Subgroups — #102213

Order 512.

406 conjugacy classes of subgroups.

The subgroup lattice has 249 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33289	C₂	1	1	yes	maximal, characteristic
2	#9896	C₂	1	1	yes	maximal, characteristic
3	#9896	C₂	1	1	yes	maximal, characteristic
4	#9896	C₂	1	1	yes	maximal, characteristic
5	#9691	C₂	1	1	yes	maximal, characteristic
6	#8784	C₂	1	1	yes	maximal, characteristic
7	C₈×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	#2868	C₂²	1	1	yes	characteristic
9	#2868	C₂²	1	1	yes	characteristic
10	#1990	—	2	2	no	—
11	#1990	C₄	1	4	yes	—
12	#1842	C₂²	1	1	yes	characteristic
13	#1842	C₂²	1	1	yes	characteristic
14	#1834	—	2	1	no	—
15	#1834	—	2	1	no	—
16	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₄²×C₈	C₂²	1	1	yes	characteristic
19	C₂×#333	—	2	2	no	—
20	C₂×C₈²	—	2	1	no	—
21	#1296	—	2	2	no	—
22	#1296	—	2	2	no	—
23	#1296	—	2	2	no	—
24	#1277	—	2	2	no	—
25	#1273	—	2	1	no	—
26	C₂×C₄⋊Q₈	—	2	1	no	—
27	C₂×C₄⋊Q₈	—	2	1	no	—
28	C₂×C₄⋊Q₈	—	2	1	no	—
29	C₂×C₄⋊Q₈	—	2	1	no	—
30	C₂×C₄⋊Q₈	—	2	1	no	—
31	C₂×C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₈	—	2	1	no	—
33	C₂×C₄⋊C₈	—	2	1	no	—
34	C₂×C₄⋊C₈	—	2	1	no	—
35	C₂×C₄⋊C₈	—	2	1	no	—
36	C₂×C₄⋊C₈	—	2	1	no	—
37	C₂×C₄⋊C₈	—	2	1	no	—
38	C₂×C₄⋊C₈	D₄	1	2	yes	—
39	C₂×C₄⋊C₈	D₄	1	2	yes	—
40	C₂×C₄×C₈	—	2	1	no	—
41	C₂×C₄×C₈	—	2	1	no	—
42	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	2	yes	—
45	C₂×C₄×C₈	D₄	1	1	yes	characteristic
46	#384	—	2	1	no	—
47	#384	—	2	1	no	—
48	#384	—	2	1	no	—
49	#384	—	2	1	no	—
50	#384	—	2	1	no	—
51	#384	—	2	1	no	—
52	#384	—	2	1	no	—
53	#384	C₂×C₄	1	2	yes	—
54	C₄³	C₂³	1	1	yes	characteristic
55	#340	—	2	4	no	—
56	#340	—	2	4	no	—
57	#336	—	2	1	no	—
58	#336	—	2	1	no	—
59	#333	—	4	4	no	—
60	C₈²	—	4	2	no	—
61	C₂²×Q₈	—	2	2	no	—
62	C₂²×Q₈	—	4	1	no	—
63	C₂²×C₈	—	2	2	no	—
64	C₂²×C₈	—	2	2	no	—
65	C₂²×C₈	—	4	1	no	—
66	C₂²×C₈	—	4	1	no	—
67	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
69	C₄⋊Q₈	—	4	4	no	—
70	C₄⋊Q₈	—	4	2	no	—
71	C₄⋊Q₈	—	4	2	no	—
72	C₄⋊Q₈	—	4	2	no	—
73	C₄⋊Q₈	—	4	2	no	—
74	C₄⋊Q₈	—	8	2	no	—
75	C₄⋊Q₈	—	8	2	no	—
76	C₄⋊Q₈	—	8	2	no	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	2	2	no	—
79	C₂×C₄⋊C₄	—	2	2	no	—
80	C₂×C₄⋊C₄	—	2	2	no	—
81	C₂×C₄⋊C₄	—	2	2	no	—
82	C₂×C₄⋊C₄	—	4	1	no	—
83	C₂×C₄⋊C₄	—	4	1	no	—
84	C₂×C₄⋊C₄	—	4	1	no	—
85	C₂×C₄⋊C₄	—	4	1	no	—
86	C₂×C₄⋊C₄	—	4	1	no	—
87	C₂×C₄⋊C₄	—	4	1	no	—
88	C₂×C₄⋊C₄	—	4	1	no	—
89	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₄	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	#41	1	1	yes	characteristic
95	C₂×C₄²	#41	1	1	yes	characteristic
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	4	2	no	—
99	C₄⋊C₈	—	4	2	no	—
100	C₄⋊C₈	—	4	2	no	—
101	C₄⋊C₈	—	4	2	no	—
102	C₄⋊C₈	—	4	2	no	—
103	C₄⋊C₈	—	4	2	no	—
104	C₄⋊C₈	SD₁₆	1	4	yes	—
105	C₄⋊C₈	Q₁₆	1	4	yes	—
106	C₄×C₈	—	2	2	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	—	2	1	no	—
110	C₄×C₈	—	4	2	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	C₄×C₈	SD₁₆	1	2	yes	—
116	C₄×C₈	Q₁₆	1	2	yes	—
117	C₂×Q₈	—	4	4	no	—
118	C₂×Q₈	—	8	2	no	—
119	C₂×Q₈	—	8	2	no	—
120	C₂×Q₈	—	8	2	no	—
121	C₂×Q₈	—	8	2	no	—
122	C₂×Q₈	—	8	2	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	—	4	2	no	—
