Subgroups — #121773

Order 512.

380 conjugacy classes of subgroups.

The subgroup lattice has 277 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17803	C₂	1	1	yes	maximal, characteristic
2	#8070	C₂	1	1	yes	maximal, characteristic
3	#8058	C₂	1	1	yes	maximal, characteristic
4	#7928	C₂	1	1	yes	maximal, characteristic
5	#7894	C₂	1	1	yes	maximal, characteristic
6	#7884	C₂	1	1	yes	maximal, characteristic
7	#7863	C₂	1	1	yes	maximal, characteristic
8	#2920	C₂²	1	1	yes	characteristic
9	#2920	C₂²	1	1	yes	characteristic
10	C₂×#432	C₂²	1	1	yes	characteristic
11	C₂×#431	C₂²	1	1	yes	characteristic
12	C₂×#431	C₂²	1	1	yes	characteristic
13	C₂×#431	C₂²	1	1	yes	characteristic
14	#2283	C₂²	1	1	yes	characteristic
15	#1572	—	2	2	no	—
16	#1571	C₄	1	2	yes	complemented
17	#1570	C₄	1	2	yes	complemented
18	#1564	—	2	2	no	—
19	#1300	—	2	2	no	—
20	#1298	—	2	2	no	—
21	#1282	—	2	2	no	—
22	#1279	—	2	2	no	—
23	#1278	—	2	2	no	—
24	#1275	—	2	2	no	—
25	C₂²×C₄²	C₂³	1	1	yes	characteristic
26	#432	—	2	1	no	—
27	#432	—	2	1	no	—
28	#432	—	2	1	no	—
29	#432	—	2	1	no	—
30	#432	—	2	1	no	—
31	#432	—	2	1	no	—
32	#432	—	2	1	no	—
33	#432	—	2	1	no	—
34	#431	—	2	1	no	—
35	#431	—	2	1	no	—
36	#431	—	2	1	no	—
37	#431	—	2	1	no	—
38	#431	—	2	1	no	—
39	#431	—	2	1	no	—
40	#431	—	2	1	no	—
41	#431	—	2	1	no	—
42	#431	—	2	1	no	—
43	#431	—	2	1	no	—
44	#431	—	2	1	no	—
45	#431	—	2	1	no	—
46	#431	—	2	1	no	—
47	#431	C₂×C₄	1	1	yes	characteristic
48	#431	C₂×C₄	1	1	yes	characteristic
49	#431	D₄	1	2	yes	—
50	#431	Q₈	1	2	yes	—
51	#423	—	2	1	no	—
52	#423	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₈	—	2	1	no	—
54	C₂×C₄⋊C₈	—	2	1	no	—
55	C₂×C₄⋊C₈	—	2	1	no	—
56	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
57	#407	—	2	1	no	—
58	#407	—	2	1	no	—
59	C₂×C₂²⋊C₈	—	2	1	no	—
60	C₂×C₂²⋊C₈	—	2	1	no	—
61	C₂×C₂²⋊C₈	—	2	1	no	—
62	C₂×C₂²⋊C₈	D₄	1	2	yes	—
63	C₄×M₁₆	—	2	1	no	—
64	C₄×M₁₆	D₄	1	2	yes	—
65	C₂×#97	—	2	1	no	—
66	C₂×C₄×C₈	—	2	1	no	—
67	C₂×C₄×C₈	—	2	1	no	—
68	C₂×C₄×C₈	D₄	1	2	yes	—
69	#383	—	2	1	no	—
70	#382	—	2	1	no	—
71	#381	—	2	1	no	—
72	#380	Q₈	1	2	yes	—
73	C₄×C₄⋊C₄	—	2	1	no	—
74	C₄×C₂²⋊C₄	—	2	1	no	—
75	#376	—	2	1	no	—
76	#335	—	2	4	no	—
77	#334	—	2	4	no	—
78	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
80	C₂³×C₄	#41	1	1	yes	characteristic
81	C₂×M₁₆	—	2	2	no	—
82	C₂×M₁₆	—	2	2	no	—
83	C₂²×C₈	—	2	2	no	—
84	C₂²×C₈	—	2	2	no	—
85	C₂²×C₈	—	2	2	no	—
86	C₂²×C₈	—	4	1	no	—
87	C₂×C₄⋊C₄	—	2	2	no	—
88	C₂×C₂²⋊C₄	—	2	2	no	—
89	C₂×C₄²	—	4	1	no	—
90	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₄²	#41	1	1	yes	characteristic
96	C₂×C₄²	#41	1	1	yes	characteristic
97	C₂×C₄²	#41	1	1	yes	characteristic
98	C₂×C₄²	#41	1	1	yes	characteristic
99	C₂×C₄²	#41	1	1	yes	characteristic
100	C₂×C₄²	#41	1	1	yes	characteristic
101	C₂×C₄²	#41	1	1	yes	characteristic
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	2	1	no	—
105	C₄⋊C₈	—	2	1	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	4	1	no	—
111	C₄⋊C₈	—	4	1	no	—
112	C₄⋊C₈	—	4	1	no	—
113	C₄⋊C₈	—	4	1	no	—
114	C₄⋊C₈	—	4	1	no	—
115	C₂²⋊C₈	—	2	1	no	—
116	C₂²⋊C₈	—	2	1	no	—
117	C₂²⋊C₈	—	2	2	no	—
118	C₂²⋊C₈	—	2	2	no	—
119	C₂²⋊C₈	—	2	2	no	—
120	C₂²⋊C₈	—	2	2	no	—
121	C₂²⋊C₈	—	4	1	no	—
122	C₂²⋊C₈	—	4	1	no	—
123	C₂²⋊C₈	—	4	1	no	—
124	C₂²⋊C₈	—	4	1	no	—
125	C₂²⋊C₈	—	4	1	no	—
126	C₂²⋊C₈	—	4	1	no	—
127	C₂²⋊C₈	—	4	1	no	—
128	#97	—	2	2	no	—
129	#97	—	4	1	no	—
130	#97	—	4	1	no	—
131	#97	—	4	1	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	—	4	1	no	—
136	C₄×C₈	—	4	1	no	—
