Subgroups — #134376

Order 512.

378 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#446	C₂	1	1	yes	maximal, characteristic
2	#12898	C₂	1	1	yes	maximal, characteristic
3	#12653	C₂	1	1	yes	maximal, characteristic
4	C₂×#1338	C₂	1	2	yes	maximal
5	#7170	C₂	1	2	yes	maximal
6	C₂×#446	C₄	1	1	yes	characteristic
7	C₂×#446	C₂²	1	1	yes	characteristic
8	C₄²×Q₈	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₁₆	—	2	1	no	—
10	C₂×C₄⋊C₁₆	—	2	1	no	—
11	C₂×C₄⋊C₁₆	—	2	1	no	—
12	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
13	C₂×#346	C₂²	1	2	yes	—
14	#1771	—	2	1	no	—
15	#1771	—	2	1	no	—
16	#1770	C₄	1	1	yes	characteristic
17	#1770	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₈	—	2	1	no	—
19	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
20	C₄×#97	—	2	1	no	—
21	C₄²×C₈	C₂²	1	1	yes	characteristic
22	#1374	—	2	1	no	—
23	#1338	—	2	4	no	—
24	C₂×C₄×Q₈	C₈	1	1	yes	characteristic
25	C₂×C₄×Q₈	C₈	1	1	yes	characteristic
26	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₆	—	2	2	no	—
28	C₂²×C₁₆	—	2	2	no	—
29	#446	—	2	2	no	—
30	#446	—	2	2	no	—
31	#446	—	2	1	no	—
32	#446	—	2	1	no	—
33	#446	D₄	1	2	yes	—
34	#446	Q₈	1	2	yes	—
35	C₂×C₄⋊C₈	—	2	1	no	—
36	C₂×C₄⋊C₈	—	2	1	no	—
37	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×#97	—	2	1	no	—
40	C₂×#97	—	2	1	no	—
41	C₂×#97	—	2	1	no	—
42	C₂×#97	—	2	1	no	—
43	C₂×C₄×C₈	—	2	1	no	—
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	D₄	1	1	yes	characteristic
46	C₂×C₄×C₈	D₄	1	1	yes	characteristic
47	C₂×C₄×C₈	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	D₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
50	C₄×C₄⋊C₄	—	2	1	no	—
51	C₄×C₄⋊C₄	—	2	1	no	—
52	C₄×C₄⋊C₄	—	2	1	no	—
53	C₄×C₄⋊C₄	C₈	1	1	yes	characteristic
54	C₄×C₄⋊C₄	C₈	1	1	yes	characteristic
55	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₄³	—	2	1	no	—
57	C₄³	C₂×C₄	1	1	yes	characteristic
58	C₄⋊C₁₆	—	2	4	no	—
59	C₄⋊C₁₆	—	2	2	no	—
60	C₄⋊C₁₆	—	2	2	no	—
61	C₄⋊C₁₆	—	4	2	no	—
62	C₄⋊C₁₆	—	4	2	no	—
63	#346	—	2	2	no	—
64	#346	—	2	2	no	—
65	#336	—	2	1	no	—
66	#336	—	2	1	no	—
67	#336	—	2	1	no	—
68	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₈	—	2	1	no	—
70	C₂²×C₈	—	2	1	no	—
71	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₄×Q₈	—	2	1	no	—
76	C₄×Q₈	—	2	1	no	—
77	C₄×Q₈	—	4	2	no	—
78	C₄×Q₈	—	4	2	no	—
79	C₄×Q₈	C₄⋊C₄	1	2	yes	—
80	C₄×Q₈	M₁₆	1	2	yes	—
81	C₂×C₄⋊C₄	—	2	1	no	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄²	—	2	1	no	—
88	C₂×C₄²	—	2	1	no	—
89	C₂×C₄²	—	2	1	no	—
90	C₂×C₄²	—	2	1	no	—
91	C₂×C₄²	—	2	1	no	—
92	C₂×C₄²	—	2	1	no	—
93	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
96	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
97	C₂×C₄²	M₁₆	1	1	yes	characteristic
98	C₂×C₄²	M₁₆	1	1	yes	characteristic
99	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₁₆	—	2	4	no	—
101	C₂×C₁₆	—	4	2	no	—
102	C₂×C₁₆	—	4	2	no	—
103	C₂×C₁₆	—	4	2	no	—
104	C₂×C₁₆	—	4	2	no	—
105	C₂×C₁₆	—	4	2	no	—
106	C₄⋊C₈	—	2	4	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	2	1	no	—
110	C₄⋊C₈	—	2	1	no	—
111	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
112	C₄⋊C₈	M₁₆	1	2	yes	—
113	#97	—	2	4	no	—
114	#97	—	2	2	no	—
115	#97	—	2	2	no	—
116	#97	—	4	2	no	—
117	#97	—	4	1	no	—
118	#97	—	4	1	no	—
119	C₄×C₈	—	2	1	no	—
120	C₄×C₈	—	2	1	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	2	1	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	4	1	no	—
125	C₄×C₈	—	4	2	no	—
126	C₄×C₈	SD₁₆	1	2	yes	—
127	C₄×C₈	Q₁₆	1	2	yes	—