125	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
127	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
128	C₂²×C₄	#121	1	1	yes	characteristic
129	C₂²×C₄	#121	1	1	yes	characteristic
130	C₂²×C₄	#124	1	1	yes	characteristic
131	C₂²×C₄	#127	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	8	1	no	—
144	C₂×C₈	—	8	1	no	—
145	C₂×C₈	—	8	1	no	—
146	C₂×C₈	—	8	1	no	—
147	C₂×C₈	—	8	1	no	—
148	C₂×C₈	—	8	1	no	—
149	C₂×C₈	#103	1	4	yes	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₄⋊C₄	—	8	2	no	—
153	C₄⋊C₄	—	8	2	no	—
154	C₄⋊C₄	—	8	2	no	—
155	C₄⋊C₄	—	8	2	no	—
156	C₄⋊C₄	—	8	2	no	—
157	C₄⋊C₄	—	8	2	no	—
158	C₄⋊C₄	—	8	2	no	—
159	C₄⋊C₄	—	8	2	no	—
160	C₄⋊C₄	—	8	1	no	—
161	C₄⋊C₄	—	8	1	no	—
162	C₄⋊C₄	—	8	1	no	—
163	C₄⋊C₄	—	8	1	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	2	2	no	—
166	C₄²	—	2	2	no	—
167	C₄²	C₂×SD₁₆	1	1	yes	characteristic
168	C₄²	C₂×SD₁₆	1	2	yes	—
169	C₄²	C₂×Q₁₆	1	1	yes	characteristic
170	C₄²	C₂×Q₁₆	1	2	yes	—
171	C₄²	#135	1	2	yes	—
172	C₄²	#135	1	2	yes	—
173	C₄²	#135	1	2	yes	—
174	C₄²	#135	1	2	yes	—
175	C₄²	#137	1	1	yes	characteristic
176	C₄²	#137	1	1	yes	characteristic
177	C₂³	#390	1	1	yes	characteristic
178	Q₈	—	16	4	no	—
179	Q₈	—	16	4	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	8	2	no	—
187	C₂×C₄	—	8	2	no	—
188	C₂×C₄	—	8	2	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	#413	1	2	yes	—
193	C₂×C₄	C₂×#103	1	1	yes	characteristic
194	C₂×C₄	#416	1	2	yes	—
195	C₂×C₄	#419	1	1	yes	characteristic
196	C₂×C₄	C₄×SD₁₆	1	2	yes	—
197	C₂×C₄	C₄×Q₁₆	1	2	yes	—
198	C₂×C₄	#461	1	1	yes	characteristic
199	C₂×C₄	#461	1	1	yes	characteristic
200	C₂×C₄	#462	1	1	yes	characteristic
201	C₂×C₄	#462	1	1	yes	characteristic
202	C₂×C₄	#465	1	2	yes	—
203	C₂×C₄	#467	1	2	yes	—
204	C₂×C₄	#471	1	2	yes	—
205	C₂×C₄	#472	1	2	yes	—
206	C₂×C₄	#488	1	2	yes	—
207	C₂×C₄	#492	1	1	yes	characteristic
208	C₂×C₄	#494	1	1	yes	characteristic
209	C₂×C₄	#495	1	1	yes	characteristic
210	C₂×C₄	#495	1	1	yes	characteristic
211	C₂×C₄	#495	1	1	yes	characteristic
212	C₂×C₄	#495	1	1	yes	characteristic
213	C₈	—	2	4	no	—
214	C₈	—	4	2	no	—
215	C₈	—	4	4	no	—
216	C₈	—	4	4	no	—
217	C₈	—	16	1	no	—
218	C₈	—	16	1	no	—
219	C₂²	#1958	1	1	yes	characteristic
220	C₂²	#1967	1	1	yes	characteristic
221	C₂²	#1968	1	1	yes	characteristic
222	C₂²	#1969	1	1	yes	characteristic
223	C₂²	#1977	1	1	yes	characteristic
224	C₂²	#1978	1	1	yes	characteristic
225	C₂²	#1981	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	16	2	no	—
230	C₄	—	16	2	no	—
231	C₄	#1542	1	2	yes	—
232	C₄	#1554	1	2	yes	—
233	C₄	#1671	1	2	yes	—
234	C₄	#1672	1	2	yes	—
235	C₄	#1675	1	2	yes	—
236	C₄	#1676	1	2	yes	—
237	C₄	#1707	1	1	yes	characteristic
238	C₄	#1708	1	1	yes	characteristic
239	C₄	#1710	1	1	yes	characteristic
240	C₄	#1711	1	1	yes	characteristic
241	C₂	#9693	1	1	yes	characteristic
242	C₂	#9729	1	1	yes	characteristic
243	C₂	#9730	1	1	yes	characteristic
244	C₂	#9769	1	1	yes	characteristic
245	C₂	#9795	1	1	yes	characteristic
246	C₂	#9800	1	1	yes	characteristic
247	C₂	#9821	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Maximal subgroups

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

subgroup	index
#33289	2
#9896	2
#9691	2
#8784	2
C₈×C₄⋊C₈	2