	C₂²	#2188	1	1	yes	characteristic
256	C₂²	#2192	1	1	yes	characteristic
257	C₂²	#2193	1	1	yes	characteristic
258	C₄	—	2	4	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	4	2	no	—
262	C₄	—	8	2	no	—
263	C₄	—	16	1	no	—
264	C₄	—	16	2	no	—
265	C₄	—	16	2	no	—
266	C₄	#1645	1	1	yes	characteristic
267	C₄	#1653	1	1	yes	characteristic
268	C₄	#1878	1	1	yes	characteristic
269	C₄	#1882	1	1	yes	characteristic
270	C₄	C₂×#358	1	1	yes	characteristic
271	C₄	#2139	1	1	yes	characteristic
272	C₄	#2235	1	1	yes	characteristic
273	C₄	#2236	1	1	yes	characteristic
274	C₂	#8987	1	1	yes	characteristic
275	C₂	#12384	1	1	yes	characteristic
276	C₂	#12386	1	1	yes	characteristic
277	C₂	#12408	1	1	yes	characteristic
278	C₂	#12410	1	1	yes	characteristic
279	C₂	#12993	1	1	yes	characteristic
280	C₂	#12996	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#447
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
21. #1676	C₄	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#107	2
C₂×#1676	2
#13282	2
#13276	2
C₂×Q₈⋊C₁₆	2
#7274	2
#7171	2