128	C₄×C₈	C₂×D₄	1	1	yes	characteristic
129	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
130	C₄×C₈	#41	1	1	yes	characteristic
131	C₄×C₈	#41	1	1	yes	characteristic
132	#95	—	2	1	no	—
133	#95	—	2	1	no	—
134	#95	—	2	1	no	—
135	C₂×Q₈	—	2	1	no	—
136	C₂×Q₈	—	4	1	no	—
137	C₂×Q₈	—	4	1	no	—
138	C₂×Q₈	C₄⋊C₈	1	2	yes	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
144	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
145	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
146	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
149	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	4	1	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	#103	1	2	yes	—
164	C₂×C₈	#121	1	1	yes	characteristic
165	C₂×C₈	#121	1	1	yes	characteristic
166	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
167	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
168	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
169	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
170	C₂×C₈	AGL(1,ℤ/8)	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	—	4	1	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	C₄⋊C₈	1	2	yes	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	4	1	no	—
188	C₄²	—	4	2	no	—
189	C₄²	—	4	2	no	—
190	C₄²	—	4	2	no	—
191	C₄²	—	4	2	no	—
192	C₄²	—	4	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	#103	1	2	yes	—
195	C₄²	#104	1	2	yes	—
196	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
197	C₄²	#117	1	1	yes	characteristic
198	C₄²	C₂×M₁₆	1	1	yes	characteristic
199	C₄²	C₂×M₁₆	1	1	yes	characteristic
200	C₁₆	—	4	4	no	—
201	C₁₆	—	8	2	no	—
202	C₂³	C₂×C₂²⋊C₈	1	1	yes	characteristic
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	Q₈⋊C₈	1	2	yes	—
229	C₂×C₄	Q₈⋊C₈	1	2	yes	—
230	C₂×C₄	#349	1	2	yes	—
231	C₂×C₄	#349	1	2	yes	—
232	C₂×C₄	#350	1	2	yes	—
233	C₂×C₄	#380	1	1	yes	characteristic
234	C₂×C₄	#407	1	1	yes	characteristic
235	C₂×C₄	C₂×#103	1	1	yes	characteristic
236	C₂×C₄	#417	1	1	yes	characteristic
237	C₂×C₄	C₂×#104	1	1	yes	characteristic
238	C₂×C₄	#421	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
240	C₂×C₄	#423	1	1	yes	characteristic
241	C₂×C₄	#431	1	1	yes	characteristic
242	C₂×C₄	#432	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	2	no	—
246	C₈	—	4	2	no	—
247	C₈	—	4	1	no	—
248	C₈	—	4	2	no	—
249	C₈	#460	1	1	yes	characteristic
250	C₈	#462	1	1	yes	characteristic
251	C₈	#475	1	1	yes	characteristic
252	C₈	#477	1	1	yes	characteristic
253	C₂²	C₂×Q₈⋊C₈	1	1	yes	characteristic
254	C₂²	#1477	1	1	yes	characteristic
255	C₂²	#1797	1	1	yes	characteristic
256	C₂²	C₂×#349	1	1	yes	characteristic
257	C₂²	C₂×#349	1	1	yes	characteristic
258	C₂²	C₂×#350	1	1	yes	characteristic
259	C₂²	#2117	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	8	1	no	—
270	C₄	—	8	1	no	—
271	C₄	#1338	1	4	yes	complemented
272	C₄	#1487	1	1	yes	characteristic
273	C₄	#1489	1	1	yes	characteristic
274	C₄	#1803	1	1	yes	characteristic
275	C₄	#1806	1	1	yes	characteristic
276	C₂	#8564	1	1	yes	characteristic
277	C₂	C₂×#1338	1	1	yes	characteristic
278	C₂	#11611	1	1	yes	characteristic
279	C₂	#11626	1	1	yes	characteristic
280	C₂	#11628	1	1	yes	characteristic
281	C₂	#12785	1	1	yes	characteristic
282	C₂	#12787	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ 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
271. C₄	#1338	semidirect

Maximal subgroups

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

subgroup	index
C₄×#446	2
#12898	2
#12653	2
C₂×#1338	2
#7170	2