Subgroups — #134404

Order 512.

360 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21276	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1332	C₂	1	2	yes	maximal
4	#8806	C₂	1	1	yes	maximal, characteristic
5	#7171	C₂	1	2	yes	maximal
6	C₂×C₄×Q₁₆	C₂²	1	1	yes	characteristic
7	#2295	—	2	1	no	—
8	C₂×C₄⋊C₁₆	—	2	1	no	—
9	C₂×C₄×C₁₆	C₂²	1	2	yes	—
10	#1969	C₄	1	1	yes	characteristic
11	#1969	C₄	1	1	yes	characteristic
12	#1942	C₄	1	1	yes	characteristic
13	#1942	C₄	1	1	yes	characteristic
14	#1933	—	2	1	no	—
15	#1933	—	2	1	no	—
16	#1842	—	2	1	no	—
17	C₄×#107	C₂²	1	1	yes	characteristic
18	C₄×#103	—	2	1	no	—
19	C₄²×C₈	C₂²	1	1	yes	characteristic
20	C₂×#335	—	2	1	no	—
21	C₂×#335	C₂²	1	2	yes	—
22	#1374	—	2	1	no	—
23	#1332	—	2	4	no	—
24	C₂²×Q₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Q₈	—	2	1	no	—
26	C₂²×C₁₆	—	2	2	no	—
27	C₄×Q₁₆	—	2	1	no	—
28	C₄×Q₁₆	—	2	1	no	—
29	C₄×Q₁₆	D₄	1	2	yes	—
30	C₄×Q₁₆	Q₈	1	2	yes	—
31	C₂×#107	C₂×C₄	1	1	yes	characteristic
32	C₂×#107	C₂×C₄	1	1	yes	characteristic
33	C₂×#107	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₈	—	2	2	no	—
35	C₂×C₄⋊C₈	—	4	1	no	—
36	C₂×C₄⋊C₈	—	4	1	no	—
37	C₂×#103	—	2	1	no	—
38	C₂×#103	—	2	1	no	—
39	C₂×#103	—	2	1	no	—
40	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
46	#391	—	2	1	no	—
47	#391	—	2	1	no	—
48	#389	—	2	1	no	—
49	#389	—	2	1	no	—
50	#387	—	4	1	no	—
51	#387	—	4	1	no	—
52	C₄×C₄⋊C₄	—	2	1	no	—
53	C₄×C₄⋊C₄	—	4	1	no	—
54	#376	—	4	1	no	—
55	C₄³	D₄	1	1	yes	characteristic
56	C₄⋊C₁₆	—	2	4	no	—
57	C₄×C₁₆	—	2	2	no	—
58	C₄×C₁₆	—	2	2	no	—
59	#340	—	2	1	no	—
60	#340	—	2	1	no	—
61	#340	—	2	1	no	—
62	#335	—	2	2	no	—
63	#335	—	2	2	no	—
64	#335	—	4	2	no	—
65	C₂²×Q₈	—	2	1	no	—
66	C₂×Q₁₆	—	2	1	no	—
67	C₂×Q₁₆	—	2	2	no	—
68	C₂×Q₁₆	—	2	2	no	—
69	C₂×Q₁₆	C₄⋊C₄	1	2	yes	—
70	C₂²×C₈	—	4	2	no	—
71	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₄×Q₈	—	2	2	no	—
76	C₄×Q₈	—	2	2	no	—
77	C₄×Q₈	—	4	1	no	—
78	C₄×Q₈	—	4	1	no	—
79	C₂×C₄⋊C₄	—	2	1	no	—
80	C₂×C₄⋊C₄	—	2	1	no	—
81	C₂×C₄⋊C₄	—	2	1	no	—
82	C₂×C₄⋊C₄	—	4	1	no	—
83	C₂×C₄⋊C₄	—	4	1	no	—
84	C₂×C₄⋊C₄	—	4	1	no	—
85	C₂×C₄²	—	4	1	no	—
86	C₂×C₄²	—	4	1	no	—
87	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄²	D₈	1	1	yes	characteristic
90	C₂×C₄²	D₈	1	1	yes	characteristic
91	C₂×C₄²	SD₁₆	1	1	yes	characteristic
92	C₂×C₄²	SD₁₆	1	1	yes	characteristic
93	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
94	C₂×C₁₆	—	2	4	no	—
95	C₂×C₁₆	—	4	2	no	—
96	C₂×C₁₆	—	4	2	no	—
97	#107	—	2	1	no	—
98	#107	—	2	2	no	—
99	#107	—	2	2	no	—
100	#107	C₄⋊C₄	1	2	yes	—
101	C₄⋊C₈	—	4	2	no	—
102	C₄⋊C₈	—	4	2	no	—
103	C₄⋊C₈	—	8	2	no	—
104	C₄⋊C₈	—	8	2	no	—
105	#103	—	2	2	no	—
106	#103	—	2	2	no	—
107	#103	—	4	2	no	—
108	#103	—	4	2	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	—	2	1	no	—
113	C₄×C₈	—	2	1	no	—
114	C₄×C₈	—	4	1	no	—
115	C₄×C₈	C₂×D₄	1	1	yes	characteristic
116	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
117	C₄×C₈	#41	1	1	yes	characteristic
118	C₄×C₈	#41	1	1	yes	characteristic
119	#95	—	4	1	no	—
120	#95	—	4	1	no	—
121	#95	—	4	1	no	—
122	#95	—	4	1	no	—
123	#95	—	4	1	no	—
124	#95	—	4	1	no	—
125	#95	—	4	1	no	—
126	C₂×Q₈	—	2	2	no	—
127	C₂×Q₈	—	4	1	no	—
128	C₂×Q₈	—	4	1	no	—
129	C₂×Q₈	—	4	1	no	—
130	C₂×Q₈	—	4	1	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂²×C₄	—	4	1	no	—
133	C₂²×C₄	—	4	1	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	#102	1	1	yes	characteristic
137	C₂²×C₄	#102	1	1	yes	characteristic