137	C₄×C₈	—	4	1	no	—
138	C₄×C₈	—	4	1	no	—
139	C₄×C₈	C₂²⋊C₄	1	2	yes	—
140	C₄×C₈	C₂²⋊C₄	1	2	yes	—
141	#95	—	2	2	no	—
142	#95	—	4	1	no	—
143	#95	—	4	1	no	—
144	#95	—	4	1	no	—
145	C₂⁴	C₂²⋊Q₈	1	1	yes	characteristic
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	#117	1	1	yes	characteristic
151	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
152	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
153	C₂²×C₄	#121	1	1	yes	characteristic
154	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
156	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
157	C₂²×C₄	#123	1	1	yes	characteristic
158	C₂²×C₄	#123	1	1	yes	characteristic
159	C₂²×C₄	#123	1	1	yes	characteristic
160	C₂²×C₄	#123	1	1	yes	characteristic
161	C₂²×C₄	#123	1	1	yes	characteristic
162	C₂²×C₄	#125	1	1	yes	characteristic
163	C₂²×C₄	#125	1	1	yes	characteristic
164	C₂²×C₄	#125	1	1	yes	characteristic
165	C₂²×C₄	#126	1	1	yes	characteristic
166	C₂²×C₄	#127	1	1	yes	characteristic
167	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
168	M₁₆	—	4	4	no	—
169	M₁₆	—	8	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	4	4	no	—
173	C₂×C₈	—	4	4	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	8	2	no	—
180	C₂×C₈	—	8	2	no	—
181	C₄⋊C₄	—	4	4	no	—
182	C₂²⋊C₄	—	4	4	no	—
183	C₄²	—	8	2	no	—
184	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₄²	#117	1	1	yes	characteristic
186	C₄²	C₄×D₄	1	1	yes	characteristic
187	C₄²	C₄×Q₈	1	1	yes	characteristic
188	C₄²	#121	1	1	yes	characteristic
189	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
190	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
191	C₄²	#123	1	1	yes	characteristic
192	C₄²	#123	1	1	yes	characteristic
193	C₄²	#123	1	1	yes	characteristic
194	C₄²	#124	1	1	yes	characteristic
195	C₄²	#124	1	1	yes	characteristic
196	C₄²	#125	1	1	yes	characteristic
197	C₄²	#126	1	1	yes	characteristic
198	C₄²	#126	1	1	yes	characteristic
199	C₄²	#126	1	1	yes	characteristic
200	C₂³	—	2	2	no	—
201	C₂³	—	4	1	no	—
202	C₂³	—	4	1	no	—
203	C₂³	#387	1	1	yes	characteristic
204	C₂³	#398	1	1	yes	characteristic
205	C₂³	#399	1	1	yes	characteristic
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	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	—	8	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	#381	1	1	yes	characteristic
228	C₂×C₄	#387	1	1	yes	characteristic
229	C₂×C₄	#387	1	1	yes	characteristic
230	C₂×C₄	#387	1	1	yes	characteristic
231	C₂×C₄	#390	1	1	yes	characteristic
232	C₂×C₄	#391	1	1	yes	characteristic
233	C₂×C₄	#397	1	1	yes	characteristic
234	C₂×C₄	#397	1	1	yes	characteristic
235	C₂×C₄	#398	1	1	yes	characteristic
236	C₂×C₄	#399	1	1	yes	characteristic
237	C₂×C₄	#400	1	1	yes	characteristic
238	C₂×C₄	#400	1	1	yes	characteristic
239	C₈	—	4	2	no	—
240	C₈	—	4	2	no	—
241	C₈	—	8	2	no	—
242	C₈	—	8	2	no	—
243	C₈	—	8	2	no	—
244	C₈	—	8	2	no	—
245	C₈	—	8	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	8	2	no	—
255	C₂²	#1445	1	1	yes	characteristic
256	C₂²	#2098	1	2	yes	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	16	2	no	—
268	C₄	#1976	1	2	yes	—
269	C₄	#2103	1	2	yes	—
270	C₂	—	2	2	no	—
271	C₂	—	4	1	no	—
272	C₂	—	4	1	no	—
273	C₂	#11364	1	1	yes	characteristic
274	C₂	#11368	1	1	yes	characteristic
275	C₂	#11369	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
16. #1571	C₄	semidirect
17. #1570	C₄	semidirect

Maximal subgroups

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

subgroup	index
#17803	2
#8070	2
#8058	2
#7928	2
#7894	2
#7884	2
#7863	2