138	C₂²×C₄	#102	1	1	yes	characteristic
139	C₂²×C₄	#102	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
142	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
143	Q₁₆	—	2	4	no	—
144	Q₁₆	—	4	2	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	8	2	no	—
153	C₂×C₈	—	8	2	no	—
154	C₂×C₈	—	8	2	no	—
155	C₂×C₈	#104	1	2	yes	—
156	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₈	#117	1	1	yes	characteristic
158	C₂×C₈	#121	1	1	yes	characteristic
159	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
160	C₂×C₈	#131	1	2	yes	—
161	C₂×C₈	#131	1	2	yes	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	4	1	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	1	no	—
167	C₄⋊C₄	—	4	1	no	—
168	C₄⋊C₄	—	4	1	no	—
169	C₄⋊C₄	—	4	1	no	—
170	C₄⋊C₄	—	8	2	no	—
171	C₄⋊C₄	—	8	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	2	1	no	—
179	C₄²	—	4	1	no	—
180	C₄²	—	4	2	no	—
181	C₄²	—	4	2	no	—
182	C₄²	—	8	1	no	—
183	C₄²	—	8	1	no	—
184	C₄²	SD₃₂	1	2	yes	—
185	C₄²	Q₃₂	1	2	yes	—
186	C₄²	#121	1	1	yes	characteristic
187	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
188	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
189	C₄²	#137	1	1	yes	characteristic
190	C₁₆	—	4	4	no	—
191	C₂³	C₂×#102	1	1	yes	characteristic
192	Q₈	—	4	2	no	—
193	Q₈	—	4	2	no	—
194	Q₈	—	8	1	no	—
195	Q₈	—	8	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	8	1	no	—
210	C₂×C₄	—	8	1	no	—
211	C₂×C₄	—	8	1	no	—
212	C₂×C₄	—	8	1	no	—
213	C₂×C₄	—	8	1	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	—	8	1	no	—
218	C₂×C₄	—	8	1	no	—
219	C₂×C₄	#358	1	2	yes	—
220	C₂×C₄	#358	1	2	yes	—
221	C₂×C₄	#359	1	2	yes	—
222	C₂×C₄	#380	1	1	yes	characteristic
223	C₂×C₄	#408	1	1	yes	characteristic
224	C₂×C₄	#408	1	1	yes	characteristic
225	C₂×C₄	#417	1	1	yes	characteristic
226	C₂×C₄	C₂×#104	1	1	yes	characteristic
227	C₂×C₄	#421	1	1	yes	characteristic
228	C₂×C₄	#459	1	1	yes	characteristic
229	C₂×C₄	#461	1	1	yes	characteristic
230	C₂×C₄	#474	1	1	yes	characteristic
231	C₂×C₄	#476	1	1	yes	characteristic
232	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
233	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
234	C₂×C₄	#509	1	1	yes	characteristic
235	C₂×C₄	#510	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	2	2	no	—
239	C₈	—	4	1	no	—
240	C₈	—	16	2	no	—
241	C₈	#424	1	2	yes	—
242	C₈	#433	1	2	yes	—
243	C₂²	#1480	1	1	yes	characteristic
244	C₂²	#1492	1	1	yes	characteristic
245	C₂²	#1802	1	1	yes	characteristic
246	C₂²	C₂×#358	1	1	yes	characteristic
247	C₂²	#2139	1	1	yes	characteristic
248	C₂²	C₂×#359	1	1	yes	characteristic
249	C₂²	#2143	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	8	2	no	—
257	C₄	—	8	2	no	—
258	C₄	—	16	1	no	—
259	C₄	—	16	1	no	—
260	C₄	#1483	1	1	yes	characteristic
261	C₄	#1495	1	1	yes	characteristic
262	C₄	#1798	1	1	yes	characteristic
263	C₄	#1806	1	1	yes	characteristic
264	C₄	#2207	1	1	yes	characteristic
265	C₄	#2209	1	1	yes	characteristic
266	C₄	#2225	1	1	yes	characteristic
267	C₄	#2227	1	1	yes	characteristic
268	C₂	#8567	1	1	yes	characteristic
269	C₂	#11646	1	1	yes	characteristic
270	C₂	#11648	1	1	yes	characteristic
271	C₂	#11672	1	1	yes	characteristic
272	C₂	#11674	1	1	yes	characteristic
273	C₂	#12812	1	1	yes	characteristic
274	C₂	#12815	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#102
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#21276	2
C₄×C₄⋊C₁₆	2
C₂×#1332	2
#8806	2
#7171	2